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+A Maximum Principle for Optimal Control Problems involving
+Sweeping Processes with a Nonsmooth Set
+M. d. R. de Pinho, M. Margarida A. Ferreira ∗and
+Georgi Smirnov †
+February 1, 2023
+Abstract
+We generalize a Maximum Principle for optimal control problems involving sweeping systems
+previously derived in [14] to cover the case where the moving set may be nonsmooth. Noteworthy,
+we consider problems with constrained end point. A remarkable feature of our work is that we rely
+upon an ingenious smooth approximating family of standard differential equations in the vein of that
+used in [10].
+Keywords: Sweeping Process Optimal Control, Maximum Principle, Approximations
+1
+Introduction
+In recent years, there has been a surge of interest in optimal control problems involving the controlled
+sweeping process of the form
+˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)), u(t) ∈ U,
+x(0) ∈ C0.
+(1.1)
+In this respect, we refer to, for example, [3], [4], [5], [8], [9], [16], [23], [10] (see also accompanying correction
+[11]), [6], [15] and [14]. Sweeping processes first appeared in the seminal paper [18] by J.J. Moreau as a
+mathematical framework for problems in plasticity and friction theory. They have proved of interest to
+tackle problems in mechanics, engineering, economics and crowd motion problems; to name but a few,
+see [1], [5], [16], [17] and [21]. In the last decades, systems in the form (1.1) have caught the attention and
+interest of the optimal control community. Such interest resides not only in the range of applications but
+also in the remarkable challenge they rise concerning the derivation of necessary conditions. This is due
+to the presence of the normal cone NC(t)(x(t)) in the dynamics. Indeed, the presence of the normal cone
+renders the discontinuity of the right hand of the differential inclusion in (1.1) destroying a regularity
+property central to many known optimal control results.
+Lately, there has been several successful attempts to derive necessary conditions for optimal control
+problems involving (1.1). Assuming that the set C is time independent, necessary conditions for optimal
+control problems with free end point have been derived under different assumptions and using different
+techniques. In [10], the set C has the form C = {x :
+ψ(x) ≤ 0} and an approximating sequence of
+optimal control problems, where (1.1) is approximated by the differential equation
+˙xγk(t) = f(t, xγk(t), u(t)) − γkeγkψ(xγk (t))∇ψ(xγk(t)),
+(1.2)
+for some positive sequence γk → +∞, is used. Similar techniques are also applied to somehow more
+general problems in [23]. A useful feature of those approximations is explored in [12] to define numerial
+schemes to solve such problems.
+∗MdR de Pinho and MMA Ferreira are at Faculdade de Engenharia da Universidade do Porto, DEEC, SYSTEC. Portugal,
+mrpinho, [email protected]
+†G. Smirnov is at Universidade do Minho, Dep. Matem´atica, Physics Center of Minho and Porto Universities (CF-UM-
+UP), Campus de Gualtar, Braga, Portugal, [email protected]
+1
+arXiv:2301.13620v1  [math.OC]  31 Jan 2023
+
+More recently, an adaptation of the family of approximating systems (1.2) is used in [14] to generalize
+the results in [10] to cover problems with additional end point constraints and with a moving set of the
+form C(t) = {x : ψ(t, x) ≤ 0}.
+In this paper we generalize the Maximum Principle proved in [14] to cover problems with possibly
+nonsmooth sets. Our problem of interest is
+(P)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Minimize φ(x(T))
+over processes (x, u) such that
+˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)), a.e. t ∈ [0, T],
+u(t) ∈ U,
+a.e. t ∈ [0, T],
+(x(0), x(T)) ∈ C0 × CT ⊂ C(0) × C(T),
+where T > 0 is fixed, φ : Rn → R, f : [0, T] × Rn × Rm → Rn, U ⊂ Rm and
+C(t) :=
+�
+x ∈ Rn : ψi(t, x) ≤ 0, i = 1, . . . , I
+�
+(1.3)
+for some functions ψi : [0, T] × Rn → R, i = 1, . . . , I.
+The case where I = 1 in (1.3) and ψ1 is C2 is covered in [14]. Here, we assume I > 1 and that
+the functions ψi are also C2. Although going from I = 1 in (1.3) to I > 1 may be seen as a small
+generalization, it demands a significant revision of the technical approach and, plus, the introduction
+of a constraint qualification. This is because the set (1.3) may be nonsmooth. We focus on sets (1.3),
+satisfying a certain constraint qualification, introduced in assumption (A1) in section 2 below. This is,
+indeed, a restriction on the nonsmoothness of (1.3). A similar problem with nonsmooth moving set is
+considered in [15]. Our results cannot be obtained from the results of [15] and do not generalize them.
+This paper is organized in the following way. In section 2, we introduce the main notation and we state
+and discuss the assumptions under which we work. In this same section, we also introduce the family of
+approximating systems to ˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)) and establish a crucial convergence result,
+Theorem 2.2. In section 3, we dwell on the approximating family of optimal control problems to (P)
+and we state the associated necessary conditions. The Maximum Principle for (P) is then deduced and
+stated in Theorem 4.1, covering additionally, problems in the form of (P) where the end point constraint
+x(T) ∈ CT is absent. Before finishing, we present an illustrative example of our main result, Theorem
+4.1.
+2
+Preliminaries
+In this section, we introduce a summary of the notation and state the assumptions on the data of (P)
+enforced throughout. Furthermore, we extract information from the assumptions establishing relations
+crucial for the forthcoming analysis.
+Notation
+For a set S ⊂ Rn, ∂S, cl S and int S denote the boundary, closure and interior of S.
+If g : Rp → Rq, ∇g represents the derivative and ∇2g the second derivative. If g : R × Rp → Rq, then
+∇xg represents the derivative w.r.t. x ∈ Rp and ∇2
+xg the second derivative, while ∂tg(t, x) represents the
+derivative w.r.t. t ∈ R.
+The Euclidean norm or the induced matrix norm on Rp×q is denoted by |·|. We denote by Bn the
+closed unit ball in Rn centered at the origin. The inner product of x and y is denoted by ⟨x, y⟩. For
+some A ⊂ Rn, d(x, A) denotes the distance between x and A. We denote the support function of A at z
+by S(z, A) = sup{⟨z, a⟩ | a ∈ A}
+The space L∞([a, b]; Rp) (or simply L∞ when the domains are clearly understood) is the Lebesgue
+space of essentially bounded functions h : [a, b] → Rp. We say that h ∈ BV ([a, b]; Rp) if h is a function
+of bounded variation. The space of continuous functions is denoted by C([a, b]; Rp).
+2
+
+Standard concepts from nonsmooth analysis will also be used. Those can be found in [7], [19] or [22],
+to name but a few. The Mordukhovich normal cone to a set S at s ∈ S is denoted by NS(s) and ∂f(s)
+is the Mordukhovich subdifferential of f at s (also known as limiting subdifferential).
+For any set A ⊂ Rn, cone A is the cone generated by the set A.
+We now turn to problem (P). We first state the definition of admissible processes for (P) and then
+we describe the assumptions under which we will derive our main results.
+Definition 2.1 A pair (x, u) is called an admissible process for (P) when x is an absolutely continuous
+function and u is a measurable function satisfying the constraints of (P).
+Assumptions on the data of (P)
+A1: The function ψi, i = 1, . . . , I, are C2. The graph of C(·) is compact and it is contained in the
+interior of a ball rBn+1, for some r > 0. There exist constants β > 0, η > 0 and ρ ∈]0, 1[ such that
+ψi(t, x) ∈ [−β, β] =⇒ |∇xψi(t, x)|> η forall (t, x) ∈ [0, T] × Rn,
+(2.1)
+and, for I(t, x) = {i = 1, . . . , I | ψi(t, x) ∈] − 2β, β]},
+⟨∇xψi(t, x), ∇xψj(t, x)⟩ ≥ 0, i, j ∈ I(t, x).
+(2.2)
+Moreover, if i ∈ I(t, x), then
+�
+j∈I(t,x)\{i}
+|⟨∇xψi(t, x), ∇xψj(t, x)⟩| ≤ ρ|∇xψi(t, x)|2
+(2.3)
+and
+ψi(t, x) ≤ −2β =⇒ ∇ψi(t, x) = 0 for i = 1, . . . I.
+(2.4)
+A2: The function f is continuous, x → f(t, x, u) is continuously differentiable for all (t, u) ∈ [0, T]×Rm.
+The constant M > 0 is such that |f(t, x, u)|≤ M and |∇xf(t, x, u)|≤ M for all (t, x, u) ∈ rBn+1×U.
+A3: For each (t, x), the set f(t, x, U) is convex.
+A4: The set U is compact.
+A5: The sets C0 and CT are compact.
+A6: There exists a constant Lφ such that |φ(x) − φ(x′)|≤ Lφ|x − x′| for all x, x′ ∈ Rn.
+Assumption (A1) concerns the functions ψi defining the set C and it plays a crucial role in the analysis.
+All ψi are assumed to be smooth with gradients bounded away from the origin when ψi takes values in
+a neighorhood of zero. Moreover, the boundary of C may be nonsmooth at the intersection points of the
+level sets
+�
+x : ψi(t, x) = 0
+�
+. However, nonsmoothness at those corner points is restricted to (2.2) which
+excludes the cases where the angle between the two gradients of the functions defining the boundary of
+C is obtuse; see figure 1.
+On the other hand, (2.3) guarantees that the Gramian matrix of the gradients of the functions
+taking values near the boundary of C(t) is diagonally dominant and, hence, the gradients are linearly
+independent.
+In many situations, as in the example we present in the last section, we can guarantee the fulfillment
+of (A1), in particular (2.4), replacing the function ψi by
+˜ψi(t, x) = h ◦ ψi(t, x),
+(2.5)
+3
+
+c
+Y1=0
+Y2=0
+▽ Y2
+Not allowed
+Allowed
+▽ Y1
+C
+Y2=0
+▽ Y2
+▽ Y1
+Y1=0
+Figure 1: Examples of two diferent sets C. On the left size, a set that does not satisfies (2.2). On the
+right side, the set C is nonsmooth and it fulfils (2.2).
+where
+h(z) =
+�
+�
+�
+z
+if
+z > −β,
+hs(z)
+if
+−2β ≤ z ≤ −β,
+−2β
+if
+z < −2β,
+Here, h is an C2 function, with hs an increasing function defined on [−2β, −β]. For example, h may be
+a cubic polinomial with positive derivative on the interval ] − 2β, −β[. For all t ∈ [0, T], set
+˜C(t) :=
+�
+x ∈ R :
+˜ψi(t, x) ≤ 0, i = 1, . . . , I
+�
+.
+It is then a simple matter to see that
+C(t) = ˜C(t) for all t ∈ [0, T].
+and that the functions ˜ψi(·) satisfy the assumption (A1).
+The assumption that the graph of C(·) is compact and contained in the interior of a ball is introduced
+to avoid technicalities in our forthcoming analysis. In applied problems, this may be easily side tracked
+by considering the intersection of the graph of C(·) with a tube around the optimal trajectory.
+We now proceed introducing an approximation family of controlled systems to (1.1). Let x(·) be a
+solution to the differential inclusion
+˙x(t) ∈ f(t, x(t), U) − NC(t)(x(t)).
+Under our assumptions, measurable selection theorems assert the existence of measurable functions u
+and ξi such that u(t) ∈ U, ξi(t) ≥ 0 a.e. t ∈ [0, T], ξi(t) = 0 if ψi(t, x(t)) < 0, and
+˙x(t) = f(t, x(t), u(t)) −
+I
+�
+i=1
+ξi(t)∇xψi(t, x(t)) a.e. t ∈ [0, T].
+Considering the trajectory x, some observations are called for. Let µ be such that
+max
+�
+(|∇xψi(t, x)||f(t, x, u)|+|∂tψi(t, x)|) + 1 :
+t ∈ [0, T], u ∈ U, x ∈ C(t) + Bn, i = 1, . . . , I} ≤ µ.
+The properties of the graph of C(·) in (A1) guarantee the existence of such maximum.
+4
+
+Consider now some t such that, for some j ∈ {1, . . . I}, ψj(t, x(t)) = 0 and ˙x(t) exists. Since the
+trajectory x is always in C, we have (see (2.2))
+0 = d
+dtψj(t, x(t)) = ⟨∇xψj(t, x(t)), ˙x(t)⟩ + ∂tψj(t, x(t))
+= ⟨∇xψj(t, x(t)), f(t, x(t), u(t))⟩ − ξj(t)|∇xψj(t, x(t))|2
+−
+�
+i∈I(t,x(t))\{j}
+ξi(t)⟨∇xψi(t, x(t)), ∇xψj(t, x(t))⟩ + ∂tψj(t, x(t))
+≤ ⟨∇xψj(t, x(t)), f(t, x(t), u(t))⟩ − ξj(t)|∇xψj(t, x(t))|2+∂tψj(t, x(t)),
+and, hence (see (2.1)),
+ξj(t) ≤
+1
+|∇xψj(t, x(t))|2 (⟨∇xψj(t, x(t)), f(t, x(t), u(t))⟩ + ∂tψj(t, x(t))) ≤ µ
+η2 .
+Define the function
+µ(γ) = 1
+γ log
+� µ
+η2γ
+�
+,
+γ > 0,
+consider a sequence {σk} such that σk ↓ 0 and choose another sequence {γk} with γk ↑ +∞ and
+C(t) ⊂ int Ck(t) = int
+�
+x : ψi(t, x) − σk ≤ µk, i = 1, . . . , I
+�
+,
+where
+µk = µ(γk).
+Let xk be a solution to the differential equation
+˙xk(t) = f(t, xk(t), uk(t)) −
+I
+�
+i=1
+γkeγk(ψi(t,xk(t))−σk)∇xψi(t, xk(t))
+(2.6)
+for some uk(t) ∈ U a.e. t ∈ [0, T]. Take any t ∈ [0, T] such that ˙xk(t) exists and ψj(t, xk(t)) − σk = µk.
+5
+
+Assume k is such that j ∈ I(t, xk(t)). Then, whenever γk is sufficiently large, we have
+d
+dtψj(t, xk(t)) = ⟨∇xψj(t, xk(t)), f(t, xk(t), uk(t))⟩
+− γkeγk(ψj(t,xk(t))−σk)|∇xψj(t, xk(t))|2
+−
+�
+i∈I(t,xk(t))\{j}
+γkeγk(ψi(t,xk(t))−σk)⟨∇xψi(t, xk(t)), ∇xψj(t, xk(t))⟩
+−
+�
+i̸∈I(t,xk(t))
+γkeγk(ψi(t,xk(t))−σk)⟨∇xψi(t, xk(t)), ∇xψj(t, xk(t))⟩
++ ∂tψj(t, xk(t))
+≤ ⟨∇xψj(t, xk(t)), f(t, xk(t), uk(t))⟩
+− γkeγk(ψj(t,xk(t))−σk)|∇xψj(t, xk(t))|2
+−
+�
+i̸∈I(t,xk(t))
+γkeγk(ψi(t,xk(t))−σk)⟨∇xψi(t, xk(t)), ∇xψj(t, xk(t))⟩
++ ∂tψj(t, xk(t))
+≤ ⟨∇xψj(t, xk(t)), f(t, xk(t), uk(t))⟩
+− γkeγk(ψj(t,xk(t))−σk)|∇xψj(t, xk(t))|2
++
+�
+i̸∈I(t,xk(t))
+γkeγk(−2β−σk)|⟨∇xψi(t, xk(t)), ∇xψj(t, xk(t))⟩|
++ ∂tψj(t, xk(t))
+≤ µ − 1
+2 − η2γkeγkµk
+= −1
+2.
+Above, we have used the definition of µ and the inequality
+�
+i̸∈I(t,xk(t))
+γkeγk(−2β−σk)|⟨∇xψi(t, xk(t)), ∇xψj(t, xk(t))⟩|≤ 1
+2,
+which holds for γk sufficiently large.
+Now, if xk(0) ∈ Ck(0), we assure that xk(t) ∈ Ck(t), for all t ∈ [0, T], and
+γkeγk(ψj(t,xk(t))−σk) ≤ γkeγkµk = µ
+η2 .
+(2.7)
+It follows that, for k sufficienttly large, we have
+| ˙xk(t)|≤ (const).
+We are now a in position to state and prove our first result, Theorem 2.2 below. This is in the vein of
+Theorem 4.1 in [23] (see also Lemma 1 in [10] when ψ is independent of t and convex) deviating from it
+in so far as the approximating sequence of control systems (2.6) differs from the one introduced in [10]1.
+The proof of Theorem 2.2 relies on (2.7).
+Theorem 2.2 Let {(xk, uk)}, with uk(t) ∈ U a.e., be a sequence of solutions of Cauchy problems
+�
+�
+�
+�
+�
+˙xk(t)
+=
+f(t, xk(t), uk(t)) −
+I
+�
+i=1
+γkeγk(ψi(t,xk(t))−σk)∇xψi(t, xk(t)),
+xk(0)
+=
+bk ∈ Ck(0).
+(2.8)
+1See also Theorem 2.2 in [14]
+6
+
+If bk → x0, then there exists a subsequence {xk} (we do not relabel) converging uniformly to x, a unique
+solution to the Cauchy problem
+˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)),
+x(0) = x0,
+(2.9)
+where u is a measurable function such that u(t) ∈ U a.e. t ∈ [0, T].
+If, moreover, all the controls uk are equal, i.e., uk = u, then the subsequence converges to a unique
+solution of (2.9), i.e., any solution of
+˙x(t) ∈ f(t, x(t), U) − NC(t)(x(t)),
+x(0) = x0 ∈ C(0)
+(2.10)
+can be approximated by solutions of (2.8).
+Proof Consider the sequence {xk}, where (xk, uk) solves (2.8). Recall that xk(t) ∈ Ck(t) for all
+t ∈ [0, T], and
+| ˙xk(t)|≤ (const)
+and
+ξi
+k(t) = γkeγk(ψi(t,xk(t))−σk) ≤ (const).
+(2.11)
+Then there exist subsequences (we do not relabel) weakly-∗ converging in L∞ to some v and ξi. Hence
+xk(t) = x0 +
+� t
+0
+˙xk(s)ds −→ x(t) = x0 +
+� t
+0
+v(s)ds, ∀ t ∈ [0, T],
+for an absolutely continuous function x. Obviously, x(t) ∈ C(t) for all t ∈ [0, T]. Considering the sequence
+{xk}, recall that
+˙xk(t) ∈ f(t, xk(t), U) −
+I
+�
+i=1
+ξi
+k(t)∇xψi(t, xk(t)).
+(2.12)
+Inclusion (2.12) is equivalent to
+⟨z, ˙xk(t)⟩ ≤ S(z, f(t, xk(t), U)) −
+I
+�
+i=1
+ξi
+k(t)⟨z, ∇xψi(t, xk(t))⟩,
+∀ z ∈ Rn.
+Integrating this inequality, we get
+�
+z, xk(t + τ) − xk(t)
+τ
+�
+≤ 1
+τ
+� t+τ
+t
+�
+S(z, f(s, xk(s), U)) −
+I
+�
+i=1
+ξi
+k(s)⟨z, ∇xψi(s, xk(s))⟩
+�
+ds
+= 1
+τ
+� t+τ
+t
+�
+S(z, f(s, xk(s), U)) −
+I
+�
+i=1
+ξi
+k(s)⟨z, ∇xψi(s, x(s))⟩
++
+I
+�
+i=1
+ξi
+k(s)⟨z, ∇xψi(s, x(s)) − ∇xψi(s, xk(s))⟩
+�
+ds.
+(2.13)
+Passing to the limit as k → ∞, we obtain
+�
+z, x(t + τ) − x(t)
+τ
+�
+≤ 1
+τ
+� t+τ
+t
+�
+S(z, f(s, x(s), U)) −
+I
+�
+i=1
+ξi(s)⟨z, ∇xψi(s, x(s))⟩
+�
+ds.
+(2.14)
+7
+
+Let t ∈ [0, T] be a Lebesgue point of x and ξ. Passing in the last inequality to the limit as τ ↓ 0, it leads
+to
+⟨z, ˙x(t)⟩ ≤ S(z, f(t, x(t), U)) −
+I
+�
+i=1
+ξi(t)⟨z, ∇xψi(t, x(t))⟩.
+Since z ∈ Rn is an arbitrary vector and the set f(t, x(t), U) is convex, we conclude that
+˙x(t) ∈ f(t, x(t), U) −
+I
+�
+i=1
+ξi(t)∇xψi(t, x(t)).
+By the Filippov lemma there exists a measurable control u(t) ∈ U such that
+˙x(t) = f(t, x(t), u(t)) −
+I
+�
+i=1
+ξi(t)∇xψi(t, x(t)).
+Furthermore, observe that ξi is zero if ψi(t, x(t)) < 0. If for some u such that u(t) ∈ U a.e., uk = u for
+all k, then the sequence xk converges to the solution of
+˙x(t) = f(t, x(t), u(t)) −
+I
+�
+i=1
+ξi(t)∇xψi(t, x(t)).
+Indeed, to see this, it suffices to pass to the limit as k → ∞ and then as τ ↓ 0, in the equality
+xk(t + τ) − xk(t)
+τ
+= 1
+τ
+� t+τ
+t
+�
+f(s, xk(s), u(s)) −
+I
+�
+i=1
+ξi
+k(s)∇xψi(s, xk(s))
+�
+ds.
+We now prove the uniqueness of the solution. We follow the proof of Theorem 4.1 in [23]. Notice,
+however, that we now consider a special case and not the general case treated in [23]. Suppose that there
+exist two different solutions of (2.9): x1 and x2. We have
+1
+2
+d
+dt|x1(t) − x2(t)|2= ⟨x1(t) − x2(t), ˙x1(t) − ˙x2(t)⟩
+= ⟨x1(t) − x2(t), f(t, x1(t), u(t)) − f(t, x2(t), u(t))⟩
+−
+�
+x1(t) − x2(t),
+I
+�
+i=1
+ξi
+1(t)∇ψi(t, x1(t)) −
+I
+�
+i=1
+ξi
+2(t)∇ψi(t, x2(t))
+�
+.
+(2.15)
+If, for all i, ψi(t, x1(t)) < 0 and ψi(t, x2(t)) < 0, then ξi
+1(t) = ξi
+2(t) = 0 and we obtain
+1
+2
+d
+dt|x1(t) − x2(t)|2≤ Lf|x1(t) − x2(t)|2.
+Suppose that ψj(t, x1(t)) = 0. Then by the Taylor formula we get
+ψj(t, x2(t)) = ψj(t, x1(t)) + ⟨∇xψj(t, x1(t)), x2(t) − x1(t)⟩
++ 1
+2⟨x2(t) − x1(t), ∇2
+xψj(t, θx2(t) + (1 − θ)x1(t))(x2(t) − x1(t))⟩,
+(2.16)
+where θ ∈ [0, 1]. Since ψj(t, x2(t)) ≤ 0, we have
+⟨∇xψj(t, x1(t)), x2(t) − x1(t)⟩
+≤ −1
+2⟨x2(t) − x1(t), ∇2
+xψj(t, θx2(t) + (1 − θ)x1(t))(x2(t) − x1(t))⟩
+≤ (const)|x1(t) − x2(t)|2.
+(2.17)
+8
+
+Now, if ψj(t, x2(t)) = 0, we deduce in the same way that
+⟨∇xψj(t, x2(t)), x1(t) − x2(t)⟩ ≤ (const)|x1(t) − x2(t)|2.
+Thus we have
+1
+2
+d
+dt|x1(t) − x2(t)|2≤ (const)|x1(t) − x2(t)|2.
+Hence |x1(t) − x2(t)|= 0.
+2
+3
+Approximating Family of Optimal Control Problems
+In this section we define an approximating family of optimal control problems to (P) and we state the
+corresponding necessary conditions.
+Let (ˆx, ˆu) be a global solution to (P) and consider sequences {γk} and {σk} as defined above. Let
+ˆxk(·) be the solution to
+�
+�
+�
+�
+�
+˙x(t) = f(t, x(t), ˆu(t)) −
+I
+�
+i=1
+γkeγk(ψi(t,x(t))−σk)∇xψi(t, x(t)),
+x(0) = ˆx(0).
+(3.1)
+Set ϵk = |ˆxk(T) − ˆx(T)|. It follows from Theorem 2.2 that ϵk ↓ 0. Take α > 0 and define the problem
+(P α
+k )
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Minimize φ(x(T)) + |x(0) − ˆx(0)|2+α
+� T
+0
+|u(t) − ˆu(t)|dt
+over processes (x, u) such that
+˙x(t) = f(t, x(t), u(t)) −
+I
+�
+i=1
+∇xeγk(ψi(t,x(t))−σk) a.e. t ∈ [0, T],
+u(t) ∈ U
+a.e. t ∈ [0, T],
+x(0) ∈ C0,
+x(T) ∈ CT + ϵkBn,
+Clearly, the problem (P α
+k ) has admissible solutions. Consider the space
+W = {(c, u) | c ∈ C0, u ∈ L∞ with u(t) ∈ U}
+and the distance
+dW ((c1, u1), (c2, u2)) = |c1 − c2|+
+� T
+0
+|u1(t) − u2(t)|dt.
+Endowed with dW , W is a complete metric space. Take any (c, u) ∈ W and a solution y to the Cauchy
+problem
+�
+�
+�
+�
+�
+˙y(t)
+=
+f(t, y(t), u(t)) −
+I
+�
+i=1
+∇xeγk(ψi(t,y(t))−σk) a.e. t ∈ [0, T],
+y(0)
+=
+c.
+Under our assumptions, the function
+(c, u) → φ(y(T)) + |c − ˆx(0)|2+α
+� T
+0
+|u − ˆu| dt
+9
+
+is continuous on (W, dW ) and bounded below. Appealing to Ekeland’s Theorem we deduce the existence
+of a pair (xk, uk) solving the following problem
+(APk)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Minimize Φ(x, u) = φ(x(T)) + |x(0) − ˆx(0)|2+α
+� T
+0
+|u(t) − ˆu(t)|dt
++ϵk
+�
+|x(0) − xk(0)|+
+� T
+0
+|u(t) − uk(t)|dt
+�
+,
+over processes (x, u) such that
+˙x(t) = f(t, x(t), u(t)) −
+I
+�
+i=1
+∇xeγk(ψi(t,x(t))−σk) a.e. t ∈ [0, T],
+u(t) ∈ U
+a.e. t ∈ [0, T],
+x(0) ∈ C0,
+x(T) ∈ CT + ϵkBn,
+Lemma 3.1 Take γk → ∞, σk → 0 and ϵk → 0 as defined above. For each k, let (xk, uk) be the solution
+to (APk). Then there exists a subsequence (we do not relabel) such that
+uk(t) → ˆu(t) a.e.,
+xk → ˆx uniformly in [0, T].
+Proof We deduce from Theorem 2.2 that {xk} uniformly converges to an admissible solution ˜x to (P).
+Since U and C0 are compact, we have U ⊂ KBm and C0 ⊂ KBn. Without loss of generality, uk weakly-∗
+converges to a function ˜u ∈ L∞([0, T], U). Hence it weakly converges to ˜u in L1. From optimality of the
+processes (xk, uk) we have
+φ(xk(T)) + |xk(0) − ˆx(0)|2+α
+� T
+0
+|uk(t) − ˆu(t)|dt
+≤ φ(ˆxk(T)) + ϵk
+�
+|ˆxk(0) − xk(0)|+
+� T
+0
+|uk(t) − ˆu(t)|dt
+�
+≤ φ(ˆxk(T)) + 2K(1 + T)ϵk.
+Since (ˆx, ˆu) is a global solution of the problem, passing to the limit, we get
+φ(˜x(T)) + |˜x(0) − ˆx(0)|2+α
+� T
+0
+|˜u(t) − ˆu(t)|dt
+≤ lim
+k→∞(φ(xk(T)) + |xk(0) − ˆx(0)|2) + α lim inf
+k→∞
+� T
+0
+|uk(t) − ˆu(t)|dt
+≤ lim
+k→∞ φ(ˆxk(T)) = φ(ˆx(T)) ≤ φ(˜x(T)).
+Hence ˜x(0) = ˆx(0), ˜u = ˆu a.e., and uk converges to ˆu in L1, and some subsequence converges to ˆu almost
+everywhere (we do not relabel).
+2
+We now finish this section with the statement of the optimality necessary conditions for the family of
+problems (APk). These can be seen as a direct consequence of Theorem 6.2.1 in [22].
+Proposition 3.2 For each k, let (xk, uk) be a solution to (APk). Then there exist absolutely continous
+functions pk and scalars λk ≥ 0 such that
+(a) (nontriviality condition)
+λk + |pk(T)|= 1,
+(3.2)
+10
+
+(b) (adjoint equation)
+˙pk = −(∇xfk)∗pk + �I
+i=1 γkeγk(ψi
+k−σk)∇2
+xψi
+kpk
++ �I
+i=1 γ2
+keγk(ψi
+k−σk)∇xψi
+k⟨∇xψi
+k, pk⟩,
+(3.3)
+where the superscript ∗ stands for transpose,
+(c) (maximization condition)
+max
+u∈U {⟨f(t, xk, u), pk⟩ − αλk|u − ˆu|−ϵkλk|u − uk|}
+(3.4)
+is attained at uk(t), for almost every t ∈ [0, T],
+(d) (transversality condition)
+(pk(0), −pk(T)) ∈ λk (2(xk(0) − ˆx(0)) + ϵkBn, ∂φ(xk(T)))
++NC0(xk(0)) × NCT +ϵkBn(xk(T)).
+(3.5)
+To simplify the notation above, we drop the t dependance in pk, ˙pk, xk, uk, ˆx and ˆu. Moreover, in
+(b), we write ψk instead of ψ(t, xk(t)), fk instead of f(t, xk(t), uk(t)). The same holds for the derivatives
+of ψ and f.
+4
+Maximum Principle for (P)
+In this section, we establish our main result, a Maximum Principle for (P). This is done by taking limits
+of the conclusions of Proposition 3.2, following closely the analysis done in the proof of [10, Theorem 2].
+Observe that
+1
+2
+d
+dt|pk(t)|2 = −⟨∇xfkpk, pk⟩ +
+I
+�
+i=1
+γkeγk(ψi
+k−σk)⟨∇2
+xψi
+kpk, pk⟩
++
+I
+�
+i=1
+γ2
+keγk(ψi
+k−σk)⟨∇xψi
+k, pk⟩2
+≥ −⟨∇xfkpk, pk⟩ +
+I
+�
+i=1
+γkeγk(ψi
+k−σk)⟨∇2
+xψi
+kpk, pk⟩
+≥ −M|pk|2+
+I
+�
+i=1
+γkeγk(ψi
+k−σk)⟨∇2
+xψi
+kpk, pk⟩,
+where M is the constant of (A2). Taking into account hypothesis (A1) and (2.7) we deduce the existence
+of a constant K0 > 0 such that
+1
+2
+d
+dt|pk(t)|2≥ −K0|pk(t)|2.
+This last inequality leads to
+|pk(t)|2 ≤ e2K0(T −t)|pk(T)|2≤ e2K0T |pk(T)|2.
+Since, by (a) of Proposition 3.2, |pk(T)|≤ 1, we deduce from the above that there exists M0 > 0 such
+that
+|pk(t)| ≤ M0.
+(4.1)
+Now, we claim that the sequence { ˙pk} is uniformly bounded in L1. To prove our claim, we need to establish
+bounds for the three terms in (3.3). Following [10] and [14], we start by deducing some inequalities that
+will be of help.
+11
+
+Denote Ik = I(t, xk(t)) and Sj
+k = sign
+�
+⟨∇xψj
+k, pk⟩
+�
+. We have
+I
+�
+j=1
+d
+dt
+���⟨∇xψj
+k, pk⟩
+���
+=
+I
+�
+j=1
+�
+⟨∇2
+xψj
+k ˙xk, pk⟩ + ⟨∂t∇xψj
+k, pk⟩ + ⟨∇xψj
+k, ˙pk⟩
+�
+Sj
+k
+=
+I
+�
+j=1
+�
+⟨pk, ∇2
+xψj
+kfk⟩ −
+I
+�
+i=1
+γkeγk(ψi
+k−σk)⟨pk, ∇2ψj
+k∇xψi
+k⟩
+�
+Sj
+k
++
+I
+�
+j=1
+�
+⟨∂t∇xψj
+k, pk⟩ − ⟨∇xψj
+k, (∇xfk)∗pk⟩
+�
+Sj
+k
++
+I
+�
+j=1
+� I
+�
+i=1
+γkeγk(ψi
+k−σk)⟨∇xψj
+k, ∇2
+xψi
+kpk⟩
+�
+Sj
+k
++
+I
+�
+i=1
+I
+�
+j=1
+γ2
+keγk(ψi
+k−σk)⟨∇xψj
+k, ∇xψi
+k⟩⟨∇xψi
+k, pk⟩Sj
+k
+Observe that (see (2.3) and (2.4))
+I
+�
+i=1
+I
+�
+j=1
+γ2
+keγk(ψi
+k−σk)⟨∇xψj
+k, ∇xψi
+k⟩⟨∇xψi
+k, pk⟩Sj
+k
+=
+I
+�
+i=1
+�
+j∈Ik
+γ2
+keγk(ψi
+k−σk)⟨∇xψj
+k, ∇xψi
+k⟩⟨∇xψi
+k, pk⟩Sj
+k
+=
+�
+i̸∈Ik
+γ2
+keγk(ψi
+k−σk) �
+j∈Ik
+⟨∇xψj
+k, ∇xψi
+k⟩⟨∇xψi
+k, pk⟩Sj
+k
++
+�
+i∈Ik
+γ2
+keγk(ψi
+k−σk)
+�
+�|∇xψi
+k|2+
+�
+j∈Ik\{i}
+⟨∇xψj
+k, ∇xψi
+k⟩Sj
+k Si
+k
+�
+� |⟨∇xψi
+k, pk⟩|
+=
+�
+i∈Ik
+γ2
+keγk(ψi
+k−σk)
+�
+�|∇xψi
+k|2+
+�
+j∈Ik\{i}
+⟨∇xψj
+k, ∇xψi
+k⟩Sj
+k Si
+k
+�
+� |⟨∇xψi
+k, pk⟩|
+≥ (1 − ρ)
+�
+i∈Ik
+γ2
+keγk(ψi
+k−σk)|∇xψi
+k|2|⟨∇xψi
+k, pk⟩|
+= (1 − ρ)
+I
+�
+i=1
+γ2
+keγk(ψi
+k−σk)|∇xψi
+k|2|⟨∇xψi
+k, pk⟩|.
+Using this and integrating the previous equality, we deduce the existence of M1 > 0 such that:
+� T
+0
+I
+�
+i=1
+γ2
+keγk(ψi
+k−σk)|∇xψi
+k|2|⟨∇xψi
+k, pk⟩|dt ≤ M1.
+(4.2)
+We are now in a position to show that
+� T
+0
+I
+�
+i=1
+γ2
+keγk(ψi
+k−σk)|∇xψi
+k|
+��⟨∇xψi
+k, pk⟩
+�� dt
+12
+
+is bounded. For simplicity, set Li
+k(t) = γ2
+keγk(ψi
+k−σk)|∇xψi
+k|
+��⟨∇xψi
+k, pk⟩
+��. Notice that
+I
+�
+i=1
+� T
+0
+Li
+k(t)dt =
+I
+�
+i=1
+��
+{t:|∇xψi
+k|<η}
+Li
+k(t) dt +
+�
+{t:|∇xψi
+k|≥η}
+Li
+k(t)dt
+�
+.
+Using (A1) and (4.2), we deduce that
+I
+�
+i=1
+� T
+0
+Li
+k(t) dt ≤
+I
+�
+i=1
+�
+γ2
+ke−γk(β+σk)η2 max
+t |pk(t)|
+�
++
+I
+�
+i=1
+�
+γ2
+k
+�
+{t:|∇xψi
+k|≥η}
+eγk(ψi
+k−σk) |∇xψi
+k|2
+|∇xψi
+k|
+��⟨∇xψi
+k, pk⟩
+�� dt
+�
+≤ γ2
+kI e−γk(β+σk)η2M0
++ 1
+η
+I
+�
+i=1
+�� T
+0
+γ2
+keγk(ψi
+k−σk)|∇xψi
+k|2��⟨∇xψi
+k, pk⟩
+�� dt
+�
+≤ η2M0I + M1
+η ,
+for k large enough. Summarizing, there exists a M2 > 0 such that
+I
+�
+i=1
+γ2
+k
+� T
+0
+eγk(ψi
+k−σk)|∇ψi
+k|
+��⟨∇ψi
+k, pk⟩
+�� dt
+≤ M2.
+(4.3)
+Mimicking the analysis conducted in Step 1, b) and c) of the proof of Theorem 2 in [10] and taking into
+account (b) of Proposition 3.2 we conclude that there exist constants N1 > 0 such that
+� T
+0
+| ˙pγk(t)| dt ≤ N1,
+(4.4)
+for k sufficiently large, proving our claim.
+Before proceeding, observe that it is a simple matter to assert the existence of a constant N2 such
+that
+I
+�
+i=1
+� T
+0
+γ2
+keγk(ψi
+k−σk)|⟨∇ψi
+k, pγk⟩|dt ≤ N2.
+(4.5)
+This inequality will be of help in what follows.
+Let us now recall that
+ξi
+k(t) = γkeγk(ψi(t,xk(t))−σk)
+and that the second inequality in (2.11) holds. We turn to the analysis of Step 2 in the proof of Theorem
+2 in [10] (see also [14]). Adapting those arguments, we can conclude the existence of some function p ∈
+BV ([0, T], Rn) and, for i = 1, . . . , I, functions ξi ∈ L∞([0, T], R) with ξi(t) ≥ 0 a. e. t, ξi(t) = 0, t ∈ Ii
+b,
+where
+Ii
+b =
+�
+t ∈ [0, T] : ψi(t, ˆx(t)) < 0
+�
+,
+and finite signed Radon measures ηi, null in Ii
+b, such that, for any z ∈ C([0, T], Rn)
+� T
+0
+⟨z, dp⟩ = −
+� T
+0
+⟨z, (∇ ˆf)∗p⟩dt +
+I
+�
+i=1
+�� T
+0
+ξi⟨z, ∇2 ˆψip⟩dt +
+� T
+0
+⟨z, ∇ ˆψi(t)⟩dηi
+�
+,
+where ∇ ˆψi(t) = ∇ψi(t, ˆx(t)). The finite signed Radon measures ηi are weak-∗ limits of
+γ2
+keγk(ψi
+k−σk)⟨∇ψi
+k(xk(t), pk(t)⟩dt.
+13
+
+Observe that the measures
+⟨∇ψi(ˆx(t), p(t)⟩dηi(t)
+(4.6)
+are nonnegative.
+For each i = 1, . . . , I, the sequence ξi
+k is weakly-∗ convergent in L∞ to ξi ≥ 0. Following [14], we
+deduce from (4.5) that, for each i = 1, . . . , I,
+� T
+0
+|ξi⟨∇x ˆψi, p⟩|dt = lim
+k→∞
+� T
+0
+|ξi
+k⟨∇x ˆψi, p⟩|dt
+≤ lim
+k→∞
+�� T
+0
+ξi
+k|⟨∇x ˆψi, p⟩ − ⟨∇xψi
+k, pk⟩|dt +
+� T
+0
+ξi
+k|⟨∇xψi
+k, pk⟩|dt
+�
+≤ lim
+k→∞
+����ξi
+k
+���
+L∞
+���⟨∇x ˆψi, p⟩ − ⟨∇xψi
+k, pk⟩
+���
+L1 + N2
+γk
+�
+= 0.
+It turns out that
+ξi⟨∇x ˆψi, p⟩ = 0 a.e..
+(4.7)
+Consider now the sequence of scalars {λk}. It is an easy matter to show that there exists a subsequence
+of {λk} converging to some λ ≥ 0. This, together with the convergence of pk to p, allows us to take limits
+in (a) and (c) of Proposition 3.2 to deduce that
+λ + |p(T)|= 1
+and
+⟨p(t), f(t, ˆx(t), u)⟩ − αλ|u − ˆu(t)|≤ ⟨p(t), f(t, ˆx(t), ˆu(t))⟩ ∀u ∈ U, a.e. t ∈ [0, T].
+It remains to take limits of the transversality conditions (d) in Proposition 3.2. First, observe that
+CT + ϵkBn = {x : d(x, CT ) ≤ ϵk} .
+From the basic properties of the Mordukhovich normal cone and subdifferential (see [19], section 1.3.3)
+we have
+NCT +ϵkBn(xk(T)) ⊂ cl cone ∂d(xk(T), CT )
+and
+NCT (ˆx(T)) = cl cone ∂d(ˆx(T), CT ).
+Passing to the limit as k → ∞ we get
+(p(0), −p(T)) ∈ NC0(ˆx(0)) × NCT (ˆx(T)) + {0} × λ ∂φ(ˆx(T)).
+Finally, and mimicking Step 3 in the proof of Theorem 2 in [10], we remove the dependence of the
+conditions on the parameter α. This is done by taking further limits, this time considering a sequence of
+αj ↓ 0.
+We then summarize our conclusions in the following Theorem.
+Theorem 4.1 Let (ˆx, ˆu) be the optimal solution to (P). Suppose that assumption A1–A6 are satisfied.
+For i = 1, · · · , I, set
+Ii
+b = {t ∈ [0, T] : ψi(t, ˆx(t)) < 0}.
+There exist λ ≥ 0, p ∈ BV ([0, T], Rn), finite signed Randon measures ηi, null in Ii
+b, for i = 1, · · · , I,
+ξi ∈ L∞([0, T], R), with i = 1, · · · , I, where ξi(t) ≥ 0 a. e. t and ξi(t) = 0, t ∈ Ii
+b, such that
+a) λ + |p(T)|̸= 0,
+14
+
+b) ˙ˆx(t) = f(t, ˆx(t), ˆu(t)) −
+I
+�
+i=1
+ξi(t)∇x ˆψi(t),
+c) for any z ∈ C([0, T]; Rn)
+� T
+0
+⟨z(t), dp(t)⟩ = −
+� T
+0
+⟨z(t), (∇x ˆf(t))∗p(t)⟩dt
++
+I
+�
+i=1
+�� T
+0
+ξi(t)⟨z(t), ∇2
+x ˆψi(t)p(t)⟩dt +
+� T
+0
+⟨z(t), ∇x ˆψi(t)⟩dηi
+�
+,
+where ∇ ˆf(t) = ∇xf(t, ˆx(t), ˆu(t)),
+∇ ˆψi(t) = ∇ψi(t, ˆx(t)) and ∇2 ˆψi(t) = ∇2ψi(t, x(t)),
+d) ξi(t)⟨∇xψi(t, ˆx(t)), p(t)⟩ = 0, a.e. t for all i = 1, . . . , I,
+e) for all i = 1, . . . , I, the meaures ⟨∇ψi(ˆx(t), p(t)⟩dηi(t) are nonnegative,
+f) ⟨p(t), f(t, ˆx(t), u)⟩ ≤ ⟨p(t), f(t, ˆx(t), ˆu(t))⟩ for all u ∈ U, a.e. t,
+g)
+(p(0), −p(T)) ∈ NC0(ˆx(0)) × NCT (ˆx(T)) + {0} × λ∂φ(ˆx(T)).
+Noteworthy, condition e) is not considered in any of our previous works.
+We now turn to the free end point case, i. e., to the problem
+(Pf)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Minimize φ(x(T))
+over processes (x, u) such that
+˙x(t) ∈ f(t, x(t), u(t)) − NC(t)(x(t)), a.e. t ∈ [0, T],
+u(t) ∈ U,
+a.e. t ∈ [0, T],
+x(0) ∈ C0 ⊂ C(0).
+Problem (Pf) differs from (P) because x(T) is not constrained to take values in CT . We apply Theorem
+4.1 to (Pf). Since x(T) is free, we deduce from (f) in the above Theorem that −p(T) = λ∂φ(ˆx(T)).
+Suppose that λ = 0.
+Then p(T) = 0 contradicting the nontriviality condition (a) of Theorem 4.1.
+Without loss of generality, we then conclude that the conditions of Theorem 4.1 hold with λ = 1. We
+summarize our findings in the following Corollary.
+Corollary 4.2 Let (ˆx, ˆu) be the optimal solution to (Pf). Suppose that assumption A1–A6 are satisfied.
+For i = 1, · · · , I, set
+Ii
+b = {t ∈ [0, T] : ψi(t, ˆx(t)) < 0}.
+There exist p ∈ BV ([0, T], Rn), finite signed Randon measures ηi, null in Ii
+b, for i = 1, · · · , I, ξi ∈
+L∞([0, T], R), with i = 1, · · · , I, where ξi(t) ≥ 0 a.e. t and ξi(t) = 0 for t ∈ Ii
+b, such that
+a) ˙ˆx(t) = f(t, ˆx(t), ˆu(t)) −
+I
+�
+i=1
+ξi(t)∇x ˆψi(t),
+15
+
+b) for any z ∈ C([0, T]; Rn)
+� T
+0
+⟨z(t), dp(t)⟩ = −
+� T
+0
+⟨z(t), (∇x ˆf(t))∗p(t)⟩dt
++
+I
+�
+i=1
+�� T
+0
+ξi(t)⟨z(t), ∇2
+x ˆψi(t)p(t)⟩dt +
+� T
+0
+⟨z(t), ∇x ˆψi(t)⟩dηi
+�
+,
+where ∇ ˆf(t) = ∇xf(t, ˆx(t), ˆu(t)),
+∇ ˆψi(t) = ∇ψi(t, ˆx(t)) and ∇2 ˆψi(t) = ∇2ψi(t, x(t)),
+c) ξi(t)⟨∇xψi(t, ˆx(t)), p(t)⟩ = 0 for a.e. t and for all i = 1, . . . , I,
+d) for all i = 1, . . . , I, the meaures ⟨∇ψi(ˆx(t), p(t)⟩dηi(t) are nonnegative,
+e) ⟨p(t), f(t, ˆx(t), u)⟩ ≤ ⟨p(t), f(t, ˆx(t), ˆu(t))⟩ for all u ∈ U, a.e. t,
+f)
+(p(0), −p(T)) ∈ NC0(ˆx(0)) × {0} + {0} × ∂φ(ˆx(T)).
+5
+Example
+Let us consider the following problem
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Minimize
+− x(T)
+over processes ((x, y, z), u) such that
+�
+�
+˙x(t)
+˙y(t)
+˙z(t)
+�
+� ∈
+�
+�
+0
+σ
+0
+0
+0
+0
+0
+0
+0
+�
+�
+�
+�
+x
+y
+z
+�
+� +
+�
+�
+0
+u
+0
+�
+� − NC(x, y, z),
+u ∈ [−1, 1],
+(x, y, z)(0) = (x0, y0, z0),
+(x, y, z)(T) ∈ CT ,
+where
+• 0 < σ ≪ 1,
+• C = {(x, y, z) | x2 + y2 + (z + h)2 ≤ 1, x2 + y2 + (z − h)2 ≤ 1}, 2h2 < 1,
+• (x0, y0, z0) ∈ intC, with x0 < −δ, y0 = 0 and z0 > 0,
+• CT = {(x, y, z) | x ≤ 0, y ≥ 0, δy − y2x ≤ δy2} ∩ C, where
+δ < y2|x0|
+y1
+, with y1 =
+�
+1 − x2
+0 − (z0 + h)2 and y2 =
+�
+1 − h2.
+We choose T > 0 small and, nonetheless, sufficiently large to guarantee that, when σ = 0, the system
+can reach the interior of CT but not the segment {(x, 0, 0) | x ∈ [−δ, 0]}. Since σ and T are small, it
+follows that the optimal trajectory should reach CT at the face δy − y2x = δy2 of CT .
+16
+
+To significantly increase the value of the x(T), the optimal trajectory needs to live on the boundary
+of C for some interval of time. Then, before reaching and after leaving the boundary of C, the optimal
+trajectory lives in the interior of C. Since δ is small, the trajectory cannot reach CT from any point of
+the sphere x2 +y2 +(z +h)2 = 1 with z > 0. This means that, while on the boundary of C the trajectory
+should move on the sphere x2 + y2 + (z + h)2 = 1 untill reaching the plane z = 0 and then it moves on
+the intersection of the two spheres.
+While in the interior of C, the control can change sign from −1 to 1 or from 1 to −1. Certainly, the
+control should be 1 right before reaching the boundary and −1 right before arriving at CT . Changes of
+the control from 1 to −1 or −1 to 1 before reaching the boundary translate into time waste and leads to
+smaller values of x(T). It then follows that the optimal control should be of the form
+u(t) =
+�
+1,
+t ∈ [0, ˜t],
+−1,
+t ∈ ]˜t, T],
+(5.1)
+for some value ˜t ∈]0, T[.
+After the modification (2.5), the data of the problem satisfy the conditions under which Theorem 4.1
+holds. We now show that the conclusions of Theorem 4.1 completly identify the structure (5.1) of the
+optimal control.
+From Theorem 4.1 we deduce the existence of λ ≥ 0, p, q, r ∈ BV ([0, T], R), finite signed Randon
+measures η1 and η2, null respectively in
+I1
+b =
+�
+(x, y, z) | x2 + y2 + (z + h)2 − 1 < 0
+�
+and
+I2
+b =
+�
+(x, y, z) | x2 + y2 + (z − h)2 − 1 < 0
+�
+,
+ξi ∈ L∞([0, T], R), with i = 1, 2, where ξi(t) ≥ 0 a. e. t and ξi(t) = 0, t ∈ Ii
+b, such that
+(i)
+�
+�
+˙x(t)
+˙y(t)
+˙z(t)
+�
+� =
+�
+�
+0
+σ
+0
+0
+0
+0
+0
+0
+0
+�
+�
+�
+�
+x
+y
+z
+�
+� +
+�
+�
+0
+u
+0
+�
+� − 2ξ1
+�
+�
+x
+y
+z + h
+�
+� − 2ξ2
+�
+�
+x
+y
+z − h
+�
+�
+(ii)
+d
+�
+�
+p
+q
+r
+�
+� =
+�
+�
+0
+0
+0
+−σ
+0
+0
+0
+0
+0
+�
+�
+�
+�
+p
+q
+r
+�
+� dt
++2(ξ1 + ξ2)
+�
+�
+p
+q
+r
+�
+� dt + 2
+�
+�
+x
+y
+z + h
+�
+� dη1 + 2
+�
+�
+x
+y
+z − h
+�
+� dη2,
+(iii)
+�
+�
+p
+q
+r
+�
+� (T) =
+�
+�
+λ
+0
+0
+�
+� + µ
+�
+�
+y2
+−δ
+0
+�
+� , where µ ≥ 0,
+(iv)
+ξ1(xp + yq + (z + h)r) = 0, ξ2(xp + yq + (z − h)r) = 0,
+(v)
+the meaures (xp + yq + (z + h)r)dη1 and (xp + yq + (z − h)r)dη2
+are nonnegative,
+(vi)
+maxu∈[−1,1] uq = ˆuq.
+where ˆu is the optimal control.
+Let t1 be the instant of time when the trajectory reaches the shere x2 + y2 + (z + h)2 = 1, t2 the
+instant of time when the trajectory reaches the intersection of the two spheres and t3 be the instant of
+time the trajectory leaves the boundary of C. We have 0 < t1 < t2 < t3 < T.
+Next we show that the multiplier q changes sign only once and so identifing the structure (5.1) of the
+optimal control in a unique way. We start by looking at the case when t = T. We have
+�
+p
+q
+�
+(T) =
+�
+λ
+0
+�
++ µ
+�
+y2
+−δ
+�
+.
+17
+
+Starting from t = T, let us go backwards in time until the instant t3 when the trajectory leaves the
+boundary of C. If q(T) = 0, then p(T) = λ > 0 and we would have q(t) > 0 for t ∈]t3, T[ (see (ii) above),
+which is impossible. We then have p(T) > 0 and q(T) < 0 and, in ]t3, T[, since σ is small, the vector
+(p(t), q(t)) does not change much. At t = t3, the vector (p, q) has a jump and such jump can only occur
+along the vector (x(t3), y(t3)). Therefore, we have p(t3 − 0) > 0 and q(t3 − 0) < 0.
+Let us now consider t ∈]t2, t3[. We have the following
+1. when t ∈ [t2, t3], we have z = 0;
+2. condition (i) above implies that ξ1 = ξ2 = ξ, ξ > 0 since, otherwise the motion along x2+y2 = 1−h2
+would not be possible;
+3. from 0 = d
+dt(x2 + y2) = σ2xy − 8ξx2 + 2uy − 8ξy2 we get ξ = σxy+uy
+4(1−h2);
+4. condition (iv) implies that r = 0 leading to xp + yq = 0. Since x < 0, y > 0, then q = 0 implies
+p = 0;
+5. condition (ii) implies dη1 = dη2 = dη;
+6. 0 = d(xp + yq) = uqdt + 4(1 − h2)dη ⇒ dη
+dt = −
+uq
+4(1−h2);
+7. from the above analysis we deduce that
+˙p = σxy + uy
+(1 − h2) p −
+xuq
+(1 − h2),
+˙q = −σp +
+σxy
+(1 − h2) q.
+Thus, (p, q) is a solution to a linear system and it can never be equal to zero. It follows that q
+cannot be zero because q = 0 implies p = 0. Since q ̸= 0, we have q > 0.
+Let us consider the case when t = t2. We claim that
+(p(t2 − 0), q(t2 − 0)) ̸= (0, 0).
+Seeking a contradiction, assume that it is (p(t2 − 0), q(t2 − 0)) = (0, 0). Then we have
+(p(t2 + 0), q(t2 + 0)) = (0, 0) + (2x2(t2), 2y2(t2))(dη1 + dη2)
+and such jump has to be normal to (x(t2), y(t2)) since r(t2 + 0) = 0 (see (iv)). It follows that (x2(t2) +
+y2(t2))(dη1 + dη2) = 0 and, since x2(t2) + y2(t2) > 0, we get dη1 + dη2 = 0, proving our claim.
+We now consider t ∈]t1, t2[. It is easy to see that ξ2 = 0 and dη2 = 0. We also deduce that
+1. 0 = d
+dt(x2+y2+(z+h)2) = 2σxy+2uy−4ξ1y2−4ξ1x2−4ξ1(z+h)2 which implies that ξ1 = σxy+uy
+2
+;
+2. also 0 = d(xp + yq + (z + h)r) = uqdt + 2dη1 implies that dη1
+dt = − uq
+2 ;
+3. from the above we deduce that
+˙p = (σxy + uy)p − xuq,
+˙q = −σp + σxyq.
+Thus (p, q) is a solution to a linear system and never is equal to zero. Second equation implies that
+if q = 0 then ˙q ̸= 0. Hence q > 0.
+18
+
+Now we need to consider t = t1. We claim that
+(p(t1 − 0), q(t1 − 0), r(t1 − 0)) ̸= (0, 0, 0).
+Let us then assume that it is (p(t1−0), q(t1−0), r(t1−0)) = (0, 0, 0). It then follows that (p(t1+0), q(t1+
+0), r(t1 +0)) = (0, 0, 0)+(2x(t1)dη1, 2y(t1)dη1, 2(z(t1)+h)dη1). We now show that there is no such jump.
+Set r(t1 − 0) = r0. Then it follows from (iv) that (x(t1) · 0 + y(t1) · 0 + (z(t1) + h))r0 = 0 which implies
+that r0 = 0. We also have (x2(t1) + y2(t1) + (z(t1) + h)2)dη1 = 0 from (v). But this implies that dη1 = 0.
+Consequently, the multipliers do not exhibit a jump at t1.
+From the previous analysis we deduce that q should be positive almost everywhere on the boundary. It
+then follows that to find the optimal solution we have to analyze admissible trajectories with the controls
+with the structure (5.1) and choose the optimal value of ˜t.
+Acknowledgements
+The authors gratefully thank the support of Portuguese Foundation for Science and Technology (FCT)
+in the framework of the Strategic Funding UIDB/04650/2020.
+Also we thank the support by the ERDF - European Regional Development Fund through the Oper-
+ational Programme for Competitiveness and Internationalisation - COMPETE 2020, INCO.2030, under
+the Portugal 2020 Partnership Agreement and by National Funds, Norte 2020, through CCDRN and
+FCT, within projects To Chair (POCI-01-0145-FEDER-028247), Upwind (PTDC/EEI-AUT/31447/2017
+- POCI-01-0145-FEDER-031447) and Systec R&D unit (UIDB/00147/2020).
+References
+[1] Addy K, Adly S, Brogliato B, Goeleven D, A method using the approach of Moreau and Pana-
+giotopoulos for the mathematical formulation of non-regular circuits in electronics, Nonlinear
+Anal. Hybrid Syst., vol. 1, 30–43, (2013), https://doi.org/10.1016/j.nahs.2006.04.00.
+[2] Arroud
+C
+and
+Colombo
+G,
+Necessary
+conditions
+for
+a
+nonclassical
+control
+problem
+with state constraints,
+20th IFAC World Congress,
+Toulouse,
+France,
+July 9-14,
+2017,
+https://doi.org/10.1016/j.ifacol.2017.08.110.
+[3] Arroud C and Colombo G, A maximum principle for the controlled sweeping process, Set-Valued
+Var. Anal 26, 607–629 (2018) DOI: 10.1007/s11228-017-0400-4.
+[4] Brokate M, Krejˇc´ı P Optimal control of ODE systems Involving a rate independent variational in-
+equality, Disc. Cont. Dyn. Syst. Ser. B, vol. 18 (2) 331–348 (2013), doi: 10.3934/dcdsb.2013.18.331.
+[5] Cao TH, Mordukhovich B, Optimality conditions for a controlled sweeping process with applica-
+tions to the crowd motion model, Disc. Cont. Dyn. Syst. Ser. B, vol. 22, 267–306 (2017).
+[6] Cao
+TH,
+Colombo
+G,
+Mordukhovich
+B,
+Nguyen
+D.,
+Optimization
+of
+fully
+con-
+trolled
+sweeping
+processes,
+Journal
+of
+Differential
+Equations,
+295,
+138–186
+(2021)
+https://doi.org/10.1016/j.jde.2021.05.042
+[7] Clarke F, Optimization and nonsmooth analysis, John Wiley, New York (1983).
+[8] Colombo G, Palladino M, The minimum time function for the controlled Moreau’s sweeping
+process, SIAM, vol. 54, no. 4,2036– 2062 (2016), https://doi.org/10.1137/15M1043364.
+[9] Colombo G,Henrion R, Hoang ND, Mordukhovich BS, Optimal control of the sweeping process
+over polyhedral controlled sets, Journal of Differential Equations, vol. 260, 4, 3397–3447, (2016),
+https://doi.org/10.1016/j.jde.2015.10.039.
+19
+
+[10] de Pinho MdR, Ferreira MMA, Smirnov G, Optimal Control involving Sweeping Processes, Set-
+Valued Var. Anal 27, 523–548, (2019), https://doi.org/10.1007/s11228-018-0501-8.
+[11] de Pinho MdR, Ferreira MMA, Smirnov G, Correction to: Optimal Control Involving Sweeping
+Processes, Set-Valued Var. Anal 27, 1025–1027 (2019) https://doi.org/10.1007/s11228-019-00520-
+5.
+[12] de Pinho MdR, Ferreira MMA, Smirnov G, Optimal Control with Sweeping Processes: Numerical
+Method, J Optim Theory Appl 185, 845– 858 (2020) https://doi.org/10.1007/s10957-020-01670-5
+[13] de Pinho MdR, Ferreira MMA, Smirnov G, Optimal Control Involving Sweeping Processes with
+End Point Constraints, 2021 60th IEEE Conference on Decision and Control (CDC), 2021, 96–
+101(2019) doi: 10.1109/CDC45484.2021.9683291
+[14] de Pinho MdR, Ferreira MMA, Smirnov G, Necessary conditions for optimal control problems
+with sweeping systems and end point constraints, Optimization, to appear (2022).
+[15] Hermosilla C, Palladino M, Optimal Control of the Sweeping Process with a Non-Smooth Moving
+Set, SIAM j. Cont. Optim., to appear (2022).
+[16] Kunze M, Monteiro Marques MDP, An Introduction to Moreau’s sweeping process. Impacts in
+Mechanical Systems, Lecture Notes in Physics, vol. 551, 1–60, (2000).
+[17] Maury B, Venel J (2011), A discrete contact model for crowd motion, ESAIM: M2AN 45 1,
+145–168.
+[18] Moreau JJ, On unilateral constraints, friction and plasticity, In: Capriz G., Stampacchia G. (Eds.)
+New Variational Techniques in Mathematical Physics, CIME ciclo Bressanone 1973. Edizioni
+Cremonese, Rome, 171–322 (1974).
+[19] Mordukhovich B, Variational analysis and generalized differentiation. Basic Theory. Fundamental
+Principles of Mathematical Sciences 330, Springer-Verlag, Berlin (2006).
+[20] Mordukhovich B, Variational analysis and generalized differentiation II. Applications, Fundamen-
+tal Principles of Mathematical Sciences 330, Springer-Verlag, Berlin (2006).
+[21] Thibault L, Moreau sweeping process with bounded truncated retraction, J. Convex Anal, vol.
+23, pp. 1051–1098 (2016).
+[22] Vinter RB, Optimal Control, Birkh¨auser, Systems and Control: Foundations and Applications,
+Boston MA (2000).
+[23] Zeidan V, Nour C, Saoud H, A nonsmooth maximum principle for a controlled noncon-
+vex sweeping process, Journal of Differential Equations, vol. 269 (11), 9531–9582 (2020),
+https://doi.org/10.1016/j.jde.2020.06.053
+20
+

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+arXiv:2301.13056v1  [math.AG]  30 Jan 2023
+EQUIVARIANT ORIENTED HOMOLOGY OF THE AFFINE
+GRASSMANNIAN
+CHANGLONG ZHONG
+Abstract. We generalize the property of small-torus equivariant K-homology of the affine Grass-
+mannian to general oriented (co)homology theory in the sense of Levine and Morel. The main tool
+we use is the formal affine Demazure algebra associated to the affine root system. More precisely, we
+prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the
+GKM condition. We also show that its dual, the small-torus equivariant homology, is isomorphic to
+the centralizer of the equivariant oriented cohomology of a point in the the formal affine Demazure
+algebra.
+0. Introduction
+Let h be an oriented cohomology theory in the sense of Levine and Morel. Let G be a semi-simple
+linear algebraic group over C with maximal torus T and a Borel subgroup B. Let GrG be the affine
+Grassmannian of G. T is called the small torus, in contrary to the big torus Ta of GrG. The theory of
+hTa(GrG) when h is the equivariant cohomology or the K-theory, is studied by Kostant and Kumar
+in [KK86, KK90]. It is dual to the so-called affine nil-Hecke algebra (equivariant cohomology case)
+or the affine 0-Hecke algebra. Alternatively, the affine nil-Hecke algebra and the affine 0-Hecke
+algebra can be called the equivariant homology and the equivariant K-homology theory.
+The small torus equivariant homology theory HT(GrG) of the affine Grassmannian was first
+studied by Peterson [P97]. Moreover, he raised a conjecture (without a proof) saying that HT(GrG)
+is isomorphic to the quantum cohomology QHT (G/B)of G/B. This conjecture, together with its
+partial flag variety version, is proved by Lam-Shimozono in [LS10]. One key step is the identification
+of HT(G/B) with the centralizer of HT (pt) in HT (Ga/Ba) where Ga is the Kac-Moody group
+associated to the affine root system and Ba is its Borel subgroup.
+For K-theory, similar property was expected to hold.
+In [LSS10], the authors study the K-
+theoretic Peterson subalgebra, i.e., the centralizer of the ring KT (pt) in the small-torus affine
+0-Hecke algebra, i.e., the equivariant K-homology KT (Ga/Ba). It is proved that this algebra is iso-
+morphic to KT (GrG). One of the main tools is the small-torus GKM condition of the T-equivariant
+K-cohomology. In [LLMS18], some evidence was provided in supporting the K-theory Peterson
+Conjecture. In [K18], using the study of semi-infinite flag variety, Kato proves this conjecture.
+More precisely, he embeds quantum K-theory of flag variety and certain localization of the Peter-
+son subalgebra into T-equivariant K-theory of semi-infinite flag variety, and proves that their image
+coincide.
+In all the work mentioned above, the Peterson subalgebra plays key roles. In this paper, we
+generalize the construction of the Peterson subalgebra into general oriented cohomology theory
+h. Associated to such theory, there is a formal group law F over the coefficient ring R = h(pt).
+Associated to F and a Kac-Moody root system, in [CZZ16, CZZ19, CZZ15, CZZ20], the author
+generalized Kostant-Kumar’s construction and defined the formal affine Demazure algebra (FADA).
+It is a non-commutative algebra generated by the divided difference operators. Its dual give an
+algebraic model for hTa(Ga/Ba). Since Levine-Morel’s oriented cohomology theory is only defined
+1
+
+2
+C. ZHONG
+for smooth projective varieties, in this paper we do not intend to generalize the geometric theory.
+Instead, we only work with the algebraic model, i.e., the FADA associated to h.
+Following the same idea as the work mentioned above in cohomology and K-theory, we look at
+the small-torus (the torus T) version, which is very similar as the big torus case Ta. We define
+the small torus FADA, DWa. In this paper, our first main result (Theorem 4.3) shows that the
+algebraic models for hT (Ga/Ba) and hT (GrG), i.e., D∗
+Wa and (D∗
+Wa)W , satisfy the small torus GKM
+condition. Based on that, we prove the second main result (Theorem 5.5), which shows that the
+dual of hT (GrG), denoted by DQ∨ (Q∨ being the coroot lattice), coincides with the centralizer of
+hT (pt) in the FADA DWa. This defines the Peterson subalgebra associated to h.
+Our result generalizes and extends properties for equivariant cohomology and K-theory. More-
+over, our method is uniform and does not reply on the specific oriented cohomology theory. As
+an application of this construction, we define actions of the FADA (of the big and small torus) on
+the algebraic models fo hTa(GrG) and hT (GrG). This is called the left Hecke action. For finite flag
+varieties case it is studied in [MNS22] using geometric arguments (see also [B97, K03, T09, LZZ20]).
+For connective K-theory (which specializes to cohomology and K-theory), we compute the recursive
+formulas for certain basis in hT (GrG) (Theorem 2.3).
+It is natural to consider generalizing Kato’s construction to this case, that is, invert Schubert
+classes in DQ∨ corresponding to tλ ∈ Q∨
+<. This localization for K-theory was proved to be isomor-
+phic to QKT (G/B). For h beyong singular cohomology and K-theory, however, the first obstruction
+is that there is no ‘quantum’ oriented cohomology theory defined. The other obstruction is that
+the divided difference operators do not satisfy braid relations.
+This was a key step in Kato’s
+construction (see [K18, Theorem 1.7]). The author plans to investigate this in a future paper.
+This paper is organized as follows: In §1 we recall the construction of the FADA for the big torus
+Ta, and in §2 we compute the recursive formulas via the left Hecke action. In §3 we we repeat the
+construction for the small torus and indicates the difference from the big torus case. In §4 we prove
+that dual of the small torus FADA satisfies the small torus GKM condition, and in §5 we define
+the Peterson subalgebra and show that it coincides with the centralizer of hT (pt). In the appendix
+we provide some computational result in the ˆA1 case.
+Notations. Let G ⊃ B ⊃ T be such that G is simple, simply connected algebraic group over C
+with a Borel subgroup B and a torus T. Let Ga ⊃ Ba ⊃ Ta where Ga is the affine Kac-Moody group
+with Borel subgroup Ba and the affine torus Ta. Let P be the maximal parabolic group scheme so
+that Ga/P = GrG is the affine Grassmannian. Let T ∗ (resp. T ∗
+a ) be the group of characters of T
+(resp. Ta), then T ∗
+a = T ∗ ⊕ Zδ.
+Let W be the Weyl group of G, I = {α1, ..., αn} be the simple roots, Q = ⊕iZαi ⊂ T ∗ be the root
+lattice, Q∨ = ⊕iZα∨
+i be the coroot lattice, θ be the longest element, δ is the null root, α0 = −θ + δ
+be the extra simple root. Denote Ia = {α0, ..., αn}. For each λ ∈ Q∨, let tλ be the translation
+acting on Q. We then have tλ1tλ2 = tλ1+λ2, and wtλw−1 = tw(λ), w ∈ W. Let Q∨
+≤ be the set of
+antidominant coroots, Q∨
+< be the set of strictly antidominant coroots (i.e., (λ, αi) < 0 ∀i ∈ I). Let
+Wa = W ⋉ Q∨ be the affine Weyl group, ℓ be the length function on Wa, and w0 ∈ W be the
+longest element.
+Let Φ be the set of roots for W, Φa = Zδ + Φ be the set of real affine roots, and Φ±
+a , Φ±
+be the corresponding set of positive/negative roots for the corresponding systems. Let inv(w) =
+w−1Φ+
+a ∩ Φ−
+a . We have
+Φ+
+a = {α + kδ|α ∈ Φ+, k = 0 or α ∈ Φ, k > 0}.
+
+AFFINE GRASSMANNIAN
+3
+Let W −
+a be the minimal length representatives of Wa/W. There is a bijection
+W −
+a → Q∨,
+w �→ λ, if wW = tλW.
+Moreover, W −
+a ∩ Q∨ = {tλ|λ ∈ Q≤}. The action of α + kδ on µ + mδ ∈ Q ⊕ Zδ is given by
+sα+kδ(µ + mδ) = µ + mδ − ⟨µ, α∨⟩(α + kδ).
+In particular, for λ ∈ Q∨, w ∈ W, µ ∈ Q, we have sα+kδ = sαtkα∨, wtλ(µ) = w(µ).
+We say the set of reduced sequences Iw, w ∈ Wa is W-compatible if Iw = Iu ∪ Iv for w = uv, u ∈
+W −
+a , v ∈ W.
+1. FADA for the big torus
+In this section, we recall the construction of the formal affine Demazure algebra (FADA) for the
+affine root system. All the construction can be found in [CZZ20].
+1.1.
+Let F be a one dimensional formal group law over a domain R with characteristic 0. Following
+from [LM07] that there is an oriented cohomology h whose associated formal group law is F. In
+this paper we won’t need any geometric property of this h, since our treatment is pure algebraic
+and self-contained.
+Example 1.1. Let F = Fc = x + y − cxy be the connective formal group law (for connective
+K-theory) over R = Z[c]. Specializing to c = 0 or c = 1, one obtains the additive or multiplicative
+formal group law. One of the simplest formal group laws beyond Fc is the hyperbolic formal group
+law considered in [LZZ20]:
+F(x, y) = x + y − cxy
+1 + axy
+, R = Z[c, a].
+Let ˆS be the formal group algebra of T ∗
+a defined in [CPZ13]. That is,
+ˆS = R[[xµ|µ ∈ T ∗
+a ]]/JF ,
+where JF is the closure of the ideal generated x0 and xµ1+µ2 − F(xµ1, xµ2), µ1, µ2 ∈ T ∗
+a . Indeed,
+after fixing a basis of T ∗
+a ∼= Zn+1, ˆS is isomorphic to the power series ring R[[x1, ..., xn+1]].
+Remark 1.2. If F = Fc is the connective formal group law, one can just replace ˆS by R[xµ|µ ∈
+T ∗
+a ]/JF . In other words, in this case one can use the polynomial ring instead of the power series ring.
+For instance, if c = 0, then ˆS ∼= SymR(T ∗
+a ), xµ �→ µ. If c ∈ R×, then ˆS ∼= R[T ∗
+a ], xµ �→ c−1(1 − e−µ).
+Throughout this paper, whenever we specializes to Fc, we assume that ˆS is the polynomial version.
+1.2.
+Define ˆQ = ˆS[ 1
+xα , α ∈ Φa]. The Weyl groups Wa acts on ˆQ, so we can define the twisted group
+algebra ˆQWa := ˆQ ⋊ R[Wa],, which is a free left ˆQ-module with basis denoted by ηw, w ∈ Wa and
+the product cηwc′ηw′ = cw(c′)ηww′, c, c′ ∈ ˆQ.
+For each α ∈ Φa, define κα =
+1
+xα +
+1
+x−α ∈ ˆS. If F = Fc, then κα = c. For each simple root
+αi, we define the Demazure element ˆXαi =
+1
+xαi (1 − ηsi). It is easy to check that ˆX2
+α = κα ˆXα. For
+simplicity, denote ηi = ηαi = ηsi, x±i = x±αi, ˆXi = ˆXαi, i ∈ Ia. If Iw = (i1, ..., ik), ij ∈ Ia is a
+reduced sequence of w ∈ Wa, we define ˆXIw correspondingly. It is well known that they depends
+on the choice of Iw, unless F = Fc.
+Write
+(1)
+ˆXIw =
+�
+v≤w
+ˆaIw,vηv,
+ηw =
+�
+v≤w
+ˆbw,Iv ˆXIv,
+ˆaIw,v ∈ ˆQ, ˆbw,Iv ∈ ˆS,
+
+4
+C. ZHONG
+then we have ˆbw,Iw = �
+α∈inv(w) xα =
+1
+ˆaIw,w .
+Let ˆDWa be the subalgebra of ˆQWa generated by ˆS and ˆXi, i ∈ Ia. This is called the formal
+affine Demazure algebra (FADA) for the big torus. It is easy to see that ˆXIw, w ∈ Wa is a ˆQ-basis
+of ˆQWa, and it is proved in [CZZ20] that it is also a basis of the left ˆS-module ˆDWa. Note that
+W ⊂ ˆDWa via the map si �→ ηi = 1 − xi ˆXi ∈ ˆDWa.
+Remark 1.3. It is not difficult to derive that there is a residue description of the coefficients in
+the expression of elements of ˆDWa as linear combinations of ηw. Such description was first given in
+[GKV97]. See [ZZ17] for more details.
+1.3.
+We define the duals of left modules:
+ˆQ∗
+Wa = Hom ˆ
+Q( ˆQWa, ˆQ) = Hom(Wa, ˆQ),
+ˆD∗
+Wa = Hom ˆS( ˆDWa, ˆS).
+Dual to the elements ηw, ˆXIw ∈ ˆDWa ⊂ ˆQWa, we have ˆfw, ˆX∗
+Iw ∈ ˆD∗
+Wa ⊂ ˆQ∗
+Wa.
+The product
+structure on ˆQ∗
+Wa is defined by ˆfw ˆfv = δw,v ˆfw, with the unit given by 1 = �
+w∈Wa ˆfw. Note that
+here we usually use � to denote a sum of (possibly) infinitely many terms, and � to denote a
+finite sum.
+Lemma 1.4. We have
+ˆD∗
+Wa = { ˆf ∈ ˆQ∗
+Wa| ˆf( ˆDWa) ⊂ ˆS}.
+Proof. Denote the RHS by Z1. It is clear that ˆD∗
+Wa is contained in Z1 since ˆXIv generate ˆDWa,
+ˆX∗
+Iw generate ˆD∗
+Wa, and ˆX∗
+Iw( ˆXIv) = δw,v. Conversely, let ˆf = �
+ℓ(w)≥k cw ˆfw ∈ Z1. If ℓ(u) = k,
+then from (1), we have
+ˆf( ˆXIu) =
+�
+ℓ(w)≥k
+cw ˆfw(
+�
+v≤u
+ˆaIu,vηv) = cuˆaIu,u ∈ ˆS.
+Denote ˆf ′ := ˆf −�
+ℓ(u)=k cuˆaIu,u ˆX∗
+Iu. Note that ˆX∗
+Iu = �
+w∈Wa ˆbw,Iufw and ˆbu,IuˆaIu,u = 1, so for any
+u with ℓ(u) = k, we have ˆf ′(ηu) = cu − cuˆaIu,u ˆX∗
+Iu(ηu) = cu − cu = 0, so ˆf ′ is a linear combination
+of ˆfw, ℓ(w) ≥ k + 1. Repeating this process, we get that ˆf ∈ ˆD∗
+Wa.
+□
+1.4.
+There is an ˆQ-linear action of ˆQWa on ˆQ∗
+Wa, defined by
+(z • ˆf)(z′) = ˆf(z′z),
+z, z′ ∈ ˆQWa, ˆf ∈ ˆQ∗
+Wa.
+This is called the right Hecke action. We have
+cηw • c′ ˆfw′ = c′w′w−1(c) ˆfw′w−1, c, c′ ∈ ˆQ.
+It follows from Lemma 1.4 and similar reason as in [CZZ19, §10] that this induces an action of ˆDWa
+on ˆD∗
+Wa. Moreover, it induces an action of W ⊂ ˆDWa on ˆQ∗
+Wa and ˆD∗
+Wa. By definition it is easy to
+get
+ˆXα •
+�
+w∈Wa
+cw ˆfw =
+�
+w∈Wa
+cw − csw(α)w
+xw(α)
+ˆfw.
+(2)
+The following proposition is proved in the finite case in [CZZ19, Lemma 10.2, Theorem 10.7].
+Proposition 1.5. The subset ˆD∗
+Wa ⊂ ˆQ∗
+Wa satisfies the following (big-torus) GKM condition:
+ˆD∗
+Wa = { ˆf ∈ ˆQ∗
+Wa| ˆf(ηw) ∈ ˆS and ˆf(ηw − ηsαw) ∈ xα ˆS, ∀α ∈ Φa}.
+
+AFFINE GRASSMANNIAN
+5
+Proof. Denote the RHS by Z2. Let ˆf ∈ ˆD∗
+Wa, we know ˆXα • ˆf ∈ ˆD∗
+Wa. Then (2) implies that ˆf
+satisfies the condition defining Z2, so ˆD∗
+Wa ⊂ Z2.
+For the other direction, we first show that ˆD∗
+Wa is a maximal ˆDWa-submodule of ˆS∗
+Wa :=
+Hom(Wa, ˆS).
+This can be proved as follows: if M ⊂ ˆS∗
+Wa is a ˆDWa-module, for any ˆf ∈ M,
+we have ˆXI • ˆf ∈ M ⊂ ˆS∗
+Wa, so ˆXI • ˆf(ηe) = ˆf( ˆXI) ∈ ˆS, so f ∈ ˆD∗
+Wa. One then can show that
+the subset Z2 is a ˆDWa-module, which follows from the same proof as in the finite case in [CZZ19,
+Theorem 10.2]. Since ˆD∗
+Wa is a maximal submodule, we have Z2 ⊂ ˆD∗
+Wa. The proof is finished.
+□
+1.5.
+We can similarly define the non-commutative ring ˆQQ∨ = ˆQ ⋊ R[Q∨] with a ˆQ-basis ηtλ, λ ∈
+Q∨. Then there is a canonical map of left ˆQ-modules:
+pr : ˆQWa → ˆQQ∨,
+cηtλw �→ cηtλ,
+w ∈ W, λ ∈ Q∨, c ∈ ˆQ.
+Define ˆDWa/W = pr( ˆDWa) ⊂ ˆQQ∨. Indeed, this is the same as the relative Demazure module
+defined in [CZZ19, §11].
+We can also consider the ˆQ-dual ˆQ∗
+Q∨ and the ˆS-dual ˆD∗
+Wa/W. The elements dual to ηtλ ∈ ˆQQ∨
+are denoted by ˆftλ.
+The projection pr then induces embeddings pr∗ : ˆQ∗
+Q∨ ֒→ ˆQ∗
+Wa and pr∗ :
+ˆDWa/W ֒→ ˆD∗
+Wa. It is easy to see that
+pr∗( ˆftλ) =
+�
+v∈W
+ˆftλv.
+Moreover, similar as in the finite case [CZZ19, Lemma 11.7], we have
+pr∗( ˆQ∗
+Q∨) = ( ˆQ∗
+Wa)W ,
+pr∗( ˆD∗
+Wa/W) = ( ˆD∗
+Wa)W .
+Indeed, elements of pr∗( ˆQ∗
+Q∨) = ( ˆQ∗
+Wa)W are precisely the elements ˆf ∈ ˆQ∗
+Wa satisfying ˆf(ηtλw −
+ηtλ) = 0 for any λ ∈ Q∨, w ∈ W. It follows from similar reason as [CZZ19, Corollary 8.5, Lemma
+11.5] that if Iw, w ∈ Wa is W-compatible, then ˆbuv,Iw = ˆbu,Iw for any v ∈ W. We then have
+Lemma 1.6. Assume the sequences Iw, w ∈ Wa is W-compatible, then pr(XIw), w ∈ W −
+a is a basis
+of ˆDWa/W , and { ˆX∗
+Iw, w ∈ W −
+a } is a ˆQ-basis of ( ˆQ∗
+Wa)W and a ˆS-basis of ( ˆD∗
+Wa)W .
+Note that ( ˆD∗
+Wa)W is the algebraic model for hTa(GrG) and the embedding ( ˆD∗
+Wa)W ⊂ ˆD∗
+Wa is
+the algebraic model for the pull-back hTa(GrG) → hTa(Ga/Ba).
+1.6.
+Similar as the finite case in [LZZ20, §3], there is another action of ˆQWa on ˆQ∗
+Wa by
+aηv ⊙ b ˆfw = av(b) ˆfvw, a, b ∈ ˆQ, w, v ∈ Wa.
+This is called the left Hecke action. It is easy to see that it commutes with the •-action. Note
+however that the ⊙-action is not ˆQ-linear.
+Lemma 1.7. The ⊙ action of ˆQWa on ˆQ∗
+Wa induces an action of ˆDWa on ˆD∗
+Wa.
+Proof. We have
+ˆXα ⊙
+�
+w
+cw ˆfw = 1
+xα
+(1 − ηα) ⊙
+�
+w
+cw ˆfw =
+�
+w
+cw − sα(csαw)
+xα
+ˆfw.
+let dw,α = cw−sα(csαw)
+xα
+. We show that dw,α satisfy the big-torus GKM condition, that is, dw,α −
+dsβw,α ∈ xβ ˆS for any β.
+
+6
+C. ZHONG
+Denote cw − csαw = xαp, p ∈ ˆS and x−α = −xα + x2
+αq, q ∈ ˆS. If β = α, then we have
+dw,α − dsβw,α
+=
+cw − sα(csαw) − csαw + sα(cw)
+xα
+= xαp + sα(cw) − sα(cw − xαp)
+xα
+=
+p + x−αsα(p)
+xα
+= p − sα(p) + xαq,
+which is clearly a multiple of xα. If β ̸= α, then
+dw,α − dsβw,α = cw − sα(csαw) − (csβw − sα(csαsβw))
+xα
+= cw − sα(csαw) − csβw + sα(csαsβw)
+xα
+.
+Since xα, xβ are coprime [CZZ20, Lemma 2.2], it suffices to prove the numerator is divisible by
+xβ. Note cw − csβw is already divisible by xβ. Furthermore, cw − csαsβw = cw − cssα(β)sαw, so it is
+divisible by ssα(β). Therefore, −sα(csαw) + sα(csαsβw) is divisible by sα(xsα(β)) = xβ. The proof is
+finished.
+□
+Consequently, the ⊙-action of ˆDWa on ˆD∗
+Wa restricts to an action on ( ˆD∗
+Wa)W .
+1.7.
+Indeed, there is a characteristic map
+c : ˆS → ˆD∗
+Wa, z �→ z • 1,
+whose geometric model is the map sending a character of the torus to the first Chern class of the
+associated line bundle over the flag variety [CZZ15, §10]. We then have a map
+φ : ˆS ⊗ ˆSWa ˆS → ˆD∗
+Wa, a ⊗ b �→ ac(b) =
+�
+w
+aw(b) ˆfw.
+This is proved to be an isomorphism in some cases. It is easy to see that for any z ∈ DWa, there
+are the following commutative diagrams
+ˆS ⊗ ˆSWa ˆS
+φ
+�
+z· ⊗id
+�
+ˆD∗
+Wa
+z⊙
+�
+ˆS ⊗ ˆSWa ˆS
+φ
+� ˆD∗
+Wa
+,
+ˆS ⊗ ˆSWa ˆS
+φ
+�
+id ⊗z·
+�
+ˆD∗
+Wa
+z•
+�
+ˆS ⊗ ˆSWa ˆS
+φ
+� ˆD∗
+Wa
+.
+2. Equivariant connective K-theory of the affine Grassmannian
+As an application of the left Hecke action, we derive the recursive formulas for this action on
+bases in connective K-theory of GrG. In this section only, assume F = Fc. Our results specialize
+to equivariant K-theory (resp. equivariant cohomology) by letting c = 1 (resp. c = 0). In both
+cases, our results are only known for flag varieties of finite root systems. Since ˆXi do not satisfy
+the braid relations, the result of this section do not generalize to general F.
+2.1.
+Denote ǫw = (−1)ℓ(w) and cw = cℓ(w). We have x−α =
+xα
+cxα−1 and κα = c for any α, and ˆXIw
+can be denoted by ˆXw.
+Note that there is another operator ˆYi = ˆYαi = c − ˆXαi such that ˆY 2
+αi = c ˆYαi and braid rela-
+tions are satisfied. This is the algebraic model of the composition hTa(Ga/Ba) → hTa(Ga/Pi) →
+hTa(Ga/Ba) where Pi is the minimal parabolic subgroup corresponding to αi ∈ Ia. Moreover, we
+have
+ˆXw =
+�
+v≤w
+ǫvcwc−1
+v
+ˆYv.
+
+AFFINE GRASSMANNIAN
+7
+Most properties of ˆXw are also satisfied by ˆYw, except for Lemma 1.6. Indeed, ˆY ∗
+w, w ∈ W −
+a is not
+W-invariant.
+Denote xΦ = �
+α∈Φ− xα. It is well known that Yw0 = �
+w∈W ηw 1
+xΦ . Moreover, the map Yw0 •
+:
+ˆD∗
+Wa → ( ˆD∗
+Wa)W is the algebraic model for the map hTa(Ga/Ba) → hTa(GrG). We first compute
+the image of the two bases via this map.
+Lemma 2.1. Let F = Fc. For any w ∈ Wa and u = u1u2, u1 ∈ W −
+a , u2 ∈ W, we have
+Yw0 • ˆX∗
+u1u2 = ǫu2cw0c−1
+u2 ˆX∗
+u1, Yw0 • ˆY ∗
+w =
+�
+v1v2≥w,v1∈W −
+a ,v2∈W
+ǫwǫv2cv1w0c−1
+w ˆX∗
+v1.
+In particular, Yw0 • ˆY ∗
+w, w ∈ W −
+a is a basis of ( ˆD∗
+Wa)W if and only if c ∈ R×.
+Proof. For each v ∈ Wa, write v = v1v2, v1 ∈ W −
+a , v2 ∈ W. From ˆXwYw0 = 0, w ∈ W, we have
+(Yw0• ˆX∗
+u1u2)( ˆXv1v2) = ˆX∗
+u1u2( ˆXv1v2Yw0) = δv2,e ˆX∗
+u1u2( ˆXv1
+�
+w′≤w0
+ǫw′cw0c−1
+w′ ˆXw′) = δv2,eδv1,u1ǫu2cw0c−1
+u2 .
+This proves the first identity. For the second one, it is easy to see that ˆY ∗
+w = �
+v≥w ǫwcvc−1
+w ˆX∗
+v. So
+Yw0 • ˆY ∗
+w = Yw0 •
+�
+v≥w
+ǫwcvc−1
+w ˆX∗
+v =
+�
+v1v2≥w,v1∈W −
+a ,v2∈W
+ǫwǫv2cv1w0c−1
+w ˆX∗
+v1.
+This proves the second identity.
+The transition matrix between ˆX∗
+v, v ∈ W −
+a
+and Yw0 • ˆY ∗
+w, w ∈ W −
+a
+is upper triangular with
+diagonal entries ǫwcw0, so the last statement follows.
+□
+2.2.
+Before computing the ⊙-action, we need to prove some identities in ˆDWa.
+Lemma 2.2. Let F = Fc. Writing ηu = �
+v≤u ˆbu,v ˆXv = �
+v≤u ˆbY
+u,v ˆYv, then
+ˆbsiu,v
+=
+�
+si(ˆbu,v),
+siv > v;
+(1 − cxi)si(ˆbu,v) − xisi(ˆbu,siv),
+siv < v.
+ˆbY
+siu,v
+=
+�
+(1 − cxi)si(ˆbY
+u,v),
+siv > v;
+xisi(ˆbY
+u,siv) + si(ˆbY
+u,v),
+siv < v.
+Proof. We prove the first one, and the second one follows similarly. Denote iWa = {v ∈ Wa|siv > v}.
+We have
+ηsiu
+=
+ηiηu = ηi
+�
+v∈iWa
+ˆbu,v ˆXv + ˆbu,siv ˆXsiv =
+�
+v∈iWa
+si(ˆbu,v)ηi ˆXv + si(ˆbu,siv)ηi ˆXsiv
+=
+�
+v∈iWa
+si(ˆbu,v)(1 − xi ˆXi) ˆXv + si(ˆbu,siv)(1 − xi ˆXi) ˆXsiv
+=
+�
+v∈iWa
+si(ˆbu,v)( ˆXv − xi ˆXsiv) + si(ˆbu,siv) ˆXsiv − cxisi(ˆbu,siv) ˆXv
+=
+�
+v∈iWa
+si(ˆbu,v) ˆXv + (si(ˆbu,siv)(1 − cxi) − xisi(ˆbu,v)) ˆXsiv.
+The conclusion then follows.
+□
+Note that if v ∈ W −
+a
+and siv < v, then siv ∈ W −
+a . We have the following recursive formula,
+whose proof follows from the definition and Lemma 2.2.
+
+8
+C. ZHONG
+Theorem 2.3. For F = Fc, with i ∈ Ia, we have
+ˆX−i ⊙ ˆX∗
+v
+=
+�
+0,
+siv > v,
+c ˆX∗
+v + ˆX∗
+siv,
+siv < v,
+ˆY−i ⊙ ˆY ∗
+v
+=
+�
+0,
+siv > v,
+c ˆY ∗
+v + ˆY ∗
+siv,
+siv < v.
+Here
+ˆX−i = ηw0 ˆXiηw0 =
+1
+x−i
+(1 − ηi), ˆY−i = ηw0 ˆYiηw0 = 1
+xi
++
+1
+x−i
+ηi.
+Consequently, if v ∈ W −
+a , we have
+ˆY−i ⊙ (Yw0 • Y ∗
+v ) =
+�
+0,
+siv > v,
+c(Yw0 • ˆY ∗
+v ) + (Yw0 • ˆY ∗
+siv),
+siv < v.
+Proof. We have
+ˆX−i ⊙ ˆX∗
+v = ( 1
+x−i
+−
+1
+x−i
+ηi) ⊙
+�
+u≥v
+ˆbu,v ˆfu =
+�
+u
+ˆbu,v
+x−i
+ˆfu −
+�
+u
+si(ˆbu,v)
+x−i
+ˆfsiu =
+�
+u
+ˆbu,v − si(ˆbsiu,v)
+x−i
+ˆfu.
+Plugging the formula in Lemma 2.2, we obtain the formula.
+The formula for ˆY−i ⊙ ˆY ∗
+v follows similarly. From the commutativity of the two actions • and ⊙,
+one obtains the last statement.
+□
+3. FADA for the small torus
+We repeat the construction of FADA for the small torus, which is very similar as above.
+3.1.
+Let S be the formal group algebra associated to T ∗, that is, it is (non-canonically) isomorphic
+a power series ring of rank n. When the formal group law F = Fc, we can again take the polynomial
+version, i.e., see Remark 1.2. Let Q = S[ 1
+xα , α ∈ Φ], QWa = Q ⋊ R[Wa], QQ∨ = Q ⋊ R[Q∨]. For
+any α ∈ Φ, let κα =
+1
+xα +
+1
+x−α and κα0 =
+1
+x−θ + 1
+xθ . We have the projection
+pr : QWa → QQ∨, ηtλw �→ ηtλ,
+w ∈ W.
+Define
+Xα = 1
+xα
+(1 − ηα),
+Xα0 =
+1
+x−θ
+(1 − ηs0),
+α ∈ Φ.
+For simplicity, denote x±i = x±αi, Xi = Xαi, ηi = ηsi, X0 = Xα0. They satisfy relations similar
+as that of ˆXi. One can define XIw for any reduced sequence Iw of w, which depends only on w if
+F = Fc.
+Remark 3.1. Consider K-theory, in which case F = Fc with c = 1. Our −X−αi is the Ti in
+[LSS10, LLMS18]. Our 1 − Xαi coincides with the Di in [K18]. For cohomology, c = 0, κα = 0,
+and our Xi is the Ai in [P97, Proposition 2.11] and [L06].
+Lemma 3.2. We have pr(zXi) = 0 if z ∈ QWa, i ∈ I.
+Proof. Let z = pηw, p ∈ Q, w ∈ Wa , then
+pr(zXi) = pr(pηwXi) = pr(
+p
+w(xi)(ηw − ηwsi)) =
+p
+w(xi)(pr(ηw) − pr(ηwsi)) = 0.
+□
+
+AFFINE GRASSMANNIAN
+9
+Define DWa to be the subalgebra of QWa generated by S and Xi, i ∈ Ia, and DWa/W = pr(DWa).
+Then DWa is a free left S-module with basis XIw, w ∈ Wa. Denote XIw = pr(XIw), w ∈ W −
+a .
+Lemma 3.3. If Iw, w ∈ Wa are W-compatible, then the set {XIw|w ∈ W −
+a } is a basis of the left
+S-module DWa/W .
+Proof. They follow easily from Lemma 3.2. See [CZZ19, Lemma 11.3].
+□
+The projection p : T ∗
+a → T ∗, µ + kδ �→ µ induces projections ˆS → S, ˆQ → Q and ˆQWa → QWa.
+Clearly p( ˆXαi) = Xαi and p( ˆXIw) = XIw, so p( ˆDWa) = DWa. More explicitly, we have
+ˆXIw =
+�
+v≤w
+ˆaIw,vηv ∈ ˆQWa,
+XIw =
+�
+v≤w
+aIw,vηv ∈ QWa,
+p(ˆaIw,v) = aIw,v ∈ Q,
+ηw =
+�
+v≤w
+ˆbw,Iv ˆXIv ∈ ˆQWa,
+ηw =
+�
+v≤w
+bw,IvXIv ∈ QWa,
+p(ˆbw,Iv) = bw,Iv ∈ S.
+Note that the embedding i : Q → ˆQ induces a section QWa → ˆQWa of p. However, it does not map
+DWa to ˆDWa. For example, X0 is mapped to x−θ+δ
+x−θ
+ˆX0 which does not belong to ˆDWa.
+3.2.
+As before, we can take the duals, which will give us Q-modules Q∗
+Wa, Q∗
+Q∨, and S-modules
+D∗
+Wa, D∗
+Wa/W . The elements dual to
+ηw, XIw ∈ DWa ⊂ QWa, ηtλ, XIw ∈ DWa/W ⊂ QQ∨,
+are denoted by
+fw, X∗
+Iw ∈ D∗
+Wa ⊂ Q∗
+Wa, ftλ, X∗
+Iw ∈ D∗
+Wa/W ⊂ Q∗
+Q∨,
+correspondingly. Note that the notation ftλ can be thought as in Q∗
+Wa and Q∗
+Q∨, just like ηtλ can
+be thought as in QWa and QQ∨. Similar as Proposition 1.5, we have
+(3)
+D∗
+Wa = {f ∈ Q∗
+Wa|f(DWa) ⊂ S}.
+Moreover, by definition, the dual map pr∗ : Q∗
+Q∨ → Q∗
+Wa satisfies
+pr∗(ftλ) =
+�
+w∈W
+ftλw.
+Following from the definition, we have
+ˆX∗
+Iw =
+�
+v≥w
+ˆbv,Iw ˆfv ∈ ˆD∗
+Wa,
+X∗
+Iw =
+�
+v≥w
+bv,Iwfv ∈ D∗
+Wa.
+Since p(ˆbv,Iw) = bv,Iw, so the map q :
+ˆQ∗
+Wa → Q∗
+Wa, �
+w aw ˆfw �→ �
+w p(aw)fw induces a map
+q : ˆD∗
+Wa → D∗
+Wa such that q( ˆX∗
+Iw) = X∗
+Iw. Moreover, since
+p∗(X∗
+Iw)( ˆXIv) = X∗
+Iw(p( ˆXIv)) = X∗
+Iw(XIv) = δw,v,
+so p∗(X∗
+Iw) = ˆX∗
+Iw. Note that neither q nor p∗ are isomorphisms, since the domains and targets are
+modules over different rings.
+Similar as Lemma 1.6, we have
+Lemma 3.4. If Iw, w ∈ Wa are W-compatible, then the set X∗
+Iw, w ∈ W −
+a form a basis of (Q∗
+Wa)W
+and of (D∗
+Wa)W , respectively.
+
+10
+C. ZHONG
+Lemma 3.5. Assume that {Iw, w ∈ Wa} is W-compatible. For any w ∈ W, u ∈ W −
+a , we have
+X∗
+Iu =
+�
+λ∈Q∨
+btλw,Iuftλ ∈ Q∗
+Q∨.
+Proof. For any λ ∈ Q∨, write
+ηtλw =
+�
+u∈W −
+a ,v∈W
+btλw,Iu∪IvXIu∪Iv.
+By Lemma 3.2, we have
+ηtλ = pr(ηtλw) =
+�
+u∈W −
+a ,v∈W
+btλw,Iu∪Iv pr(XIu∪Iv) =
+�
+u∈W −
+a
+btλw,Iu pr(XIu) =
+�
+u∈W −
+a
+btλw,IuXIu.
+Therefore,
+X∗
+Iu =
+�
+λ∈Q∨
+btλw,Iuftλ ∈ Q∗
+Q∨.
+□
+This lemma implies that we have pr∗(X∗
+Iu) = X∗
+Iu, u ∈ W −
+a .
+3.3.
+There is a •-action of QWa on Q∗
+Wa, defined similar as the big torus case.
+Lemma 3.6. The •-action of QWa on Q∗
+Warestricts to an action of DWa on D∗
+Wa.
+Proof. Since DWa is a S-module with basis XIu, u ∈ Wa, so for any w, v ∈ Wa, i ∈ Ia, we have
+XIvXi = �
+u cIv∪si,IuXIu with cIv∪si,Iu ∈ S. We have
+(Xi • X∗
+Iw)(XIv) = X∗
+Iw(XIvXi) = cIv∪si,Iw ∈ S.
+By (3), Xi • X∗
+Iw ∈ D∗
+Wa.
+□
+Lemma 3.7. We have
+D∗
+Wa ⊂ {f ∈ Q∗
+Wa|f(ηw) ∈ S, and f(ηw − ηsαw) ∈ xαS, ∀α ∈ Φ, w ∈ Wa}.
+One of the main results of this paper is to study how different the two sets are, that is, to derive
+the small torus GKM condition.
+Proof. Since ηw ∈ DWa, then it follows from (3) that f(ηw) ∈ S. Let i ∈ I and f = �
+w∈Wa awfw ∈
+D∗
+Wa with aw = f(ηw) ∈ S. We have
+Xi•f = 1
+xi
+(1−ηi)•
+�
+w
+awfw =
+�
+w
+aw
+w(xi)fw−
+�
+w
+aw
+wsi(xi)fwsi =
+�
+w
+aw − awsi
+w(xi)
+=
+�
+w
+aw − asw(αi)w
+xw(αi)
+fw.
+By Lemma 3.6, Xi • f ∈ D∗
+Wa, so f(ηw − ηsβw) =
+aw−asβw
+xβ
+∈ S for any β ∈ Φ.
+□
+3.4.
+We can similarly define the ⊙ action
+aηw ⊙ bfv = aw(b)fwv, w, v ∈ Wa, a, b ∈ Q.
+It is easy to see that the ⊙ and the • actions commute with each other.
+Lemma 3.8. For any ˆz ∈ ˆQWa, ˆf ∈ ˆQ∗
+Wa, we have
+p(ˆz) ⊙ q( ˆf) = q(ˆz ⊙ ˆf).
+In particular, the ⊙-action of QWa on Q∗
+Wa induces an action of DWa on D∗
+Wa.
+
+AFFINE GRASSMANNIAN
+11
+Proof. Write ˆz = ˆaηv, ˆf = ˆb ˆfw, ˆa,ˆb ∈ ˆQ, w, v ∈ Wa and suppose p(ˆa) = a, p(ˆb) = b, then
+p(ˆz) ⊙ q( ˆf) = aηv ⊙ bfw = av(b)fvw = q(ˆav(ˆb) ˆfvw) = q(ˆaηv ⊙ ˆb ˆfw) = q(ˆz ⊙ ˆf).
+For the second part, note that p : ˆDWa → DWa and q : ˆD∗
+Wa → D∗
+Wa are both surjective. Given
+z ∈ DWa and f ∈ D∗
+Wa, suppose z = p(ˆz) and f = q( ˆf) for some ˆz ∈ ˆDWa and ˆf ∈ ˆD∗
+Wa, then
+z ⊙ f = p(ˆz) ⊙ q( ˆf) = q(ˆz ⊙ ˆf) ∈ q( ˆD∗
+Wa) = D∗
+Wa.
+□
+Remark 3.9. If F = Fc, then all results in §2 holds for X∗
+w and the corresponding Y ∗
+w.
+4. The small-torus GKM condition
+In this section, we study the small-torus GKM condition on the equivariant oriented cohomology
+of the affine flag variety and of the affine Grassmannian.
+4.1.
+For each α ∈ Φ, we define
+Zα =
+1
+x−α
+(1 − ηtα∨) ∈ QWa.
+Lemma 4.1. For each α ∈ Φ, we have Zα ∈ DWa.
+Proof. It suffices to show that Zα is contained in the subalgebra of DWa generated by S and Xα. So
+we assume the root system is the affine root system of SL2 with simple roots α1 = α, α0 = −α + δ.
+Then tα∨ = s0s1. We have ηs1 = 1 − xαX1, ηs0 = 1 − x−αX0, so ηs0s1 = 1 − x−αX0 − x−αX1 +
+x2
+−αX0X1. Therefore,
+Zα =
+1
+x−α
+(1 − ηs0s1) = X0 + X1 − x−αX0X1 ∈ DWa.
+□
+Example 4.2. Suppose the root system is ˆA1 with two simple roots α1 = α, α0 = −α + δ.
+(1) If F = Fc with c = 0, then we have Zα = X0 + X1 + αX0X1.
+(2) If F = Fc with c = 1, then we have Zα = X0 + X1 + (eα − 1)X0X1.
+Since DWa acts on D∗
+Wa, so we know that Zα acts on D∗
+Wa. Note that
+Zk
+α = 1
+xkα
+(1 − ηtα∨)k.
+4.2.
+We are now ready to prove the first main result of this paper.
+Theorem 4.3.
+(1) The subset D∗
+Wa ⊂ Q∗
+Wa consists of elements satisfying the following small-
+torus GKM condition:
+f
+�
+(1 − ηtα∨)dηw
+�
+∈ xd
+αS, and f
+�
+(1 − ηtα∨)d−1(1 − ηsα)��w
+�
+∈ xd
+αS, ∀α ∈ Φ, w ∈ Wa, d ≥ 1.
+(2) The subset (D∗
+Wa)W ⊂ (Q∗
+Wa)W consists of elements satisfying the following small-torus
+Grassmannian condition:
+f
+�
+(1 − ηtα∨)dηw
+�
+∈ xd
+αS, ∀α ∈ Φ, w ∈ Wa, d ≥ 1.
+
+12
+C. ZHONG
+Our proof follows similarly as that of [LSS10, Theorem 4.3]. The key improvement is that we
+don’t need to prove Propositions 4.4 and 4.5 of loc.it., since we can use the operators Zα. However,
+for the convenience of the readers, we include an appendix, which gives all coefficients of bw,Iv in
+the ˆA1 case. They can be used to show that X∗
+Iw satisfy the small torus GKM condition.
+Proof. (1).
+We prove that elements of D∗
+Wa satisfy the small-torus GKM condition.
+Let f =
+�
+w cwfw ∈ D∗
+Wa, we have
+Zα •
+�
+w
+cwfw =
+�
+w
+(
+cw
+w(x−α)fw −
+cw
+wt−α∨(x−α)fwt−α∨) =
+�
+w
+cw − ctw(α∨)w
+x−w(α)
+fw ∈ D∗
+Wa.
+Note that
+xα
+x−α is invertible in S. Therefore, denoting w(α) = β, by (3), we have f((1 − ηtβ∨)ηw) ∈
+xβS for any β ∈ Φ.
+Moreover, denote dw =
+cw−cwtα∨
+x−w(α) , then dwtα∨ =
+cwtα∨ −cwtα∨ tα∨
+x−wtα∨ (α)
+=
+cwtα∨ −cwt2α∨
+x−w(α)
+. Therefore, We
+have
+Z2
+α • f
+=
+Zα • Zα •
+�
+w
+cwfw =
+�
+w
+(dw − dwtα∨
+w(x−α)
+)fw =
+�
+w
+cw − 2cwtα∨ + cwt2α∨
+w(x−α)2
+fw
+=
+�
+w
+cw − 2ctw(α∨)w + ct2w(α∨)w
+x2
+−w(α)
+fw =
+�
+w
+1
+x2
+−w(α)
+f((1 − ηtw(α∨))2ηw)fw.
+Denoting w(α) = β, we see that f((1−ηtβ∨)2ηw) ∈ x2
+βS. Inductively, we see that f((1−ηtα∨ )dηw) ∈
+xd
+αS for all d ≥ 1.
+Similarly, if one applies Zd−1
+α
+Xα ∈ DWa on f, which gives Zd−1
+α
+Xα • f ∈ D∗
+Wa, one will see that
+f satisfies the second condition.
+For the rest of the proof and for that of (2), it is identical to that of [LSS10, Theorem 4.3], so it
+is skipped.
+□
+Corollary 4.4. The subset D∗
+Wa/W ⊂ Q∗
+Q∨ consists of elements satisfying the following small torus
+Grassmannian condition:
+f((1 − ηt∨
+α)dηtλ) ∈ xd
+αS, ∀α ∈ Φ, d ≥ 1, λ ∈ Q∨.
+Proof. This follows from the identity pr∗(ftλ) = �
+v∈W ftλv.
+□
+5. The Peterson subalgebra
+In this section, we embed DWa/W into DWa and show that it coincides with the centralizer of S
+in DWa. This is called the Peterson subalgebra, which gives the algebraic model for the equivariant
+oriented ‘homology’ of the affine Grassmannian.
+5.1.
+We have a canonical ring embedding (and also a Q-module embedding)
+k : QQ∨ → QWa,
+pηtλ �→ pηtλ,
+such that pr ◦k = idQWa. It is easy to see that the dual map k∗ : Q∗
+Wa → Q∗
+Q∨ satisfies
+(4)
+k∗(ftλu) = δu,eftλ,
+u ∈ W.
+For K-theory, our map k is the map k : KT (GrG) → K in [LSS10, §5.2], and k∗ is the wrong-way
+map ̟ of [LSS10, $4.4].
+The following lemma generalizes [P97], [L06, Theorem 4.4] for the cohomology case, and [LSS10,
+Lemma 4.6] for the K-theory case.
+
+AFFINE GRASSMANNIAN
+13
+Lemma 5.1. The map k∗ induces a map k∗ : D∗
+Wa → D∗
+Wa/W . Consequently, the map k induces a
+map k : DWa/W → DWa.
+Proof. Given f ∈ D∗
+Wa, then f satisfies the small-torus GKM condition Theorem 4.3, that is,
+f((1 − ηtα∨)dηtλu) ∈ xd
+αS,
+∀u ∈ W, λ ∈ Q∨, α ∈ Φ, d ≥ 1.
+Therefore,
+k∗(f)((1 − ηtα∨)dηtλ) = f
+�
+k((1 − ηtα∨)dηtλ)
+�
+= f((1 − ηt∨
+α)dηtλ) ∈ xd
+αS.
+Therefore, by Corollary 4.4, k∗(f) ∈ D∗
+Wa/W .
+□
+Remark 5.2. It would be interesting to find a direct proof of the fact that k maps DWa/W to
+DWa. One possible choice is to find the small torus residue condition of DWa similar to the residue
+condition of [GKV97] (see [ZZ17]).
+Example 5.3. Note that this result is not true for the big torus case, that is, k( ˆDWa/W ) is not
+contained in ˆDWa. For example, in the ˆA1 case, we have
+pr(X0) = pr(
+1
+x−α+δ
+(1 − ηtα∨s1)) =
+1
+x−α+δ
+(1 − ηtα∨) ∈ ˆDWa/W ,
+and
+k(pr(X0)) =
+1
+x−α+δ
+(1 − ηtα) =
+1
+x−α+δ
+(1 − ηs0s1) ̸∈ ˆDWa.
+Lemma 5.4. If Iw, w ∈ W is W-compatible, then k∗(X∗
+Iu) = X∗
+Iu for any u ∈ W −
+a .
+Proof. By (4) and Lemma 3.5, we have
+k∗(X∗
+Iu) = k∗(
+�
+λ∈Q∨,w∈W
+btλw,Iuftλw) =
+�
+λ∈Q∨
+btλ,Iuftλ = X∗
+Iu.
+□
+5.2.
+Let CDWa(S) be the centralizer of S in DWa. Our second main result is the following, which
+generalizes [LSS10, Lemma 5.2] in the K-theory case and [P97, §9.3] in the cohomology case (proved
+in [LS10, Theorem 6.2]).
+Theorem 5.5. We have CDWa(S) = k(QQ∨) ∩ DWa = k(DWa/W ).
+Proof. We look at the first identity. Since tλ(p) = p for any p ∈ S, so it is clear that QQ∨ ∩ DWa ⊂
+CDWa(S). Conversely, let z = �
+w∈Wa cwηw ∈ CDWa(S), then for any µ ∈ T ∗, we have
+0 = xµz − zxµ =
+�
+w∈Wa
+cw(xµ − xw(µ))ηw.
+Therefore, for any cw ̸= 0, we have µ = w(µ) for all µ ∈ T ∗. we can take µ to be W-regular, which
+shows that cw ̸= 0 only when w = tλ for some λ ∈ Q∨. So z ∈ k(QQ∨). The first identity is proved.
+We now look at the second identity. It follows from Lemma 5.1 that k(DWa/W) ⊂ k(QQ∨)∩DWa.
+For the other inclusion, note that ηtλ ∈ DWa is a Q-basis of k(QQ∨). Given any z = �
+λ∈Q∨ pληtλ ∈
+k(QQ∨) ∩ DWa, pλ ∈ Q, then pr(z) ∈ pr(DWa) = DWa/W , and
+k ◦ pr(z) = k ◦ pr(
+�
+λ
+pληtλ) = k(
+�
+λ
+pληtλ) =
+�
+λ
+pληtλ = z.
+Therefore, k(QQ∨) ∩ DWa ⊂ k(DWa/W ). The second identity is proved.
+□
+
+14
+C. ZHONG
+Definition 5.6. We define the Peterson subalgebra to be DQ∨ = k(DWa/W ).
+Let Iw, w ∈ Wa be W-compatible. Since DWa/W is a free S-module with basis XIw, w ∈ W −
+a , so
+k(XIw) form a basis of DQ∨. This is the algebraic model for the oriented homology of the affine
+Grassmannian GrG. The following result generalizes [LSS10, Theorem 5.3] in K-theory.
+Theorem 5.7. The ring DQ∨ is a Hopf algebra, and the embedding DQ∨ → QQ∨ is an Hopf-algebra
+homomorphism.
+Proof. The coproduct structure on QWa is defined as △ : QWa → QWa ⊗Q QWa, ηw �→ ηw ⊗ ηw. It is
+easy to see that this induces a coproduct structure on QQ∨, and by [CZZ16], it induces a coproduct
+structure on DWa. Therefore, it induces a coproduct structure on DQ∨. The product structure is
+induced by that of QQ∨, The antipode is s : QQ∨ → QQ∨, ηtλ �→ ηt−λ. It is then routine to check
+that DQ∨ is a Hopf algebra and the embedding to QQ∨ is an embedding of Hopf algebras.
+□
+Remark 5.8. For K-theory, we know the Hecke algebra is contained DWa.
+It is proved by
+Berenstein-Kazhdan [BK19] that certain localization of the Hecke algbra is a Hopf algebra.
+It
+is not difficult to see that it is compatible with the Hopf algebra structure of DQ∨.
+5.3.
+The following theorem generalizes [LSS10, Theorem 5.4] in the K-theory case and [LS10,
+Theorem 6.2] in the cohomology case.
+Theorem 5.9. Assume Iw, w ∈ Wa is W-compatible. If u ∈ W −
+a , then we have
+k(XIu) = XIu +
+�
+v∈Wa\W −
+a
+cIu,IvXIv, cIu,Iv ∈ S.
+Proof. If w ∈ W −
+a , by Lemma 5.4, we have
+X∗
+Iw(k(XIu)) = k∗(X∗
+Iw)((XIu)) = X∗
+Iw(XIu) = δw,u,
+Therefore,
+k(XIu) =
+�
+v∈Wa
+X∗
+Iv(k(XIu))XIv = XIu +
+�
+v∈Wa\W −
+a
+cIu,IvXIv.
+□
+Example 5.10. Consider the ˆA1 case, then there are two simple roots α1 = α, α0 = −α + δ. By
+direct computation, we have
+(1) k(X0) = X0 + X1 − x−αX01.
+(2) k(X10) = X10 − x−α
+xα X01.
+(3) k(X010) = X010 + X101 − x−αX1010.
+Corollary 5.11. Assume Iw, w ∈ Wa is W-compatible. Let u, v ∈ W −
+a . Write
+XIuXIv =
+�
+w∈Wa
+dIw
+Iu,IvXIw, XIuXIv =
+�
+w∈W −
+a
+dIw
+Iu,IvXIw,
+then
+d
+Iw3
+Iu,Iv =
+�
+w2∈Wa
+cIu,Iw2d
+Iw3
+Iw2,Iv.
+Proof. We have
+k(
+�
+w∈W −
+a
+dIw
+Iu,IvXIw)
+=
+k(XIuXIv) = k(XIu)k(XIv) = k(XIu)
+�
+w1∈Wa
+cIv,Iw1XIw1
+
+AFFINE GRASSMANNIAN
+15
+=
+�
+w1∈Wa
+cIv,Iw1k(XIu)XIw1 =
+�
+w1,w2∈Wa
+cIv,Iw1cIu,Iw2XIw2XIw1.
+Let w3 ∈ W −
+a . By [CZZ19, Theorem 8.2], we know that X∗
+Iw3(XIw2XIw1) = 0 unless w1 ∈ W −
+a , in
+which case cIv,Iw1 = δKr
+v,w1 by Theorem 5.9. Therefore, applying X∗
+Iw3, w3 ∈ W −
+a , and using Lemma
+5.4, we get
+d
+Iw3
+Iu,Iv
+=
+X∗
+Iw3(
+�
+w∈W −
+a
+dIw
+Iu,IvXIw) = k∗(X∗
+Iw3)(
+�
+w∈W −
+a
+dIw
+Iu,IvXIw)
+=
+X∗
+Iw3(k(
+�
+w∈W −
+a
+dIw
+Iu,IvXIw)) = X∗
+Iw3(
+�
+w1,w2∈Wa
+cIv,Iw1cIu,Iw2XIw2XIw1)
+=
+�
+w2∈Wa
+cIu,Iw2X∗
+Iw3(XIw2XIv) =
+�
+w2∈Wa
+cIu,Iw2d
+Iw3
+Iw2,Iv.
+□
+6. Appendix: Restriction formula in the ˆA1 case
+In this Appendix, we perform some computation in the ˆA1 case.
+6.1.
+In this case, there are two simple roots, α1 = α, α0 = −α + δ, and any w ∈ Wa has a unique
+reduced decomposition, so XIw, YIw can be denoted by Xw, Yw, respectively. Moreover, X2
+i = καXi.
+We use the notation as in [LSS10, §4.3]. Let
+σ0 = e, σ2i = (s1s0)i = t−iα∨, σ−2i = (s0s1)i = tiα∨, σ2i+1 = s0σ2i, σ−(2i+1) = s1σ−2i, i ≥ 1,
+and W −
+a = {σi|i ≥ 0}. Denote µ = − x−1
+x1 . So if F = Fc with c = 0, then µ = 1, and if F = Fc with
+c = 1, then µ = eα if one identifies xα with 1 − e−α.
+Let S≤a be the sum h0 + h1 + · · · + ha of homogeneous symmetric functions. Denote Si
+≤a to be
+S≤a(x, x, · · · , x) where there are i copies of x. For instance, S3
+≤3(x) = 1 + 3x + 6x2 + 10x3. We
+have the following identities:
+Si
+≤a(x) = xSi
+≤a−1(x) + Si−1
+≤a (x),
+Si
+≤a(x) =
+a
+�
+j=0
+xj
+�
+j + i − 1
+i − 1
+�
+.
+Then the following identities can be verified by direct computation for lower k and then continued
+with induction:
+ησ2k
+=
+1 + x2k
+1 Xσ2k +
+�
+1≤j≤k−1
+x2j
+1 (S2j
+≤k−j(µ−1)Xσ2j + S2j
+≤k−j−1(µ−1)Xσ−2j)
+−
+�
+1≤i≤k
+x2i−1
+1
+S2i−1
+≤k−i(µ−1)(Xσ2i−1 + Xσ−2i+1),
+ησ−2k
+=
+1 + x2k
+−1Xσ−2k +
+�
+1≤j≤k−1
+x2j
+−1(S2j
+≤k−j−1(µ)Xσ2j + S2j
+≤k−j(µ)Xσ−2j)
+−
+�
+1≤i≤k
+x2i−1
+−1 S2i���1
+≤k−i(µ)(Xσ2i−1 + Xσ−2i+1),
+ησ−2k−1
+=
+1 − x2k+1
+1
+Xσ−2k−1 +
+�
+1≤j≤k
+x2j
+1 S2j
+≤k−j(µ−1)(Xσ2j + Xσ−2j)
+
+16
+C. ZHONG
+−
+�
+1≤i≤k
+x2i−1
+1
+�
+S2i−1
+≤k−i(µ−1)Xσ2i−1 + S2i−1
+≤k−i+1(µ−1)Xσ−2i+1
+�
+,
+ησ2k+1
+=
+1 − x2k+1
+−1
+Xσ2k+1 +
+�
+1≤j≤k
+x2j
+−1S2j
+≤k−j(µ)(Xσ2j + Xσ−2j)
+−
+�
+1≤i≤k
+x2i−1
+−1
+�
+S2i−1
+≤k−i+1(µ)Xσ2i−1 + S2i−1
+≤k−i(µ)Xσ−2i+1
+�
+.
+For F = Fc with c = 1, that is, in the K-theory case, these identities specializes to the corresponding
+ones in [LSS10, (4.5), (4.6)] after identifying our −X−αi with Ti in [LSS10] (see Remark 3.1). By
+using these identities, following the same idea as in [LSS10, §4.3], one can prove that X∗
+Iw satisfy
+the small torus GKM conditions in Theorem 4.3.
+Acknowledge. The author would like to thank Cristian Lenart, Changzheng Li and Gufang Zhao
+for helpful discussions.
+References
+[BK19] A. Berenstein, D. Kazhcan, Hecke-Hopf algebras, Advances in Mathematics, 353 (2019) 312-395. 5.8
+[CPZ13] B. Calm´es, V. Petrov, K. Zainoulline, Invariants, torsion indices and oriented cohomology of complete flags,
+Annales scientifiques de l’ ´Ecole normale sup´erieure (4) 46(3), 405–448 (2013). 1.1
+[CZZ16] B. Calm`es, K. Zainoulline, and C. Zhong, A coproduct structure on the formal affine Demazure algebra,
+Mathematische Zeitschrift, 282 (2016) (3), 1191-1218. 0, 5.2
+[CZZ19] B. Calm`es, K. Zainoulline, and C. Zhong, Push-pull operators on the formal affine Demazure algebra and
+its dual, Manuscripta Mathematica, 160 (2019), no. 1-2, 9-50. 0, 1.4, 1.4, 1.4, 1.5, 3.1, 5.3
+[CZZ15] B. Calm`es, K. Zainoulline, and C. Zhong, Equivariant oriented cohomology of flag varieties, Documenta
+Mathematica, Extra Volume: Alexander S. Merkurjev’s Sixtieth Birthday (2015), 113-144. 0, 1.7
+[CZZ20] B. Calm`es, K. Zainoulline, and C. Zhong, Formal affine Demazure and Hecke algebras associated to Kac-
+Moody root systems, Algebra Representation Theory, 23 (2020), no.3, 1031-1050. 0, 1, 1.2, 1.6
+[B97] M. Brion, Equivariant Chow groups for torus actions, Transformation Groups, 2(3): 225-267, 1997. 0
+[GKV97] V. Ginzburg, M. Kapranov, and E. Vasserot, Residue construction of Hecke algebras, Advances in Mathe-
+matics 128 (1997), no. 1, 1-19. 1.3, 5.2
+[K18] S. Kato, Loop structure on equivariant K-theory of semi-infinite flag manifolds, arXiv:1805.01718. 0, 3.1
+[KK86] B. Kostant and S. Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G∗, Advances
+in Mathematics. 62 (1986), no. 3, 187-237. 0
+[KK90] B. Kostant and S. Kumar, T -equivariant K-theory of generalized flag varieties, Journal of Differential Ge-
+ometry 32 (1990), 549–603.
+[K03] A. Knutson, A Schubert calculus recurrence from the noncomplex W-action on G/B,arXiv:0306304. 0
+0
+[L06] T. Lam, Schubert polynomials for the affine Grassmannian, Journal of the American Mathematical Society, 21
+(1), 3.1, 5.1
+[LLMS18] T. Lam, C. Li, L. Mihalcea, M. Shimozono, A conjectural Peterson isomorphism in K-theory, Journal of
+Algebra, 513:326–343, 2018. 0, 3.1
+[LSS10] T. Lam, A. Schilling, M. Shimozono, K-theory Schubert calculus of the affine Grassmannian, Compositio
+Mathematica, 146 (2010), no. 4, 811–852. 0, 3.1, 4.2, 5.1, 5.2, 5.2, 5.3, 6.1
+[LS10] T. Lam, M. Shimozono, Quantum cohomology of G/P and homology of affine Grassmannian, Acta Mathe-
+matica, 204(1):49–90, 2010. 0, 5.2, 5.3
+[LZZ20] C. Lenart, K. Zainoulline, C. Zhong, Parabolic Kazhdan-Lusztig basis, Schubert classes and equivariant
+oriented cohomology, Journal of the Institute of Mathematics of Jussieu, 19 (2020), no. 6, 1889-1929. 0, 1.1, 1.6
+[LM07] M. Levine and F. Morel, Algebraic cobordism, Springer Monographs in Mathematics. Springer-Verlag, Berlin,
+2007. 1.1
+[MNS22] L.C. Mihalcea, H. Naruse, C. Su, Left Demazure-Lusztig operators on equivariant (quantum) cohomology
+and K-theory, International Mathematics Research Notices, 2022, no. 16, 12096–12147. 0
+[P97] D. Peterson, Quantum cohomology of G/P, Lecture at MIT, 1997. 0, 3.1, 5.1, 5.2
+(4) 53 (2020), no. 3, 663–711.
+
+AFFINE GRASSMANNIAN
+17
+[T09] J. Tymoczko, Divided difference operators for partial flag varieties, arXiv:0912.2545 0
+[ZZ17] G. Zhao and C. Zhong, Geometric representations of the formal affine Hecke algebra, Advances in Mathemat-
+ics, 317 (2017), 50-90. 1.3, 5.2
+State University of New York at Albany, 1400 Washington Ave, CK399, Albany, NY, 12222
+Email address: [email protected]
+

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+ 
+ 
+On the Smoothness of the Solution to the Two-Dimensional Radiation 
+Transfer Equation 
+ 
+Dean Wang 
+ 
+The Ohio State University 
+201 West 19th Avenue, Columbus, Ohio 43210  
+ 
+[email protected] 
+ 
+ 
+ABSTRACT 
+ 
+In this paper, we deal with the differential properties of the scalar flux 𝜙(𝑥) defined over a 
+two-dimensional bounded convex domain, as a solution to the integral radiation transfer 
+equation. Estimates for the derivatives of 𝜙(𝑥) near the boundary of the domain are given 
+based on Vainikko’s regularity theorem. A numerical example is presented to demonstrate the 
+implication of the solution smoothness on the convergence behavior of the diamond difference 
+method.  
+ 
+KEYWORDS: Integral Equation, Radiation Transfer, Regularity, Numerical Convergence 
+ 
+ 
+1. INTRODUCTION 
+ 
+Consider the integral equation of the second kind 
+ 
+𝜙(𝑥) = ∫ K(𝑥, 𝑦)𝜙(𝑦)𝑑𝑦
+!
++ 𝑓(𝑥) ,  
+𝑥 ∈ G ,  
+ 
+ 
+ (1) 
+ 
+where G ⊂ ℝ" , 𝑛 ≥ 1,  is an open bounded domain and the kernel K(𝑥, 𝑦)  is weakly singular, i.e., 
+|K(𝑥, 𝑦)| ≤ 𝐶|𝑥 − 𝑦|#$, 0 ≤ 𝜈 ≤ 𝑛. Weakly singular integral equations arise in many physical applications 
+such as elliptic boundary problems and particle transport.  
+ 
+The standard integro-differential equation of radiation transfer can be reformulated as a weakly singular 
+integral equation. The one-group radiation transfer problem in a three-dimensional (3D) convex domain 
+reads as follows: find a function 𝜙: G: × Ω → ℝ% such as  
+ 
+∑
+Ω&
+'((*,,)
+'*!
+.
+&/0
++ 𝜎(𝑥)𝜙(𝑥, Ω) =
+1"(*)
+23 ∫ 𝑠(𝑥, Ω, Ω′)𝜙(𝑥, Ω4)𝑑Ω′
+,
++ 𝑓(𝑥, Ω) , 
+𝑥 ∈ G ,   (2) 
+ 
+ 
+𝜙(𝑥, Ω) = 𝜙56(𝑥, Ω),  𝑥 ∈ 𝜕G,  Ω ∙ 𝑛D(𝑥) < 0 ,  
+ 
+ 
+(3) 
+ 
+where Ω denotes the direction of radiation transfer, 𝜕G is the boundary of the domain G ⊂ ℝ., 𝜎 is the 
+extinction coefficient (or macroscopic total cross section in neutron transport), 𝜎7  is the scattering 
+coefficient (or macroscopic scattering cross section), 𝑠  is the phase function of scattering with 
+∫ 𝑠(𝑥, Ω, Ω′)𝑑Ω′
+,
+= 4𝜋, 𝑓 is the external source function, and 𝑛D is the unit normal vector of the domain 
+surface. Note that 𝜎7(𝑥) ≤ 𝜎(𝑥) and 𝑠(𝑥, Ω, Ω4) = 𝑠(𝑥, Ω′, Ω) under physical conditions.  
+ 
+
+Dean Wang  
+ 
+Assuming the isotropic scattering, i.e., 𝑠(𝑥, Ω, Ω4) = 1, 𝑓(𝑥, Ω) =
+8(*)
+23 , and 𝜙56(𝑥, Ω) =
+(#$(*)
+23
+, we can 
+obtain the so-called Peierls integral equation of radiation transfer for the scalar flux 𝜙(𝑥) as follows: 
+ 
+𝜙(𝑥) =
+0
+23 ∫
+1"(9):–&((,*)
+|*–9|,
+𝜙(𝑦)𝑑𝑦
+6
++
+0
+23 ∫
+:–&((,*)
+|*–9|, 𝑓(𝑦)𝑑𝑦
+6
+  
+ 
++
+0
+23 ∫
+:–&((,*)
+|*–9|, H
+*–9
+|*–9| ∙ 𝑛(𝑥)H 𝜙56(𝑦)𝑑𝑆9
+'6
+  , 
+ 
+                   (4) 
+ 
+𝜏(𝑥, 𝑦) = ∫
+𝜎(𝑟– 𝜉Ω)𝑑𝜉
+|*–9|
+=
+ , 
+ 
+ 
+ 
+          (5) 
+ 
+where 𝑑𝑆 is the differential element of the domain surface, 𝜏(𝑥, 𝑦) is the optical path between 𝑥 and 𝑦. One 
+can find detailed derivation in [6,7].  
+ 
+For simplicity, we assume 𝜎 and 𝜎7 are constant over the domain. Then Eq. (4) can be simplified as 
+ 
+𝜙(𝑥) = ∫ K(𝑥, 𝑦)𝜙(𝑦)𝑑𝑦
+6
++
+0
+1" ∫ K(𝑥, 𝑦)𝑓(𝑦)𝑑𝑦
+6
+  
+ 
++
+0
+1" ∫
+K(𝑥, 𝑦) H
+*–9
+|*–9| ∙ 𝑛(𝑥)H 𝜙56(𝑦)𝑑𝑆9
+'6
+ , 
+ 
+ 
+         (6) 
+ 
+where the 3D radiation kernel is given as 
+ 
+K(𝑥, 𝑦) =
+1":–-|(–*|
+23|*–9|,  . 
+ 
+ 
+ 
+                  (7) 
+ 
+The boundary integral term in the above equation can produce singularities in the solution. We omit its 
+discussion in this paper. In other words, we only consider the problem with the vacuum boundary condition, 
+i.e., 𝜙56 = 0. Thus, Eq. (6) can be treated as the weakly integral equation of the second kind. 
+ 
+Since K(𝑥, 𝑦) has a singularity at 𝑥 = 𝑦, the solution of a weakly integral equation is generally not a smooth 
+function and its derivatives at the boundary would become unbounded from a certain order. There was 
+extensive research on the smoothness (regularity) properties of the solutions to weakly integral equations 
+[1,2], especially those early work in neutron transport theory done in the former Soviet Union [3,4]. It is 
+believed that Vladimirov first proved that the scalar flux 𝜙(𝑥)  possesses the property |𝜙(𝑥 + ℎ) −
+𝜙(𝑥)|~ℎlogℎ for the one-group transport problem with isotropic scattering in a bounded domain [3]. 
+Germogenova analyzed the local regularity of the angular flux 𝜙(𝑥, Ω)  in a neighborhood of the 
+discontinuity interface and obtained an estimate of the first derivative, which has the singularity near the 
+interface [4]. Pitkaranta derived a local singular resolution showing explicitly the behavior of 𝜙(𝑥) near 
+the smooth portion of the boundary [5]. Vainikko introduced weighted spaces and obtained sharp estimates 
+of pointwise derivatives near the smooth boundary for multidimensional weakly singular integral equations 
+[6].  
+ 
+There exists some previous research work on the regularity of the integral radiation transfer solutions [7,8]. 
+However, the 2D kernel used in those studies is physically incorrect. In this paper, we rederive the 2D 
+kernel by directly integrating the 3D kernel with respect to the third dimension. We examine the differential 
+properties of the new 2D kernel and provide estimates of pointwise derivatives of the scalar flux according 
+to Vainikko’s regularity theorem for the weakly integral equation of the second kind.       
+ 
+
+Smoothness of the Radiation Transfer Solution 
+ 
+The remainder of the paper is organized as follows. In Sect. 2, we derive the 2D kernel for the integral 
+radiation transfer equation. We examine the derivatives of the kernel and show that they satisfy the 
+boundedness condition of Vainikko’s regularity theorem in Sect. 3. Then the estimates of local regularity 
+of the scalar flux near the boundary of the domain are given. Sect. 4 presents numerical results to 
+demonstrate that the rate of convergence of numerical methods can be affected by the smoothness of the 
+exact solution. Concluding remarks are given in Sect. 5. 
+ 
+ 
+2. TWO-DIMENSIONAL RADIATION TRANSFER EQUATION 
+ 
+In this section, we derive the 2D integral radiation transfer equation from its 3D form, Eq. (6). In 3D, 𝑑𝑦 =
+𝑑𝑦0𝑑𝑦>𝑑𝑦.  and |𝑥– 𝑦| = S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)> + (𝑥.– 𝑦.)> . Let 𝜌 = S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)> , then 
+|𝑥– 𝑦| = S𝜌> + (𝑥.– 𝑦.)>. In a 2D domain 𝐺 ⊂ ℝ>, the solution function 𝜙(𝑥) only depends on 𝑥0 and 𝑥> 
+in Cartesian coordinates. Therefore, we only need to find the 2D radiation kernel, which can be obtained 
+by integrating out 𝑦. as follows: 
+ 
+K(𝑥, 𝑦) = ∫
+1":–-|(–*|
+23|*–9|, 𝑑𝑦.
+?
+#?
+  
+ 
+=
+1"
+23 ∫
+:
+/-01,2((3–*3),
+@,%(*3–93),
+𝑑𝑦.
+?
+#?
+ .                      
+   (8) 
+ 
+To proceed, we introduce the variables 𝑡 = 𝜎S𝜌> + (𝑥.– 𝑦.)> and 𝑧 = 𝑦. − 𝑥.. Then we substitute 𝑑𝑦. =
+𝑑z =
+A
+1BA,–1,@, 𝑑𝑡	into	the	above	equation	to	have 
+ 
+K(𝑥, 𝑦) =
+1"
+23 ∫
+:/4
+5,
+-,
+𝑑z
+?
+#?
+=
+1"
+>3 ∫
+:/4
+5,
+-,
+𝑑z
+?
+=
+  
+ 
+=
+1"
+>3 ∫
+:/4
+5,
+-,
+A
+1BA,–1,@, 𝑑𝑡
+?
+=
+  
+ 
+=
+1"1
+>3 ∫
+:/4
+ABA,–1,@, 𝑑𝑡
+?
+1@
+ =
+1"1
+>3 ∫
+:/4
+ABA,#1,|*–9|, 𝑑𝑡
+?
+1|*–9|
+ .                                      (9) 
+ 
+Note that the 2D radiation kernel is always positive. By replacing the 3D kernel of Eq. (7) with the above 
+one, Eq. (6) becomes the 2D integral radiation transfer equation. Notice that the surface integral in the last 
+term on the right-hand side of Eq. (6) should be replaced with the line integral in the 2D domain.  
+ 
+Now we show that the 2D kernel K(𝑥, 𝑦) has a singularity at 𝜌 = 0 (i.e., 𝑥 = 𝑦) as follows: 
+ 
+K(𝑥, 𝑦) =
+1"1
+>3 ∫
+:/4
+ABA,–1,@, 𝑑𝑡
+?
+1@
+>
+1"1
+>3 ∫
+:/4
+A, 𝑑𝑡
+?
+1@
+  
+ 
+=
+1"1
+>3 e
+C/-1
+1@ − Γ(0, 𝜎𝜌)g ,                                                            (10) 
+ 
+where Γ(0, 𝑎) = ∫
+:/4
+A 𝑑𝑡
+?
+D
+, is the incomplete gamma function. The singular behavior of K(𝑥, 𝑦) near 𝜌 =
+0 is dominated by the first term 
+C/-1
+1@  in the brackets since the gamma function tends to infinity much slower.  
+
+Dean Wang  
+ 
+Remark 2.1. It should be noted that the 2D kernel defined by Eq. (9) is equivalent to the more conventional 
+one defined by the Bickley-Naylor functions [9]. Johnson and Pitkaranta derived a 2D kernel for neutron 
+transport by reformulating the standard integro-differential equation on the 2D plane [7]. The kernel 
+obtained is, K(𝑥, 𝑦) =
+:–|(–*|
+|*–9|  (assuming 𝜎 = 1), which is however mathematically correct but physically 
+incorrect. Hennebach et al. also used the same 2D kernel for analyzing the radiation transfer solutions [8]. 
+In addition, the integral equations in other geometries such as slab or sphere can be obtained by following 
+the same approach, and they can be found in [10]. 
+ 
+Applying Banach’s fixed-point theorem, we can prove the existence and uniqueness of the solution in the 
+2D domain by showing that ∫ K(𝑥, 𝑦)𝑑𝑦
+!
+	is bounded below unity as follows. 
+ 
+∫ K(𝑥, 𝑦)𝑑𝑦
+6
+= ∫ i
+1"1
+>3 ∫
+:/4
+ABA,–1,@, 𝑑𝑡
+?
+1@
+j 𝑑𝑦
+6
+  
+ 
+=
+1"1
+>3 ∫ 𝑑𝑦 ∫
+:/4
+ABA,–1,@, 𝑑𝑡
+?
+1@
+6
+  
+ 
+=
+1"1
+>3 ∫ 𝜌𝑑𝜑𝑑𝜌 ∫
+:/4
+ABA,#1,@, 𝑑𝑡
+?
+1@
+6
+ , 
+ 
+(11) 
+ 
+where 𝜑 is the azimuthal angle. By extending the above bounded domain to the whole space, we have 
+ 
+∫ K(𝑥, 𝑦)𝑑𝑦
+6
+<
+1"1
+>3 ∫
+2𝜋𝜌𝑑𝜌
+?
+=
+∫
+:/4
+ABA,–1,@, 𝑑𝑡
+?
+1@
+  
+ 
+= 𝜎7𝜎 ∫
+𝜌𝑑𝜌
+?
+=
+∫
+:/4
+ABA,–1,@, 𝑑𝑡
+?
+1@
+ . 
+ 
+ (12) 
+ 
+Denoting 𝜁 = 𝜎𝜌, Eq. (12) is simplified as 
+ 
+∫ K(𝑥, 𝑦)𝑑𝑦
+6
+<
+1"
+1 ∫
+𝜁𝑑𝜁
+?
+=
+∫
+:/4
+ABA,–E, 𝑑𝑡
+?
+E
+  
+ 
+=
+1"
+1 ∫
+∫
+:/4
+A
+E
+BA,#E, 𝑑𝑡𝑑𝜁
+?
+E
+?
+=
+ =
+1"
+1 ∫
+:/4
+A 𝑑𝑡 ∫
+E
+BA,–E, 𝑑𝜁
+A
+=
+?
+=
+  
+ 
+=
+1"
+1 ∫
+e#F𝑑𝑡
+?
+=
+  
+ 
+=
+1"
+1 ≤ 1 .                                                                      (13) 
+ 
+Notice we have changed the order of integration to solve the integral. It is apparent that for there exists a 
+unique solution, the physical condition, 𝜎7 ≤ 𝜎, must be satisfied. 
+ 
+ 
+3. SMOOTHNESS OF THE SOLUTIONS 
+ 
+We first introduce Vainikko’s regularity theorem [6], which provides a sharp characterization of 
+singularities for the general weakly integral equation of the second kind. Then we analyze the 
+differentiational properties of the 2D radiation kernel and show that the derivatives are properly bounded. 
+Finally, Vainikko’s theorem is used to give the estimates of pointwise derivatives of the radiation solution.  
+
+Smoothness of the Radiation Transfer Solution 
+ 
+3.1. Vainikko’s Regularity Theorem 
+ 
+Before we state the theorem, we introduce the definition of weighted spaces ℂG,$(G) [6].  
+ 
+Weighted space ℂ𝒎,𝝂(𝐆). For a 𝜆 ∈ ℝ, introduce a weight function  
+ 
+𝑤J = r
+1																																,			𝜆 < 0
+(1 + |log𝜚(𝑥)|)#0	, 𝜆 = 0
+𝜚(𝑥)J																							,			𝜆 > 0
+ ,  
+𝑥 ∈ G  
+ 
+            (14) 
+ 
+where G ⊂ ℝ" is an open bounded domain and 𝜚(𝑥) = inf
+9∈'6|𝑥 − 𝑦| is the distance from 𝑥 to the boundary 
+𝜕G. Let 𝑚 ∈ ℕ, 𝜈 ∈ ℝ and 𝜈 < 𝑛. Define the space ℂG,$(G) as the set of all 𝑚 times continuously 
+differentiable functions 𝜙: G → 	ℝ such that 
+ 
+‖𝜙‖G,$ = ∑
+sup
+*∈6
+{𝑤|L|–("–$)|𝐷L𝜙(𝑥)|}
+|L|MG
+< ∞ . 
+ 
+ 
+ (15) 
+ 
+In other words, a 𝑚 times continuously differentiable function 𝜙 on G belongs to ℂG,$(G) if the growth of 
+its derivatives near the boundary can be estimated as follows: 
+ 
+|𝐷L𝜙(𝑥)| ≤ 𝑐 r
+1																								,			|𝛼| < 𝑛– 𝜈
+1 + |log𝜚(𝑥)|	,			|𝛼| = 𝑛– 𝜈
+𝜚(𝑥)"#$#|L|				,			|𝛼| > 𝑛– 𝜈
+ , 
+𝑥 ∈ G,  |𝛼| ≤ 𝑚 , 
+ 
+ (16) 
+ 
+where 𝑐 is a constant. The space ℂG,$(G), equipped with the norm ‖∙‖G,$, is a complete Banach space.  
+ 
+After defining the weighted space, we introduce the smoothness assumption about the kernel in the 
+following form: the kernel K(𝑥, 𝑦) is 𝑚 times continuously differentiable on (G × G)\{𝑥 = 𝑦} and there 
+exists a real number 𝜈 ∈ (−∞, 𝑛) such that the estimate  
+ 
+H𝐷*L𝐷*%9
+N
+K(𝑥, 𝑦)H ≤ 𝑐 r
+1																													,			𝜈 + |𝛼| < 0
+1 + „log|𝑥 − 𝑦|„	,			𝜈 + |𝛼| = 0
+|𝑥 − 𝑦|#$#|L|						,			𝜈 + |𝛼| > 0
+ ,      𝑥, 𝑦 ∈ G  
+               (17) 
+ 
+where 
+𝐷*L = …
+'
+'*6†
+L6 ⋯ …
+'
+'*7†
+L7 , 
+ 
+ 
+ 
+ 
+(18) 
+ 
+𝐷*%9
+N
+= …
+'
+'*6 +
+'
+'96†
+N6 ⋯ …
+'
+'*7 +
+'
+'97†
+N7 , 
+ 
+ 
+             (19) 
+ 
+holds for all multi-indices 𝛼 = (𝛼0, ⋯ , 𝛼") ∈ ℤ%" and 𝛽 = (𝛽0, ⋯ , 𝛽") ∈ ℤ%" with |𝛼| + |𝛽| ≤ 𝑚. Here the 
+following usual conventions are adopted: |𝛼| = 𝛼0 + ⋯ + 𝛼", and |𝑥| = S𝑥0
+> + ⋯ + 𝑥">. 
+ 
+Now we present Vainikko’s theorem in characterizing the regularity properties of a solution to the weakly 
+integral equation of the second kind [6].  
+ 
+Theorem 3.1. Let G ⊂ ℝ" be an open bounded domain, 𝑓 ∈ ℂG,$(G) and let the kernel K(𝑥, 𝑦) satisfy the 
+condition (17). If the integral equation (1) has a solution, 𝜙 ∈ L?(G) then 𝜙 ∈ ℂG,$(G).  
+
+Dean Wang  
+ 
+ 
+Remark 3.1. The solution does not improve its properties near the boundary 𝜕G, remaining only in 
+ℂG,$(G), even if 𝜕G is of class ℂ? and, 𝑓 ∈ ℂ?(G). A proof can be found in [6]. More precisely, for any 𝑛 
+and 𝜈 (𝜈 < 𝑛) there are kernels K(𝑥, 𝑦) satisfying (17) and such that Eq. (1) is uniquely solvable and, for a 
+suitable 𝑓 ∈ ℂ?(G), the normal derivatives of order 𝑘 of the solution behave near 𝜕G as log𝜚(𝑥) if 𝑘 =
+𝑛– 𝜈, and as 𝜚(𝑥)"#$#O for 𝑘 > 𝑛– 𝜈. 
+ 
+3.2. Smoothness of the Radiation Transfer Solution 
+ 
+To apply the results of Theorem 3.1 to the 2D integral radiation transfer equation, we need to analyze the 
+kernel K(𝑥, 𝑦) and show it satisfying the condition (17), i.e., H𝐷*L𝐷*%9
+N
+K(𝑥, 𝑦)H ≤ 𝑐|𝑥 − 𝑦|#0#|L|. We can 
+simply set |𝛽| = 0 without loss of generality for our problem.  
+ 
+|𝜶| = 𝟎:                             
+K(𝑥, 𝑦) =
+1"1
+>3 ∫
+:/4
+ABA,–1,@, 𝑑𝑡
+?
+1@
+<
+1"1
+>3 ∫
+:/-1
+ABA,–1,@, 𝑑𝑡
+?
+1@
+  
+ 
+=
+1"1:/-1
+>3
+∫
+0
+ABA,–1,@, 𝑑𝑡
+?
+1@
+ =
+1"1:/-1
+>3
+3
+>1@ =
+1":/-|(–*|
+2|*–9|   
+ 
+≤ 𝑐|𝑥– 𝑦|#0 .                                                                    (20) 
+ 
+|𝜶| = 𝟏: Let 𝜁 = 𝜎𝜌 = 𝜎|𝑥– 𝑦| = 𝜎S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)>, then K(𝑥, 𝑦) =
+1"1
+>3 ∫
+:/4
+ABA,–E, 𝑑𝑡
+?
+E
+, and  
+ 
+|𝐷*K(𝑥, 𝑦)| = H
+'
+'E K(𝑥, 𝑦) 'E
+'*H = H
+'
+'E K(𝑥, 𝑦)H H
+'E
+'*H , 
+                  (21) 
+ 
+where, 
+ 
+H
+'E
+'*6H = 𝜎 •
+(*6–96)
+B(*6–96),%(*,–9,),• ≤ 𝜎 , 
+       (22) 
+ 
+H
+'E
+'*,H = 𝜎 •
+(*,–9,)
+B(*6–96),%(*,–9,),• ≤ 𝜎 . 
+       (23) 
+ 
+Apparently, we only need to find the upper bound of •
+'
+'E ∫
+:/4
+ABA,–E, 𝑑𝑡
+?
+E
+• ≤ 𝑐𝜁#>, which is shown in the 
+following. First, we simplify the integral ∫
+:/4
+ABA,–E, 𝑑𝑡
+?
+E
+ as 
+ 
+∫
+:/4
+ABA,–E, 𝑑𝑡
+?
+E
+=
+0
+E, ∫
+A:/4
+BA,–E, 𝑑𝑡
+?
+E
+−
+0
+E, ∫
+:/4BA,–E,
+A
+𝑑𝑡
+?
+E
+  
+ 
+=
+P6(E)
+E
+−
+0
+E, ∫
+:/4BA,–E,
+A
+𝑑𝑡
+?
+E
+ , 
+(24) 
+ 
+where 𝐾0(𝜁) is the modified Bessel function of the second kind, and 𝐾0(𝜁)~
+0
+E when 𝜁 → 0 [11].  
+ 
+•
+'
+'E ∫
+:/4
+ABA,–E, 𝑑𝑡
+?
+E
+•  
+
+Smoothness of the Radiation Transfer Solution 
+ 
+ 
+= •−
+P6(E)
+E, +
+P68(E)
+E
++
+>
+E3 ∫
+:/4BA,–E,
+A
+𝑑𝑡
+?
+E
+−
+0
+E ∫
+:/4
+ABA,–E, 𝑑𝑡
+E
+?
+• . 
+ 
+ 
+       (25) 
+ 
+Notice the third term on the right-hand side of Eq. (25), ∫
+:/4BA,–E,
+A
+𝑑𝑡
+?
+E
+→ 1 as 𝜁 → 0.  It is not difficult to 
+find that the first three terms will cancel out when 𝜁 → 0. Then we obtain  
+ 
+•
+'
+'E ∫
+:/4
+ABA,–E, 𝑑𝑡
+?
+E
+• ≤ •
+0
+E ∫
+:/4
+ABA,–E, 𝑑𝑡
+E
+?
+•  
+ 
+≤
+3
+>
+:/9
+E, =
+3
+>1,
+:/-|(–*|
+|*–9|,  .                                                            (26) 
+ 
+Notice that here we have used the upper bound of Eq. (20). Now we arrive at the desired result for  |𝛼| = 1: 
+ 
+|𝐷*K(𝑥, 𝑦)| ≤ 𝑐|𝑥– 𝑦|#> .                                                       (27) 
+ 
+|𝜶| = 𝟐	(and	larger): we can follow the same procedure to find |𝐷*LK(𝑥, 𝑦)| ≤ 𝑐|𝑥 − 𝑦|#0#|L|. 
+ 
+Finally, we conclude that the 2D radiation kernel satisfies the condition (17). Therefore, by Theorem 3.1, 
+the estimates of derivatives of the scalar flux 𝜙(𝑥) for radiation transfer are the same as for the general 
+weakly integral equation of the second kind:  
+ 
+|𝐷L𝜙(𝑥)| ≤ 𝑐 r
+1																								,			|𝛼| < 1
+1 + |log𝜚(𝑥)|	,			|𝛼| = 1
+𝜚(𝑥)0#|L|									,			|𝛼| > 1
+ , 
+𝑥 ∈ G .   
+            (28) 
+ 
+Remark 3.2. The first derivative of the solution 𝜙(𝑥)	behaves as log𝜚(𝑥) and becomes unbounded as 
+approaching the boundary. The derivatives of order 𝑘 behave as 𝜚(𝑥)0#O for 𝑘 > 1. As mentioned in 
+Remark 3.1, these pointwise estimates cannot be improved by adding more strong smoothness on the data 
+and domain boundary. We point out that the lack of smoothness in the exact solution could adversely affect 
+the convergence rate of spatial discretization schemes for solving the radiation transfer equation [12-14]. 
+According to the regularity results, it is expected that the asymptotic convergence rate of the spatial 
+discretization error of finite difference methods would be around 1 in the 𝐿? or 𝐿0 norm.  
+ 
+ 
+4. NUMERICAL RESULTS 
+ 
+In this section, we demonstrate how the regularity of the exact solution will impact the numerical 
+convergence rate by solving the SN neutron transport equation in its original integro-differential form, using 
+the classic second-order diamond difference (DD) method. The model problem is a 1cm × 1cm square with 
+the vacuum boundary condition. Thus, there will be no complication from the boundary condition. The S12 
+level-symmetric quadrature set is used for angular discretization. 
+ 
+We analyze the following four cases: Case 1: ΣA = 1, Σ7 = 0; Case 2: ΣA = 1, Σ7 = 0.8; Case 3: ΣA = 10, 
+Σ7 = 0, and Case 4: ΣA = 10, Σ7 = 0.9. For all the cases, the external source 𝑓 = 1, is infinitely 
+differentiable, i.e., 𝑓 ∈ ℂ?(G). Cases 1 and 3 are pure absorption problems, while Case 3 is optically 
+thicker. It is interesting to note that the solutions are only determined by the external source for these two 
+cases. Cases 2 and 4 include the scattering effects, while Case 4 is optically thicker and more diffusive. 
+Both the scattering and external source contribute to the solution. The flux L1 errors as a function of mesh 
+
+Dean Wang  
+ 
+size and the rates of convergences are summarized in Table I. The error distributions on the mesh 
+160 × 160 are plotted in Fig. 1. The reference solution for each case is obtained on a very fine mesh, 
+5120 × 5120. 
+ 
+ 
+Table I. Flux L1 errors and convergence rates. 
+ 
+Mesh 
+(𝑵 × 𝑵) 
+Case 1 
+Case 2 
+Case 3 
+Case 4 
+Error 
+Rate 
+Error 
+Rate 
+Error 
+Rate 
+Error 
+Rate 
+10 × 10 
+2.87E-03 
+ 
+3.59E-03 
+ 
+2.31E-03 
+ 
+9.29E-03 
+ 
+20 × 20 
+7.95E-04 
+1.85 
+1.01E-03 
+1.83 
+8.12E-04 
+1.51 
+2.56E-03 
+1.86 
+40 × 40 
+2.90E-04 
+1.45 
+3.73E-04 
+1.44 
+2.31E-04 
+1.82 
+5.89E-04 
+2.12 
+80 × 80 
+1.14E-04 
+1.35 
+1.44E-04 
+1.37 
+5.19E-05 
+2.15 
+1.37E-04 
+2.10 
+160 × 160 
+5.04E-05 
+1.17 
+6.32E-05 
+1.19 
+1.32E-05 
+1.97 
+3.53E-05 
+1.96 
+320 × 320 
+2.46E-05 
+1.03 
+3.06E-05 
+1.04 
+3.61E-06 
+1.87 
+9.39E-06 
+1.91 
+640 × 640 
+1.31E-05 
+0.91 
+1.63E-05 
+0.91 
+1.11E-06 
+1.71 
+2.70E-06 
+1.80 
+1280 × 1280 
+6.26E-06 
+1.07 
+7.76E-06 
+1.07 
+3.87E-07 
+1.51 
+8.51E-07 
+1.66 
+ 
+ 
+                            Case 1  
+ 
+        
+ 
+                            Case 2           
+ 
+ 
+                            Case 3  
+ 
+        
+ 
+                            Case 4           
+   
+Figure 1. Flux error distribution on the mesh 𝟏𝟔𝟎 × 𝟏𝟔𝟎. 
+
+×10-4
+4
+Flux L1 Error
+3
+2
+0
+150
+100
+150
+100
+50
+50
+0
+0×10-4
+4
+Flux L1 Error
+3
+2
+0
+150
+100
+150
+100
+50
+50
+0
+0×10-4
+4
+3
+Flux L1 Error
+2
+0
+150
+150
+100
+100
+50
+50
+0
+0×10-4
+6
+Flux L1 Error
+4
+2
+0
+150
+150
+100
+100
+50
+50
+0
+0Smoothness of the Radiation Transfer Solution 
+ 
+It is evident that the convergence rate decreases as the mesh refines, and the errors are much larger at the 
+boundary. The “noisier” distributions in Cases 1 and 2 are due to the ray effects of the discrete ordinates 
+(SN) method, which are more pronounced in the optically thin problem. The convergence behavior is similar 
+between the cases with and without the scattering, indicating that the source term plays a significant role in 
+defining the irregularity of the solution. Cases 3 and 4 show the improved convergence rate as compared to 
+Cases 1 and 2 because the exponential function e–1|*–9| makes the kernel less singular as the total cross 
+section 𝜎 increases. In addition, Case 4 has a slightly better rate of convergence than Case 3 on fine meshes 
+(e.g., 1.84 vs. 1.75 on 640 × 640), because the transport problem becomes more like an elliptic diffusion 
+problem [17], and the diffusion solution in general has better regularity. It should be pointed out that in 
+Case 3, the convergence rate is only 1.51 on the coarse mesh. It is because for the pure absorption case, the 
+DD method becomes unstable when the mesh size is larger than 
+>Q!
+1 , where 𝜇& is the direction cosine of the 
+radiation transfer direction. However, it is more stable for the scattering case. 
+ 
+Remark 4.1. The error of the DD can be estimated by „𝜙& − 𝜙&
+R„ ≤ 𝐶ℎ&
+>‖𝜙′′‖?, where 𝜙& is the exact 
+solution at cell 𝑗, 𝜙&
+R is its numerical result, and ℎ& is the mesh size [15]. Although this optimal error 
+estimate is obtained for the 1D slab geometry, one can expect the same to be true in two dimensions. As 
+given by Eq. (28), the second derivative 𝜙44 will be bounded in the interior of the domain, while it would 
+behave as 𝜙44~ℎ&
+#0 near the boundary. Therefore, it is expected that the convergence rate of the DD would 
+decrease with refining the mesh, and asymptotically tend to 𝑂(ℎ). If the solution is sufficiently smooth 
+(e.g., a manufactured smooth solution), the DD would maintain its second order of accuracy on any mesh 
+size [16].  
+ 
+Remark 4.2. The scattering does not appear to play a role in defining the smoothness of the solution. For 
+the problem without the external source, if there exists a nonsmooth incoming flux on the boundary, then 
+the scattering may not be able to regularize the solution either, since the irregularity caused by the incoming 
+flux, which is defined by the surface integral term of Eq. (4), has nothing to do with the scattering and the 
+solution flux 𝜙.    
+ 
+ 
+5. CONCLUSIONS 
+ 
+We have derived the two-dimensional integral radiation transfer equation and examined the differential 
+properties of the integral kernel for fulfilling the boundedness conditions of Vainikko’s theorem. We use 
+the theorem to estimate the derivatives of the radiation transfer solution near the boundary of the domain. 
+It is noted that the first derivative of the scalar flux  𝜙(𝑥)	becomes unbounded when approaching the 
+boundary. The derivatives of order 𝑘 behave as 𝜚(𝑥)0#O for 𝑘 > 1, where 𝜚(𝑥) is the distance to the 
+boundary. A numerical example is presented to demonstrate that the irregularity of the exact solution will 
+reduce the rate of convergence of numerical solutions. The convergence rate improves as the optically 
+thickness of the problem increases. It is interesting to note that the scattering does not help smoothen the 
+solution. However, it does play a crucial role in transforming the transport problem into an elliptic diffusion 
+problem in the asymptotic diffusion limit. We are currently extending the analysis to the boundary integral 
+transport problem in considering nonzero incoming boundary conditions and corner effects. In addition, it 
+would be interesting to study the convergence behavior of weak solutions.  
+ 
+ 
+REFERENCES 
+ 
+1. S. G. Mikhlin, S. Prossdorf, Singular Integral Operators, Springer-Verlag (1986). 
+
+Dean Wang  
+ 
+2. S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, Oxford 
+(1965). 
+3. V. S. Vladimirov, Mathematical Problems in the One-Velocity Theory of Particle Transport, (Translated 
+from Transactions of the V. A. Steklov Mathematical Institute, 61, 1961), Atomic Energy of Canada 
+Limited (1963). 
+4. T. A. Germogenova, “Local properties of the solution of the transport equation,” Dokl. Akad. Nauk 
+SSSR, 187(5), pp. 978-981 (1969). 
+5. J. Pitkaranta, “Estimates for the Derivatives of Solutions to Weakly Singular Fredholm Integral 
+Equations,” SIAM J. Math. Anal., 11(6), pp. 952-968 (1980). 
+6. G. Vainikko, Multidimensional Weakly Singular Integral Equations, Springer-Verlag, Berlin 
+Heidelberg (1993). 
+7. C. Johnson and J. Pitkaranta, “Convergence of A Fully Discrete Scheme for Two-Dimensional Neutron 
+Transport,” SIAM J. Math. Anal., 20(5), pp. 951-966 (1983). 
+8. E. Hennebach, P. Junghanns, G. Vainikko, “Weakly Singular Integral Equations with Operator-Valued 
+Kernels and An Application to Radiation Transfer Problems,” Integr. Equat. Oper. Th., 22, pp. 37-64 
+(1995). 
+9. E. E. Lewis and W. F. Miller, Jr., Computational Methods of Neutron Transport, American Nuclear 
+Society (1993). 
+10. G. J. Bell and S. Glasstone, Nuclear Reactor Theory, Van Nostrand Reinhold Company, New York 
+(1970). 
+11. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and 
+Mathematical Tables, Dover, New York (1970). 
+12. N. K. Madsen, “Convergence of Singular Difference Approximations for the Discrete Ordinate 
+Equations in 𝑥– 𝑦 Geometry,” Math. Comput., 26(117), 45-50 (1972). 
+13. E. W. Larsen, “Spatial Convergence Properties of the Diamond Difference Method in x, y Geometry,” 
+Nucl. Sci. Eng., 80, 710-713 (1982). 
+14. Y. Wang and J. C. Ragusa, “On the Convergence of DGFEM Applied to the Discrete Ordinates 
+Transport Equation for Structured and Unstructured Triangular Meshes,” Nucl. Sci. Eng., 163, 56-72 
+(2009). 
+15. D. Wang, “Error Analysis of Numerical Methods for Thick Diffusive Neutron Transport Problems on 
+Shishkin Mesh,” Proceedings of International Conference on Physics of Reactors 2022 (PHYSOR 
+2022), Pittsburgh, PA, USA, May 15-20, 2022, pp. 977-986 (2022). 
+16. D. Wang, et al., “Solving the SN Transport Equation Using High Order Lax-Friedrichs WENO Fast 
+Sweeping Methods,” Proceedings of International Conference on Mathematics and Computational 
+Methods Applied to Nuclear Science and Engineering 2019 (M&C 2019), Portland, OR, USA, August 
+25-29, 2019, pp. 61-72 (2019). 
+17. D. Wang and T. Byambaakhuu, “A New Proof of the Asymptotic Diffusion Limit of the SN Neutron 
+Transport Equation,” Nucl. Sci. Eng., 195, 1347-1358 (2021). 
+ 
+

59AyT4oBgHgl3EQfcfe1/content/tmp_files/load_file.txt

@@ -0,0 +1,451 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf,len=450
+page_content='On the Smoothness of the Solution to the Two-Dimensional Radiation Transfer Equation  Dean Wang  The Ohio State University 201 West 19th Avenue, Columbus, Ohio 43210  wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='12239@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='edu  ABSTRACT  In this paper, we deal with the differential properties of the scalar flux 𝜙(𝑥) defined over a two-dimensional bounded convex domain, as a solution to the integral radiation transfer equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Estimates for the derivatives of 𝜙(𝑥) near the boundary of the domain are given based on Vainikko’s regularity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' A numerical example is presented to demonstrate the implication of the solution smoothness on the convergence behavior of the diamond difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  KEYWORDS: Integral Equation, Radiation Transfer, Regularity, Numerical Convergence  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' INTRODUCTION  Consider the integral equation of the second kind  𝜙(𝑥) = ∫ K(𝑥, 𝑦)𝜙(𝑦)𝑑𝑦 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' + 𝑓(𝑥) , 𝑥 ∈ G ,  (1)  where G ⊂ ℝ" , 𝑛 ≥ 1,  is an open bounded domain and the kernel K(𝑥, 𝑦)  is weakly singular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', |K(𝑥, 𝑦)| ≤ 𝐶|𝑥 − 𝑦|#$, 0 ≤ 𝜈 ≤ 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Weakly singular integral equations arise in many physical applications such as elliptic boundary problems and particle transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  The standard integro-differential equation of radiation transfer can be reformulated as a weakly singular integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=" The one-group radiation transfer problem in a three-dimensional (3D) convex domain reads as follows: find a function 𝜙: G: × Ω → ℝ% such as  ∑ Ω& '((*,,) '*!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' &/0 + 𝜎(𝑥)𝜙(𝑥, Ω) = 1"(*) 23 ∫ 𝑠(𝑥, Ω, Ω′)𝜙(𝑥, Ω4)𝑑Ω′ , + 𝑓(𝑥, Ω) , 𝑥 ∈ G ,   (2)  𝜙(𝑥, Ω) = 𝜙56(𝑥, Ω),  𝑥 ∈ 𝜕G,  Ω ∙ 𝑛D(𝑥) < 0 ,  (3)  where Ω denotes the direction of radiation transfer, 𝜕G is the boundary of the domain G ⊂ ℝ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 𝜎 is the extinction coefficient (or macroscopic total cross section in neutron transport), 𝜎7  is the scattering coefficient (or macroscopic scattering cross section), 𝑠  is the phase function of scattering with ∫ 𝑠(𝑥, Ω, Ω′)𝑑Ω′ , = 4𝜋, 𝑓 is the external source function, and 𝑛D is the unit normal vector of the domain surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Note that 𝜎7(𝑥) ≤ 𝜎(𝑥) and 𝑠(𝑥, Ω, Ω4) = 𝑠(𝑥, Ω′, Ω) under physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Dean Wang  Assuming the isotropic scattering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 𝑠(𝑥, Ω, Ω4) = 1, 𝑓(𝑥, Ω) = 8(*) 23 , and 𝜙56(𝑥, Ω) = (#$(*) 23 , we can obtain the so-called Peierls integral equation of radiation transfer for the scalar flux 𝜙(𝑥) as follows:  𝜙(𝑥) =  0  23 ∫  1"(9):–&((, )  | –9|,  𝜙(𝑦)𝑑𝑦  6  +  0  23 ∫  :–&((, )  | –9|, 𝑓(𝑦)𝑑𝑦  6  +  0  23 ∫  :–&((, )  | –9|, H –9  | –9| 𝑛(𝑥)H 𝜙56(𝑦)𝑑𝑆9  \'6  ,  (4)  𝜏(𝑥, 𝑦) = ∫  𝜎(𝑟– 𝜉Ω)𝑑𝜉  | –9|  =  ,  (5)  where 𝑑𝑆 is the differential element of the domain surface, 𝜏(𝑥, 𝑦) is the optical path between 𝑥 and 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' One can find detailed derivation in [6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  For simplicity, we assume 𝜎 and 𝜎7 are constant over the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (4) can be simplified as  𝜙(𝑥) = ∫ K(𝑥, 𝑦)𝜙(𝑦)𝑑𝑦 6 + 0 1" ∫ K(𝑥, 𝑦)𝑓(𝑦)𝑑𝑦 6  +  0  1" ∫  K(𝑥, 𝑦) H –9  | –9| 𝑛(𝑥)H 𝜙56(𝑦)𝑑𝑆9  \'6  ,  (6)  where the 3D radiation kernel is given as  K(𝑥, 𝑦) =  1":– |(– |  23| –9|,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  (7)  The boundary integral term in the above equation can produce singularities in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' We omit its discussion in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' In other words, we only consider the problem with the vacuum boundary condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 𝜙56 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (6) can be treated as the weakly integral equation of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Since K(𝑥, 𝑦) has a singularity at 𝑥 = 𝑦, the solution of a weakly integral equation is generally not a smooth function and its derivatives at the boundary would become unbounded from a certain order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' There was extensive research on the smoothness (regularity) properties of the solutions to weakly integral equations [1,2], especially those early work in neutron transport theory done in the former Soviet Union [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' It is believed that Vladimirov first proved that the scalar flux 𝜙(𝑥)  possesses the property |𝜙(𝑥 + ℎ) − 𝜙(𝑥)|~ℎlogℎ for the one-group transport problem with isotropic scattering in a bounded domain [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Germogenova analyzed the local regularity of the angular flux 𝜙(𝑥, Ω)  in a neighborhood of the discontinuity interface and obtained an estimate of the first derivative, which has the singularity near the interface [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Pitkaranta derived a local singular resolution showing explicitly the behavior of 𝜙(𝑥) near the smooth portion of the boundary [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Vainikko introduced weighted spaces and obtained sharp estimates of pointwise derivatives near the smooth boundary for multidimensional weakly singular integral equations [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  There exists some previous research work on the regularity of the integral radiation transfer solutions [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' However, the 2D kernel used in those studies is physically incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' In this paper, we rederive the 2D kernel by directly integrating the 3D kernel with respect to the third dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' We examine the differential properties of the new 2D kernel and provide estimates of pointwise derivatives of the scalar flux according to Vainikko’s regularity theorem for the weakly integral equation of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Smoothness of the Radiation Transfer Solution  The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 2, we derive the 2D kernel for the integral radiation transfer equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' We examine the derivatives of the kernel and show that they satisfy the boundedness condition of Vainikko’s regularity theorem in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Then the estimates of local regularity of the scalar flux near the boundary of the domain are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 4 presents numerical results to demonstrate that the rate of convergence of numerical methods can be affected by the smoothness of the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Concluding remarks are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' TWO DIMENSIONAL RADIATION TRANSFER EQUATION  In this section, we derive the 2D integral radiation transfer equation from its 3D form, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' In 3D, 𝑑𝑦 = 𝑑𝑦0𝑑𝑦>𝑑𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' and |𝑥– 𝑦| = S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)> + (𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='– 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' )> .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Let 𝜌 = S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)> , then |𝑥– 𝑦| = S𝜌> + (𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='– 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=')>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' In a 2D domain 𝐺 ⊂ ℝ>, the solution function 𝜙(𝑥) only depends on 𝑥0 and 𝑥> in Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Therefore, we only need to find the 2D radiation kernel, which can be obtained by integrating out 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' as follows:  K(𝑥, 𝑦) = ∫  1":– |(– |  23| –9|, 𝑑��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  #?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  =  1"  23 ∫  :  / 01,2((3– 3),  @,%( 3–93),  𝑑𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  #?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  (8)  To proceed, we introduce the variables 𝑡 = 𝜎S𝜌> + (𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='– 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' )> and 𝑧 = 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' − 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='. Then we substitute 𝑑𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' = 𝑑z = A 1BA,–1,@, 𝑑𝑡 into the above equation to have  K(𝑥, 𝑦) =  1"  23 ∫  :/4  5, ,  𝑑z  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  #?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  =  1"  >3 ∫  :/4  5, ,  𝑑z  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  =  =  1"  >3 ∫  :/4  5, ,  A  1BA,–1,@, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  =  = 1"1 >3 ∫ :/4 ABA,–1,@, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 1@ = 1"1 >3 ∫ :/4 ABA,#1,|*–9|, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 1|*–9| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (9)  Note that the 2D radiation kernel is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' By replacing the 3D kernel of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (7) with the above one, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (6) becomes the 2D integral radiation transfer equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Notice that the surface integral in the last term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (6) should be replaced with the line integral in the 2D domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Now we show that the 2D kernel K(𝑥, 𝑦) has a singularity at 𝜌 = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 𝑥 = 𝑦) as follows:  K(𝑥, 𝑦) =  1"1  >3 ∫  :/4  ABA,–1,@, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  1@  >  1"1  >3 ∫  :/4  A, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  1@  = 1"1 >3 e C/-1 1@ − Γ(0, 𝜎𝜌)g ,                                                            (10)  where Γ(0, 𝑎) = ∫ :/4 A 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' D , is the incomplete gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The singular behavior of K(𝑥, 𝑦) near 𝜌 = 0 is dominated by the first term C/-1 1@  in the brackets since the gamma function tends to infinity much slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Dean Wang  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' It should be noted that the 2D kernel defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (9) is equivalent to the more conventional one defined by the Bickley-Naylor functions [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Johnson and Pitkaranta derived a 2D kernel for neutron transport by reformulating the standard integro-differential equation on the 2D plane [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The kernel obtained is, K(𝑥, 𝑦) = :–|(–*| |*–9|  (assuming 𝜎 = 1), which is however mathematically correct but physically incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Hennebach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' also used the same 2D kernel for analyzing the radiation transfer solutions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' In addition, the integral equations in other geometries such as slab or sphere can be obtained by following the same approach, and they can be found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Applying Banach’s fixed-point theorem, we can prove the existence and uniqueness of the solution in the 2D domain by showing that ∫ K(𝑥, 𝑦)𝑑𝑦 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' is bounded below unity as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  ∫ K(𝑥, 𝑦)𝑑𝑦  6  = ∫ i  1"1  >3 ∫  :/4  ABA,–1,@, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  1@  j 𝑑𝑦  6  =  1"1  >3 ∫ 𝑑𝑦 ∫  :/4  ABA,–1,@, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  1@  6  =  1"1  >3 ∫ 𝜌𝑑𝜑𝑑𝜌 ∫  :/4  ABA,#1,@, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  1@  6  ,  (11)  where 𝜑 is the azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' By extending the above bounded domain to the whole space, we have  ∫ K(𝑥, 𝑦)𝑑𝑦  6  <  1"1  >3 ∫  2𝜋𝜌𝑑𝜌  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  =  ∫  :/4  ABA,–1,@, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  1@  = 𝜎7𝜎 ∫  𝜌𝑑𝜌  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  =  ∫  :/4  ABA,–1,@, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  1@  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  (12)  Denoting 𝜁 = 𝜎𝜌, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (12) is simplified as  ∫ K(𝑥, 𝑦)𝑑𝑦  6  <  1"  1 ∫  𝜁𝑑𝜁  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  =  ∫  :/4  ABA,–E, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  E  =  1"  1 ∫  ∫  :/4  A  E  BA,#E, 𝑑𝑡𝑑𝜁  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  E  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  =  =  1"  1 ∫  :/4  A 𝑑𝑡 ∫  E  BA,–E, 𝑑𝜁  A  =  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  =  =  1"  1 ∫  e#F𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  =  = 1" 1 ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (13)  Notice we have changed the order of integration to solve the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' It is apparent that for there exists a unique solution, the physical condition, 𝜎7 ≤ 𝜎, must be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' SMOOTHNESS OF THE SOLUTIONS  We first introduce Vainikko’s regularity theorem [6], which provides a sharp characterization of singularities for the general weakly integral equation of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Then we analyze the differentiational properties of the 2D radiation kernel and show that the derivatives are properly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Finally, Vainikko’s theorem is used to give the estimates of pointwise derivatives of the radiation solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Smoothness of the Radiation Transfer Solution  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Vainikko’s Regularity Theorem  Before we state the theorem, we introduce the definition of weighted spaces ℂG,$(G) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Weighted space ℂ𝒎,𝝂(𝐆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' For a 𝜆 ∈ ℝ, introduce a weight function  𝑤J = r 1                                ,   𝜆 < 0 (1 + |log𝜚(𝑥)|)#0 , 𝜆 = 0 𝜚(𝑥)J                       ,   𝜆 > 0 , 𝑥 ∈ G  (14)  where G ⊂ ℝ" is an open bounded domain and 𝜚(𝑥) = inf 9∈\'6|𝑥 − 𝑦| is the distance from 𝑥 to the boundary 𝜕G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Let 𝑚 ∈ ℕ, 𝜈 ∈ ℝ and 𝜈 < 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Define the space ℂG,$(G) as the set of all 𝑚 times continuously differentiable functions 𝜙: G →  ℝ such that  ‖𝜙‖G,$ = ∑  sup ∈6  {𝑤|L|–("–$)|𝐷L𝜙(𝑥)|}  |L|MG  < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  (15)  In other words, a 𝑚 times continuously differentiable function 𝜙 on G belongs to ℂG,$(G) if the growth of its derivatives near the boundary can be estimated as follows:  |𝐷L𝜙(𝑥)| ≤ 𝑐 r 1                        ,   |𝛼| < 𝑛– 𝜈 1 + |log𝜚(𝑥)| ,   |𝛼| = 𝑛– 𝜈 𝜚(𝑥)"#$#|L|    ,   |𝛼| > 𝑛– 𝜈 , 𝑥 ∈ G,  |𝛼| ≤ 𝑚 ,  (16)  where 𝑐 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The space ℂG,$(G), equipped with the norm ‖∙‖G,$, is a complete Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  After defining the weighted space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' we introduce the smoothness assumption about the kernel in the following form: the kernel K(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 𝑦) is 𝑚 times continuously differentiable on (G × G)\\{𝑥 = 𝑦} and there exists a real number 𝜈 ∈ (−∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 𝑛) such that the estimate  H𝐷*L𝐷*%9 N K(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 𝑦)H ≤ 𝑐 r 1                             ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='   𝜈 + |𝛼| < 0 1 + „log|𝑥 − 𝑦|„ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='   𝜈 + |𝛼| = 0 |𝑥 − 𝑦|#$#|L|      ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='   𝜈 + |𝛼| > 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='      𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=" 𝑦 ∈ G (17)  where  𝐷 L = …  '  ' 6†  L6 ⋯ …  '  ' 7†  L7 ," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content="  (18)  𝐷 %9  N  = …  '  ' 6 +  '  '96†  N6 ⋯ …  '  ' 7 +  '  '97†  N7 ," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  (19)  holds for all multi-indices 𝛼 = (𝛼0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 𝛼") ∈ ℤ%" and 𝛽 = (𝛽0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' ⋯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 𝛽") ∈ ℤ%" with |𝛼| + |𝛽| ≤ 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Here the following usual conventions are adopted: |𝛼| = 𝛼0 + ⋯ + 𝛼", and |𝑥| = S𝑥0 > + ⋯ + 𝑥">.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Now we present Vainikko’s theorem in characterizing the regularity properties of a solution to the weakly integral equation of the second kind [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Let G ⊂ ℝ" be an open bounded domain, 𝑓 ∈ ℂG,$(G) and let the kernel K(𝑥, 𝑦) satisfy the condition (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' If the integral equation (1) has a solution, 𝜙 ∈ L?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (G) then 𝜙 ∈ ℂG,$(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Dean Wang  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The solution does not improve its properties near the boundary 𝜕G, remaining only in ℂG,$(G), even if 𝜕G is of class ℂ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' and, 𝑓 ∈ ℂ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' A proof can be found in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' More precisely, for any 𝑛 and 𝜈 (𝜈 < 𝑛) there are kernels K(𝑥, 𝑦) satisfying (17) and such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (1) is uniquely solvable and, for a suitable 𝑓 ∈ ℂ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (G), the normal derivatives of order 𝑘 of the solution behave near 𝜕G as log𝜚(𝑥) if 𝑘 = 𝑛– 𝜈, and as 𝜚(𝑥)"#$#O for 𝑘 > 𝑛– 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Smoothness of the Radiation Transfer Solution  To apply the results of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='1 to the 2D integral radiation transfer equation, we need to analyze the kernel K(𝑥, 𝑦) and show it satisfying the condition (17), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', H𝐷*L𝐷*%9 N K(𝑥, 𝑦)H ≤ 𝑐|𝑥 − 𝑦|#0#|L|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' We can simply set |𝛽| = 0 without loss of generality for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  |𝜶| = 𝟎:  K(𝑥, 𝑦) =  1"1  >3 ∫  :/4  ABA,–1,@, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  1@  <  1"1  >3 ∫  :/ 1  ABA,–1,@, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  1@  =  1"1:/ 1  >3  ∫  0  ABA,–1,@, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  1@  =  1"1:/ 1  >3  3  >1@ =  1":/ |(– |  2| –9|  ≤ 𝑐|𝑥– 𝑦|#0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (20)  |𝜶| = 𝟏: Let 𝜁 = 𝜎𝜌 = 𝜎|𝑥– 𝑦| = 𝜎S(𝑥0– 𝑦0)> + (𝑥>– 𝑦>)>, then K(𝑥, 𝑦) = 1"1 >3 ∫ :/4 ABA,–E, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=" E , and  |𝐷 K(𝑥, 𝑦)| = H  '  'E K(𝑥, 𝑦) 'E  ' H = H  '  'E K(𝑥, 𝑦)H H  'E  ' H ,  (21)  where,  H  'E  ' 6H = 𝜎 ( 6–96)  B( 6–96),%( ,–9,), ≤ 𝜎 ,  (22)  H  'E  ' ,H = 𝜎 ( ,–9,)  B( 6–96),%( ,–9,), ≤ 𝜎 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content="  (23)  Apparently, we only need to find the upper bound of • ' 'E ∫ :/4 ABA,–E, 𝑑𝑡 ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' E • ≤ 𝑐𝜁#>, which is shown in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' First, we simplify the integral ∫ :/4 ABA,–E, 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' E as  ∫  :/4  ABA,–E, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  E  =  0  E, ∫  A:/4  BA,–E, 𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  E  −  0  E, ∫  :/4BA,–E,  A  𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  E  =  P6(E)  E  −  0  E, ∫  :/4BA,–E,  A  𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  E  ,  (24)  where 𝐾0(𝜁) is the modified Bessel function of the second kind, and 𝐾0(𝜁)~ 0 E when 𝜁 → 0 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content="  ' 'E ∫ :/4 ABA,–E, 𝑑𝑡 ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' E Smoothness of the Radiation Transfer Solution  = −  P6(E)  E, +  P68(E)  E  +  >  E3 ∫  :/4BA,–E,  A  𝑑𝑡  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  E  −  0  E ∫  :/4  ABA,–E, 𝑑𝑡  E  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  (25)  Notice the third term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (25), ∫ :/4BA,–E, A 𝑑𝑡 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' E → 1 as 𝜁 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' It is not difficult to find that the first three terms will cancel out when 𝜁 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=" Then we obtain  ' 'E ∫ :/4 ABA,–E, 𝑑𝑡 ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' E  ≤  0 E ∫ :/4 ABA,–E, 𝑑𝑡 E ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' ≤ 3 > :/9 E, = 3 >1, :/-|(–*| |*–9|,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (26)  Notice that here we have used the upper bound of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Now we arrive at the desired result for  |𝛼| = 1:  |𝐷*K(𝑥, 𝑦)| ≤ 𝑐|𝑥– 𝑦|#> .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (27)  |𝜶| = 𝟐 (and larger): we can follow the same procedure to find |𝐷*LK(𝑥, 𝑦)| ≤ 𝑐|𝑥 − 𝑦|#0#|L|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Finally, we conclude that the 2D radiation kernel satisfies the condition (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Therefore, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='1, the estimates of derivatives of the scalar flux 𝜙(𝑥) for radiation transfer are the same as for the general weakly integral equation of the second kind:  |𝐷L𝜙(𝑥)| ≤ 𝑐 r 1                        ,   |𝛼| < 1 1 + |log𝜚(𝑥)| ,   |𝛼| = 1 𝜚(𝑥)0#|L|         ,   |𝛼| > 1 , 𝑥 ∈ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (28)  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The first derivative of the solution 𝜙(𝑥) behaves as log𝜚(𝑥) and becomes unbounded as approaching the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The derivatives of order 𝑘 behave as 𝜚(𝑥)0#O for 𝑘 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' As mentioned in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='1, these pointwise estimates cannot be improved by adding more strong smoothness on the data and domain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' We point out that the lack of smoothness in the exact solution could adversely affect the convergence rate of spatial discretization schemes for solving the radiation transfer equation [12-14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' According to the regularity results, it is expected that the asymptotic convergence rate of the spatial discretization error of finite difference methods would be around 1 in the 𝐿?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' or 𝐿0 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' NUMERICAL RESULTS  In this section, we demonstrate how the regularity of the exact solution will impact the numerical convergence rate by solving the SN neutron transport equation in its original integro-differential form, using the classic second-order diamond difference (DD) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The model problem is a 1cm × 1cm square with the vacuum boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Thus, there will be no complication from the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The S12 level-symmetric quadrature set is used for angular discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  We analyze the following four cases: Case 1: ΣA = 1, Σ7 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Case 2: ΣA = 1, Σ7 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Case 3: ΣA = 10, Σ7 = 0, and Case 4: ΣA = 10, Σ7 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' For all the cases, the external source 𝑓 = 1, is infinitely differentiable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 𝑓 ∈ ℂ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Cases 1 and 3 are pure absorption problems, while Case 3 is optically thicker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' It is interesting to note that the solutions are only determined by the external source for these two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Cases 2 and 4 include the scattering effects, while Case 4 is optically thicker and more diffusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Both the scattering and external source contribute to the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The flux L1 errors as a function of mesh  Dean Wang  size and the rates of convergences are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The error distributions on the mesh 160 × 160 are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The reference solution for each case is obtained on a very fine mesh, 5120 × 5120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Flux L1 errors and convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Mesh  (𝑵 × 𝑵)  Case 1  Case 2  Case 3  Case 4  Error  Rate  Error  Rate  Error  Rate  Error  Rate  10 × 10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='87E 03  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='59E 03  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='31E 03  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='29E 03  20 × 20  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='95E 04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
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+page_content='01E 03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='83  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='12E 04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='51  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='56E 03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='86  40 × 40  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='90E 04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='45  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
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+page_content='31E 04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='82  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='89E 04  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='12  80 × 80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='14E 04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='44E 04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='37  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='19E 05  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='37E 04  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='10  160 × 160  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='04E 05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='17  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='32E 05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='32E 05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='97  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='53E 05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='96  320 × 320  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='46E 05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='03  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='06E 05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='04  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='61E 06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='87  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='39E 06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='91  640 × 640  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='31E 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='63E 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='11E 06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='71  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='70E 06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='80  1280 × 1280  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='26E 06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='07  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='76E 06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='07  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='87E 07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='51  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='51E 07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='66  Case 1  Case 2  Case 3  Case 4  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Flux error distribution on the mesh 𝟏𝟔𝟎 × 𝟏𝟔𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  ×10-4 4 Flux L1 Error 3 2 0 150 100 150 100 50 50 0 0×10-4 4 Flux L1 Error 3 2 0 150 100 150 100 50 50 0 0×10-4 4 3 Flux L1 Error 2 0 150 150 100 100 50 50 0 0×10-4 6 Flux L1 Error 4 2 0 150 150 100 100 50 50 0 0Smoothness of the Radiation Transfer Solution  It is evident that the convergence rate decreases as the mesh refines, and the errors are much larger at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The “noisier” distributions in Cases 1 and 2 are due to the ray effects of the discrete ordinates (SN) method, which are more pronounced in the optically thin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The convergence behavior is similar between the cases with and without the scattering, indicating that the source term plays a significant role in defining the irregularity of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Cases 3 and 4 show the improved convergence rate as compared to Cases 1 and 2 because the exponential function e–1|*–9| makes the kernel less singular as the total cross section 𝜎 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' In addition, Case 4 has a slightly better rate of convergence than Case 3 on fine meshes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='84 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='75 on 640 × 640), because the transport problem becomes more like an elliptic diffusion problem [17], and the diffusion solution in general has better regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' It should be pointed out that in Case 3, the convergence rate is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='51 on the coarse mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' It is because for the pure absorption case, the DD method becomes unstable when the mesh size is larger than >Q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 1 , where 𝜇& is the direction cosine of the radiation transfer direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' However, it is more stable for the scattering case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The error of the DD can be estimated by „𝜙& − 𝜙& R„ ≤ 𝐶ℎ& >‖𝜙′′‖?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', where 𝜙& is the exact solution at cell 𝑗, 𝜙& R is its numerical result, and ℎ& is the mesh size [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Although this optimal error estimate is obtained for the 1D slab geometry, one can expect the same to be true in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' As given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (28), the second derivative 𝜙44 will be bounded in the interior of the domain, while it would behave as 𝜙44~ℎ& #0 near the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Therefore, it is expected that the convergence rate of the DD would decrease with refining the mesh, and asymptotically tend to 𝑂(ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' If the solution is sufficiently smooth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', a manufactured smooth solution), the DD would maintain its second order of accuracy on any mesh size [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The scattering does not appear to play a role in defining the smoothness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' For the problem without the external source, if there exists a nonsmooth incoming flux on the boundary, then the scattering may not be able to regularize the solution either, since the irregularity caused by the incoming flux, which is defined by the surface integral term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' (4), has nothing to do with the scattering and the solution flux 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' CONCLUSIONS  We have derived the two-dimensional integral radiation transfer equation and examined the differential properties of the integral kernel for fulfilling the boundedness conditions of Vainikko’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' We use the theorem to estimate the derivatives of the radiation transfer solution near the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' It is noted that the first derivative of the scalar flux  𝜙(𝑥) becomes unbounded when approaching the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The derivatives of order 𝑘 behave as 𝜚(𝑥)0#O for 𝑘 > 1, where 𝜚(𝑥) is the distance to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' A numerical example is presented to demonstrate that the irregularity of the exact solution will reduce the rate of convergence of numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' The convergence rate improves as the optically thickness of the problem increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' It is interesting to note that the scattering does not help smoothen the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' However, it does play a crucial role in transforming the transport problem into an elliptic diffusion problem in the asymptotic diffusion limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' We are currently extending the analysis to the boundary integral transport problem in considering nonzero incoming boundary conditions and corner effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' In addition, it would be interesting to study the convergence behavior of weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  REFERENCES  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Mikhlin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Prossdorf, Singular Integral Operators, Springer-Verlag (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  Dean Wang  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, Oxford (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Vladimirov, Mathematical Problems in the One-Velocity Theory of Particle Transport, (Translated from Transactions of the V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Steklov Mathematical Institute, 61, 1961), Atomic Energy of Canada Limited (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Germogenova, “Local properties of the solution of the transport equation,” Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Nauk SSSR, 187(5), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 978-981 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Pitkaranta, “Estimates for the Derivatives of Solutions to Weakly Singular Fredholm Integral Equations,” SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 11(6), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 952-968 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Vainikko, Multidimensional Weakly Singular Integral Equations, Springer-Verlag, Berlin Heidelberg (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Johnson and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Pitkaranta, “Convergence of A Fully Discrete Scheme for Two-Dimensional Neutron Transport,” SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 20(5), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 951-966 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Hennebach, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Junghanns, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Vainikko, “Weakly Singular Integral Equations with Operator-Valued Kernels and An Application to Radiation Transfer Problems,” Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Equat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 37-64 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Lewis and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Miller, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', Computational Methods of Neutron Transport, American Nuclear Society (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Bell and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Glasstone, Nuclear Reactor Theory, Van Nostrand Reinhold Company, New York (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Abramowitz and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Dover, New York (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content='  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Madsen, “Convergence of Singular Difference Approximations for the Discrete Ordinate Equations in 𝑥– 𝑦 Geometry,” Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 26(117), 45-50 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Larsen, “Spatial Convergence Properties of the Diamond Difference Method in x, y Geometry,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 80, 710-713 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Wang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Ragusa, “On the Convergence of DGFEM Applied to the Discrete Ordinates Transport Equation for Structured and Unstructured Triangular Meshes,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 163, 56-72 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Wang, “Error Analysis of Numerical Methods for Thick Diffusive Neutron Transport Problems on Shishkin Mesh,” Proceedings of International Conference on Physics of Reactors 2022 (PHYSOR 2022), Pittsburgh, PA, USA, May 15-20, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 977-986 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', “Solving the SN Transport Equation Using High Order Lax-Friedrichs WENO Fast Sweeping Methods,” Proceedings of International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering 2019 (M&C 2019), Portland, OR, USA, August 25-29, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 61-72 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Wang and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Byambaakhuu, “A New Proof of the Asymptotic Diffusion Limit of the SN Neutron Transport Equation,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}
+page_content=', 195, 1347-1358 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfcfe1/content/2301.00285v1.pdf'}

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+SeedTree: A Dynamically Optimal and
+Local Self-Adjusting Tree
+Arash Pourdamghani1, Chen Avin2, Robert Sama3, Stefan Schmid1,4
+1TU Berlin, Germany 2School of Electrical and Computer Engineering, Ben Gurion University of the Negev, Israel
+3Faculty of Computer Science, University of Vienna, Austria 4Fraunhofer SIT, Germany
+Abstract—We consider the fundamental problem of design-
+ing a self-adjusting tree, which efficiently and locally adapts
+itself towards the demand it serves (namely accesses to the
+items stored by the tree nodes), striking a balance between
+the benefits of such adjustments (enabling faster access) and
+their costs (reconfigurations). This problem finds applications,
+among others, in the context of emerging demand-aware and
+reconfigurable datacenter networks and features connections to
+self-adjusting data structures. Our main contribution is SeedTree,
+a dynamically optimal self-adjusting tree which supports local
+(i.e., greedy) routing, which is particularly attractive under highly
+dynamic demands. SeedTree relies on an innovative approach
+which defines a set of unique paths based on randomized item
+addresses, and uses a small constant number of items per node.
+We complement our analytical results by showing the benefits
+of SeedTree empirically, evaluating it on various synthetic and
+real-world communication traces.
+Index Terms—Reconfigurable datacenters, Online algorithms,
+Self-adjusting data structure
+I. INTRODUCTION
+This paper considers the fundamental problem of designing
+self-adjusting trees: trees which adapt themselves towards the
+demand they serve. Such self-adjusting trees need to strike
+an efficient tradeoff between the benefits of such adjustments
+(better performance in the future) and their costs (reconfigura-
+tion overheads now). The problem is motivated by the fact that
+workloads in practice often feature much temporal and spatial
+structure, which may be exploited by self-adjusting optimiza-
+tions [1], [2]. Furthermore, such adjustments are increasingly
+available, as researchers and practitioners are currently making
+great efforts to render networked and distributed systems more
+flexible, supporting dynamic reconfigurations, e.g., by leverag-
+ing programmability (via software-defined networks) [3], [4],
+network virtualization [5], or reconfigurable optical commu-
+nication technologies [6].
+In particular, we study the following abstract model (appli-
+cations will follow): we consider a binary tree which serves
+access requests, issued at the root of the tree, to the items
+stored by the nodes. Each node (e.g., server) stores up to
+c items (e.g., virtual machines), where c is a parameter
+indicating the capacity of a node. We consider an online
+perspective where items are requested over time. An online
+algorithm aims to optimize the tree in order to minimize
+the cost of future access requests (defined as the path length
+This project has received funding from the European Research Council
+(ERC) under grant agreement No. 864228 (AdjustNet), 2020-2025.
+between root and accessed item), while minimizing the number
+of items moving up or down in the tree: the reconfigurations.
+We call each movement a reconfiguration, and keep track of
+its cost. In particular, the online algorithm which does not
+know the future access requests, aims to be competitive with
+an optimal offline algorithm that knows the entire request
+sequence ahead of time. In other words, we are interested in
+an online algorithm with minimum competitive ratio [7] over
+any (even worst-case) request sequence.
+Self-adjusting trees are not only one of the most fundamen-
+tal topological structures of their own merit, they also have
+interesting applications. For example, such trees are a crucial
+building block for more general self-adjusting networks: Avin
+et al. [8] recently showed that multiple trees optimized indi-
+vidually for a single root, can be combined to build general
+communication networks which provide low degree and low
+distortion. The design of a competitive self-adjusting tree as
+studied in this paper, is hence a stepping stone.
+Self-adjusting trees also feature interesting connections to
+self-adjusting data structures (see §VI for a detailed discus-
+sion), for some of which designing and proving constant-
+competitive online algorithms is still an open question [9].
+Interestingly, a recent result shows that constant-competitive
+online algorithms exist for self-adjusting balanced binary trees
+if one maintains a global map of the items in the tree; it was
+proposed to store such a map centrally, at a logical root [10].
+In this paper, we are interested in the question whether
+this limitation can be overcome, and whether a competitive
+decentralized solution exist.
+Our main contribution is a dynamically optimal self-
+adjusting tree, SeedTree*, which achieves a constant compet-
+itive ratio by keeping recently accessed items closer to the
+root, ensuring a working set theorem [9]. Our result also im-
+plies weaker notions such as key independent optimality [11]
+(details will follow). SeedTree further supports local (that is,
+greedy and hence decentralized) routing, which is particularly
+attractive in dynamic networks, by relying on an innovative
+and simple routing approach that enables nodes to take local
+forwarding decisions: SeedTree hashes items to i.i.d. random
+addresses and defines a set of greedy paths based on these
+addresses. A main insight from our work is that a constant
+competitive ratio with locality property can be achieved if
+*The name is due to the additional capacity in nodes of the tree, which
+resembles seeds in fruits of a tree.
+arXiv:2301.03074v1  [cs.DS]  8 Jan 2023
+
+Fig. 1: A depiction of SeedTree with capacity 2. Large circles
+represent nodes (nodes) of the system, and small circles
+represent items. The number inside each small circle is the
+hash of the corresponding item.
+nodes feature small constant capacities, that is, by allowing
+nodes to store a small constant number of items. Storing more
+than a single item on a node is often practical, e.g., on a server
+or a peer [12], and it is common in hashing data structures with
+collision [13], [14]. We also evaluate SeedTree empirically,
+both on synthetic traces with ranging temporal locality and
+also data derived from Facebook datacenter networks [1],
+showing how tuning parameters of the SeedTree can lower
+the total (and access) cost for various scenarios.
+The remainder of the paper is organized as follows. §II in-
+troduces our model and preliminaries. We present and analyze
+our online algorithm in §III, and transform it to the matching
+model of datacenter networks in §IV. After discussing our
+empirical evaluation results in §V, we review related works
+in §VI and conclude our contributions in §VII.
+II. MODEL AND PRELIMINARIES
+This section presents our model and introduces preliminar-
+ies used in the design of SeedTree.
+Items and nodes. We assume a set of items V
+=
+(v1, . . . , vn), and a set of nodes S = (s1, . . . )† arranged as a
+binary tree T. We call the node s1 the root, which is at depth
+0 in the tree T, and a node sj is at depth ⌊log j⌋.
+Each node can store c items, where c is a parameter
+indicating the capacity of a node. In our model, we assume
+that c is a constant. The assignment of items to nodes can
+change over time. We say a node is full if it contains c items,
+and empty if it contains no item (See an example in Figure 1).
+We define the level of item v at time t, levelt(v), as the
+depth of the node containing v. For example, if item v is at
+node sj at time t, we have levelt(v) = ⌊log j⌋.
+Request Sequence and Working Set. Items are requested
+over time in an online manner, modeled as a request sequence
+σ = (σ1, . . . , σm), where σt = v ∈ V means item v is
+requested at time t. We are sometimes interested in the recency
+of item requests, particularly the size of the working set.
+Formally, we define wst(σ, v) as the working set of item v in
+at time t in the request sequence σ. The working set wst(σ, v)
+is a set of unique items requested since the last request to the
+item v before time t. We define a rank of item v at time t,
+rankt(v), as the size of working set of the item v at time t.
+†We assume the set of nodes to be arbitrarily large, as the exact number
+of nodes will be determined based on their used capacity.
+Costs and Competitive Ratio. We partition costs incurred
+by an algorithm, ALG, into two parts, the cost of finding an
+item: the access cost, and the cost of reconfigurations: the
+reconfiguration cost. The search for any item starts at the root
+node and ends at the node containing the item. Based on our
+assumption of constant capacity, we assume the cost of search
+inside a node to be negligible. Furthermore, assuming the local
+routing property, we find an item by traversing a single path
+in our tree; hence the access cost for an access request σi,
+CA
+ALG(σi), equals the level at which the item is stored.
+In our model, a reconfiguration consists of moving an item
+one level up or one level down in the tree, plus potentially
+additional lookups inside a node. We denote the total recon-
+figuration cost after an access request σi by CR
+ALG(σi). Hence,
+the total cost of each access request is CA
+ALG(σi)+CR
+ALG(σi),
+and the total cost of the algorithm on the whole request
+sequence is: CALG(σ) = �m
+i=1 CA
+ALG(σi) + CR
+ALG(σi). The
+objective of SeedTree is to operate at the lowest possible cost,
+or more specifically, as close as possible to the cost of an
+optimal offline algorithm, OPT.
+Definition 1 (Competitive ratio). Given an online algorithm
+ALG and an optimal offline algorithm OPT, the (strict)
+competitive ratio is defined as: ρALG = maxσ
+CALG(σ)
+COP T (σ)
+Furthermore, we say an algorithm has (strict) access com-
+petitive ratio considering only the access cost of the online
+algorithm ALG (not including the reconfiguration cost).
+In this paper, we prove that SeedTree is dynamically optimal.
+It means that the cost of our algorithm matches the cost of the
+optimal offline algorithm asymptotically.
+Definition 2 (Dynamic optimality). Algorithm ALG is dy-
+namically optimal if it has constant competitive ratio, i.e.,
+ρALG = O(1).
+MRU trees. We define a specific class of self-adjusting
+trees, MRU trees. An algorithm maintains a MRU tree if it
+keeps items at a similar level to their ranks.
+Definition 3 (MRU tree). An algorithm has the MRU(0)
+property if for any item v inside its tree and at any given time
+t, the equality levelt(v) = ⌊log ⌈ rankt(v)
+c
+⌉⌋ holds.
+Similarly, we say an algorithm maintains an MRU(β) if it
+ensures the relaxed bound of levelt(v) ≤ ⌊log ⌈ rankt(v)
+c
+⌉⌋ + β
+for any item v in the tree.
+III. ONLINE SEEDTREE
+This section presents SeedTree, an online algorithm that
+is dynamically optimal in expectation. This algorithm build
+upon uniformly random generated addresses, and allows for
+local routing, while ensuring dynamic optimality. Details of
+the algorithm are as follows: Algorithm 1 always starts from
+the root node. Upon receiving an access request to an item v
+it performs a local routing (described in Procedure LocalRout-
+ing) based on the uniformly random binary address generated
+for the node v, which uniquely determines the path of v in the
+tree. We call the i-th bit of the address of v by H(v, i). Let
+us assume that the local routing for node v ends in level ℓ.
+
+100
+001
+101
+010
+010
+011
+0
+111)
+(110
+011
+001
+100
+110
+110
+111(a) Item 001 moves up, node-by-node, until it
+reaches the root.
+(b) The first try of push-down failed, because
+node 100 is full.
+(c) After finding non-full node, items are
+pushed down node-by-node.
+Fig. 2: An example of steps taken in Algorithm 1, starting from the state of SeedTree in Figure 1, which has a capacity equal
+to 2. In this example, the request is an access request to the item with the hash value 001 (the purple circle). Subfigure 2a
+shows the move-to-the-root phase, and Subfigures 2b and 2c depict the push-down phase.
+Procedure LocalRouting(s,v)
+1 if H(v, level(s)) equals 0 then
+2
+Return the left child of s.
+3 else
+4
+Return the right child of s.
+Then SeedTree performs the following two-phase reconfig-
+uration. These two phases are designed to ensure the level of
+items remains in the same range as their rank (details will
+follow), and the number of items remains the same at each
+level.
+1) Move-to-the-root: This phase moves the accessed item to
+the node at the lowest level possible, the root of the tree.
+The movement of the item is step-by-step, and it keeps
+all the other items in their previous node (we keep the
+item in a temporary buffer if a node on the path was full).
+This phase is depicted in Figure 2a by zig-zagged purple
+arrows.
+2) Push-down: In this phase, our algorithm starts from the
+root node, selects an item in the node (including the
+item that has just moved to this node) uniformly at
+random, and moves this item one level down to the new
+node selected in the LocalRouting procedure. The same
+procedure is continued for the new node until reaching
+level ℓ, the level of the accessed item. If the node at
+level ℓ was non-full, the re-establishment of balance
+was successful. Otherwise, if this attempt is failed, the
+algorithm reverses the previous push downs back to the
+root, and starts again, until an attempt is successful. As
+an example, the failed attempt of this phase is depicted
+by dashed red edges in Figure 2b and the last successful
+one by curved blue arrows in Figure 2c.
+Algorithm 1 always terminates, as there is always the chance
+that the item which has been moved to root is selected among
+all candidates, and we know that the node which that item is
+taken from is not full. We now state the main theorem of the
+paper that proves the dynamic optimality of SeedTree.
+Theorem 1. SeedTree is dynamically optimal for any given
+capacity c ≥ 1.
+Algorithm 1: Online SeedTree
+Input: Accessed item v.
+1 Set s as the root.
+2 while s does not contain v do
+3
+s = LocalRouting(s,v).
+4 Call the current level of v as ℓ.
+5 Set s as the root, and move item v to s.
+6 while balance is not fixed do
+7
+Call the current node s.
+8
+while level of s is less than ℓ do
+9
+Take an item in node s, uniformly at random, call it
+v.
+10
+s = LocalRouting(s,v).
+11
+Add item v to the node s.
+12
+if the last chosen node is full then
+13
+Reverse the push-down back to the root.
+The proof of Theorem 1 is at the end of the section. The first
+step towards the proof is showing that the number of items in
+each level remains the same. It is true because after removing
+an item at a certain level, the algorithm adds an item to the
+same level as a result of the push-down phase.
+Observation 1. SeedTree keeps the number of items the same
+at each level.
+The rest of the analysis is based on the assumption that the
+algorithm was initialized with a fixed fractional occupancy
+0 < f < 1 of the capacity of each level, i.e., in level i, the
+initial tree has exactly ⌊c · f · 2i⌋ items. At the end of this
+section, we will see that f = 1
+2 works best for our analysis.
+However, we emphasize that having 0 < f < 1 suffices for
+SeedTree to run properly.
+The second observation is a result of Observation 1. As the
+number of items remains the same in each level (based on
+Observation 1) at most a fraction f of all nodes are full. In
+the lowest level, the number of full nodes might be even lower;
+hence the probability of a uniformly random node being full
+is at most f when we go to the next request.
+Observation 2. Algorithm 1 ensures that the probability of
+any uniformly random chosen node in SeedTree to be full,
+after serving each access request, is at most f.
+
+100
+001
+001
+010
+010
+011
+111
+110
+011
+100
+110
+110100
+001
+001
+010
+010
+011
+111
+110
+011
+100
+110
+110001
+100
+001
+010
+010
+011
+101
+111
+110
+011
+100
+110
+110According to Algorithm 1, items are selected uniformly at
+random inside a node. In the following lemma, we show that
+a node in a certain level is also selected uniformly at random,
+which enables the rest of the proof.
+Lemma 1. Nodes selected on the final path of the push-down
+phase with a level lower than ℓ are selected uniformly at
+random.
+Proof. Let us denote the probability of ℓ′-th node on the path
+(the node at level ℓ′, denoted by sℓ′) being the selected node
+is
+1
+2ℓ′ . Our proof goes by induction. For the basis, ℓ′ = 0, it
+is true since we only have one node, the root. Now assume
+that in the final path of push down, we want to see the
+probability of reaching the current node, sℓ′. Based on the
+induction assumption, we know that the parent of sℓ′, the node
+sℓ′−1, has been selected uniformly at random, with probability
+1
+2ℓ′−1 . Based on Line 9 of Algorithm 1, an item is selected
+from those inside sℓ′−1 uniformly at random, plus having the
+independence guarantee of our hash function that generated
+address of the selected item, we can conclude the decision to
+go to left or right from sℓ′−1 was also uniformly at random,
+hence the probability of reach sℓ′ is
+1
+2ℓ′−1 · 1
+2 =
+1
+2ℓ′ . Note
+that the above-mentioned choices are independent of whether
+or not the descents sℓ′−1 are full or not. Hence the choice
+is independent of (possible) previous failed attempts of the
+push-down phase (which might happen due to having a full
+node at level ℓ), i.e., the previous attempts do not affect the
+probability of choosing the node sℓ′.
+An essential element of the proof of Theorem 1 is that the
+rank and level of items are related to each other. Lemma 2
+describes one of the aspects of this relation.
+Lemma 2. During the execution of the SeedTree, for items v
+and u at time t, if rankt(v) > rankt(u) then E[levelt(v)] >
+E[levelt(u)].
+Proof. Having rankt(v) > rankt(u), we know that u was
+accessed more recently than v. Let us consider time t′, the
+last time u was accessed. Since the rank of v is strictly larger
+than the rank of u, and as u was moved to the root at time t′,
+we know that levelt′(v) > levelt′(u).
+Items u and v might reach the same level after time t′, but
+it is not a must. We consider the level that they first met as
+a random variable, Luv. We denote Luv = −1 if u and v
+never appear on the same level after time t′. Let us quantify
+the difference in the expected level of u and v, using the law
+of total expectation:
+E[levelt(v)] − E[levelt(u)]
+=
+⌊log ⌈ n
+c ⌉⌋
+�
+k=−1
+Pr(Luv = k) · (E[levelt(v)|Luv = k]
+−E[levelt(u)|Luv = k])
+For the case Luv = −1, we know that u and v never reached
+the same level, and the following is always true:
+E[levelt(v)|Lv,u = −1] > E[levelt(u)|Lv,u = −1]
+For k ≥ 0, let us consider the time t′′ when u and v meet
+at the same level, i.e levelt′′(u) = levelt′′(v). After items u
+and v meet for the first time, their expected progress is the
+same. More precisely, consider the current subtree of the node
+containing v at time t′′, and call it T ′. Since the item addresses
+are chosen uniformly at random, the expected number of times
+that T ′ is a subtree of a node containing v, equals the number
+of times that T ′ might be a subtree of node containing u in
+the same level. Hence the expected increase in the level for
+both items u and v stays the same from time t′′ onward.
+Next, we explain why the number of items accessed at a
+higher level is limited in expectation for any given item.
+Lemma 3. For a given item v at time t, there are at most
+2 · rankws
+t (v) items accessed at a higher level since the last
+time v was accessed, in expectation.
+Proof. Given Lemma 2, the proof is along the lines of the
+proof of Lemma 4 from [10]. We removed the details of the
+proof due to space constraints.
+Now we prove the items in the tree maintained by the online
+SeedTree are not placed much farther from their position in
+a tree that realizes the exact working set property. This in
+turn allows us to approximate the total cost of the online
+SeedTree in comparison to the optimal offline algorithm with
+the same capacity. The approximation factor, 2 − log(f), is
+intuitive: with less capacity in each level (lower values of
+levels’ fractional occupancy), we need to put items further
+down.
+Lemma 4. SeedTree is MRU(2 − log(f)) in expectation.
+Proof. For any given item v and time t, we show that
+E[levelt(v)] ≤ ⌊log ⌈ rankt(v)
+c
+⌉⌋ + 2 − log(f) remains true,
+considering move-to-the-root and push-down phases. As can
+be seen in Line 8 of Algorithm 1, the item v might move down
+if the current level of v is lower than the level of the accessed
+item.
+Let us denote the increase in the level from time t′ to time
+t by a random variable D(t′, t). We express this increase in
+terms of an indicator random variable I(t′, t, ℓ) which denotes
+whether item v went down from level ℓ during [t′, t] or not.
+We know that:
+D(t′, t) =
+�
+ℓ
+I(t′, t, ℓ)
+Let K denote the number of items accessed from a higher
+level, and let us write K = k1 + · · · + k⌈ n
+c ⌉, where kℓ means
+that kℓ such accesses happened when item v was at level ℓ.
+For the level ℓ, based on the Observation 1 and Lemma 1 and
+the fact that each level contains f · c · 2ℓ items, we conclude
+v is being selected after kℓ − 1 accesses with probability (1 −
+1
+f·c·2ℓ )k−1 · (
+1
+f·c·2ℓ ).
+I(t′, t, ℓ) = min(1,
+K
+�
+kℓ=0
+(1 −
+1
+f · c · 2ℓ )kℓ−1 · (
+1
+f · c · 2ℓ ))
+
+= min(1, (
+1
+f · c · 2ℓ ) ·
+K
+�
+kℓ=0
+(1 −
+1
+f · c · 2ℓ )kℓ−1)
+≤ min(1, (
+K
+f · c · 2ℓ ))
+Going back to our original goal of finding how many levels
+an item goes down during a time period [t′, t], we have:
+E[D(t′, t)] ≤
+�
+ℓ
+E[min(1, (
+K
+f · c · 2ℓ ))]
+= log(E[K]
+f · c ) + 1 = log(E[K]) − log(c) − log(f) + 1
+The last equality comes from the fact that for ℓ
+=
+log( E[K]
+f·c ), we have
+K
+f·c·2ℓ ≤ 1, and for all larger values of
+ℓ, the value will decrease exponentially with factors of two.
+From Lemma 3 we know that the expected value of K is
+less than equal to 2·rankt(v); therefore, the expected increase
+is:
+E[D(t′, t)] ≤ log(2 · rankt(v)) − log(c) − log(f) + 1
+= log(rankt(v)) − log(c) + 2 − log(f)
+The following lemma shows the relation between the total
+cost of the online SeedTree and fractional occupancy f. The
+relation is natural: as f becomes smaller, the chance of finding
+a non-full node becomes larger, and thus fewer attempts are
+needed to find a non-full node.
+Lemma 5. The expected cost of SeedTree is less than equal
+to 2 · (⌈
+1
+(1−f)⌉ + 1) times the access cost.
+Proof. Let us consider the accessed item v at level ℓ. In the
+first part of the algorithm, the move-to-the-root phase costs
+the same as the access, which is equal to traversing ℓ edges.
+As the probability of a node being non-full is 1 − f based on
+Observation 2, and as the choice of nodes is uniform based
+on Observation 1, only ⌈
+1
+1−f ⌉ iterations are needed during the
+push-down phase for finding a non-full node, each at cost 2·ℓ.
+Hence, given the linearity of expectation, we have:
+E[CALG] = E[CAccess
+ALG + CMove-to-the-root
+ALG
++ CPush-down
+ALG
+]
+≤ 2 · (1 + ⌈
+1
+1 − f ⌉) · ℓ = 2 · (1 + ⌈
+1
+1 − f ⌉) · CAccess
+ALG
+We now describe why working set optimality is enough for
+dynamic optimality, given that reconfigurations do not cost
+much (which is proved in Lemma 5). Hence, any other form
+of optimality, such as key independent optimality or finger
+optimality is guaranteed automatically [11].
+Lemma 6. For any given c, an MRU(0) algorithm is (1+e)
+access competitive.
+Proof. The proof relies on the potential function argument. We
+describe a potential function at time t by φt, and show that
+the change in the potential from time t to t + 1 is ∆φt→t+1.
+Our potential function at time t, counts the number of
+items that are misplaced in the tree of the optimal offline
+algorithm OPT with regard to their rank. (As the definition
+of MRU(0) indicates, there exists no inversion in such a tree,
+that is why we only focus on the number of inversions in
+OPT.) Concretely, we say a pair (v, u) is an inversion if
+rankt(v) < rankt(u) but levelt(v) > levelt(u). We denote
+the number of items that have an inversion with item v at time
+t by invt(v), and define Bt(v) = 1 +
+invt(v)
+c·2levelt(v) . Furthermore,
+define Bt = �n
+v=1 Bt(v). We define the potential function at
+time t as φt = log Bt. We assume that the online SeedTree
+rearranges its required items in the tree before the optimal
+algorithm’s rearrangements. Let us first describe the change
+in potential due to rearrangement in the online SeedTree after
+accessing item σt = v. This change has the following effects:
+1) Rank of the accessed item, v, has been set to 1.
+2) Rank of other items in the tree might have been increased
+by at most 1.
+Since the relative rank of items other than v does not change
+because of the second effect, it does not affect the number
+of inversions and hence the potential function. Therefore, we
+focus on the first effect. Since OPT has not changed its
+configuration, for all items u that are being stored in a lower
+level than v in the OPT, a single inversion is created, therefore
+we have Bt+1(u) = Bt(u) +
+1
+c·2levelc(u) . For the accessed
+item v, as its rank has changed to one, all of its inversions
+get deleted. The number of inversions for other items, except
+v, remains the same. Let us denote the number of items
+with lower level than v at time t by Lt(v) and partition the
+�n
+i=1 Bt+1(i) into three parts as we discussed (v, items stored
+in a lower level than v, and other items denoted by set Ot(v)):
+n
+�
+i=1
+Bt+1(i) = Bt+1(v) ·
+�
+i∈Lt(v)
+Bt+1(i) ·
+�
+i∈Ot(v)
+Bt+1(i)
+By rewriting Bt+1(i) in terms of Bt(i), we get:
+n
+�
+i=1
+Bt+1(i) = 1 ·
+�
+i∈Lt(v)
+(Bt(i) +
+1
+c · 2levelt(i) ) ·
+�
+i∈Ot(v)
+Bt(i)
+Now let us look at potential due the first effect from time
+t to t + 1 by ∆φ1
+t→t+1, and describe it in more detail:
+∆φ1
+t→t+1 = log Bt+1 − log Bt = log Bt+1
+Bt
+= log
+n�
+i=1
+Bt+1(i)
+n�
+i=1
+Bt(i)
+= log(
+1
+Bt(v) ·
+�
+Lt(v)
+(Bt(i) +
+1
+c·2levelt(i) )
+�
+Lt(v)
+Bt(i)
+)
+≤ log(
+1
+Bt(v) · e|Lt(v)|)
+
+in which the last inequality comes from the fact that |Lt(v)| =
+c · 2levelt(i) and also the inequality that:
+|Lt(v)|
+�
+i=1
+(Bt(i) +
+1
+|Lt(v)|) ≤
+|Lt(v)|
+�
+i=1
+(Bt(i) + Bt(i)
+|Lt(v)|)
+= (1 +
+1
+|Lt(v)|)|Lt(v)| ·
+|Lt(v)|
+�
+i=1
+Bt(i) ≤ e|Lt(v)| ·
+|Lt(v)|
+�
+i=1
+Bt(i)
+Now
+let
+us
+focus
+on
+Bt(v),
+and
+first
+assume
+that
+⌊log ⌈ rankt(v)
+c
+⌉⌋ > levelt(v). We want to find the maximum
+number of items that might cause inversion with the accessed
+item v.
+Among all c · 2⌊log ⌈ rankt(v)
+c
+⌉⌋ − 1 items that v might have
+higher rank them, at most c · 2levelt(v) − 1 have lower level in
+the OPT tree. Hence we have:
+Bt(v) = (c · 2⌊log ⌈ rankt(v)
+c
+⌉⌋ − 1) − (c · 2levelt(v) − 1)
+c · 2levelt(v)
+≥ (2⌊log ⌈ rankt(v)
+c
+⌉⌋ − 1)
+2levelt(v)
+− 1
+≥ 2⌊log ⌈ rankt(v)
+c
+⌉⌋
+2levelt(v)+1
+= 2⌊log ⌈ rankt(v)
+c
+⌉⌋−levelt(v)−1
+hence the change in potential due to the first effect is:
+∆φ1
+t→t+1 ≤ log(
+1
+2⌊log ⌈ rankt(v)
+c
+⌉⌋−levelt(v)−1 · elevelt(v))
+= log(2(1+log e)·levelt(v)−⌊log ⌈ rankt(v)
+c
+⌉⌋)
+= (1 + log e) · levelt(v) − ⌊log ⌈rankt(v)
+c
+⌉⌋
+For the case ⌊log ⌈ rankt(v)
+c
+⌉⌋ < levelt(v), we use the fact that
+Bt
+v > 1, from the first inequality below:
+∆φt→t+1 = log( 1
+Btv
+· elevelt(v))
+≤ log(2log e·levelt(v)) = log e · levelt(v)
+= (1 + log e) · levelt(v) − ⌊log ⌈rankt(v)
+c
+⌉⌋
+Hence, in both cases of ⌊log ⌈ rankt(v)
+c
+⌉⌋ being larger or smaller
+than levelt(v), we have ∆φt→t+1 ≤ (1 + log e) · levelt(v) −
+⌊log ⌈ rankt(v)
+c
+⌉⌋.
+We then show changes in the potential because of OPT’s
+reconfiguration. Details of the computations are omitted due
+to space constraints, but they are similar to the changes in
+potential due to rearrangements in the ON’s algorithm, and
+the result is that each OPT’s movement costs less than log e.
+Summing up changes in the potential after ON’s and
+OPT’s reconfiguration, assuming OPT has done wt move-
+ments at time t, we end up with:
+∆φt→t+1 = (1+log e)·levelt(v)−⌊log ⌈rankt(v)
+c
+⌉⌋+w·log e
+And hence the cost of the online algorithm MRU(0) at time
+t is at most:
+Ct
+MRU(0) = Ct
+Amortized + ∆φt
+= ⌊log ⌈rankt(v)
+c
+⌉⌋ + (1 + log e) · levelt(v)
+−⌊log ⌈rankt(v)
+c
+⌉⌋+wt ·log e ≤ (1+log e)·(levelt(v)+wt)
+And then summing up the cost of the MRU(0) and OPT for
+the whole request sequence, we will get:
+CON =
+�
+t
+Ct
+ON ≤
+�
+t
+(1 + log e) · (levelt(v) + wt)
+= (1 + log e) · COP T
+In which the last equality comes from the fact that OPT also
+needs to access the item, and as we assumed an additional wt
+reconfigurations.
+As the first application of Lemma 6 we prove a lower bound
+on the cost of any online algorithm that only depends on the
+size of the working set of accessed items in the sequence.
+Theorem 2. Any online algorithm maintaining a self-adjusting
+complete binary tree with capacity c > 1 on a request
+sequence σ = σ1, . . . σm, requires an access cost of at least
+�m
+i=1⌊log ⌈ rankt(σi)
+c
+⌉⌋
+(1+e)
+.
+Proof. This proof is an extension and improvement of the
+proof from [10] for any values of c > 2. A result of Lemma 6
+is that even an optimal algorithm cannot be better than
+1
+(1+e)
+the MRU(0), otherwise contradicting Lemma 6. As the cost
+of each access to the item σi is ⌊log ⌈ rankt(σi)
+c
+⌉⌋ in MRU(0),
+we can conclude the total cost of any algorithm should be
+larger than
+�m
+i=1⌊log ⌈ rankt(σi)
+c
+⌉⌋
+(1+e)
+.
+Lemma 7. Any MRU(β) tree is β·(1+e)-access competitive.
+Proof. Lemma 6 shows that an MRU(0) is (1 + e)-access
+competitive. Any item which was in level k in MRU(0), is
+in level k + β in MRU(β). As an MRU(β) algorithm keeps
+items with rankc(0) at level(0), and because for any k ≥ 1,
+we have k +β ≤ βk, we obtain that MRU(β) is (β)·(1+e)-
+access competitive.
+We conclude this section by proving our main theorem,
+dynamic optimality of online SeedTree.
+proof of Theorem 1. Combining Lemma 4, Lemma 5 and
+Lemma 7 yields that the upper bound for competitiveness is
+(1+e)·(2·(1+⌈
+1
+1−f ⌉))·(2−log(f)). The fractional occupancy
+f = 1/2 in the above formula is the optimal value for f, which
+gives us the 43-competitive ratio.
+We need to point out that the above calculation is just an
+upper bound on the competitive ratio. As we will discuss in
+§V, the best results are usually achieved with a slightly higher
+value of f, which we hypothesize might be because of an
+overestimation of items’ depth in our theoretical analysis.
+
+IV. APPLICATION IN RECONFIGURABLE DATACENTERS
+SeedTree provides a fundamental self-adjusting structure
+which is useful in different settings. For example, it may
+be used to adapt the placement of containers in virtualized
+settings, in order to reduce communication costs. However,
+SeedTree can also be applied in reconfigurable networks in
+which links can be adapted. In the following, we describe
+how to use SeedTree in such a use case in more detail. In
+particular, we consider reconfigurable datacenters in which the
+connectivity between racks, or more specifically Top-of-the-
+Rack (ToR) switches, can be adjusted dynamically, e.g., based
+on optical circuit switches [6]. An optical switch provides a
+matching between racks, and accordingly, the model is known
+as a matching model in the literature [15]. In the following,
+we will show how a SeedTree with capacity c and fractional
+occupancy of f = 1
+c can be seen in terms of 2 + c matchings,
+and how reconfigurations can be transformed to the matching
+model‡. We group these matchings into two sets:
+• Topological matchings: consists of 2 static matchings,
+embedding the underlying binary tree of SeedTree. The
+first matching represents edges between a node and its left
+child (with the ID twice the ID of the node), and similarly
+the second matching for the right children (with the ID
+twice plus one of the ID of their parents). An example is
+depicted with solid edges in Figure 3.
+• Membership matchings: has c dynamic matchings, con-
+necting nodes to items inside them. If a node has more
+than one item, the corresponding order of items to match-
+ings is arbitrary. An example is shown with dotted edges
+in Figure 3.
+Having the matchings in place, let us briefly discuss how
+search and reconfiguration operations are implemented. A
+search for an item starts at the node with ID 001, the root
+node. We then check membership matchings of this node. If
+they map to the item, we have found the node which contains
+the item, and our search was successful. Otherwise, we follow
+the edge determined by the hash of the item, going to the
+new possible node hosting the item. We repeat the process of
+checking membership matchings and going along topological
+matchings until we find the item. The item will be found, as
+it is stored in one of the nodes in the path determined by its
+hash value. Each step of moving an item can be implemented
+in the matching mode with only one edge removal and one
+edge addition in membership matchings.
+V. EXPERIMENTAL EVALUATION
+We
+complement
+our
+analytical
+results
+by
+evaluating
+SeedTree on multiple datasets. Concretely, we are interested
+in answering the following questions:
+Q1 How does the access cost of our algorithm compare
+to the statically-optimal algorithm (optimized based on
+frequencies) and a demand-oblivious algorithm?
+‡The matching model considers perfect matchings only, however, in
+practice imperfect matchings can be enforced by ignore rules in switches.
+Fig. 3: A transformation from the example SeedTree shown in
+Figure 1, which has capacity c = 2 and fractional occupancy
+of f = 1
+2. The disco balls on top represent the reconfigurable
+switches, and below are datacenter racks. Solid edges show
+structural matchings, and dotted edges represent membership
+matchings.
+Q2 How does additional capacity improve the performance
+of the online SeedTree, given fixed fractional occupancy
+of each level?
+Q3 What is the best initial fractional occupancy for the online
+SeedTree, given a fixed capacity?
+Answers to these questions would help developers tune pa-
+rameters of the SeedTree based on their requirements and
+needs. Before going through results, we describe the setup
+that we used: Our code is written in Python 3.6 and we
+used seaborn 0.11 [16] and Matplotlib 3.5 [17] libraries for
+visualization. Our programs were executed on a machine with
+2x Intel Xeons E5-2697V3 SR1XF with 2.6 GHz, 14 cores
+each, and a total of 128 GB DDR4 RAM.
+A. Input
+• Real-world dataset: Our real-world dataset is communi-
+cations between servers inside three different Facebook
+clusters, obtained from [1]. We post-processed this dataset
+for single-source communications. Among all possible
+sources, we chose the most frequent source.
+• Synthetic dataset: We use the Markovian model dis-
+cussed in [1], [18] for generating sequences based on a
+temporal locality parameter which ranges from 0 (uni-
+form distribution, no locality) to 0.9 (high temporal
+locality). Our synthetic input consists of 65, 535 items
+and 1 million requests. For generating such a dataset, we
+start from a random sample of items. We post-process
+this sequence, overwriting each request with the previous
+request with the probability determined by our temporal
+locality parameter. After that, we execute the second post-
+processing to ensure that exactly 65, 535 items are in the
+final trace.
+B. Algorithm setup
+We use SHA-512 [19] from the hashlib-library as the hash
+function in our implementation, approximating the uniform
+distribution for generating addresses of items. In order to store
+items in a node we used a linked list, and when we move an
+item to a node that is already full with other items, items
+are stored in a temporary buffer. We assume starting from a
+pre-filled tree with items, a tree which respects the fractional
+occupancy parameter.
+
+001
+010
+011
+100
+101
+110
+111
+001
+010
+011
+100
+101
+110
+111(a)
+(b)
+(c)
+Fig. 4: Improvements in the performance of SeedTree by fine-tuning parameters. Figures are generated using the synthetic
+dataset with various locality values. (4a) Comparing the access cost of the SeedTree with fractional occupancy f = 1
+2 to the
+best possible static algorithm and the demand-oblivious algorithm, all given capacity c = 4. Access costs are divided by 100
+thousands. (4b) The effect of increasing capacity of nodes and temporal locality of input on the total cost of the algorithm.
+The fractional occupancy is set to f = 1
+2 for all capacities. Total costs are divided by 1 million for this plot. (4c) Tradeoff
+between the total cost and the fractional occupancy, given a range of temporal localities. The capacity of nodes is set to 12.
+The number in each cell represents the cost, which are divided by 1 million.
+(a)
+(b)
+Fig. 5: Improvements in the normalized access cost of the
+algorithm by changing SeedTree parameters. These results
+are obtained based on communications of the most frequent
+source from three clusters of the real-world dataset. Costs are
+normalized by the cost of the demand-oblivious algorithm. (5a)
+Changes in the normalized cost by varying capacity. Fractional
+occupancy is set to f = 1
+2. (5a) Changes in the normalized cost
+by varying fractional occupancy. Gray dots show the minimum
+values. Capacity of nodes is set to 12.
+In our experiments, we range the capacities (c) from 2 to 16,
+and the fractional occupancies (f) from 0.16 to 0.83. Due to
+the random nature of our algorithms and input generations, we
+repeat each experiment up to 100 times to ensure consistency
+in our results.
+C. Results
+The performance of SeedTree improves significantly with
+the increased temporal locality, as can be seen in Figure 4.
+Furthermore, we have the following empirical answers to
+questions proposed at the beginning of this section:
+A1: The SeedTree improves the access cost significantly, with
+increased temporal locality, as shown in Figures 4a,
+which compares the access cost of SeedTree to static and
+demand-oblivious algorithms.
+A2: As the Figures 4b and 5a show, increasing capacity
+reduces the cost of the algorithm. However, as we can
+see, this increase slows down beyond capacity to 8, and
+hence this value can be considered as the best option for
+practical purposes.
+A3: As discussed at the end of the §III and can be seen
+in Figures 4c and 5b, the lowest cost can be achieved
+with fractions higher or lower than 1
+2, but f = 1
+2 is near
+optimal in most scenarios.
+VI. ADDITIONAL RELATED WORK
+Self-adjusting lists and trees have already been studied
+intensively in the context of data structures. The pioneering
+work is by Sleator and Tarjan [20], who initiated the study of
+the dynamic list update problems and who also introduced the
+move-to-front algorithm, inspiring many deterministic [21],
+[22] and randomized [23]–[26] approaches for datastructures,
+as well as other variations of the problem [27].
+Self-adjusting binary search trees also aim to keep recently
+used elements close to the root, similarly to our approach
+in this paper (a summary of results is in Table I). However,
+adjustments in binary search trees are based on rotations rather
+than the movement of items between different nodes. One
+of the well-known self-adjusting binary search trees is the
+splay tree [9], although it is still unknown whether this tree is
+dynamically optimal; the problem is still open also for recent
+variations such as Zipper Tree [31], Multi Splay Tree [32]
+and Chain Splay [33] which improve the O(log n) competitive
+ratio of the splay tree to O(log log n). For Tango Trees [29],
+a matching Ω(log log n) lower bound is known. We also
+know that if we allow for free rotations after access, dynamic
+
+Cluster C
+Cluster A
+Cluster B
+0.95
+Cost
+0.90
+Normalized
+0.85
+0.80
+0.75
+0.16
+0.25
+0.33
+0.5
+0.66
+0.83
+Fractional Occupancy120
+100
+Cos
+80
+Access
+60
+40
+SeedTree
+Oblivious Algorithm
+20
+Static Algorithm
+0.15
+0.3
+0.45
+0.6
+0.75
+0.9
+0
+Temporal Locality60
+Total Cost
+50
+40
+30
+20
+10
+.5
+03
+汇
+6
+5
+06
+8
+9
+LO
+4
+Z
+S
+6
+Capacity36.7
+29.0
+21.3
+5.5
+0.16
+51.7
+44.2
+13.4
+Occupancy
+27.4
+5.1
+0.25
+49.2
+42.0
+34.7
+20.1
+12.6
+47.5
+40.5
+33.4
+26.4
+19.2
+12.1
+4.9
+0.33
+0.5
+45.2
+38.5
+31.8
+25.0
+18.2
+11.4
+4.7
+Fractional
+24.5
+17.8
+4.5
+0.66
+44.4
+37.8
+31.1
+11.1
+0.75
+44.7
+38.0
+31.3
+24.6
+17.9
+11.2
+4.5
+46.6
+32.6
+25.7
+18.6
+11.6
+4.6
+0.83
+39.6
+0.0
+0.15
+0.3
+0.6
+0.75
+0.9
+0.45
+Temporal LocalityCluster A
+Cluster C
+Cluster B
+0.95
+Normalized Cost
+0.90
+0.85
+0.80
+0.75
+2
+6
+10
+12
+8
+14
+16
+4
+CapacityData Structure
+Operation
+Ratio
+Search
+Splay Tree [9]
+Rotation
+O(log n)
+Yes
+Greedy Future [28]
+Rotation
+O(log n)
+Yes
+Tango Tree [29]
+Rotation
+θ(log log n)
+Yes
+Adaptive Huffman [30]
+Subtree swap
+θ(1)
+No
+Push-down Tree [10]
+Item swap
+θ(1)
+No
+SeedTree
+Item movement
+θ(1)
+Yes
+TABLE I: Comparison of properties of self-adjusting tree data
+structures. The best known competitive ratio (to this date)
+is in terms of the data structure’s respective cost model and
+optimal offline algorithm. We note that none of the above trees
+considers additional capacity, except for our model.
+optimally becomes possible [34]. We also point out that some
+of these structures, in particular, multi splay tree and chain
+splay, benefitted from additional memory as well, however,
+there it is used differently, namely toward saving additional
+attributes for each node. Another variation which was first
+proposed by Lucas [28] in 1988 is called Greedy Future. This
+tree first received attention as an offline binary search tree
+algorithm [35], [36], but then an O(log n) amortized time
+in online settings was suggested by Fox [37]. Greedy Future
+has motivated researchers to take a geometric view of online
+binary search trees [36], [38]. We note that in contrast to binary
+search trees, our local tree does not require an ordering of the
+items in the left and right subtrees of a node.
+Self-adjusting trees have also been explored in the context
+of coding, where for example adaptive Huffman coding [30],
+[39]–[42] is used to minimize the depth of most frequent items.
+The reconfiguration cost, however, is different: in adaptive
+Huffman algorithms, two subtrees might be swapped at the
+cost of one.
+A few data structures have tried to achieve a better compet-
+itive ratio by expanding and altering binary search trees (see
+Table II for a summary): The first example, PokeTree [43],
+adds extra pointers between the internal nodes of the tree and
+achieves an O(log log n) competitive ratio in comparison to
+an optimal binary search tree. There are also self-adjusting
+data structures based on skip lists [44], [45], which have been
+introduced as an alternative for balanced trees that enforce
+probabilistic balancing instead. A biased version of skip lists
+was considered in [46], and later on, a statically optimal
+variation was given in [47] and a dynamic optimal version
+in a restricted model in [48]. Another example is Iacono’s
+working set structure [49] which combines a series of self-
+adjusting balanced binary search trees and deques, achieving
+a worst-case running time of O(log n), however, it lacks the
+dynamic optimality property. We are not aware of any work
+exploring augmentations to improve the competitive ratio of
+these data structures.
+Our work is also motivated by emerging self-adjusting
+datacenter networks. Recent optical communication technolo-
+gies enable datacenters to be reconfigured quickly and fre-
+quently [8], [18], [50]–[58], see [59] for a recent survey. The
+datacenter application mentioned in our paper is based on the
+matching model proposed by [15]. Recently [60] introduced
+Data Structure
+Structure
+Ratio
+Iacono’s structure [49]
+Trees & deques
+O(log n)
+Skip List [44]
+Linked lists
+O(log n)
+PokeTree [43]
+Tree & dynamic links
+O(log log n)
+SeedTree
+Tree
+θ(1)
+TABLE II: Comparison with other self-adjusting data struc-
+tures that support local-search. The best known competitive
+ratio (to this date) is in terms of the data structure’s respective
+cost model and optimal offline algorithm. We note that none
+of the other data structures considers capacity in their design.
+an online algorithm for constructing self-adjusting networks
+based on this model, however the authors do not provide
+dynamic optimality proof for their method.
+It has been shown that demand-aware and self-adjusting
+datacenter networks can be built from individual trees [61],
+called ego-trees, which are used in many network designs [8],
+[50], [62], [63], and also motivate our model. However, until
+now it was an open problem how to design self-adjusting
+and constant-competitive trees that support local routing and
+adjustments, a desirable property in dynamic settings.
+Last but not least, our work also features interesting con-
+nections to peer-to-peer networks [12], [64]. It is known that
+consistent hashing with previously assigned and fixed capaci-
+ties allows for significantly improved load balancing [13], [14],
+which has interesting applications and is used, e.g., in Vimeo’s
+streaming service [65] and in Google’s cloud service [13].
+Although these approaches benefit from data structures with
+capacity, these approaches are not demand-aware.
+VII. CONCLUSION AND FUTURE WORK
+This paper presented and evaluated a self-adjusting and
+local tree, SeedTree, which adapts towards the workload in
+an online, constant-competitive manner. SeedTree supports a
+capacity augmentation approach, while providing local rout-
+ing, which can be useful for other self-adjusting structures and
+applications as well. We showed a transformation of our algo-
+rithm into the matching model for application in reconfigurable
+datacenters, and evaluated our algorithm on synthetic and real-
+world communication traces. The code used for our experi-
+mental evaluation is available at github.com/inet-tub/SeedTree.
+We believe that our work opens several interesting avenues
+for future research. In particular, while we so far focused on
+randomized approaches, it would be interesting to explore de-
+terministic variants of SeedTree. Furthermore, while trees are
+a fundamental building block toward more complex networks
+(as they, e.g., arise in datacenters today), it remains to design
+and evaluate networks based on SeedTree.
+REFERENCES
+[1] C. Avin, M. Ghobadi, C. Griner, and S. Schmid, “On the complexity of
+traffic traces and implications,” in ACM SIGMETRICS, 2020.
+[2] T. Benson, A. Anand, A. Akella, and M. Zhang, “Understanding data
+center traffic characteristics,” ACM SIGCOMM CCR, 2010.
+[3] O. Michel, R. Bifulco, G. Retvari, and S. Schmid, “The programmable
+data plane: Abstractions, architectures, algorithms, and applications,” in
+ACM CSUR, 2021.
+
+[4] W. Kellerer, P. Kalmbach, A. Blenk, A. Basta, M. Reisslein, and
+S. Schmid, “Adaptable and data-driven softwarized networks: Review,
+opportunities, and challenges,” in IEEE PIEEE, 2019.
+[5] A. Fischer, J. F. Botero, M. T. Beck, H. de Meer, and X. Hesselbach,
+“Virtual network embedding: A survey,” IEEE Commun. Surv. Tutor.,
+2013.
+[6] M. N. Hall, K.-T. Foerster, S. Schmid, and R. Durairajan, “A survey of
+reconfigurable optical networks,” in OSN, 2021.
+[7] A. Borodin and R. El-Yaniv, Online computation and competitive
+analysis.
+cambridge university press, 2005.
+[8] C. Avin, K. Mondal, and S. Schmid, “Demand-aware network designs
+of bounded degree,” in DISC, 2017.
+[9] D. D. Sleator and R. E. Tarjan, “Self-adjusting binary search trees,” J.
+ACM, 1985.
+[10] C. Avin, K. Mondal, and S. Schmid, “Push-down trees: Optimal self-
+adjusting complete trees,” in IEEE/ACM, TON, 2022.
+[11] J. Iacono, “Key-independent optimality,” Algorithmica, 2005.
+[12] I. Stoica, R. T. Morris, D. Liben-Nowell, D. R. Karger, M. F. Kaashoek,
+F. Dabek, and H. Balakrishnan, “Chord: a scalable peer-to-peer lookup
+protocol for internet applications,” IEEE/ACM Trans. Netw., 2003.
+[13] V. S. Mirrokni, M. Thorup, and M. Zadimoghaddam, “Consistent
+hashing with bounded loads,” in ACM-SIAM SODA, 2018.
+[14] A. Aamand, J. B. T. Knudsen, and M. Thorup, “Load balancing with
+dynamic set of balls and bins,” in ACM SIGACT STOC, 2021.
+[15] C. Griner, J. Zerwas, A. Blenk, S. Schmid, M. Ghobadi, and C. Avin,
+“Cerberus: The power of choices in datacenter topology design (a
+throughput perspective),” in ACM SIGMETRICS, 2021.
+[16] M. L. Waskom, “seaborn: statistical data visualization,” J. of Open
+Source Softw., 2021.
+[17] J. D. Hunter, “Matplotlib: A 2d graphics environment,” Comput. Sci.
+Eng., 2007.
+[18] C. Avin, M. Bienkowski, I. Salem, R. Sama, S. Schmid, and P. Schmidt,
+“Deterministic self-adjusting tree networks using rotor walks,” in IEEE
+ICDCS, 2022.
+[19] C. Dobraunig, M. Eichlseder, and F. Mendel, “Analysis of SHA-512/224
+and SHA-512/256,” IACR Cryptol. ePrint Arch., 2016.
+[20] D. D. Sleator and R. E. Tarjan, “Amortized efficiency of list update and
+paging rules,” Commun. ACM, 1985.
+[21] S. Albers, “A competitive analysis of the list update problem with
+lookahead,” MFCS, 1994.
+[22] S. Kamali and A. L´opez-Ortiz, “A survey of algorithms and models for
+list update,” in LNTCS, 2013.
+[23] S. Albers and M. Janke, “New bounds for randomized list update in the
+paid exchange model,” in STACS, 2020.
+[24] S. Albers, B. Von Stengel, and R. Werchner, “A combined bit and
+timestamp algorithm for the list update problem,” Inf. Process. Lett.,
+1995.
+[25] T. Garefalakis, “A new family of randomized algorithms for list access-
+ing,” in ESA, 1997.
+[26] N. Reingold, J. R. Westbrook, and D. D. Sleator, “Randomized compet-
+itive algorithms for the list update problem,” Algorithmica, 1994.
+[27] S. Albers and S. Lauer, “On list update with locality of reference,” in
+ICALP, 2008.
+[28] J. M. Lucas, Canonical forms for competitive binary search tree algo-
+rithms.
+Rutgers University, 1988.
+[29] E. D. Demaine, D. Harmon, J. Iacono, and M. Patrascu, “Dynamic
+optimality - almost,” in IEEE FOCS, 2004.
+[30] G. V. Cormack and R. N. Horspool, “Algorithms for adaptive huffman
+codes,” Inf. Process. Lett., 1984.
+[31] P. Bose, K. Dou¨ıeb, V. Dujmovi´c, and R. Fagerberg, “An o (log log n)-
+competitive binary search tree with optimal worst-case access times,” in
+SWAT, 2010.
+[32] C. C. Wang, J. Derryberry, and D. D. Sleator, “O (log log n)-competitive
+dynamic binary search trees,” in ACM-SIAM SODA, 2006.
+[33] G. F. Georgakopoulos, “Chain-splay trees, or, how to achieve and prove
+loglogn-competitiveness by splaying,” Inf. Process. Lett., 2008.
+[34] A. Blum, S. Chawla, and A. Kalai, “Static optimality and dynamic
+search-optimality in lists and trees,” in ACM-SIAM SODA, 2002.
+[35] J. I. Munro, “On the competitiveness of linear search,” in ESA, 2000.
+[36] E. D. Demaine, D. Harmon, J. Iacono, D. M. Kane, and M. Patrascu,
+“The geometry of binary search trees,” in ACM-SIAM SODA, 2009.
+[37] K. Fox, “Upper bounds for maximally greedy binary search trees,” in
+WADS, 2011.
+[38] J. Iacono, “In pursuit of the dynamic optimality conjecture,” in Space-
+Efficient Data Structures, Streams, and Algorithms, 2013.
+[39] D. E. Knuth, “Dynamic huffman coding,” J. Algorithms, 1985.
+[40] R. L. Milidi´u, E. S. Laber, and A. A. Pessoa, “Bounding the compression
+loss of the FGK algorithm,” J. Algorithms, 1999.
+[41] A. Moffat, “Huffman coding,” ACM CSUR, 2019.
+[42] J. S. Vitter, “Design and analysis of dynamic huffman codes,” J. of the
+ACM, 1987.
+[43] J. Kujala and T. Elomaa, “Poketree: A dynamically competitive data
+structure with good worst-case performance,” in ISAAC, 2006.
+[44] W. Pugh, “Skip lists: A probabilistic alternative to balanced trees,”
+Commun. ACM, 1990.
+[45] C. Avin, I. Salem, and S. Schmid, “Working set theorems for routing in
+self-adjusting skip list networks,” in IEEE INFOCOM, 2020.
+[46] A. Bagchi, A. L. Buchsbaum, and M. T. Goodrich, “Biased skip lists,”
+Algorithmica, 2005.
+[47] V. Ciriani, P. Ferragina, F. Luccio, and S. Muthukrishnan, “A data
+structure for a sequence of string accesses in external memory,” ACM
+Trans. Algorithms, 2007.
+[48] P. Bose, K. Dou¨ıeb, and S. Langerman, “Dynamic optimality for skip
+lists and b-trees,” in ACM-SIAM SODA, 2008.
+[49] J. Iacono, “Alternatives to splay trees with o(log n) worst-case access
+times,” in ACM-SIAM SODA, 2001.
+[50] C. Avin, K. Mondal, and S. Schmid, “Demand-aware network design
+with minimal congestion and route lengths,” in IEEE INFOCOM, 2019.
+[51] H. Ballani, P. Costa, R. Behrendt, D. Cletheroe, I. Haller, K. Jozwik,
+F. Karinou, S. Lange et al., “Sirius: A flat datacenter network with
+nanosecond optical switching,” in ACM SIGCOMM, 2020.
+[52] K. Chen, A. Singla, A. Singh, K. Ramachandran, L. Xu, Y. Zhang,
+X. Wen, and Y. Chen, “Osa: An optical switching architecture for data
+center networks with unprecedented flexibility,” IEEE/ACM TON, 2014.
+[53] F. Douglis, S. Robertson, E. Van den Berg, J. Micallef, M. Pucci,
+A. Aiken, M. Hattink, M. Seok, and K. Bergman, “Fleet—fast lanes for
+expedited execution at 10 terabits: Program overview,” IEEE Internet
+Comput., 2021.
+[54] K.-T. Foerster, M. Ghobadi, and S. Schmid, “Characterizing the al-
+gorithmic complexity of reconfigurable data center architectures,” in
+ACM/IEEE ANCS, 2018.
+[55] M. Ghobadi, R. Mahajan, A. Phanishayee, N. Devanur, J. Kulkarni,
+G. Ranade, P.-A. Blanche, H. Rastegarfar et al., “Projector: Agile
+reconfigurable data center interconnect,” in ACM SIGCOMM, 2016.
+[56] J. Kulkarni, S. Schmid, and P. Schmidt, “Scheduling opportunistic links
+in two-tiered reconfigurable datacenters,” in ACM SPAA, 2021.
+[57] W. M. Mellette, R. Das, Y. Guo, R. McGuinness, A. C. Snoeren, and
+G. Porter, “Expanding across time to deliver bandwidth efficiency and
+low latency,” in USENIX NSDI, 2020.
+[58] W. M. Mellette, R. McGuinness, A. Roy, A. Forencich, G. Papen, A. C.
+Snoeren, and G. Porter, “Rotornet: A scalable, low-complexity, optical
+datacenter network,” in ACM SIGCOMM, 2017.
+[59] K.-T. Foerster and S. Schmid, “Survey of reconfigurable data center
+networks: Enablers, algorithms, complexity,” in SIGACT News, 2019.
+[60] E. Feder, I. Rathod, P. Shyamsukha, R. Sama, V. Aksenov, I. Salem,
+and S. Schmid, “Lazy self-adjusting bounded-degree networks for the
+matching model,” in IEEE INFOCOM, 2022.
+[61] C. Avin and S. Schmid, “Toward demand-aware networking: a theory
+for self-adjusting networks,” ACM SIGCOMM CCR, 2018.
+[62] ——, “Renets: Statically-optimal demand-aware networks,” in SIAM
+APOCS, 2021.
+[63] B. S. Peres, O. A. de Oliveira Souza, O. Goussevskaia, C. Avin,
+and S. Schmid, “Distributed self-adjusting tree networks,” in IEEE
+INFOCOM, 2019.
+[64] D. R. Karger, E. Lehman, F. T. Leighton, R. Panigrahy, M. S. Levine, and
+D. Lewin, “Consistent hashing and random trees: Distributed caching
+protocols for relieving hot spots on the world wide web,” in ACM STOC,
+1997.
+[65] A. Rodland, “Improving load balancing with a new consistent-hashing
+algorithm,” Vimeo Engineering Blog, Medium, 2016.
+

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@@ -0,0 +1,1893 @@
+Distributional outcome regression and its
+application to modelling continuously
+monitored heart rate and physical activity
+Rahul Ghosal1, Sujit K. Ghosh2, Jennifer A. Schrack3, Vadim Zipunnikov4
+1 Department of Epidemiology and Biostatistics, University of South Carolina
+2Department of Statistics, North Carolina State University
+3 Department of Epidemiology, Johns Hopkins Bloomberg
+School of Public Health
+4 Department of Biostatistics, Johns Hopkins Bloomberg
+School of Public Health
+January 30, 2023
+Abstract
+We propose a distributional outcome regression (DOR) with scalar and distribu-
+tional predictors. Distributional observations are represented via quantile functions
+and the dependence on predictors is modelled via functional regression coefficients.
+DOR expands existing literature with three key contributions: handling both scalar
+and distributional predictors, ensuring jointly monotone regression structure with-
+out enforcing monotonicity on individual functional regression coefficients, pro-
+viding a statistical inference for estimated functional coefficients. Bernstein poly-
+nomial bases are employed to construct a jointly monotone regression structure
+without over-restricting individual functional regression coefficients to be mono-
+tone. Asymptotic projection-based joint confidence bands and a statistical test of
+global significance are developed to quantify uncertainty for estimated functional
+regression coefficients. Simulation studies illustrate a good performance of DOR
+model in accurately estimating the distributional effects. The method is applied to
+continuously monitored heart rate and physical activity data of 890 participants of
+Baltimore Longitudinal Study of Aging. Daily heart rate reserve, quantified via a
+subject-specific distribution of minute-level heart rate, is modelled additively as a
+function of age, gender, and BMI with an adjustment for the daily distribution of
+minute-level physical activity counts. Findings provide novel scientific insights in
+epidemiology of heart rate reserve.
+Keywords: Distributional Data Analysis; Distribution-on-distribution regression; Quan-
+tile function-on-scalar Regression; BLSA; Physical Activity; Heart Rate.
+1
+arXiv:2301.11399v1  [stat.ME]  26 Jan 2023
+
+1
+Introduction
+Distributional data analysis is an emerging area of research with diverse applications in
+digital medicine and health (Augustin et al., 2017; Matabuena et al., 2021; Ghosal et al.,
+2021; Matabuena and Petersen, 2021), radiomics (Yang et al., 2020), neuroimaging (Tang
+et al., 2020) among many others. With the advent of modern medical devices and wear-
+ables, many studies collect subject-specific high frequency or high density observations
+including heart rate, physical activity (steps, activity counts), continuously monitored
+blood glucose, functional and structured brain images, and others. The central idea of
+distributional data analysis is to capture the distributional aspect in this data and model
+it within regression frameworks. Thus, distributional data analysis inherently deals with
+data objects which are distributions typically represented via histograms, densities, quan-
+tile functions or other distributional representations. Petersen et al. (2021) provide an
+in-depth overview of recent developments in this area.
+Similar to functional regression models, depending on whether the outcome or the
+predictor is distributional, there are various types of distributional regression models.
+Petersen and M¨uller (2016) and Hron et al. (2016) developed functional compositional
+methods to analyze samples of densities. For scalar outcome and distributional predictors
+represented via densities, a common idea has been to transform densities by mapping
+them to a proper Hilbert space L2 and then use existing functional regression approaches
+for modelling scalar outcomes. Petersen and M¨uller (2016) used a log-quantile density
+transformation, whereas Talsk´a et al. (2021) used a centered log-ratio transformation.
+Other approaches for modelling scalar outcomes and distributional predictors include
+scalar-on-quantile function regression (Ghosal et al., 2021), kernel-based approaches using
+quantile functions (Matabuena and Petersen, 2021) and many others (see Petersen et al.
+(2021), Chen et al. (2021) and references therein).
+In parallel, there was also a substantial work on developing models with distributional
+outcome and scalar predictors.
+Yang et al. (2020) developed a quantile function-on-
+scalar (QFOSR) regression model, where subject-specific quantile functions of data were
+modelled via scalar predictors of interest using a function-on-scalar regression approach
+(Ramsay and Silverman, 2005), which make use of data-driven basis functions called
+quantlets. One limitation of the approach is a no guarantee of underlying monotonicity
+of the predicted quantile functions. To address this, Yang (2020) extended this approach
+2
+
+using I-splines (Ramsay et al., 1988) or Beta CDFs which enforce monotonicity at the
+estimation step.
+One important limitation of this approach is enforcement of jointly
+monotone (non-decreasing) regression structure via enforcement of monotonicity on each
+individual functional regression coefficients. As we demonstrate in our application, this
+assumption could be too restrictive in real world.
+Distribution-on-distribution regression models when both outcome and predictors are
+distributions have been studied by Verde and Irpino (2010); Irpino and Verde (2013);
+Chen et al. (2021); Ghodrati and Panaretos (2021); Pegoraro and Beraha (2022). These
+models aim to understand the association between distributions within a pre-specified,
+often linear, regression structure.
+Verde and Irpino (2010); Irpino and Verde (2013)
+used an ordinary least square approach based on the squared L2 Wasserstein distance
+between distributions. Outcome quantile function QiY (p) was modelled as a non-negative
+linear combination of other quantile functions QiXj(p)s using a multiple linear regression.
+This model although useful and adequate for some applications, may not be flexible
+enough as it assumes a linear association between the distribution valued response and
+predictors, which are additionally assumed to be constant across all quantile levels p ∈
+(0, 1). Chen et al. (2021) used a geometric approach taking distributional valued outcome
+and predictor to a tangent space, where regular tools of function-on-function regression
+(Ramsay and Silverman, 2005; Yao et al., 2005) were applied.
+Pegoraro and Beraha
+(2022) used an approximation of the Wasserstein space using monotone B-spline and
+developed methods for PCA and regression for distributional data. Recently, Ghodrati
+and Panaretos (2021) developed a shape-constrained approach linking Frechet mean of
+the outcome distribution to the predictor distribution via an optimal transport map that
+was estimated by means of isotonic regression.
+Many of above-mentioned methods mainly focused on dealing with constraints en-
+forced by a specific functional representation. Developing inferential tools is somewhat
+under-developed are of distributional data analysis. Chen et al. (2021) derived the asymp-
+totic convergence rates for the estimated regression operator in their proposed method
+for Wasserstein regression. Yang et al. (2020) developed joint credible bands for distribu-
+tional effects, but monotonocity of the quantile function was not imposed. Yang (2020)
+developed a global statistical test for estimated functional coefficients in the distribu-
+tional outcome regression, however, no confidence bands was proposed to identify and
+3
+
+test local quantile effects.
+In this paper, we propose a distributional outcome regression that expands exist-
+ing literature in three main directions. First, our model includes both scalar and dis-
+tributional predictors.
+Second, it ensures jointly monotone (non-decreasing) additive
+regression structure without enforcing monotonicity of individual functional regression
+coefficients.
+Thirdly, it provides a toolbox of statistical inference tools for estimated
+functional coefficients including asymptotic projection-based joint confidence bands and
+a statistical test of global significance. We capture distributional aspect in outcome and
+predictors via quantile functions and construct a jointly monotone regression model via
+a specific shape-restricted functional regression model. The distributional effects of the
+scalar covariates are captured via functional coefficient βj(p)’s varying over quantile lev-
+els and the effect of the distributional predictor is captured via a monotone function
+h(·), similar to an optimal transport approach in Ghodrati and Panaretos (2021). In the
+special case, when there is no distributional predictor, the model resembles a quantile
+function-on-scalar regression model, but with much more flexible constraints compared
+to Yang (2020). In the absence of scalar predictors the model reduces to a distribution
+on distribution regression model, where the monotone function representing the optimal
+transport map is estimated by a non-parametric functional regression model under shape
+constraints. We use Bernstein polynomial (BP) basis functions to model the distribu-
+tional effects βj(p)s and the monotone map h(·), which are known to enjoy attractive
+and optimal shape-preserving properties (Lorentz, 2013; Carnicer and Pena, 1993). Ad-
+ditionally, BP is instrumental in constructing and enforcing a jointly monotone regression
+structure without over-restricting individual functional regression coefficients to be mono-
+tone. Finally, inferential tools are developed including joint asymptotic confidence bands
+for distributional functional effects and p-values for testing the distributional effects of
+predictors.
+As a motivating application, we study continuously monitored heart rate and physical
+activity collected in Baltimore Longitudinal Study of Aging (BLSA). We aim to study the
+association between the distribution of heart rate as a distributional outcome and age, sex
+and body mass index (BMI) while also adjusting for a key confounder, the distribution
+of minute-level physical activity aggregated over 8am-8pm time period. Figure 1 displays
+daily profiles of heart rate and physical activity between 8am-8pm for a BLSA participant
+4
+
+along with the corresponding subject-specific quantile functions.
+8
+10
+12
+14
+16
+18
+20
+40
+60
+80
+100
+120
+Time of Day
+Heartrate
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+40
+60
+80
+100
+120
+p
+Heartrate QF
+8
+10
+12
+14
+16
+18
+20
+0
+200
+600
+1000
+Time of Day
+Activity
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+200
+600
+1000
+p
+Activity QF
+Figure 1: Diurnal profile of heart rate and physical activity between 8 a.m.- 8 p.m. and
+the corresponding subject specific quantile functions for a randomly chosen subject in
+the BLSA.
+The rest of this article is organized as follows. We present our distributional modeling
+framework and illustrate the proposed estimation method in Section 2. In Section 3, we
+perform numerical simulations to evaluate the performance of the proposed method and
+provide comparisons with existing methods for distributional regression. In Section 4, we
+demonstrate application of the proposed method in modelling continuously monitored
+heart rate reserve in BLSA study. Section 5 concludes with a brief discussion of our
+proposed method and some possible extensions of this work.
+2
+Methodology
+2.1
+Modelling Framework and Distributional Representations
+We consider the scenario, where there are repeated subject-specific measurements of a
+distributional response Y along with several scalar covariates zj, j = 1, 2, . . . , q and we
+5
+
+also have a distributional predictor X. Let us denote the subject-specific response and
+covariates as Yik, Xil, zij (k = 1, . . . , n1i, l = 1, . . . , n2i), for subject i = 1, . . . , n. Here
+n1i, n2i denotes the number of repeated observations of the distributional response and
+predictor respectively for subject i. Assume Yik (k = 1, . . . , n1i) ∼ FiY (y), a subject-
+specific cumulative distribution function (cdf), where FiY (y) = P(Yik ≤ y). Then, the
+subject-specific quantile function is defined as QiY (p) = inf{y : FiY (y) ≥ p}, p ∈ [0, 1].
+The quantile function completely characterizes the distribution of the individual obser-
+vations. Given Yik s, the empirical quantile function can be calculated based on linear
+interpolation of order statistics (Parzen, 2004) and serves as an estimate of the latent
+subject specific quantile function QiY (p) (Yang et al., 2020; Yang, 2020). In particular,
+for a sample (X1, X2, . . . , Xn), letX(1) ≤ X(2) ≤ . . . , ≤ X(n) be the corresponding order
+statistics. The empirical quantile function, for p ∈ [
+1
+n+1,
+n
+n+1], is then given by,
+ˆQ(p) = (1 − w)X([(n+1)p]) + wX([(n+1)p]+1),
+(1)
+where and w is a weight satisfying (n + 1)p = [(n + 1)p] + w. Based on this formulation
+and observations Yik, Xil, we can obtain the subject specific quantile functions ˆQiY (p)
+and ˆQiX(p) which are estimators of the true quantile functions QiY (p),QiX(p). The em-
+pirical quantile functions are consistent (Parzen, 2004) and are suitable for distributional
+representaion for several attractive mathematical properties (Powley, 2013; Ghosal et al.,
+2021) without requiring any smoothing parameter selection as in density estimation.
+2.2
+Distribution-on-scalar and Distribution Regression
+We assume that the scalar covariates (z1, z2, . . . , zq) ∈ [0, 1]q without any loss of gener-
+ality (e.g., achievable by linear transformation). We posit the following distributional
+regression model, associating the distributional response QiY (p) to the scalar covariates
+zij, j = 1, 2, . . . , q, and a distributional predictor QiX(p).
+We will refer to this as a
+distribution-on-scalar and distribution regression (DOSDR) model.
+QiY (p) = β0(p) +
+q
+�
+j=1
+zijβj(p) + h(QiX(p)) + ϵi(p).
+(2)
+6
+
+Here β0(p) is a distributional intercept and βj(p) s are the distributional effects of the
+scalar covariates Zj at quantile level p.
+The unknown nonparametric function h(·)
+captures the additive effect of the distributional predictor QiX(p).
+The residual er-
+ror process ϵi(p) is assumed to be a mean zero stochastic process with an unknown
+covariance structure.
+We make the following flexible and interpretable assumptions
+on the coefficient functions βj(·), j = 0, 1, . . . , q and on h(·) which ensures the pre-
+dicted value of the response quantile function QiY (p) conditionally on the predictors,
+E(QiY (p) | zi1, zi2, . . . , ziq, qix(p)) is non-decreasing.
+Theorem 1 Let the following conditions hold in the model (2).
+1. The distributional intercept β0(p) is non-decreasing.
+2. Any additive combination of β0(p) with distributional slopes βj(p) is non-decreasing,
+i.e., β0(p) + �r
+k=1 βjk(p) is non-decreasing for any sub-sample {j1, j2, . . . , jr} ⊂
+{1, 2, . . . , q}.
+3. h(·) is non-decreasing.
+Then E(QY (p) | z1, z2, . . . , zq, qx(p)) is non-decreasing.
+Note that E(QY (p) | z1, z2, . . . , zq, qx(p)) is the predicted quantile function under the
+squared Wasserstein loss function, which is same as the squared error loss for the quan-
+tile functions. The proof is illustrated in Appendix A of the Supplementary Material.
+Assumptions (1) and (2) are much weaker and flexible than the monotonicity conditions
+of the QFOSR model in Yang (2020), where each of the function coefficients βj(p)s is re-
+quired to be monotone, whereas, we only impose monotonicity on the sum of functional
+coefficients. This is not just a technical aspect but this flexibility is important from a prac-
+tical perspective, as it allows for capturing possible non-monotone association between
+the distributional response and individual scalar predictors zj’s while still maintaining the
+required monotonicity of the predicted response quantile function. Condition (3) matches
+with the monotonicity assumption of the distributional regression model in Ghodrati and
+Panaretos (2021) and in the absence of any scalar predictors, essentially captures the op-
+timal transport map between the two distributions. Note that this optimal transport map
+is constructed after adjusting for scalars of interest - thus, it provides a model general in-
+ferential framework compared to that in Ghodrati and Panaretos (2021). Thus, the above
+7
+
+DOSDR model extends the previous inferential framework for distributional response on
+scalar and contains both the QFOSR model and the distribution-on-distribution regres-
+sion model as its submodels. More succinctly, in absence of distributional predictor we
+have,
+QiY (p) = β0(p) +
+q
+�
+j=1
+zijβj(p) + ϵi(p),
+(3)
+which is a quantile-function-on-scalar regression (QFOSR) model ensuring monotonon-
+icity under conditions (1),(2). Similarly, in absence of any scalar covariates, we have a
+distribution-on-distribution regression model
+QiY (p) = β0(p) + h(QiX(p)) + ϵi(p).
+(4)
+Model (4) is a bit more general than the one considered in Ghodrati and Panaretos
+(2021), including a transnational effect β0(p). As a technical note, in models (2) and
+(4), function h(·) is identifiable only up to an additive constant, and in particular, the
+estimable quantity is the additive effect β0(p) + h(qx(p)) for a fixed QX(p) = qx(p).
+2.3
+Estimation in DOSDR
+We follow a shape constrained estimation approach (Ghosal et al., 2022a) for estimating
+the distributional effects βj(p) and the nonparamatric function h() which naturally in-
+corporates the constraints (1)-(3) of Theorem 1 in the estimation step. The univariate
+coefficient functions βj(p) (j = 0, 1, . . . , p) are modelled in terms of univariate expansions
+of Bernstein basis polynomials as
+βj(p) =
+N
+�
+k=0
+βjkbk(p, N), where bk(p, N) =
+�N
+k
+�
+pk(1 − p)N−k, for 0 ≤ p ≤ 1.
+(5)
+The number of basis polynomials depends on the degree of the polynomial basis N (which
+is assumed to be same for all βj(·) for computational tractability in this paper). The
+Bernstein polynomials bk(p, N) ≥ 0 and �N
+k=0 bk(p, N) = 1. Wang and Ghosh (2012)
+and Ghosal et al. (2022a) illustrate that various shape constraints e.g., monotonicity,
+convexity, etc. can be reduced to linear constraints on the basis coefficients of the form
+ANβN
+j ≥ 0, where βN
+j = (βj0, βj1, . . . , βjN)T and AN is the constraint matrix chosen in
+8
+
+a way to guarantee a desired shape restriction. In particular, in our context of DOSDR,
+we need to choose constraint matrices AN in such a way which jointly ensure conditions
+(1),(2) in Theorem 1 and thus guarantee a non-decreasing predicted value of the response
+quantile function. The nonparametric function h(·) is modelled similarly using univariate
+Bernstein polynomial expansion as
+h(x) =
+N
+�
+k=0
+θkbk(x, N), where bk(x, N) =
+�N
+k
+�
+xk(1 − x)N−k, for 0 ≤ x ≤ 1.
+(6)
+Since the domain of h(·) modelled via Bernstein basis is [0, 1], the quantile functions of the
+distributional predictor QX(p) are transformed to a [0, 1] scale using linear transformation
+of the observed predictors. We make the assumption here that the distributional predic-
+tors are bounded, which is reasonable in the applications we are interested in. Henceforth,
+we assume QX(p) ∈ [0, 1] without loss of generality. Further, note that, b0(x, N) = 1,
+and since β0(p) already contains this constant term in the DOSDR model (2), including
+the constant basis while modelling h(·) will lead to model singularity. Hence we drop
+the constant basis (i.e. the first term) while modelling h(·). In particular, h(QiX(p)) is
+modelled as h(QiX(p)) = �N
+k=1 θkbk(QiX(p), N). Note that this is equivalent to imposing
+the constraint h(0) = θ0 = 0. The non-decreasing condition in (3) of Theorem 1 can
+again be specified as a linear constraint on the basis coefficients of the form Rθ ≥ 0,
+where θ = (θ1, . . . , θN)T, and R is the constraint matrix. The DOSDR model (2) can be
+reformulated in terms of basis expansions as
+QiY (p)
+=
+N
+�
+k=0
+β0kbk(p, N) +
+q
+�
+j=1
+zij
+N
+�
+k=0
+βjkbk(p, N) +
+N
+�
+k=1
+θkbk(QiX(p), N) + ϵi(p).(7)
+=
+bN(p)Tβ0 +
+q
+�
+j=1
+ZT
+ij(p)βj + bN(QiX(p))Tθ + ϵi(p).
+Here βj = (βj0, βj1, . . . , βjN)T, bN(p)T = (b0(p, N), b1(p, N), . . . , bN(p, N)), bN(QiX(p))T =
+(b1(QiX(p), N),
+b2(QiX(p), N), . . . , bN(QiX(p), N)) and ZT
+ij(p) = zij ∗ bN(p)T. Suppose that we have the
+qunatile functions QiY (p), QiX(p) evaluated on a grid P = {p1, p2, . . . , pm} ⊂ [0, 1]. De-
+note the stacked value of the quantiles for ith subject as QiY = (QiY (p1), QiY (p2), . . . , QiY (pm))T.
+9
+
+The DOSDR model in terms of Bernstein basis expansion (7) can be reformulated as
+QiY
+=
+B0β0 +
+q
+�
+j=1
+Wijβj + Siθ + ϵi,
+(8)
+where B0 = (bN(p1), bN(p2), . . . , bN(pm))T,Wij = (Zij(p1), Zij(p2), . . . , Zij(pm))T and
+Si = (bN(QiX(p1)),
+bN(QiX(p2)), . . . , bN(QiX(pm)))T and ϵi are the stacked residuals ϵi(p)s. The parameters
+in the above model are the basis coefficients ψ = (βT
+0 , βT
+1 , . . . , βT
+q , θT)T. For estimation
+of the parameters, we use the following least square criterion, which reduces to a shape
+constrained optimization problem. Namely, we obtain the estimates ˆψ by minimizing
+residual sum of squares as
+ˆψ = argmin
+ψ
+n
+�
+i=1
+||QiY − B0β0 −
+q
+�
+j=1
+Wijβj − Siθ||2
+2
+s.t
+Dψ ≥ 0.
+(9)
+The universal constraint matrix D on the basis coefficients is chosen to ensure the con-
+ditions (1),(2),(3) in Theorem 1. Later in this section, we illustrate examples how the
+constraint matrix is formed in practice. The above optimization problem (9) can be iden-
+tified as a quadratic programming problem (Goldfarb and Idnani, 1982, 1983). R package
+restriktor (Vanbrabant and Rosseel, 2019) can be used for performing the above opti-
+mization.
+Example 1: Single scalar covariate (q = 1) and a distributional predictor
+We consider the case where there is a single scalar covariate z1 (q = 1) and a dis-
+tribution predictor QX(p). In this case, the DOSDR model (2) is given by QiY (p) =
+β0(p) + zi1β1(p) + h(QiX(p)) + ϵi(p). The sufficient conditions (1)-(3) for non-decreasing
+quantile functions in this case reduces to: A) The distributional intercept β0(p) is non-
+decreasing B) β0(p) + β1(p) is non-decreasing C) h(·) is non-decreasing. Note that the
+above conditions do no enforce β1(p) to be non-decreasing. Once the coefficient func-
+tions are modelled in terms of Bernstein basis expansions as in (4) and (5), conditions
+(A)-(C) can be be enforced via the following linear restrictions on the basis coefficients
+i.e., ANβ0 ≥ 0, [AN AN](βT
+0 , βT
+1 )T ≥ 0, AN−1θ ≥ 0. Here AN is a constraint matrix
+which imposes monotonicity on functions fN(x) modelled with Bernstein polynomials
+as fN(x) = �N
+k=0 βkbk(x, N), where bk(x, N) =
+�N
+k
+�
+xk(1 − x)N−k, for 0 ≤ x ≤ 1. The
+10
+
+derivative is given by f ′
+N(x) = N �N−1
+k=0 (βk+1 − βk)bk(x, N − 1). Hence if βk+1 ≥ βk for
+k = 0, 1, . . . , N −1, fN(x) is non decreasing, which is achieved with the constraint matrix
+AN. The combined linear restrictions on the parameter ψ = (βT
+0 , βT
+1 , θT)T is given by
+Dψ ≥ 0. The matrices AN, D are given by
+AN ≡
+�
+�
+�
+�
+�
+�
+�
+�
+−1
+1
+0
+. . .
+0
+0
+−1
+1
+0
+. . .
+...
+0
+. . .
+0
+−1
+1
+�
+�
+�
+�
+�
+�
+�
+�
+, D =
+�
+�
+�
+�
+�
+AN
+0
+0
+AN
+AN
+0
+0
+0
+AN−1
+�
+�
+�
+�
+� .
+(10)
+Similar example with two scalar covariates (q = 2) and a distributional predictor is
+given in Appendix B of the Supplementary Material. Our estimation ensures that the
+shape restrictions are enforced everywhere and hence the predicted quantile functions
+are nondecreasing in the whole domain p ∈ [0, 1] as opposed to fixed quantile levels or
+design points in Ghodrati and Panaretos (2021). The order of the Bernstein polynomial
+basis N controls the smoothness of the coefficient functions βj(·) and h(·). We follow a
+truncated basis approach (Ramsay and Silverman, 2005; Fan et al., 2015), by restricting
+the number of BP basis to ensure the resulting coefficient functions are smooth. The
+optimal order of the basis functions is chosen via V -fold cross-validation method (Wang
+and Ghosh, 2012) using cross-validated residual sum of squares defined as, CV SSE =
+�V
+v=1
+�nv
+i=1 ||QiY,v − ˆQ−v
+iY,v||2
+2. Here ˆQ−v
+iY is the fitted quantile values of observation i within
+the v th fold obtained from the constrained optimization criterion (9) and trained on the
+rest (V − 1) folds.
+2.4
+Uncertainty Quantification and Joint Confidence Bands
+To construct confidence intervals, we use the result that the constrained estimator ˆψ in
+(9) is the projection of the corresponding unconstrained estimator (Ghosal et al., 2022a)
+onto the restricted space: ˆψr = argmin
+ψ∈ΘR
+||ψ − ˆψur||2
+ˆΩ, for a non-singular matrix ˆΩ. The
+restricted parameter space is given by ΘR = {ψ ∈ RKn : Dψ ≥ 0}. The DOSDR model
+(8) can be reformulated as QiY = Tiψ + ϵi, where Ti = [B0 Wi1 Wi2, . . . , Wiq Si] . The
+11
+
+unrestricted and restricted estimators are given by,
+ˆψur = argmin
+ψ∈RKn
+n
+�
+i=1
+||QiY − Tiψ||2
+2
+(11)
+ˆψr = argmin
+ψ∈ΘR
+n
+�
+i=1
+||QiY − Tiψ||2
+2
+Let us denote QT
+Y = (Q1Y , Q2Y , . . . , QnY )T and T = [TT
+1 , TT
+2 , . . . , TT
+n]T. Then we can
+write,
+1
+n||QY − Tψ||2
+2 = 1
+n||QY − T ˆψur||2
+2 + 1
+n||T ˆψur − Tψ||2
+2.
+Hence ˆψr = argmin
+ψ∈ΘR
+||ψ− ˆψur||2
+ˆΩ, where ˆΩ = 1
+n
+�n
+i=1 TT
+i Ti and Ω = E( ˆΩ) is non-singular.
+Thus, we can use the projection of the large sample distribution of √n( ˆψur − ψ0) to
+approximate the distribution of √n( ˆψr − ψ0).
+Now √n( ˆψur − ψ0) is asymptotically
+distributed as N(0, ∆) under suitable regularity conditions (Huang et al., 2004, 2002) for
+general choice of basis fucntions (holds true for finite sample sizes if ϵ(p) is Gaussian),
+where ∆ can be estimated by a consistent estimator. In particular, we use a sandwich
+covariance estimator corresponding to model QiY = Tiψ + ϵi, for estimating ∆ following
+a functional principal component analysis (FPCA) approach (Ghosal and Maity, 2022)
+for estimation of the covariance matrix of the residuals ϵi (i = 1, . . . , n). Details of this
+estimation procedure is included in Appendix C of the Supplementary Material.
+Let us consider the scenario with a single scalar covariate and distributional pre-
+dictor for simplicity of illustration. The Bernstein polynomial approximation of β1(p)
+be given by β1N(p) = �N
+k=0 βkb1k(p, N) = ρKn(p)
+′β1. Algorithm 1 in Appendix D is
+used to obtain an asymptotic 100(1 − α)% joint confidence band for the true coefficient
+function β0
+1(p), corresponding to a scalar predictor of interest. Here β0
+1(p) denotes the
+true distributional coefficient β1(p). The algorithm relies on two steps i) Use the asymp-
+totic distribution of √n( ˆψr − ψ0) to generate samples from the asymptotic distribution
+of ˆβ1r(p) (these can be used to get point-wise confidence intervals) ii) Use the gener-
+ated samples and the supremum test statistic (Meyer et al., 2015; Cui et al., 2022) to
+obtain joint confidence band for β0
+1(p). Similar strategy can also be employed for ob-
+taining an asymptotic joint confidence band for the additive effect β0(p) + h(qx(p)), for
+a fixed value of QX(p) = qx(p). Based on the joint confidence band, it is possible to
+directly test for the global distributional effects β(p) (or h(x)). The p-value for the test
+12
+
+H0 : β(p) = 0 for all p ∈ [0, 1] versus H1 : β(p) ̸= 0 for at least one p ∈ [0, 1], could be
+obtained based on the 100(1 − α)% joint confidence band for β(p). In particular, follow-
+ing Sergazinov et al. (2022), the p-value for the test can be defined as the smallest level
+α for which at least one of the 100(1 − α)% confidence intervals around β(p) (p ∈ P)
+does not contain zero. Alternatively, a nonparametric bootstrap procedure for testing the
+global effects of scalar and distributional predictors is illustrated in Appendix E of the
+Supplementary Material which could be useful for finite sample sizes and non Gaussian
+error process.
+3
+Simulation Studies
+In this Section, we investigate the performance of the proposed estimation and testing
+method for DOSDR via simulations. To this end, we consider the following data gener-
+ating scenarios.
+3.1
+Data Generating Scenarios
+Scenario A1: DOSDR, Both distributional and scalar predictor
+We consider the DOSDR model given by,
+QiY (p) = β0(p) + zi1β1(p) + h(QiX(p)) + ϵi(p).
+(12)
+The distributional effects are taken to be β0(p) = 2+3p, β1(p) = sin( π
+2p) and h(x) = ( x
+10)3.
+The scalar predictor zi1 is generated independently from a U(0, 1) distribution. The dis-
+tributional predictor QiX(p) is generated as QiX(p) = ciQN(p, 10, 1), where QN(p, 10, 1)
+denotes the pth quantile of a normal distribution N(10, 1) and ci ∼ U(1, 2). The resid-
+ual error process ϵ(p) is independently sampled from N(0, 0.1) for each p. Since we do
+not directly observe these quantile functions QiX(p), QiY (p) in practice we assume we
+have the subject-specific observations Xi = {xi1 = QiX(ui1), xi2 = QiX(ui2), . . . , xiLi1 =
+QiX(uiLi1)} and Yi = {yi1 = QiY (vi1), yi2 = QiY (vi2), . . . , yiLi2 = QiY (viLi2)}, where ui, vj
+s are independently generated from U(0, 1) distribution. For simplicity, we assume that
+Li1 = Li2 = L many subject specific observations are available for both the distributional
+outcome and the predictor. Based on the observations Xi, Yi the subject specific quantile
+13
+
+functions QiX(p) and QiY (p) are estimated based on empirical quantiles as illustrated in
+equation (1) on a grid of p values ∈ [0, 1]. We consider number of individual measure-
+ments L = 200, 400 and training sample size n = 200, 300, 400 for this data generating
+scenario. The grid P = {p1, p2, .. . . . , pm} ⊂ [0, 1] is taken to be a equi-spaced grid of
+length m = 100 in [0.005, 0.995]. A separate sample of size nt = 100 is used as a test set
+for each of the above cases.
+Scenario A2: DOSDR, Testing the effect of scalar predictor
+We consider the data generating scheme (12) in scenario A1 above and test for the distri-
+butional effect of the scalar predictor z1 using the proposed joint-confidence band based
+test in section 2. To this end we let β1(p) = d × sin( π
+2p), where the parameter d controls
+the departure from the null hypothesis H0 : β1(p) = 0 for all p ∈ [0, 1]
+versus
+H1 :
+β1(p) ̸= 0 for some p ∈ [0, 1]. The number of subject-specific measurements L is set to
+200 and sample sizes n ∈ {200, 300, 400} are considered.
+Scenario B: DOSDR, Only distributional predictor
+We consider the following distribution on distribution regression model
+QiY (p) = h(QiX(p)) + ϵi(p).
+(13)
+The distributional outcome QiY (p), the distributional predictor QiX(p) and the error
+process ϵi(p) are generated similarly as in Scenario A. The number of subject-specific
+measurements L is set to 200 and sample sizes n ∈ {200, 300, 400} are considered. This
+scenario is used to compare the performance of the proposed DOSDR method with that
+of the isotonic regression approach illustrated in Ghodrati and Panaretos (2021).
+We consider 100 Monte-Carlo (M.C) replications from simulation scenarios A1 and
+B to assess the performance of the proposed estimation method. For scenario A2, 200
+replicated datasets are used to assess type I error and power of the proposed testing
+method.
+14
+
+3.2
+Simulation Results
+Performance under scenario A1:
+We evaluate the performance of our proposed method in terms of integrated mean squared
+error (MSE), integrated squared Bias (Bias2) and integrated variance (Var).
+For the
+distributional effect β1(p), these are defined as MSE =
+1
+M
+�M
+j=1
+� 1
+0 {ˆβj
+1(p) − β1(p)}2dp,
+Bias2 =
+� 1
+0 {ˆ¯β1(p) − β1(p)}2dp, V ar =
+1
+M
+�M
+j=1
+� 1
+0 {ˆβj
+1(p) − ˆ¯β1(p)}2dp. Here ˆβj
+1(p) is the
+estimate of β1(p) from the jth replicated dataset and ˆ¯β1(p) =
+1
+M
+�M
+j=1 ˆβj
+1(p) is the M.C
+average estimate based on the M replications. Table 1 reports the squared Bias, Variance
+and MSE of the estimates of β1(p) for all cases considered in scenario A1. MSE as well
+as squared Bias and Variance are found to decrease and be negligible as sample size n
+or number of measurements L increase, illustrating satisfactory accuracy of the proposed
+estimator.
+Table 1: Integrated squared bias, variance and mean square error of estimated β1(p) over
+100 Monte-Carlo replications, Scenario A1.
+Sample Size
+L=200
+L=400
+β1(p)
+Bias2
+Var
+MSE
+Bias2
+Var
+MSE
+n= 200
+0.0001
+0.0034
+0.0035
+2.8 × 10−5
+0.0019
+0.0019
+n= 300
+1.9 × 10−5
+0.0026
+0.0026
+1.7 × 10−5
+0.0016
+0.0016
+n= 400
+2.6 × 10−5
+0.0018
+0.0018
+5.5 × 10−6
+0.0010
+0.0010
+Since, h(x) is not directly estimable in the DOSDR model (12), we consider estimation
+of the estimable additive effect γ(p) = β0(p) + h(qx(p)) at qx(p) = 1
+n
+�n
+i=1 QiX(p). The
+performance of the estimates in terms of squared Bias, variance and MSE are reported
+in Table 2, which again illustrates satisfactory performance of the proposed method in
+capturing the distributional effect of the distributional predictor QX(p).
+Table 2: Integrated squared bias, variance and mean square error of the estimated additive
+effect γ(p) = β0(p)+h(qx(p)) at qx(p) = 1
+n
+�n
+i=1 QiX(p) over 100 Monte-Carlo replications,
+Scenario A1.
+Sample Size
+L=200
+L=400
+β0(p) + h(qx(p))
+Bias2
+Var
+MSE
+Bias2
+Var
+MSE
+n= 200
+9.9 × 10−5
+0.023
+0.023
+4.6 × 10−5
+0.023
+0.023
+n= 300
+7.3 × 10−5
+0.017
+0.017
+3.2 × 10−5
+0.017
+0.017
+n= 400
+5.8 × 10−5
+0.013
+0.013
+4.8 × 10−5
+0.013
+0.013
+The estimated M.C mean for the distributional effect β1(p) and γ(p) along with their
+15
+
+respective 95% point-wise confidence intervals are displayed in Figure 2, for the case
+n = 400, L = 400.
+The M.C mean estimates are superimposed on the true curves
+and along with the narrow confidence intervals, they illustrate low variability and high
+accuracy of the estimates.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+p
+β1(p)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+4
+6
+8
+10
+p
+γ(p)
+Figure 2: Left: True distributional effect β1(p) (solid) and estimated ˆβ1(p) averaged over
+100 M.C replications (dashed) along with point-wise 95% confidence interval (dotted),
+scenario A1, n = 400, L = 400. Right: Additive effect γ(p) = β0(p) + h(qx(p)) (solid) at
+qx(p) = 1
+n
+�n
+i=1 QiX(p) and its estimate ˆγ(p) averaged over 100 M.C replications (dashed)
+along with point-wise 95% confidence interval (dotted).
+As a measure of out-of-sample prediction performance, we report the average Wasser-
+stein distance between the true quantile functions and the predicted ones in the test set
+defined as WD =
+1
+nt
+�nt
+i=1[
+� 1
+0 {Qtest
+i
+(p)− ˆQi
+test(p)}2dp]
+1
+2. Supplementary Table S1 reports
+the summary of the average Wasserstein distance across the 100 Monte-Carlo replica-
+tions. The low values of the average WD metric and their M.C standard error indicate
+a satisfactory prediction performance of the proposed method. The prediction accuracy
+appears to be improving with an increase in the number of measurements L. The perfor-
+mance of the proposed projection based joint confidence intervals for β1(p) is investigated
+in Supplementary Table S2 which reports the coverage and width of the joint confidence
+bands for β1(p) for various choices of N and for the case L = 200. It is observed that
+the nominal coverage of 95% lies within the two standard error limit of the estimated
+coverage in the all the cases, particularly for choices of N picked by our proposed cross
+16
+
+validation method.
+Performance under scenario A2:
+We assess the performance of the proposed testing method in terms of estimated type
+I error and power calculated from the Monte-Carlo replications. We set the order of
+the Bernstein polynomial basis N = 3 based on our results from previous section. The
+estimated power curve is displayed as a function of the parameter d in Supplementary
+Figure S1, using a nominal level of α = 0.05. At d = 0, the null hypothesis holds and the
+power corresponds to the type I error of the test. The nominal level α = 0.05 lies within
+its two standard error limit for all the sample sizes, illustrating that the test maintains
+proper size. For d > 0, we see the power quickly increase to 1, showing that the proposed
+test is able to capture small departures from the null hypothesis successfully.
+Performance under scenario B:
+We again consider estimation of the estimable additive effect we consider estimation of
+the estimable additive effect γ(p) = β0(p) + h(qx(p)) at qx(p) =
+1
+n
+�n
+i=1 QiX(p), which
+can be estimated by both the proposed DOSDR (2) method and the isotonic regression
+method (Ghodrati and Panaretos, 2021). Note that true β0(p) = 0, but we include a
+distributional intercept in our DOSDR model, nonetheless, as this information is not
+available to practitioners. For the isotonic regression method we directly fit the model
+(13) without any intercept. The performance of the estimates are compared in terms
+of squared Bias, variance and MSE and are reported in Table 3. We observe a similar
+performance of the proposed method with the PAVA based isotnic regression method.
+Table 3: Integrated squared bias, variance and mean square error of the estimated additive
+effect γ(p) = β0(p)+h(qx(p)) at qx(p) = 1
+n
+�n
+i=1 QiX(p) over 100 Monte-Carlo replications,
+Scenario B, from the DOSDR method and the isotonic regression method with PAVA
+(Ghodrati and Panaretos, 2021).
+Sample Size
+DOSDR
+PAVA
+β0(p) + h(qx(p))
+Bias2
+Var
+MSE
+Bias2
+Var
+MSE
+n= 200
+0.0002
+0.022
+0.022
+2.6 × 10−5
+0.022
+0.022
+n= 300
+0.0002
+0.016
+0.016
+2.4 × 10−5
+0.016
+0.016
+n= 400
+0.0002
+0.012
+0.013
+3 × 10−5
+0.012
+0.012
+The estimated M.C mean for the distributional effect γ(p) along with their respective
+17
+
+95% point-wise confidence intervals are displayed in Supplementary Figure S2, for the
+case n = 400. Again, both the method are observed to perform a good job in capturing
+γ(p).
+The proposed DOSDR method enables conditional estimation of γ(p) = β0(p) +
+h(qx(p)) on the entire domain p ∈ [0, 1], where as for the isotonic regression method,
+interpolation is required from grid level estimates. The PAVA based isotonic regression
+method failed to converge in 5% of the cases for sample size n = 200, where as, this
+issue was not faced by our proposed method. In terms of model flexibility, the isotonic
+regression method do not directly accommodate scalar predictors, or a distributional
+intercept, and keeping these points in mind our proposed method certainly provide a
+uniform and flexible approach for modelling distributional outcome, in the presence of
+both distributional and scalar predictors.
+4
+Modelling Distribution of Heart Rate in Baltimore
+Longitudinal Study of Aging
+In this section, we apply our proposed framework to continuously monitored heart rate
+and physical activity data collected in Baltimore Longitudinal Study of Aging (BLSA),
+the longest-running scientific study of aging in the United States. Specifically, the distri-
+bution of minute-level heart rate is modelled via age, sex and BMI and the distribution
+of minute-level activity counts capturing daily composition of physical activity. We set
+our study period to be 8 a.m. - 8 p.m. and calculate distributional representation of
+minute-level heart rate and (log-transformed) activity counts of BLSA participants via
+subject-specific quantile functions QiY (p) (heart rate) and QiX(p) (represented via log-
+transformed AC). For each participant, we consider only their first BLSA visit while
+obtaining the subject-specific quantile functions QiY (p), QiX(p). Our final sample con-
+stitutes of n = 890 BLSA participants, who had heart rate, physical activity and other
+covariates used for the analysis available. Supplementary Table S3 presents the descrip-
+tive statistics of the sample.
+Supplementary Figure S3 shows the subject-specific quantile functions of heart rate
+and physical activity (log-transformed, during 8 a.m.-8 p.m. time period). As a starting
+point, we study the dependence of mean heart rate on mean activity count and age, sex
+18
+
+(Male=1, Female=0) and BMI via the multiple regression model,
+µH,i = θ0 + θ1agei + θ2sexi + θ3BMIi + θ4µA,i + ϵi,
+where µH,i, µA,i are the subject specific means of heart rate and activity counts. Supple-
+mentary Table S4 reports the results of the model fit. Mean heart rate is found to be
+negatively associated with age and mean activity, and positively associated with BMI.
+The above results although useful, does not paint the whole picture about how the dis-
+tribution of hear rate depends on these biological factors and the distribution of physical
+activity. Therefore, we use the proposed DOR model
+QiY (p) = β0(p) + ageiβage(p) + BMIiβBMI(p) + sexiβsex(p) + h(QiX(p)) + ϵi(p), . (14)
+The scalar covariates age, BMI as well as activity counts are transformed to be [0, 1] scale
+using monotone linear transformations. The distributional effects of age, sex (Male=1,
+Female=0) and BMI on heart rate are captured by βage(p), βsex(p), βBMI(p), respectively.
+The monotone nonparametric function h(·) is used to link the distribution of heart rate
+and the distribution of activity counts.
+We use the proposed estimation method for
+estimation of the distributional effects βj(p)s and h(·) (h(0) = 0 is imposed). The common
+degree of the Bernstein polynomial basis used to model all the distributional coefficient
+was chosen via five-fold cross-validation method that resulted in N = 5. The estimated
+distributional effects along with their asymptotic 95% joint confidence bands using the
+proposed projection based method are displayed in Figure 3. The p-values from the joint
+confidence band based global test for the intercept and the effect of age, BMI, sex, and
+distribution of activity counts are found to be 1 × 10−6, 1 × 10−6, 5 × 10−5, 3 × 10−4 and
+1 × 10−6, respectively, resulting in the significance of all the predictors.
+The estimated distributional intercept ˆβ0(p) is monotone and represents the baseline
+distribution of heart rate.
+The estimated distributional effect of age is found to be
+significant for all p, in particular, ˆβage(p) is negative and appears to be decreasing and
+then stabilizing in p ∈ [0, 1] illustrating moderate-high levels of heart rate decrease at an
+accelerated rate with age compared to sedentary levels of activity (Antelmi et al., 2004).
+The maximal levels of heart rate (p > 0.8) are found to be decreasing with age (βage(p) <
+0) (Kostis et al., 1982; Tanaka et al., 2001; Gellish et al., 2007).
+The distributional
+19
+
+effect of BMI ˆβBMI(p) is found to be positive and increasing in p (especially at higher
+quantiles), indicating that a higher maximal heart rate is associated with a higher BMI
+after adjusting for age, sex and the daily distribution of activity counts (Foy et al., 2018).
+The estimated effect of sex (Male) ˆβsex(p) illustrates that females have higher heart rate
+(Antelmi et al., 2004; Prabhavathi et al., 2014) compared to males across all quantile
+levels after adjusting for age, BMI and PA. The lower heart rate in males compared
+to females can be attributed to size of the heart, which is typically smaller in females
+than males (Prabhavathi et al., 2014) and thus need to beat faster to provide the same
+output. The estimated monotone regression map between PA and heart rate distribution
+ˆh(x) (estimated under constraint h(0) = 0) is found to be highly nonlinear and convex,
+illustrating a non-linear dependence of heart rate on physical activity, especially at higher
+values of PA. The convex nature of the map points out an accelerated increase in the
+heart rate quantiles with an increase in the corresponding quantile levels of PA (Leary
+et al., 2002). The estimated distributional effects especially for age and gender in our
+analysis, illustrate that the distributional effects have no reason to be non-decreasing,
+as enforced in the qunatile function-on-scalar regression model in Yang (2020), which
+might lead to wrong conclusions here. The proposed DOR method is more flexible in
+this regard and enforces the monotonicity of the quantile functions without requiring the
+distributional effects to be monotone.
+We also compare the predictive performance of the proposed DOSDR model with
+that of the distribution-on-distribution regression model by Ghodrati and Panaretos
+(2021) based on isotonic regression (DODR-ISO). Supplementary Figure S4 displays
+the leave-one-out-cross-validated (LOOCV) predicted quantile functions of heart rate
+from both the methods. We define the measure LOOCV R-Squared as R2
+loocv = 1 −
+�N
+i=1
+� 1
+0 {Qi(p)− ˆ
+Qi
+loocv(p)}2dp
+�N
+i=1
+� 1
+0 {Qi(p)− ¯Q}2dp
+, where ¯Q =
+1
+N
+�N
+i=1
+� 1
+0 Qi(p)dp to compare the out-of-sample
+prediction accuracy of the two methods. The R2
+loocv value for the DOSDR and the DODR-
+ISO model are calculated to be 0.60 and 0.49 respectively. This illustrates the proposed
+DOSDR method is able to predict the heart rate quantile functions more accurately with
+the use of additional information from the biological scalar factors age, sex and BMI.
+20
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+60
+80
+100
+120
+intercept for HR
+p
+beta0_intercept
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+−30
+−10
+10
+30
+age effect on HR
+p
+beta_age
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+−20
+0
+10
+30
+bmi effect on HR
+p
+beta_bmi
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+−10
+−6
+−4
+−2
+0
+sex effect on HR
+p
+beta_sex(M)
+0
+2
+4
+6
+8
+0
+50
+100
+150
+h function
+h(x)
+Figure 3: Estimated distributional effects (solid) along with their joint 95% confidence
+bands (dotted) for age, BMI (both scaled to [0, 1]) and sex (Male) on heart rate along
+with the estimated link function h(·) (solid) (under the constraint h(0) = 0) between the
+distribution of heart rate and physical activity.
+5
+Discussion
+In this article, we have developed a flexible distributional outcome regression. The dis-
+tributional functional effects are modelled via Bernstein polynomial basis with appro-
+priate shape constraints to ensure monotonicity of the predicted quantile functions. A
+novel construction of BP-based regression structure results in imposing much less restric-
+tive compared to existing methods for modelling monotone quantile function outcome.
+21
+
+Thus, the proposed framework enables more flexible dependencies between distributional
+outcome and scalar and distributional predictors. Inferential tools are developed that
+include projection-based asymptotic joint confidence bands and a global test of statisti-
+cal significance for estimated functional regression coefficients. Numerical analysis using
+simulations illustrate an accurate performance of the estimation method. The proposed
+test is also shown to maintain the nominal test size and have a satisfactory power. An
+additional nonparametric bootstrap test provided in the supplementary material could
+be particularly useful in finite sample sizes.
+Application of DOR is demonstrated in studying the distributional association be-
+tween heart rate reserve and key demographics while adjusting for physical activity. Our
+findings provide important insights about age and gender differences in distribution of
+heart rate. Beyond the considered epidemiological application, the proposed regression
+model could be used in other epidemiological studies to more flexibly model distributional
+aspect of high frequency and high intensity data. Additionally, it can be used for estima-
+tion of treatment effects in primary or secondary endpoints quantified via distributions.
+There are multiple research directions that remain to be explored based on this cur-
+rent work. In developing our method we have implicitly assumed that there are enough
+measurements available per subject to accurately estimate quantile functions. Scenarios
+with only a few sparse measurements pose a practical challenge and will need careful
+handling. Other aspects of studies collecting distributional data such as distributional
+measurements being multilevel (Goldsmith et al., 2015) or incorporating spatio-temporal
+structure (Yang, 2020; Ghosal et al., 2022b) would be important to consider. Another
+interesting direction of research could be to extend these models beyond the additive
+paradigm, for example the single index model (Jiang et al., 2011) could be employed
+to accommodated interaction and nonlinear effects of multiple scalar and distributional
+predictors. Extending the proposed method to such more general and complex models
+would be computationally challenging, nonetheless merits future attention because of
+their potentially diverse applications.
+22
+
+Supplementary Material
+Appendix A-E along with the Supplementary Tables and Supplementary Figures refer-
+enced in this article are available online as Supplementary Material.
+Software
+Software implementation via R (R Core Team, 2018) and illustration of the proposed
+framework is available upon request from the authors.
+References
+Antelmi, I., De Paula, R. S., Shinzato, A. R., Peres, C. A., Mansur, A. J., and Grupi,
+C. J. (2004), “Influence of age, gender, body mass index, and functional capacity on
+heart rate variability in a cohort of subjects without heart disease,” The American
+journal of cardiology, 93, 381–385.
+Augustin, N. H., Mattocks, C., Faraway, J. J., Greven, S., and Ness, A. R. (2017),
+“Modelling a response as a function of high-frequency count data: The association
+between physical activity and fat mass,” Statistical methods in medical research, 26,
+2210–2226.
+Carnicer, J. M. and Pena, J. M. (1993), “Shape preserving representations and optimality
+of the Bernstein basis,” Advances in Computational Mathematics, 1, 173–196.
+Chen, Y., Lin, Z., and M¨uller, H.-G. (2021), “Wasserstein regression,” Journal of the
+American Statistical Association, 1–40.
+Cui, E., Leroux, A., Smirnova, E., and Crainiceanu, C. M. (2022), “Fast univariate
+inference for longitudinal functional models,” Journal of Computational and Graphical
+Statistics, 31, 219–230.
+Fan, Y., James, G. M., and Radchenko, P. (2015), “Functional additive regression,” The
+Annals of Statistics, 43, 2296–2325.
+23
+
+Foy, A. J., Mandrola, J., Liu, G., and Naccarelli, G. V. (2018), “Relation of obesity
+to new-onset atrial fibrillation and atrial flutter in adults,” The American journal of
+cardiology, 121, 1072–1075.
+Gellish, R. L., Goslin, B. R., Olson, R. E., McDONALD, A., Russi, G. D., and Moudgil,
+V. K. (2007), “Longitudinal modeling of the relationship between age and maximal
+heart rate.” Medicine and science in sports and exercise, 39, 822–829.
+Ghodrati, L. and Panaretos, V. M. (2021), “Distribution-on-Distribution Regression via
+Optimal Transport Maps,” arXiv preprint arXiv:2104.09418.
+Ghosal, R., Ghosh, S., Urbanek, J., Schrack, J. A., and Zipunnikov, V. (2022a), “Shape-
+constrained estimation in functional regression with Bernstein polynomials,” Compu-
+tational Statistics & Data Analysis, 107614.
+Ghosal, R. and Maity, A. (2022), “A Score Based Test for Functional Linear Concurrent
+Regression,” Econometrics and Statistics, 21, 114–130.
+Ghosal, R., Varma, V. R., Volfson, D., Hillel, I., Urbanek, J., Hausdorff, J. M., Watts,
+A., and Zipunnikov, V. (2021), “Distributional data analysis via quantile functions
+and its application to modelling digital biomarkers of gait in Alzheimer’s Disease,”
+Biostatistics.
+Ghosal, R., Varma, V. R., Volfson, D., Urbanek, J., Hausdorff, J. M., Watts, A., and
+Zipunnikov, V. (2022b), “Scalar on time-by-distribution regression and its application
+for modelling associations between daily-living physical activity and cognitive functions
+in Alzheimer’s Disease,” Scientific reports, 12, 1–16.
+Goldfarb, D. and Idnani, A. (1982), “Dual and primal-dual methods for solving strictly
+convex quadratic programs,” in Numerical analysis, Springer, pp. 226–239.
+— (1983), “A numerically stable dual method for solving strictly convex quadratic pro-
+grams,” Mathematical programming, 27, 1–33.
+Goldsmith, J., Zipunnikov, V., and Schrack, J. (2015), “Generalized multilevel function-
+on-scalar regression and principal component analysis,” Biometrics, 71, 344–353.
+24
+
+Hron, K., Menafoglio, A., Templ, M., Hruzova, K., and Filzmoser, P. (2016), “Simplicial
+principal component analysis for density functions in Bayes spaces,” Computational
+Statistics & Data Analysis, 94, 330–350.
+Huang, J. Z., Wu, C. O., and Zhou, L. (2002), “Varying-coefficient models and basis
+function approximations for the analysis of repeated measurements,” Biometrika, 89,
+111–128.
+— (2004), “Polynomial spline estimation and inference for varying coefficient models with
+longitudinal data,” Statistica Sinica, 14, 763–788.
+Irpino, A. and Verde, R. (2013), “A metric based approach for the least square regression
+of multivariate modal symbolic data,” in Statistical Models for Data Analysis, Springer,
+pp. 161–169.
+Jiang, C.-R., Wang, J.-L., et al. (2011), “Functional single index models for longitudinal
+data,” The Annals of Statistics, 39, 362–388.
+Kostis, J. B., Moreyra, A., Amendo, M., Di Pietro, J., Cosgrove, N., and Kuo, P. (1982),
+“The effect of age on heart rate in subjects free of heart disease. Studies by ambulatory
+electrocardiography and maximal exercise stress test.” Circulation, 65, 141–145.
+Leary, A. C., Struthers, A. D., Donnan, P. T., MacDonald, T. M., and Murphy, M. B.
+(2002), “The morning surge in blood pressure and heart rate is dependent on levels of
+physical activity after waking,” Journal of hypertension, 20, 865–870.
+Lorentz, G. G. (2013), Bernstein polynomials, American Mathematical Soc.
+Matabuena, M. and Petersen, A. (2021), “Distributional data analysis with accelerometer
+data in a NHANES database with nonparametric survey regression models,” arXiv.
+Matabuena, M., Petersen, A., Vidal, J. C., and Gude, F. (2021), “Glucodensities: a
+new representation of glucose profiles using distributional data analysis,” Statistical
+Methods in Medical Research, 30, 1445–1464.
+Meyer, M. J., Coull, B. A., Versace, F., Cinciripini, P., and Morris, J. S. (2015), “Bayesian
+function-on-function regression for multilevel functional data,” Biometrics, 71, 563–
+574.
+25
+
+Parzen, E. (2004), “Quantile probability and statistical data modeling,” Statistical Sci-
+ence, 19, 652–662.
+Pegoraro, M. and Beraha, M. (2022), “Projected Statistical Methods for Distributional
+Data on the Real Line with the Wasserstein Metric.” J. Mach. Learn. Res., 23, 37–1.
+Petersen, A. and M¨uller, H.-G. (2016), “Functional data analysis for density functions by
+transformation to a Hilbert space,” The Annals of Statistics, 44, 183–218.
+Petersen, A., Zhang, C., and Kokoszka, P. (2021), “Modeling Probability Density Func-
+tions as Data Objects,” Econometrics and Statistics.
+Powley, B. W. (2013), “Quantile function methods for decision analysis,” Ph.D. thesis,
+Stanford University.
+Prabhavathi, K., Selvi, K. T., Poornima, K., and Sarvanan, A. (2014), “Role of biological
+sex in normal cardiac function and in its disease outcome–a review,” Journal of clinical
+and diagnostic research: JCDR, 8, BE01.
+R Core Team (2018), R: A Language and Environment for Statistical Computing, R
+Foundation for Statistical Computing, Vienna, Austria.
+Ramsay, J. and Silverman, B. (2005), Functional Data Analysis, New York: Springer-
+Verlag.
+Ramsay, J. O. et al. (1988), “Monotone regression splines in action,” Statistical science,
+3, 425–441.
+Sergazinov, R., Leroux, A., Cui, E., Crainiceanu, C., Aurora, R. N., Punjabi, N. M., and
+Gaynanova, I. (2022), “A case study of glucose levels during sleep using fast function
+on scalar regression inference,” arXiv preprint arXiv:2205.08439.
+Talsk´a, R., Hron, K., and Grygar, T. M. (2021), “Compositional Scalar-on-Function
+Regression with Application to Sediment Particle Size Distributions,” Mathematical
+Geosciences, 1–29.
+Tanaka, H., Monahan, K. D., and Seals, D. R. (2001), “Age-predicted maximal heart
+rate revisited,” Journal of the american college of cardiology, 37, 153–156.
+26
+
+Tang, B., Zhao, Y., Venkataraman, A., Tsapkini, K., Lindquist, M., Pekar, J. J., and
+Caffo, B. S. (2020), “Differences in functional connectivity distribution after transcra-
+nial direct-current stimulation: a connectivity density point of view,” bioRxiv.
+Vanbrabant, L. and Rosseel, Y. (2019), Restricted Statistical Estimation and Inference
+for LinearModels, 0.2-250.
+Verde, R. and Irpino, A. (2010), “Ordinary least squares for histogram data based on
+wasserstein distance,” in Proceedings of COMPSTAT’2010, Springer, pp. 581–588.
+Wang, J. and Ghosh, S. K. (2012), “Shape restricted nonparametric regression with
+Bernstein polynomials,” Computational Statistics & Data Analysis, 56, 2729–2741.
+Yang, H. (2020), “Random distributional response model based on spline method,” Jour-
+nal of Statistical Planning and Inference, 207, 27–44.
+Yang, H., Baladandayuthapani, V., Rao, A. U., and Morris, J. S. (2020), “Quantile
+function on scalar regression analysis for distributional data,” Journal of the American
+Statistical Association, 115, 90–106.
+Yao, F., M¨uller, H.-G., and Wang, J.-L. (2005), “Functional linear regression analysis for
+longitudinal data,” The Annals of Statistics, 2873–2903.
+27
+
+Supplementary Material for Distributional
+outcome regression and its application to
+modelling continuously monitored heart
+rate and physical activity
+Rahul Ghosal1,∗, Sujit Ghosh2, Jennifer A. Schrack3, Vadim Zipunnikov4
+1 Department of Epidemiology and Biostatistics, University of South Carolina
+2Department of Statistics, North Carolina State University
+3 Department of Epidemiology, Johns Hopkins Bloomberg
+School of Public Health
+4 Department of Biostatistics, Johns Hopkins Bloomberg
+School of Public Health
+January 30, 2023
+1
+arXiv:2301.11399v1  [stat.ME]  26 Jan 2023
+
+1
+Appendix A: Proof of Theorem 1
+The predicted outcome quantile function is the conditional expectation of the outcome
+quantile function based on the distribution-on-scalar and distribution regression (DOSDR)
+model (2) and is given by,
+E(QY (p) | z1, z2, . . . , zq, qx(p)) = β0(p) +
+q
+�
+j=1
+zjβj(p) + h(qx(p)).
+(1)
+We will show conditions (1)-(3) are sufficient conditions to ensure E(QY (p) | z1, z2, . . . , zq, qx(p))
+is non-decreasing. Let us assume 0 ≤ zj ≤ 1, ∀j = 1, 2, . . . , J, without loss of generality.
+It is enough to show T1(p) = β0(p) + �q
+j=1 zjβj(p) and T2(p) = h(qx(p)) both are non
+decreasing. The second part is immediate as both qx(·) and h(·) (by condition (3)) are
+non decreasing. To complete the proof we only need to show T1(p) is non decreasing.
+T ′
+1(p) = β′
+0(p)+�q
+j=1 zjβ′
+j(p). Enough to show T ′
+1(p) ≥ 0 for all (z1, z2, . . . , zq) ∈ [0, 1]q.
+Note that this is a linear function in (z1, z2, . . . , zq) ∈ [0, 1]q. By the well-known Bauer’s
+principle the minimum is attained at the boundary points B = {(z1, z2, . . . , zq) : zj ∈
+{0, 1}}. Hence, the sufficient conditions are β′
+0(p) ≥ 0 and β′
+0(p) + �r
+k=1 β′
+jk(p) ≥ 0 for
+any sub-sample {j1, j2, . . . , jr} ⊂ {1, 2, . . . , q}, which follows from condition (1) and (2).
+2
+Appendix B: Example of DOSDR
+Example 2: Two scalar covariates (q = 2) and a distributional predictor
+We illustrate the estimation for DOSDR where there are two scalar covariates z1, z2
+(q = 1) and a single distribution predictor QX(p). The DOSDR model (2) is given by
+QiY (p) = β0(p)+zi1β1(p)+zi2β2(p)+h(QiX(p))+ϵi(p). The sufficient conditions (1)-(3) of
+Theorem 1 in this case reduce to : A) The distributional intercept β0(p) is non-decreasing.
+B) β0(p) + β1(p), β0(p) + β2(p), β0(p) + β1(p) + β2(p) is non-decreasing. C) h(·) is non-
+decreasing. Note that condition B) illustrates that as the number of scalar covariates
+increase we have more and more combinatorial combinations of the coefficint functions
+restricted to be non-decreasing. Similar to Example 1, Conditions (A)-(C) again become
+linear restrictions on the basis coefficients of the form Dψ ≥ 0, where the constraint
+2
+
+matrix is given by D =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+AN
+0
+0
+0
+AN
+AN
+0
+0
+AN
+0
+AN
+0
+AN
+AN
+AN
+0
+0
+0
+0
+AN−1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+As the number of restrictions increase the parameter space becomes smaller and smaller,
+which can result in a faster convergence of the optimization algorithm.
+3
+Appendix C: Estimation of Asymptotic Covariance
+Matrix
+The DOSDR model (8) in the paper was reformulated as
+QiY
+= Tiψ + ϵi, ,
+where Ti = [B0 Wi1 Wi2, . . . , Wiq Si]. Under suitable regularity conditions (Huang et al.,
+2004), √n( ˆψur − ψ0) can be shown to be asymptotically distributed as N(0, ∆) (also
+holds true for finite sample sizes if ϵ(p) is Gaussian). In reality, ∆ is unknown and we
+want to estimate ∆ by an estimator ˆ∆.
+We derive a sandwich covariance estimator
+ˆ∆ corresponding to the above model. Based on the ordinary least square optimization
+criterion for model (11) (of the paper), the unrestricted estimator is given by ˆψur =
+(TTT)−1TTQY , where QT
+Y = (Q1Y , Q2Y , . . . , QnY )T and T = [TT
+1 , TT
+2 , . . . , TT
+n]T. Hence,
+V ar( ˆψur) = (TTT)−1TTΣT(TTT)−1. Here Σ = V ar(ϵ), which is typically unknown. We
+apply an FPCA based estimation approach (Ghosal and Maity, 2022) to estimate Σ.
+Let us assume (Huang et al., 2004) the error process ϵ(p) can be decomposed as
+ϵ(p) = V (p) + wp, where V (p) is a smooth mean zero stochastic process with covariance
+kernel G(p1, p2) and wp is a white noise with variance σ2. The covariance function of the
+error process is then given by Σ(p1, p2) = cov{ϵ(p1), ϵ(p2)} = G(p1, p2)+σ2I(p1 = p2). For
+data observed on dense and regular grid P, the covariance matrix of the residual vector
+ϵi is Σm×m, the covariance kernel Σ(p1, p2) evaluated on the grid P = {p1, p2, . . . , pm}.
+We can estimate Σ(·, ·) nonparametrically using functional principal component analysis
+(FPCA) if the original residuals ϵij were available. Given ϵi(pj)s, FPCA (Yao et al., 2005)
+3
+
+can be used to get ˆφk(·), ˆλks and ˆσ2 to form an estimator of Σ(p1, p2) as
+ˆΣ(p1, p2) =
+K
+�
+k=1
+ˆλk ˆφk(p1)ˆφk(p2) + ˆσ2I(p1 = p2),
+where K is large enough such that percent of variance explained (PVE) by the selected
+eigencomponents exceeds some pre-specified value such as 99%.
+In practice, we don’t have the original residuals ϵij. Hence we fit the unconstrained
+DOSDR model (11) and and obtain the residuals eij = QiY (pj) − ˆ
+QiY (pj). Then treating
+eij as our original residuals, we can obtain ˆΣ(p1, p2) and ˆΣm×m using the FPCA approach
+outlined above. Then
+ˆ
+V ar(ϵ) = ˆΣ = diag{ˆΣm×m, ˆΣm×m, . . . , ˆΣm×m}. Ghosal and Maity
+(2022) discusses consistency of ˆΣ under standard regularity conditions. Hence an consis-
+tent estimator of the covariance matrix is given by
+ˆ
+V ar( ˆψur) = (TTT)−1TT ˆΣT(TTT)−1.
+In particular, ˆ∆n = ˆ∆/n = ˆ
+cov( ˆψur) = (TTT)−1TT ˆΣT(TTT)−1.
+4
+
+4
+Appendix D: Algorithm 1 for Joint Confidence Band
+Algorithm 1 Joint confidence band of β0
+1(p)
+1. Fit the unconstrained model and obtain the unconstrained estimator
+ˆψur =
+argmin
+ψ∈RKn
+�n
+i=1 ||QiY − Tiψ||2
+2.
+2. Fit
+the
+constrained
+model
+and
+obtain
+the
+constrained
+estimator
+ˆψr
+=
+argmin
+ψ∈ΘR
+�n
+i=1 ||QiY −Tiψ||2
+2. Obtain the constrained estimator of β0
+1(p) as ˆβ1r(p) =
+ρKn(p)
+′ ˆβ1r.
+3. Let ˆ∆n be an estimate of the asymptotic covariance matrix of the unconstrained
+estimator given by ˆ∆n = ˆ∆/n = ˆ
+cov( ˆψur)
+4. For b = 1 to B
+- generate Zb ∼ NKn( ˆψur, ˆ∆n).
+- compute the projection of Zb as ˆψr,b = argmin
+ψ∈ΘR
+||ψ − Zb||2
+ˆΩ.
+- End For
+5. For each generated sample ˆψr,b calculate estimate of β0
+1(p) as ˆβ1r,b(p) = ρKn(p)
+′ ˆβ1r,b
+(b = 1, . . . , B). Compute V ar(ˆβ1r(p)) based on these samples.
+6. For b = 1 to B
+- calculate ub = max
+p∈P
+|ˆβ1r,b(p)−ˆβ1r(p)|
+√
+V ar(ˆβ1r(p)) .
+- End For
+7. Calculate q1−α the (1 − α) empirical quantile of {ub}B
+b=1.
+8. 100(1−α)% joint confidence band for β0
+1(p) is given by ˆβ1r(p)±q1−α
+�
+V ar(ˆβ1r(p)).
+5
+Appendix E: Bootstrap Test for Global Distribu-
+tional Effects
+A practical question of interest in the DOSDR model is to directly test for the global
+distributional effect of the scalar covariates Zj or test for the distributional effect of the
+distributional predictor QX(p). In this section, we illustrate an nonparametric bootstrap
+test based on our proposed estimation method which also easily lends itself to the required
+5
+
+shape constraints of the regression problem. In particular, we obtain the residual sum of
+squares of the null and the full model and come up with the F-type test statistic defined
+as
+TD = RSSN − RSSF
+RSSF
+.
+(2)
+Here RSSN, RSSF are the residual sum of squares under the null and the full model
+respectively. For example, let us consider the case of testing
+H0 : βr(p) = 0 for all p ∈ [0, 1]
+versus
+H1 : βr(p) ̸= 0 for some p ∈ [0, 1].
+Let r = q without loss of generality. The residual sum of of squares for the full model
+is given by RSSF = �n
+i=1 ||QiY − B0 ˆβ0 − �q
+j=1 Wij ˆβj − Si ˆθ||2
+2, where the estimates are
+obtained from the optimization criterion (9) in the paper, with the constraint DFψ ≥ 0
+(denoting the constraint matrix for the full model as DF). Similarly, we have RSSN =
+�n
+i=1 ||QiY − B0 ˆβ0 − �q−1
+j=1 Wij ˆβj − Si ˆθ||2
+2, where the estimates are again obtained from
+(9) with the constraint DNψ ≥ 0. Note that, in this case the constraint matrix is denoted
+by DN and this is essentially a submatrix of DF as the conditions for monotinicity in (1)-
+(3) (Theorem 1) for the reduced model is a subset of the original constrains for the full
+model. The null distribution of the test statistic TD is nonstandard, hence we use residual
+bootstrap to approximate the null distribution. The complete bootstrap procedure for
+testing the distributional effect of a scalar predictor is presented in algorithm (2) below.
+Similar strategy could be employed for testing the distributional effect of a distributional
+predictor or multiple scalar predictors.
+6
+
+Algorithm 2 Bootstrap algorithm for testing the distributional effect of a scalar predictor
+1. Fit the full DOSDR model in the paper using the optimization criterion
+ˆψF = argmin
+ψ
+n
+�
+i=1
+||QiY − B0β0 −
+q
+�
+j=1
+Wijβj − Siθ||2
+2
+s.t
+DFψ ≥ 0.
+and calculate the residuals ei(pl) = QiY (pl) − ˆQiY (pl), for i = 1, 2, . . . , n and l =
+1, 2, . . . , m.
+2. Fit the reduced model corresponding to H0 (the null) and estimate the parameters
+using the minimization criteria,
+ˆψN = argmin
+ψ
+n
+�
+i=1
+||QiY − B0β0 −
+q−1
+�
+j=1
+Wijβj − Siθ||2
+2
+s.t
+DNψ ≥ 0.
+Denote the estimates of the distributional effects as ˆβN
+j (p) for j = 0, 1, . . . , q − 1
+and ˆhN(x).
+3. Compute test statistic TD (2) based on these null and full model fits, denote this
+as Tobs.
+4. Resample B sets of bootstrap residuals {e∗
+b,i(p)}n
+i=1 from residuals {ei(p)}n
+i=1 ob-
+tained in step 1.
+5. for b = 1 to B
+6. Generate distributional response under the reduced DOSDR model as
+Q∗
+b,iY (p) = ˆβN
+0 (p) +
+q−1
+�
+j=1
+zij ˆβN
+j (p) + ˆhN(QiX(p)) + e∗
+b,i(p).
+7. Given the bootstrap data set {QiX(p), Q∗
+b,iY (p), z1, z2, . . . , zq}n
+i=1 fit the null and the
+full model to compute the test statistic T ∗
+b .
+8. end for
+9. Calculate the p-value of the test as ˆp =
+�B
+b=1 I(T ∗
+b ≥Tobs)
+B
+.
+7
+
+6
+Supplementary Tables
+Table S1: Average Wasserstein distance (standard error) between true and predicted
+quantile functions in the test set over 100 Monte-Carlo replications, Scenario A1.
+Sample Size
+L=200
+L=400
+n= 200
+0.2587 (0.0154)
+0.1882 (0.0138)
+n= 300
+0.2568 (0.0132)
+0.1858 (0.0105)
+n= 400
+0.2554 (0.0141)
+0.1865 (0.0120)
+Table S2: Coverage of the projection-based 95% joint confidence interval for β1(p), for
+various choices of the order of the Bernstein polynomial (BP) basis, scenario A1, based
+on 100 M.C replications with L = 200. Average width of the joint confidence interval is
+given in the parenthesis. The average choices of N from cross-validation for this scenario
+are highlighted in bold.
+BP order (N)
+Sample size (n=200)
+Sample size (n=300)
+Sample size (n=400)
+2
+0.92 (0.29)
+0.9 (0.24)
+0.9 (0.20)
+3
+0.92 (0.31)
+0.94 (0.25)
+0.96 (0.22)
+4
+0.93 (0.33)
+0.93 (0.26)
+0.93 (0.23)
+Table S3: Descriptive statistics of age and BMI for the complete, male and female samples
+in the BLSA analysis.
+Characteristic
+Complete (n=890)
+Male (n=432)
+Female (n=458)
+P value
+Mean
+SD
+Mean
+SD
+Mean
+SD
+Age
+66.66
+13.35
+68.03
+13.41
+65.37
+13.17
+0.003
+BMI (kg/m2)
+27.40
+4.96
+27.52
+4.23
+27.28
+5.57
+0.45
+Table S4: Results from multiple linear regression model of mean heart rate on age, sex
+(Male), BMI and mean activity count. Reported are the estimated fixed effects along
+with their standard error and P-values.
+Dependent variable : Mean heart rate
+Value
+Std.Error
+P-value
+Intercept
+82.47
+3.458
+< 2 × 10−16∗∗∗
+age
+−0.18
+0.026
+< 1.2 × 10−11∗∗∗
+sex
+−4.19
+0.659
+< 3.2 × 10−10∗∗∗
+BMI
+0.18
+0.067
+0.0091∗∗
+Mean activity
+2.44
+0.697
+0.0005∗∗∗
+Observations
+890
+Adjusted R2
+0.142
+Note:
+∗p<0.05; ∗∗p<0.01; ∗∗∗p<0.001
+8
+
+7
+Supplementary Figures
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.2
+0.4
+0.6
+0.8
+1.0
+d
+Power
+n=200
+n=300
+n=400
+Figure S1:
+Displayed are the estimated power curves for simulation scenario A2.
+The parameter d controls the departure from the null and the power curves for n ∈
+{200, 300, 400} are shown by solid, dashed and dotted lines. The dashed horizontal line
+at the bottom corresponds to the nominal level of α = 0.05.
+9
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+2
+3
+4
+5
+6
+7
+DODSR
+p
+γ(p)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+2
+3
+4
+5
+6
+7
+PAVA
+p
+γ(p)
+Figure S2: Displayed are estimates of additive effect γ(p) = β0(p) + h(qx(p)) (solid)
+at at qx(p) =
+1
+n
+�n
+i=1 QiX(p) and its estimate ˆγ(p) averaged over 100 M.C replications
+(dashed) along with point-wise 95% confidence interval (dotted) for scenario B, n = 400.
+Left: Estimates from the proposed DOSDR method. Right: Isotonic regression method
+with PAVA.
+10
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+50
+100
+150
+200
+250
+Heartrate
+p
+Heartrate QF (raw)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+2
+4
+6
+8
+Activity
+p
+Activity QF (log)
+Figure S3: Subject-specific quantile functions of heart rate and log-transformed activity
+counts during 8 a.m.- 8 p.m. period. Color profiles show four randomly chosen partici-
+pants.
+11
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+50
+100
+150
+200
+250
+HR
+p
+Q(p)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+50
+100
+150
+200
+250
+DODSR Predicted HR
+p
+Q(p)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+50
+100
+150
+200
+250
+HR
+p
+Q(p)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+50
+100
+150
+200
+250
+PAVA Predicted HR
+p
+Q(p)
+Figure S4: Top: LOOCV predictions of quantile functions of heart rate from DOSDR
+method based on age, sex, BMI and PA distribution. Bottom: LOOCV predictions of
+quantile functions of heart rate from PAVA method (Ghodrati and Panaretos, 2021) based
+on PA distribution.
+References
+Ghodrati, L. and V. M. Panaretos (2021). Distribution-on-distribution regression via
+optimal transport maps. arXiv preprint arXiv:2104.09418.
+Ghosal, R. and A. Maity (2022). A score based test for functional linear concurrent
+regression. Econometrics and Statistics 21, 114–130.
+12
+
+Huang, J. Z., C. O. Wu, and L. Zhou (2004). Polynomial spline estimation and inference
+for varying coefficient models with longitudinal data. Statistica Sinica 14, 763–788.
+Yao, F., H.-G. M¨uller, and J.-L. Wang (2005). Functional linear regression analysis for
+longitudinal data. The Annals of Statistics, 2873–2903.
+13
+

8NE5T4oBgHgl3EQfQg5R/content/tmp_files/2301.05513v1.pdf.txt

@@ -0,0 +1,1532 @@
+1 
+ 
+Exploring the substrate-driven morphological changes 
+in Nd0.6Sr0.4MnO3 thin films 
+ 
+R S Mrinaleni 1, 2, E P Amaladass1, 2*, S Amirthapandian 1, 2, A. T. Sathyanarayana 1, 2, 
+Jegadeesan P 1, 2, Ganesan K 1, 2, R M  Sarguna 1, 2, P. N. Rao 3, Pooja Gupta 3, 4, T 
+Geetha Kumary1, 2, and S. K. Rai 3, 4, Awadhesh Mani1, 2 
+ 
+1Material Science Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, 603102, 
+India 
+2Homi Bhabha National Institute, Indira Gandhi Centre for Atomic Research, Kalpakkam 
+603102, India 
+3Synchrotrons Utilisation Section, Raja Ramanna Centre for Advanced Technology, PO 
+RRCAT, Indore, Madhya Pradesh 452013, India 
+4Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai, 
+Maharashtra 400094, India 
+*Corresponding author:  [email protected] 
+ 
+ABSTRACT 
+Manganite thin films are promising candidates for studying the strongly correlated electron 
+systems. Understanding the growth-and morphology-driven changes in the physical properties 
+of manganite thin films is vital for their applications in oxitronics. This work reports the 
+morphological, 
+structural, 
+and 
+electrical 
+transport 
+properties 
+of 
+nanostructured 
+Nd0.6Sr0.4MnO3 (NSMO) thin films fabricated using the pulsed laser deposition technique. 
+Scanning electron microscopy (SEM) imaging of the thin films revealed two prominent surface 
+morphologies: a granular and a unique crossed-nano-rod-type morphology. From X-ray 
+diffraction (XRD) and atomic force microscopy (AFM) analysis, we found that the observed 
+nanostructures resulted from altered growth modes occurring on the terraced substrate surface. 
+Furthermore, investigations on the electrical-transport properties of thin films revealed that the 
+films with crossed-nano-rod type morphology showed a sharp resistive transition near the 
+metal-to-insulator transition (MIT).  An enhanced temperature coefficient of resistance (TCR) 
+of up to one order of magnitude was also observed compared to the films with granular 
+morphology.  Such enhancement in TCR % by tuning the morphology makes these thin films 
+promising candidates for developing oxide-based temperature sensors and detectors.  
+ 
+
+2 
+ 
+INTRODUCTION 
+Nd0.6Sr0.4MnO3 (NSMO) belongs to the class of magnetic oxides RE1-xAxMnO3 (where RE= 
+La3+, Nd3+, Pr3+, Sm3+, and A = Ca2+, Sr2+, Ba2+, etc.) with perovskite (ABO3) structure which 
+exhibits a variety of magnetic phases by tuning the dopant concentration x (x = 0 to 0.9)1–3. 
+Manganites are known for their exotic properties such as the Colossal magnetoresistive (CMR) 
+phenomenon4, Metal-insulator-transition (MIT) accompanied by a magnetic transition from 
+paramagnetic (PM) to ferromagnetic (FM) state5, half-metallicity6, and tuneable in-plane and 
+out of plane magnetic anisotropy7. These properties are exploited for potential spintronics 
+applications such as spin injection devices8, Magnetic tunnel junctions9–11, and magnetic 
+storage devices (MRAMs)12. In recent times, the perovskite-manganite systems are the ideal 
+oxide candidates for developing superlattices, self-assembled nano-arrays13, nano-ribbons14, 
+nano-wires, vertically aligned nanocomposite (VAN) thin films15–19, etc. which offer enhanced 
+Low-field magnetoresistance (LFMR), switchable magnetic anisotropy and for studying other 
+interesting interface effects such as magnetic exchange bias20. Focus on growth dynamics is 
+required to tune exclusive nano-architectures in the thin film as it offers additional handles to 
+tailor its physical properties such as a high CMR %, high Curie & MIT temperature, high-
+temperature coefficient of resistance (TCR %), and enhanced magnetoresistive (MR) 
+phenomenon. The manganite system is highly sensitive to external perturbations due to the 
+strong connection between the spin-charge and lattice degrees of freedom21,22. This poses a 
+major challenge in obtaining epitaxial/patterned thin films for useful applications.  
+The pulsed laser deposition (PLD) technique has been extensively used to fabricate oxide-
+based manganite thin films. This is because it offers good stoichiometric transfer of the target 
+material onto the substrate in addition to deposition in an oxygen background. Various studies 
+have been carried out to obtain epitaxial thin films by tuning the deposition parameters such as 
+the oxygen partial pressure, substrate temperature, laser energy density, and repetition rate, 
+affecting its growth and physical properties23,24. Additionally, the growth of the thin film is 
+influenced by the substrate. The strain offered by the substrate affects the surface morphology 
+and microstructure of the manganite thin film. Different methodologies such as i) varying the 
+substrates for different lattice matching25–27 (ii) choice of substrates with different 
+crystallographic orientations with corresponding chemical terminations14 iii) varying the 
+thickness of the thin films28, and iv) high-temperature annealing17  are adopted to tune the strain 
+and morphology of the thin films. Therefore, thin films with unique morphology and long-
+range ordered nanostructures can be obtained by fine-tuning the growth parameters. Compared 
+
+3 
+ 
+to the previous works on VAN and other nanostructures of the popular manganite system La-
+Sr-Mn-O, we have observed a granular nanostructure and another distinct nanostructure with 
+crossed-nano-rods in our thin films. We have synthesized NSMO thin films using the PLD 
+technique on single-crystal SrTiO3 (100) oriented substrates (STO). The effects of PLD 
+parameters and annealing conditions on the surface morphology were investigated. Using 
+SEM, AFM, and XRD techniques, the growth mechanism leading to a specific type of nano-
+structuring in the NSMO thin films is studied. Additionally, the morphology-driven changes in 
+the temperature dependence of resistivity are investigated, and we observed a signature trend 
+in the MIT corresponding to the particular morphology. 
+EXPERIMENTAL METHODS 
+The NSMO thin films were fabricated using the PLD technique using a commercial NSMO 
+pellet as the target. Before deposition, SrTiO3 (STO) (1 0 0) single crystals substrate was 
+cleaned by boiling in de-ionized(DI) water for 3 minutes, followed by ultra-sonication in DI 
+water, acetone, and iso-propyl alcohol followed by rinsing in DI water. With the water leaching 
+procedure, the SrO terminations present in the substrate surface can be effectively dissolved 
+and removed with DI at elevated temperatures > 60 oC followed by ultra-sonication. A KrF 
+Excimer laser source (λ = 248 nm) operated with a laser energy density of 1.75 J/cm2 at 3Hz 
+was used to ablate the target. The films were deposited in an oxygen partial of 0.36 mbar with 
+substrate temperature fixed at 750 oC. After deposition, the films were in situ annealed at 750 
+oC for 2h, and the PLD chamber was maintained with O2 background pressure of 0 to 1 bar. 
+Further, the films were ex-situ annealed in a tube furnace at 950 oC in an oxygen atmosphere 
+with a flow rate of ~ 20 sccm for 2h.  
+The surface morphology of the thin films was examined using a Scanning electron microscope 
+(SEM) from Carl Zeiss, crossbeam 340, and the images were collected in inlens-duo mode at 
+3-5 kV. Atomic force microscopy (AFM) was used for 2D and 3D visualization of the surface 
+of substrates and the films. XRD studies have been carried out at Engineering Applications 
+Beamline, BL-02, Indus-2 synchrotron source, India using beam energy of 15 keV for the 
+structural characterization of the films29. The Grazing incidence (GI) and ω-2θ scans were 
+performed, and data were collected using the Dectris detector (MYTHEN2 X 1K) in reflection 
+geometry. In the GI-scan, the incident angle is kept fixed at ω = 0.5o, and the detector moves 
+along the given 2θ range. The monochromatic high-resolution mode of the beamline was used, 
+
+4 
+ 
+keeping the beam energy at 15 keV (λ = 0.826 Å). The peaks were indexed with reference to 
+the ICDD data30 (ICDD number - 01-085-6743) 
+ 
+RESULTS AND DISCUSSION:   
+1.  Morphology studies of the nanostructured thin films: 
+The NSMO thin films prepared under the above conditions possessed two prominent 
+surface morphology – granular and rod-type. Two representative films with granular 
+nanostructure and crossed-rod nanostructure were chosen to study the physical properties. 
+These two systems will be referred to as NS-G and NS-R, where NS stands for NSMO thin 
+film, and ‘G’/’R’ stands for the type of morphology. The thickness of NS-G and NS-R thin 
+films is determined to be ~ 100 nm by cross-sectional SEM. 
+Figure 1(a) shows the SEM image of NS-G thin films with granular morphology. The 
+film is uniformly covered with multifaceted grains. Figure 1(c) (i) shows the average grain size 
+ 
+ 
+0
+20
+40
+60
+80
+100
+0
+100
+200
+300
+400
+Frequency (count)
+Grain size (nm)
+ Frequency count
+ Lognormal fitting
+Avg grain size  
+= 38.9 nm  0.1 nm 
+0
+200
+400
+600
+800 1000
+0
+100
+200
+300
+400
+500
+Frequency (count)
+Length of rod (nm)
+ Frequency count
+ Lognormal fitting
+Avg length of rod  
+= 188.7 nm  1.7 nm 
+0
+20
+40
+60
+80
+100
+120
+0
+100
+200
+300
+Frequency (count)
+Width of rod (nm)
+ Frequency count
+ Lognormal fitting
+Avg width of rod  
+= 39.6 nm  0.2 nm 
+c) (i)
+(ii)
+(iii)
+Figure 1: Scanning electron microscopy images of NSMO thin films on STO. a) NS-G -
+granular morphology. b) NS-R – self-aligned-crossed-Nano-rod-morphology. c) The 
+histograms illustrate the grain size calculation for NS-G and NS-R thin film. (i) Average grain 
+size estimated for NS-G. (ii) Average rod length estimated for NS-R. (iii) Average rod width 
+estimated for NS-R. 
+
+100 nm100 nm5 
+ 
+estimated to be 38.9 nm. Figure 1(b) shows the SEM image of NS-R thin films with unique 
+surface morphology. The thin film surface is uniformly covered with nano-rods crossed at right 
+angles embedded in a matrix of NSMO containing square/rectangular pits. In NS-R, the 
+average rod length is estimated to be 188 nm with an average width of 39.6 nm, as shown in 
+the Figure 1(c), (ii) and (iii). Further, AFM measurements have been carried out on the NS-G 
+and NS-R thin films. The 2D and 3D AFM scan in Figure 2 show columnar/island-type features 
+in the NS-G thin film and crossed-rod features in the NS-R thin film.  
+2. Structural analysis of the thin film: 
+The bulk NSMO compound has an orthorhombic crystal structure belonging to the Pbnm space 
+group. In the pseudo-cubic (pc) representation, the unit cell parameter is given by apc ≈ c / 2 ≈ 
+3.849 Å. The substrate STO has a cubic crystal structure with a lattice constant aSTO = 3.905 
+Å. NSMO grown on the STO substrate experiences a tensile strain due to the lattice mismatch 
+ 
+  
+ 
+ 
+ 
+ 
+Fig. 2: a), b), 2D and c), d) 3D AFM scans of grain-type NSMO thin film (left) and rod-
+type sample NSMO thin film on STO substrate. 
+a) 
+b) 
+c) 
+d) 
+Figure 2: a), b), 2D, and c), d) 3D AFM scans of grain-type NSMO thin film (left) and rod-type 
+sample NSMO thin film on STO substrate. 
+
+nm
+20
+0
+2.0
+0
+1.8
+0.2
+1.6
+0.4
+1.4
+0.6
+1.2
+0.8
+1.0
+1.0
+0.8
+1.2
+0.6
+1.4
+0.4
+1.6
+0.2
+1.8
+0
+2.048.3 1F:Height
+2
+8
+1.6
+61
+4-
+21
+9
+nm
+2
+41
+0
+0.2
+0.4
+0.6
+8'0
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+um21. 1F:Height
+2
+51
+1.8...
+1.6
+51
+1.4
+1.2
+1.0
+51
+nm
+8'0
+0.6
+51
+0.4
+2
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+umnm
+8
+0
+0
+2.0
+0.2
+1.8
+0.4
+1.6
+0.6
+1.4
+0.8
+1.2
+1.0
+1.0
+0.8
+1.2
+0.6
+1.4
+0.4
+1.6
+0.2
+1.8
+0
+2.06 
+ 
+of 1.4 %. The GI-XRD and high-resolution XRD (HR-XRD) reflections of the films are shown 
+in Figure 3(a) and (b). The presence of multiple reflections in the GI-XRD scan of NS-G in 
+Figure 3(a) reveals that the granular thin film is polycrystalline. In NS-R, the reflections of 
+NSMO are absent, as seen in Figure 3(b). This may be due to its out-of-plane orientation with 
+respect to the substrate. At the high 2θ angle ≈ 39.1o , the (3 1 0) STO plane gets aligned, 
+resulting in high STO (3 1 0) reflection along with the NSMO (2 4 0) peak. This shows that the 
+films are well-oriented, mirroring the substrate. Though NS-G is oriented, the crystallographic 
+difference between NS-G and NS-R is attributed to the type of nano-structuring in the films.  
+ 
+Figure 3: GI-XRD scans of NSMO thin films a) NS-G b) NS-R indexed using ICDD data (* - 
+STO peaks) 
+ 
+3. Effect of ex-situ annealing on morphology: 
+To gain insight into the type of growth across these films, we compare the 
+morphological changes in the in-situ annealed and ex-situ annealed samples in Figure 4. In 
+granular thin films, no significant changes have been observed after in-situ and ex-situ 
+annealing, apart from a minor increase in grain size, as seen in Figure 4(a). Whereas the sample 
+with rod-type morphology obtained after ex-situ annealing in Figure 4(d) exhibits facetted 
+droplets embedded in a matrix with rectangular holes and rod features in the in-situ annealed 
+ 
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+1E+02
+1E+03
+1E+04
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+1E+02
+1E+03
+1E+04
+* (310)
+ (240),(332)
+ (040), (224)
+ (024), (132)
+* (200)
+ (220), (004)
+ (202), (022)
+ (020), (112)
+ (002), (110)
+Intensity (arb. units)
+2(deg)
+ NS-G
+(a)
+(b)
+* (100)
+ (240),(332)
+* (310)
+* (200)
+2(deg)
+ NS-R
+
+7 
+ 
+case, Figure 4(c). It is evident that once the initial growth mode is set, the ex-situ annealing 
+aids in increasing grain size, relieving the strain in thin films in addition to decreasing oxygen 
+defects in NSMO thin films23. We inspect the HR-XRD scans of the NS-G and NS-R thin films 
+in the in-situ and ex-situ annealed cases to verify this claim.  
+Figure 5(a) and (b) show the HR-XRD scan performed over a range of 2θ (10o – 40o) 
+for the films NS-G and NS-R after in-situ and ex-situ annealing. It is observed that the (0 0 4) 
+NSMO peak is absent in the in-situ annealed NS-G thin film, whereas upon ex-situ annealing, 
+NS-G shows improved texturing with the (0 0 4) NSMO peak close to the (0 0 2) substrate 
+peak. In the case of NS-R, along with the substrate’s (002) reflection, corresponding (0 0 l) 
+pseudo-cubic reflections from NSMO are present with significant intensity even in the in-situ 
+annealed condition. Further, as we compare HR-XRD scans of NS-G and NS-R after ex-situ 
+annealing, the NS-R thin film has increased relative intensity compared with the NS-G. 
+a)
+b)
+ 
+Pristine thin film after in-situ annealing  Further upon ex-situ annealing  
+      
+a) 
+      
+ 
+ 
+ 
+     
+b) 
+ 
+Fig. 3:  Effect ex-situ annealing on NSMO thin films with a) granular  b)rod-type 
+surface morphology 
+c)
+d)
+Figure 4: Illustration of the effect of ex-situ annealing on NSMO thin films. a) SEM image of 
+in-situ annealed granular thin film b) SEM image of the granular thin film after ex-situ 
+annealing c) SEM image of the thin film after in-situ annealing showing rods and squared 
+blocks in the encircled regions. d) SEM image of the same thin film after ex-situ annealing 
+showing crossed-rod type morphology. 
+
+100nm100 nm100 nm100nm8 
+ 
+ 
+Figure 5: High resolution-XRD scan of NSMO thin films around the STO-(200) reflection 
+inset: fine scan of NSMO (004) of NS-R sample showing double peaks – P1 and P2 (* - STO 
+peaks) 
+Therefore, NS-R is highly oriented and more crystalline, which can be attributed to its 
+epitaxial nature of growth. Additionally, the HRXRD scan of NS-R thin films after ex-situ 
+annealing shows a doublet feature at its (004) reflection. A high-resolution fine scan was 
+performed on the NS-R thin film to confirm the double peaks. Referring to the literature, we 
+found that a similar doublet feature has been reported due to strain relaxation in PSMO thin 
+films on STO substrate31. By fitting the peaks using the pseudo-Voigt function, as shown in 
+figure S1 of supplementary information, the peaks were de-convoluted to evaluate the out-of-
+plane lattice parameter (tabulated in table T1 – supplementary information). The first peak was 
+at 2θ=24.88o with a c-lattice constant of 7.66 Å, and the second peak was at 2θ=24.97o with a 
+c-lattice constant of 7.63 Å. The reduction in the c-lattice constant of the second peak shows 
+that there is compression of the lattice along the c-axis because of the tensile strain experienced 
+by the thin film due to the substrate. Such a splitting in the peak was absent in films of thickness 
+< 80 nm, indicating that this double peak is due to partial strain relaxation in the thicker film 
+initiated by ex-situ annealing.  
+Thus, from the detailed XRD studies and discussions in the previous section, it is inferred 
+that difference in initial-growth mode, and subsequent ex-situ annealing has prominently tuned 
+the resulting surface morphology of the NSMO thin films. The granular thin film NS-G has 
+multiple orientations similar to a polycrystalline system, whereas NS-R shows improved 
+crystallinity and orientation mirroring the substrate. The parameters affecting the initial growth 
+are discussed in the upcoming section. 
+ 
+22
+24
+26
+28
+10
+-1
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+7
+10
+8
+22
+24
+26
+28
+10
+-1
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+7
+10
+8
+b)
+Intensity (arb. units)
+2(deg)
+Ex-situ annealed 
+ In-situ annealed
+* (200)
+(004)
+NS- R
+P1
+a)
+NS- G
+2(deg)
+Intensity (arb. units)
+ Ex-situ annealed
+ In-situ annealed
+* (200)
+(004)
+24.8
+25.0
+25.2
+1000
+10000
+P2
+2(deg)
+
+9 
+ 
+4. Effect of PLD parameters in tuning the morphology: 
+PLD Parameters like laser energy density, oxygen partial pressure, and substrate temperature 
+highly influence the type of growth. Changes in these parameters lead to variations in the 
+energy of the ad-atoms deposited on the substrate.  To understand the role of  O2 partial pressure 
+and laser energy density during the deposition, we have prepared NSMO thin films by varying 
+these parameters. Post deposition, the films were in-situ annealed at 750 oC for 2h in an oxygen 
+background pressure of 1 bar. Ex-situ annealing was carried out subsequently.  
+Figure 6 presents the morphology of films deposited under different laser energy 
+densities varied from 1 to 1.75 J/cm2. During deposition, the oxygen partial pressure and 
+substrate temperature were maintained at 0.36 mbar and 750 oC. Figure 7 presents the 
+morphology of films obtained at different oxygen partial pressure of 0.3mbar, 0.4 mbar, and 
+0.5 mbar while the laser energy density and substrate temperature were maintained at 1 J/cm2 
+and 750 oC during deposition, respectively.  
+We found that changes in oxygen partial pressure and laser energy density did not 
+influence the surface morphology, as both type of morphologies have been observed in 
+different deposition runs with the same parameters. Further, as we have obtained granular and 
+rod-type films for the same substrate temperature of 750 oC, the role of substrate temperature 
+is also ruled out. Thus, irrespective of changes in the parameters mentioned above, thin films 
+of either granular or crossed-rod nanostructure were obtained. Therefore we suspect the 
+ 
+a)  
+b) 
+c) 
+d)    
+e) 
+f) 
+1 J/cm2
+1 J/cm2
+1.5 J/cm2
+1.5 J/cm2
+1.75 J/cm2
+1.75 J/cm2
+Figure 6: The SEM images of NSMO thin films with granular morphology (a), (b), and (c) and 
+rod morphology from (d), (e), and (f) obtained at corresponding laser energy density- 1 J/cm2, 
+1.5 J/cm2, and 1.75 J/cm2. 
+
+200 nm200nm200nm200nm200nm200nm10 
+ 
+substrate and the strain it offers to plays a vital role in altering the growth mode of the thin 
+film. 
+5. Effect of miscut angle in tuning the morphology: 
+The commercial STO substrates used here are one-sided polished, and their surface was 
+found to have a miscut. In commercially purchased wafers, the occurrence of a miscut in the 
+range of 0.05o-0.3o is well known and unavoidable due to mechanical cutting and polishing of 
+single crystal STO wafers14,32. In Figure 7(a), the as-received STO substrate, after cleaning, 
+shows clear terrace features in the AFM scan, confirming the presence of miscut on the 
+substrate surface. In a given wafer, the miscut can be in-plane or out-of-plane or both (some 
+works refer to this as miscut directions φ and θ instead of in-plane and out-of-plane, 
+respectively). The miscut angle and direction can alter the growth mode as the lattice strain is 
+anisotropic along the substrate surface and step edges6, thus resulting in different surface 
+morphology by forming anisotropic structural domains33. Several works are available in 
+literature 33–35 on the growth of manganite thin film on STO substrate with miscut. These 
+reports claim that the value of the miscut angle and appropriate adjustments in growth 
+conditions can control the number of structural domains in the thin film. As we have already 
+ruled out the possibility of growth conditions influencing the resulting morphology, we tried 
+to evaluate the value of miscut present in our STO substrates to see if it has affected the 
+resulting morphology.  
+a) 
+b) 
+c) 
+d) 
+e) 
+f) 
+ 
+0.3 mbar
+0.4 mbar
+0.5 mbar
+0.5 mbar
+0.4 mbar
+0.3 mbar
+Figure 7: The SEM images of NSMO thin films prepared at oxygen partial pressure of 0.3 mbar, 
+0.4 mbar, and 0.5 mbar. a), b), and c) are granular NSMO thin, and films with rod morphology 
+are shown in d), e), and f) at corresponding oxygen partial pressure. 
+
+200 nm200nm200nm200 nm200 nm200 nm11 
+ 
+To determine the value of miscut present in the substrates, we have followed the XRD-
+protocol from literature36. This was carried out in a BRUKER D8, Lab source XRD setup. 
+According to the protocol, a low incident angle (~0.2o) rocking-scan was initially performed to 
+ensure that the sample was aligned with the X-ray. This was done to optimize the angle of the 
+sample holder, and the offset in the 2θ value (~0.4o) was noted as ζ.  Following that, a rocking 
+scan was performed around the (200) peak of STO (46.483o), and phi & chi scans were done 
+to orient the wafer. Further, the rocking scan around the (200) peak of STO was repeated, fixing 
+the X-ray tube position. Finally, a detector scan was performed around the (200) peak of STO 
+and this time the offset in 2θ was noted as ζ’. The difference δζ, between ζ and ζ’, gives the 
+estimate of miscut. Next, the sample was rotated by 90o, and the scans mentioned above are 
+repeated in the same order. The difference between the offsets obtained this time was denoted 
+as δξ. Finally, the out-of-plane miscut angle was evaluated using equation (1).  After 
+determining miscut on various STO wafers, we found that the value out of plane miscut angle 
+varies from 0.13o up to 0.48o. 
+𝜃𝑜𝑢𝑡−𝑜𝑓−𝑝𝑙𝑎𝑛𝑒 = 𝑎𝑟𝑐𝑡𝑎𝑛 √tan2(𝛿𝜁) + tan2(𝛿𝜉)                                 (1) 
+Table 1 : This table illustrates the morphology of NSMO thin films obtained on STO substrates 
+with different values of miscut. 
+ 
+Sample 
+Miscut 
+angle 
+Granular 
+Morphology  
+Sample 
+Miscut 
+angle 
+Rod type 
+Morphology  
+NS-G 
+0.48 o 
+ 
+NS-R 
+0.31 o 
+ 
+G1 
+0.31 o 
+ 
+R1 
+0.19 o 
+ 
+G2 
+0.30 o 
+ 
+R2 
+0.25 o 
+ 
+Table T1 : This table illustrates the morphology of NSMO thin films obtained on STO 
+substrates with different values of miscut.  
+
+100nm100nm100nm100 nm100nm100nm12 
+ 
+ 
+We see from table T1 that, both granular and rod-type morphology was observed on substrates 
+with miscut angle varying from 0.13o up to 0.48o. Sample G1 with granular morphology and 
+NS-R with rod-type morphology, possess the same miscut angle of ~ 0.3o. This is very 
+interesting, as the value of the miscut angle has not influenced the altered growth modes present 
+in our samples. Therefore to comprehend the resulting morphology, we have further 
+investigated the type of growth occurring on the terraced surface. 
+6. Thin film growth on the terraced surface: 
+A miscut on the substrate is useful for epitaxial thin films37 as the steps and terrace 
+edges act as nucleation centres and result in a step flow growth mode38. But the actual processes 
+governing the step-flow growth are more complex. The basic parameters driving this type of 
+growth are the coefficient of diffusion and the height of the Ehrlich-Schwoebel (ES) barrier39. 
+The diffusion of the adatoms on the surface and their incorporation into the crystal structure 
+govern the formation of different morphologies at the surface. Additionally, the ES barrier at 
+the terrace/step edges introduces an asymmetry in the potential energy at the edge.  An adatom, 
+reaching the terrace, either nucleates or descends into the step depending on the ES barrier 
+height. Similarly, an adatom reaching below the step experiences an inverse step barrier which 
+prevents the particles from attaching to the step from below. 
+If the barrier height is appropriate, ad-atoms can properly attach themselves to the step 
+edges resulting in a step flow growth. However, the existence of the barrier makes the growth 
+on the stepped surfaces highly unstable resulting in modified surface features such as step 
+meandering, nano-columns/wire formation, spirals/mound formations, and faceted pits. In a 
+recent work by Magdalena et al.40, a simulation using the Cellular Automaton model in (2+1)D 
+gave rise to different patterns of surface morphology on vicinal surfaces. According to the 
+simulation, different processes occurred depending on the values assigned to the barrier height 
+at step edges. The adatom could either attach to the step to build the crystal by jumping/ 
+descending at the step edge or scatter away from the barrier resulting in the formation of 
+islands. For a fixed adatom flux, diffusion of adatoms takes place on the vicinal surface, and 
+probabilities are assigned for each of the processes mentioned above. Depending on the 
+probability value, various surface patterns were simulated for three cases. In case (i), for a high 
+ES barrier, the three-dimensional surface formation resulted in square/rectangular islands 
+following the cubic lattice symmetry at the middle of the terraces. In case (ii), with a reduced 
+
+13 
+ 
+ES barrier height, more atoms were trapped at the top of the step, and a new pattern of 
+nanocolumns emerged consisting of cubic formations with deep narrow cubic pits. Finally, in 
+case (iii), when the height of the barrier was adjusted such that the probability of the adatoms 
+descending the step is equal/of the same order as the probability of the adatoms jumping up to 
+the step from below, it resulted with nano-wire or a columnar growth. Further, the presence of 
+additional local sinks that alters the potential barrier also resulted in nano-columns/islands at 
+random positions.  
+Thus, we can understand that our resulting granular morphology on the miscut STO 
+substrate is precisely similar to the surface morphology resulting from the case (iii). In the STO 
+susbtrate, the presence of disoriented terraces and improperly removed SRO terminations may 
+have altered the ES barrier resulting in local sinks at the substrate surface, thus resulting in the 
+island/columnar growth. Finally, the surface morphology of the NS-R thin film resembles the 
+ 
+As received substrate 
+after cleaning 
+After treatment for  
+TiO2 termination 
+  a)  
+ 
+b)      
+ 
+ c)      
+     
+ 
+ 
+ 
+ 
+ 
+ 
+    
+ 
+d)      
+Figure 8:  AFM scan of the STO substrate a) as-received commercial substrate after 
+cleaning b) the same substrate after TiO2 termination obtained after heat treatment method 
+with a step height of ~ 0.4 Å (one-unit cell height of STO). c), d) NSMO thin films grown on 
+the corresponding substrates 
+
+10-3nm
+400
+um
+um
+0
+1.0
+1.0
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+010-3nm
+600
+0
+0
+0.2
+0.2
+0.4
+0.4
+0.6
+0.6
+0.8
+0.8
+1.0
+1.0
+um
+wn100 nm100 nm14 
+ 
+morphology they obtained in case (ii). This fact can be verified from a close inspection of the 
+surface of NS-R at high magnification in Figure 9(a). The surface morphology clearly shows 
+layer-by-layer growth with squared pits. 
+Further, in attempting to reduce the local sinks, NSMO thin films were synthesized on 
+pure TiO2 terminated substrates. The substrates are treated with DI water and then annealed at 
+high temperatures according to the protocol for TiO2-termination41. The treatment produced 
+clear step and terrace characteristics in the substrate, as observed in the AFM scan shown in 
+Figure 8(b). NSMO thin films were deposited on these substrates and, subsequently, ex-situ 
+annealed. The SEM imaging revealed that they exhibited similar rod-type morphology where 
+the rods are self-aligned and crossed at right angles embedded in a matrix of NSMO with 
+rectangular features, shown in figure 8(d). This procedure was repeated on several TiO2 - 
+terminated STO substrates, and we could reproduce the same morphology. This is because the 
+complete removal of SrO assures the absence of local sinks and suppresses the island/columnar 
+growth. However, rods in the thin film are believed to arise from droplets deposited due to high 
+laser energy density (1.75 J/cm2).  This is verified in the SEM images of in-situ annealed 
+NSMO thin film shown in Figure 4(b), where the droplets are elongated into rods upon ex-situ 
+annealing.  
+Figure 9: SEM images of NSMO thin films under high magnification. a) NSMO thin film grown 
+on as-received, cleaned STO substrate deposited b) NSMO thin film grown with laser fluence 
+on TiO2 terminated STO substrate  
+Lastly, to obtain smoother films, we have synthesized NSMO thin films on fully TiO2 
+terminated STO substrate at low laser energy density (1 J/cm2), reducing droplets' density. As 
+expected, we obtained thin films with reduced density of rods with the same type of 
+morphology. The SEM image of the film is shown in Figure 9(b), free of nano-rods. The films 
+have rectangular faceted pits, and layer-by-layer growth is evident through the holes. 
+a) 
+b) 
+ 
+
+100 nm100 nm15 
+ 
+Therefore the ES barrier plays a significant role in vicinal surfaces and can result in the 
+spontaneous ordering of adatoms resulting in unique surface nanostructures. Thus we 
+emphasize that when films are grown on a commercial substrate, the resulting morphology can 
+be either granular or rod-type depending on the potential energy landscape that depends upon 
+a wide range of parameters, including the size, shape of terraces, and type of terminations 
+present at the substrate. 
+7. Electrical-transport measurements: 
+The nanostructure plays a vital role in the transport behaviour of a manganite thin film 
+system30. To understand the transport behaviour of the nanostructured NSMO thin films, the 
+resistivity measurements are carried out using the standard 4-probe geometry 42 and plotted as 
+a function of temperature in Figure 10. It is observed that the granular film NS-G has higher 
+resistivity as compared to NS-R. Both thin films, NS-G and NS-R, exhibit the insulator-to-
+metal transition (MIT), and the transition temperature TMIT is found to be 147 K for sample N-
+G and 135 K for NS-R. The transition into the metallic regime is sharper in the case of NS-R 
+compared to NS-G thin film. The electrical transport behaviour has been analysed using 
+different theoretical models and fitted in the corresponding temperature regimes. The best fit 
+in each region is chosen based on the reduced 2 value. 
+𝜌 (𝑇) = 𝜌𝑅 𝑇𝑒𝑥𝑝 (𝐸𝑎 𝜅𝐵𝑇
+⁄
+)                                             …(2)  
+                                      𝜌 (𝑇) = 𝜌0 𝑒𝑥𝑝 (𝑇0 𝑇
+⁄ )
+1 4
+⁄
+                                                …(3) 
+                                 𝐸ℎ𝑜𝑝𝑝𝑖𝑛𝑔 = 
+𝜅𝐵 𝑇𝑜
+1 4
+⁄ 𝑇
+3 4
+⁄
+4
+                                          …(4) 
+The high-temperature insulating phase is studied using the small polaron hopping 
+(SPH) model and the variable range hopping (VRH) mechanism given by equations (2) and 
+(3), and hopping energy is calculated from equation (4) 42,43. The VRH model better fits the 
+high-temperature region (≈ 195 K to 300 K) for both films. The hopping energy in the case of 
+NS-G is 128 meV and 125 meV for NS-R, in agreement with the order of value reported for 
+manganite thin films (~100 meV) 43,44. The resistivity in the metallic region below TMIT is 
+generally fitted with an empirical equation (5). At low temperatures, in addition to the 
+temperature-independent scattering effects from defects and grain boundaries (GBs) (ρo), etc., 
+
+16 
+ 
+scattering effects due to electron-electron (ρ2), electron-magnon (ρ4.5) and electron-phonon (ρP) 
+dominate along with the strong correlation effects (ρ0.5) 45. 
+A low-temperature resistive upturn is observed below 50 K in both films. In figure S3 
+of supplementary information, the resistive upturn in the low-temperature region from 4 K up 
+to 60 K is fitted using equation (6), which considers all the scattering mechanisms mentioned 
+above. An enhanced resistive upturn is observed at low temperatures in NS-G. This is due to 
+the enhanced GB-scattering effect and the contribution from other scattering mechanisms at 
+low-temperature. The contributions from different scattering mechanisms are analysed, and the 
+values are tabulated in supplementary information Table T2. 
+The intermediate temperature regime from 90 K to 134 K in the ferromagnetic-metallic 
+state is fitted using the equation (7). The addition of the polaronic term to the resistivity gives 
+a better fitting in this region as theoretical models claim the formation of polaron near the 
+MIT46.                
+𝜌 (𝑇) = 𝜌𝑜 + 𝜌𝑚𝑇𝑚                                                 (5) 
+ 
+0
+50
+100
+150
+200
+250
+300
+0
+5
+10
+15
+20
+25
+30
+0
+100
+200
+300
+0.0
+1.5
+3.0
+4.5
+0
+100
+200
+300
+0.0
+0.2
+0.4
+0.6
+46
+.0
+
+
+
+T
+
+
+Temperature (K)
+ NS-G
+ NS-R
+ Linear fit from 
+        110K to 125K- NS-G
+ Linear fit from 
+        110K to 125K- NS-R
+33
+.1
+
+
+
+T
+
+(b)
+ Resistivity - NS -G
+ VRH fit
+ FM-metallic fit
+ Low-temp uptrun
+TIMT  147 K
+(cm)
+Temperature (K)
+(a)
+(c)
+ Resistivity - NS-R
+ VRH fit
+ FM-metallic fit
+ Low-temp uptrun
+TIMT  135 K
+(cm)
+Temperature (K)
+ Figure 10: a), b) Resistivity vs. temperature curve of NSMO thin films – NS-G and NS-R 
+showing insulator to metal transition with decreasing temperature and fitted according to 
+theoretical models in different temperature regimes c) Normalized resistivity plot of NS-G and 
+NS-R thin film. Inset: Plot of variation of TCR with respect to temperature. 
+
+17 
+ 
+An interesting feature is observed in the resistivity plots of the NS-G and NS-R thin 
+films apart from the low-temperature resistive upturn. In Figure 10(c), the resistivity of both 
+the thin films (NS-G and NS-R) has been normalized with their resistivity at 300 K, and a linear 
+fitting in the metallic region below TMIT (110 K to 125 K) is carried out to determine the slope. 
+The resistivity slope of samples with rod morphology differs from samples with granular 
+morphology up to an order of magnitude. The increase in slope value below the transition 
+temperature indicates the sharpness of the resistive transition for the samples with rod 
+morphology. This characteristic increase in slope up to an order is evident in all our thin films 
+with rod-type morphology (see supplementary figure S2). To characterize the sensitivity of 
+resistance with respect to changes in temperature, the temperature-coefficient of resistance 
+(TCR) has been evaluated using equation (7). It was found that NS-R has a higher value of 
+TCR %, ~ 12 %, compared to NS-G with TCR %, ~ 7 %.  Additionally, the samples with rod 
+morphology were found to have enhanced TCR% (supplementary information figure S2). To 
+comprehend this result, we discuss the effect of GBs on the conduction mechanism. 
+The manganite system undergoes a disorder-induced phase transition from PM to FM 
+state with decreasing temperature21. Due to phase co-existence during the transition, the 
+conduction channel is presumed to have filamentary FM paths in the PM matrix 47. Conduction 
+takes place through the percolation of current across the well-connected FM regions. In 
+addition to the FM filamentary path, the GBs also play a significant role in the conduction 
+mechanism. We refer to Verutruyen et al.’s 48  work which explores the effect of a single GB 
+in the La-Ca-Mn-O (LCMO) system. They showed that the resistivity falls sharply at the 
+transition temperature when measured on a single grain of LCMO (free of GBs). However, 
+when measured across a single GB, the resistivity initially decreased, followed by a broad 
+resistive feature near the transition temperature. Thus, in a granular system, though the 
+conduction takes place through the percolation paths of well-connected FM regions, the GBs 
+cause increased resistivity due to increased spin-dependent scattering across the GB47. The 
+above explanation is consistent with our results, where the thin film with granular morphology 
+(NS-G) shows a broad resistive transition below the transition temperature with reduced TCR 
+𝜌 (𝑇) = 𝜌𝑜 + 𝜌2𝑇2 + 𝜌4.5𝑇4.5 + 𝜌𝑃𝑇5 + 𝜌0.5𝑇0.5                      (6) 
+𝜌 (𝑇) = 𝜌𝑜 + 𝜌2𝑇2 + 𝜌4.5𝑇4.5 + 𝜌𝑃𝑇5 + 𝜌0.5𝑇0.5 + 𝜌7.5𝑇7.5               (7) 
+𝑇𝐶𝑅 % = 
+1
+𝜌 (
+𝑑𝜌
+𝑑𝑇) x 100                                                    (8) 
+
+18 
+ 
+%. If the connectivity is enhanced between the grains, a sharper decrease in the resistivity can 
+occur in the metallic regime. Remarkably, we observe that all of our thin films with rod-
+morphology show sharp resistive transition near MIT irrespective of the thickness of the film. 
+Thus, this nanostructure aids improved conduction in the FM metallic phase, leading to the 
+sharp resistive transition with enhanced TCR % comparable to that of a highly-crystalline 
+system. Attempts to enhance the TCR % have been carried out by doping with elements such 
+as Ag, as high TCR % is required for applications in sensors and infrared detectors49,50. These 
+elements precipitate as nanocomposite in the manganite system and improve the conductivity, 
+leading to a sharper resistive transition. However, in our study, we have substantiated that the 
+enhancement of TCR % is possible with proper tuning of the nanostructured morphology of 
+thin films. 
+CONCLUSION: 
+In conclusion, the PLD-grown NSMO thin films were observed to have two prominent 
+surface morphologies – granular and crossed-nano rods. The metal-to-insulator transition 
+(MIT) temperature, TMIT, was found to be 147 K for a granular NSMO (NS-G) thin film and 
+135 K for a thin film with crossed-rod morphology (NS-R). The nature of the resistive 
+transition is broad in the former, whereas the latter exhibits a sharp MIT feature. The 
+temperature coefficient of resistance (TCR) was evaluated, and NS-R thin film has a higher 
+value of TCR %, ~ 12 %, compared to NS-G with TCR % ~ 7 %. Additionally, we have 
+observed that all the films with rod-type morphology exhibit a significant enhancement in 
+TCR% up to one order of magnitude compared to the granular thin film. Thus, we have 
+demonstrated that TCR % can be enhanced with proper tuning of the nanostructures in thin 
+films, which is relevant for technological applications. The reason for such nano-structuring is 
+explored in great detail. It was found that parameters like laser energy density, O2 partial 
+pressure, and the substrate miscut angle had minimal effect. At the same time, the difference 
+in the potential landscape of the Ehrlich-Schwoebel (ES) barrier is believed to play a vital role 
+in the growth dynamics of the films. Films grown with reduced laser energy density (1 J/cm2) 
+on the TiO2 terminated substrates exhibited highly reproducible layer-by-layer growth. This 
+substantiates the presence of reduced local sinks and ES barrier height, resulting in epitaxial 
+growth of NSMO thin films. Therefore, a fine-tuning of a wide range of parameters, including 
+strain and surface terminations, is required to obtain a fine control of the ES barrier that 
+influences the growth process of thin films. This paves the way for investigation into the role 
+of the ES barrier in manganite thin film growth. Using RHEED and in-situ STM techniques, a 
+
+19 
+ 
+few groups have already attempted to experimentally determine the value of the ES barrier on 
+SrTiO3 substrates for the growth of La-Ca-Mn-O manganite system51. It would be interesting 
+to explore the relationship between the value of the ES-barrier and the type of morphology 
+experimentally in the future. 
+ 
+Author contributions 
+The division of work is as follows: NSMO thin film samples were prepared by R.S.M. SEM 
+imaging was carried out by S.A, J.P. AFM measurements were carried out by K.G. XRD 
+measurements were carried out by R.M.S, P.N.R, PG, S.K.R. Magneto-transport measurements 
+were carried out by R.S.M and E.P.A. Analysis were done by R.S.M, E.P.A, and S.A. Writing 
+was carried out by R.S.M, and all authors discussed the results and commented on the 
+manuscript. E.P.A., T.G.K and A.M. supervised this research work. 
+ 
+Conflict of interest:  
+The authors declare no conflict of interest. 
+Acknowledgments 
+One of the authors (R S Mrinaleni) would like to acknowledge the Department of Atomic 
+Energy, India for the provision of experimental facilities. We thank UGC-DAE CSR, 
+Kalpakkam node, for providing access to magnetic and magnetotransport measurement 
+systems. The authors are grateful to RRCAT, Indore, for beam line facilities.   
+Funding statement:  
+One of the authors (R S Mrinaleni) would like to acknowledge the funding support from the 
+Department of Atomic Energy, India. 
+ 
+References: 
+1. 
+E. Dagotto. Nanoscale phase seperation and CMR. 
+2. 
+Tokura, Y. Critical features of colossal magnetoresistive manganites. Reports Prog. 
+Phys. 69, 797–851 (2006). 
+3. 
+Ebata, K. et al. Chemical potential shift induced by double-exchange and polaronic 
+effects in Nd1-x Srx Mn O3. Phys. Rev. B - Condens. Matter Mater. Phys. 77, (2008). 
+
+20 
+ 
+4. 
+Haghiri-Gosnet, A. M. & Renard, J. P. CMR manganites: Physics, thin films and 
+devices. J. Phys. D. Appl. Phys. 36, (2003). 
+5. 
+Tokura, Y. & Tomioka, Y. Colossal magnetoresistive manganites. J. Magn. Magn. 
+Mater. 200, 1–23 (1999). 
+6. 
+Perna, P. et al. Tailoring magnetic anisotropy in epitaxial half metallic 
+La0.7Sr0.3MnO3 thin films. J. Appl. Phys. 110, 013919 (2011). 
+7. 
+Song, C. et al. Emergent perpendicular magnetic anisotropy at the interface of an oxide 
+heterostructure. Phys. Rev. B 104, (2021). 
+8. 
+Li, X., Lindfors-Vrejoiu, I., Ziese, M., Gloter, A. & van Aken, P. A. Impact of 
+interfacial coupling of oxygen octahedra on ferromagnetic order in 
+La0.7Sr0.3MnO3/SrTiO3 heterostructures. Sci. Rep. 7, 40068 (2017). 
+9. 
+Liu, Q. et al. Perpendicular Manganite Magnetic Tunnel Junctions Induced by 
+Interfacial Coupling. ACS Appl. Mater. Interfaces 14, 13883–13890 (2022). 
+10. 
+Liu, Q. et al. Perpendicular Manganite Magnetic Tunnel Junctions Induced by 
+Interfacial Coupling. ACS Appl. Mater. Interfaces 14, 13883–13890 (2022). 
+11. 
+Chi, X. et al. Enhanced Tunneling Magnetoresistance Effect via Ferroelectric Control 
+of Interface Electronic/Magnetic Reconstructions. ACS Appl. Mater. Interfaces 13, 
+56638–56644 (2021). 
+12. 
+Gajek, M. et al. Tunnel junctions with multiferroic barriers. Nat. Mater. 6, 296–302 
+(2007). 
+13. 
+Kim, D. H., Ning, S. & Ross, C. A. Self-assembled multiferroic perovskite–spinel 
+nanocomposite thin films: epitaxial growth, templating and integration on silicon. J. 
+Mater. Chem. C 7, 9128–9148 (2019). 
+14. 
+Sánchez, F., Ocal, C. & Fontcuberta, J. Tailored surfaces of perovskite oxide 
+substrates for conducted growth of thin films. Chemical Society Reviews vol. 43 2272–
+2285 (2014). 
+15. 
+Ning, X., Wang, Z. & Zhang, Z. Large, Temperature-Tunable Low-Field 
+Magnetoresistance in La0.7Sr0.3MnO3:NiO Nanocomposite Films Modulated by 
+Microstructures. Adv. Funct. Mater. 24, 5393–5401 (2014). 
+
+21 
+ 
+16. 
+Zhang, W., Ramesh, R., MacManus-Driscoll, J. L. & Wang, H. Multifunctional, self-
+assembled oxide nanocomposite thin films and devices. MRS Bull. 40, 736–745 
+(2015). 
+17. 
+Chen, A., Bi, Z., Jia, Q., MacManus-Driscoll, J. L. & Wang, H. Microstructure, 
+vertical strain control and tunable functionalities in self-assembled, vertically aligned 
+nanocomposite thin films. Acta Mater. 61, 2783–2792 (2013). 
+18. 
+Zhang, C. et al. Large Low-Field Magnetoresistance (LFMR) Effect in Free-Standing 
+La0.7Sr0.3MnO3 Films. ACS Appl. Mater. Interfaces 13, 28442–28450 (2021). 
+19. 
+Huang, J. et al. Exchange Bias in a La0.67Sr0.33MnO3/NiO Heterointerface 
+Integrated on a Flexible Mica Substrate. ACS Appl. Mater. Interfaces 12, 39920–39925 
+(2020). 
+20. 
+Qin, Q. et al. Interfacial antiferromagnetic coupling between SrRu O3 and L a0.7 S 
+r0.3Mn O3 with orthogonal easy axis. Phys. Rev. Mater. 2, 104405 (2018). 
+21. 
+Dagotto, E., Hotta, T. & Moreo, A. Colossal magnetoresistant materials: the key role 
+of phase separation. Phys. Rep. 344, 1–153 (2001). 
+22. 
+Krivoruchko, V. N. The Griffiths phase and the metal-insulator transition in substituted 
+manganites (Review Article). Low Temperature Physics vol. 40 586–599 (2014). 
+23. 
+Bhat, S. G. & Kumar, P. S. A. Tuning the Curie temperature of epitaxial 
+Nd0.6Sr0.4MnO3 thin films. J. Magn. Magn. Mater. 448, 378–386 (2018). 
+24. 
+Kumari, S. et al. Effects of Oxygen Modification on the Structural and Magnetic 
+Properties of Highly Epitaxial La0.7Sr0.3MnO3 (LSMO) thin films. Sci. Rep. 10, 
+(2020). 
+25. 
+Wang, H. S., Li, Q., Liu, K. & Chien, C. L. Low-field magnetoresistance anisotropy in 
+ultrathin Pr0.67Sr0.33MnO3 films grown on different substrates. Appl. Phys. Lett. 74, 
+2212–2214 (1999). 
+26. 
+Huang, J., Wang, H., Sun, X., Zhang, X. & Wang, H. Multifunctional La 0.67 Sr 0.33 
+MnO 3 (LSMO) Thin Films Integrated on Mica Substrates toward Flexible Spintronics 
+and Electronics. ACS Appl. Mater. Interfaces 10, 42698–42705 (2018). 
+27. 
+Boileau, A. et al. Textured Manganite Films Anywhere. ACS Appl. Mater. Interfaces 
+
+22 
+ 
+11, 37302–37312 (2019). 
+28. 
+Greculeasa, S. G. et al. Influence of Thickness on the Magnetic and Magnetotransport 
+Properties of Epitaxial La0.7Sr0.3MnO3 Films Deposited on STO (0 0 1). 
+Nanomaterials  vol. 11 (2021). 
+29. 
+Gupta, P. et al. BL-02: A versatile X-ray scattering and diffraction beamline for 
+engineering applications at Indus-2 synchrotron source. J. Synchrotron Radiat. 28, 
+1193–1201 (2021). 
+30. 
+Arun, B., Suneesh, M. V. & Vasundhara, M. Comparative Study of Magnetic Ordering 
+and Electrical Transport in Bulk and Nano-Grained Nd0.67Sr0.33MnO3 Manganites. 
+J. Magn. Magn. Mater. 418, 265–272 (2016). 
+31. 
+Zhang, B. et al. Effects of strain relaxation in Pr0.67Sr0.33MnO3 films probed by 
+polarization dependent X-ray absorption near edge structure. Sci. Rep. 6, 19886 
+(2016). 
+32. 
+Pai, Y. Y., Tylan-Tyler, A., Irvin, P. & Levy, J. Physics of SrTiO3-based 
+heterostructures and nanostructures: A review. Reports on Progress in Physics vol. 81 
+036503 (2018). 
+33. 
+Paudel, B. et al. Anisotropic domains and antiferrodistortive-transition controlled 
+magnetization in epitaxial manganite films on vicinal SrTiO3 substrates. Appl. Phys. 
+Lett. 117, (2020). 
+34. 
+Boschker, J. E. et al. In-plane structural order of domain engineered La0.7Sr 0.3MnO3 
+thin films. Philos. Mag. 93, 1549–1562 (2013). 
+35. 
+Konstantinović, Z., Sandiumenge, F., Santiso, J., Balcells, L. & Martínez, B. Self-
+assembled pit arrays as templates for the integration of Au nanocrystals in oxide 
+surfaces. Nanoscale 5, 1001–1008 (2013). 
+36. 
+Wang, J. et al. Quick determination of included angles distribution for miscut 
+substrate. Meas. J. Int. Meas. Confed. 89, 300–304 (2016). 
+37. 
+Scheel, H. J. Control of Epitaxial Growth Modes for High‐Performance Devices. 
+Cryst. growth Technol. 621–644 (2003). 
+38. 
+Chae, R. H., Rao, R. A., Gan, Q. & Eom, C. B. Initial Stage Nucleation and Growth of 
+
+23 
+ 
+Epitaxial SrRuO3 Thin Films on (0 0 1) SrTiO3 Substrates. J. Electroceramics 4, 345–
+349 (2000). 
+39. 
+Schwoebel, R. L. & Shipsey, E. J. Step Motion on Crystal Surfaces. J. Appl. Phys. 37, 
+3682–3686 (1966). 
+40. 
+Załuska-Kotur, Magdalena, Hristina Popova,  and V. T. Step Bunches, Nanowires and 
+Other Vicinal “Creatures”—Ehrlich–Schwoebel Effect by Cellular Automata. Crystals 
+11, 1135 (2021). 
+41. 
+Connell, J. G., Isaac, B. J., Ekanayake, G. B., Strachan, D. R. & Seo, S. S. A. 
+Preparation of atomically flat SrTiO3 surfaces using a deionized-water leaching and 
+thermal annealing procedure. Appl. Phys. Lett. 101, (2012). 
+42. 
+Miccoli, I., Edler, F., Pfnür, H. & Tegenkamp, C. The 100th anniversary of the four-
+point probe technique: the role of probe geometries in isotropic and anisotropic 
+systems. J. Phys. Condens. Matter 27, 223201 (2015). 
+43. 
+Gopalarao, T. R., Ravi, S. & Pamu, D. Electrical transport and magnetic properties of 
+epitaxial Nd0.7Sr0.3MnO3 thin films on (001)-oriented LaAlO3 substrate. J. Magn. 
+Magn. Mater. 409, 148–154 (2016). 
+44. 
+Gopalarao, T. R. & Ravi, S. Study of Electrical Transport and Magnetic Properties of 
+Nd0.7Sr0.3MnO3/Nd0.8Na0.2MnO3 Bilayer Thin Films. J. Supercond. Nov. Magn. 
+31, 1149–1154 (2018). 
+45. 
+Arun, B., Suneesh, M. V & Vasundhara, M. Comparative Study of Magnetic Ordering 
+and Electrical Transport in Bulk and Nano-Grained Nd0.67Sr0.33MnO3 Manganites. 
+J. Magn. Magn. Mater. 418, 265–272 (2016). 
+46. 
+Sudakshina, B., Supin, K. K. & Vasundhara, M. Effects of Nd-deficiency in 
+Nd0.67Ba0.33MnO3 manganites on structural, magnetic and electrical transport 
+properties. J. Magn. Magn. Mater. 542, 168595 (2022). 
+47. 
+de Andrés, A., García-Hernández, M. & Martínez, J. L. Conduction channels and 
+magnetoresistance in polycrystalline manganites. Phys. Rev. B 60, 7328–7334 (1999). 
+48. 
+Vertruyen, B. et al. Magnetotransport properties of a single grain boundary in a bulk 
+La-Ca-Mn-O material. J. Appl. Phys. 90, 5692–5697 (2001). 
+
+24 
+ 
+49. 
+Li, J. et al. Improvement of electrical and magnetic properties in 
+La0.67Ca0.33Mn0.97Co0.03O3 ceramic by Ag doping. Ceram. Int. (2022) 
+doi:10.1016/j.ceramint.2022.08.255. 
+50. 
+Jin, F. et al. La0.7Ca0.3MnO3-δ:Ag nanocomposite thin films with large temperature 
+coefficient of resistance (TCR). J. Mater. (2022) doi:10.1016/j.jmat.2022.01.010. 
+51. 
+Gianfrancesco, A. G., Tselev, A., Baddorf, A. P., Kalinin, S. V & Vasudevan, R. K. 
+The Ehrlich–Schwoebel barrier on an oxide surface: a combined Monte-Carlo and in 
+situ scanning tunneling microscopy approach. Nanotechnology 26, 455705 (2015). 
+ 
+SUPPLEMENTARY INFORMATION 
+ 
+I- 
+Deconvolution of NSMO (004) reflection: 
+ 
+Figure S1: The double peak in the HR-XRD scan of NS-R thin film is confirmed by a HR-fine 
+scan. The individual peak positions are noted as the centre of the fitted peaks P1 and P2.   
+Table ST1: The table illustrates the values of c-lattice parameters evaluated from the (004) NSMO 
+reflection. 
+24.4
+24.6
+24.8
+25.0
+25.2
+25.4
+25.6
+25.8
+0.0
+5.0x10
+4
+1.0x10
+5
+1.5x10
+5
+2.0x10
+5
+Intensity (arb. units)
+2(deg)
+ NS-R - fine scan
+ Fit Peak 1
+ Fit Peak 2
+ Cumulative Fit Peak
+Sample 
+2θ for (004) reflection  
+(o deg) 
+Calculated c-lattice 
+parameter (Å) 
+NS-G 
+25.05o 
+7.61 
+NS-R 
+24.88 o – P1  
+24.97 o – P2 
+7.66 
+7.63 
+
+25 
+ 
+II- 
+Transport studies on NSMO thin films 
+Three samples with granular morphology G-A, G-B, G-C, and rod morphology R-A, R-B, R-
+C, were selected and their resistivity was measured using 4-probe technique. The normalized-
+resistivity plot for the selected NSMO thin films are shown Figure. 4. The value of resistivity 
+is different across the NSMO thin films, since they are deposited under slightly different PLD 
+conditions but all of them exhibited MIT. Observing the nature of MIT transition in these 
+selected samples, G-A, G-B, G-C with granular morphology have a broad resistive transition 
+below their MIT temperature. The samples R-A, R-B, R-C with rod-morphology show a sharp 
+resistive transition in the FM-metallic state below their MIT temperature. The value of slope is 
+evaluated from the linear fit in the metallic region and it shows that samples with rod-type 
+morphology have increased slope up to one order as compared to the granular samples.  
+Temperature coefficient of resistance (TCR) is evaluated for these films and it is found that 
+samples G-A, G-B, G-C have peak-TCR % of 5 %, 4 %, and 8 % at 105 K, 77 K, and 121 K, 
+respectively. An enhanced TCRpeak % is obtained for samples with rod-morphology. The 
+samples R-A, R-B, R-C have peak-TCR % of 21 %, 14.5 %, and 18 % at 98 K, 80 K, and 100 
+K, respectively. 
+
+26 
+ 
+ 
+Figure S2: Plots of normalized resistivity vs. temperature of NSMO films with granular and 
+rod-type morphology. (a),(c),(e): Samples with granular morphology G-A, G-B, G-C. 
+(b),(d),(f): Samples with rod morphology R-A, R-B, R-C. A linear fit in the FM-metallic region 
+give the rate of change of resistivity with respect to temperature.  
+ 
+III- Low-temperature studies on NSMO thin films – NS-G 
+and NS-R 
+ 
+To study the low-temperature transport across the thin films with different morphology, the plot of low-
+temperature resistivity of the granular thin film NS-G and rod-type thin film NS-R is shown in figure 
+S5.  An enhanced low-temperature resistive upturn is observed in NS-G from figure. S5. Using the low-
+temperature transport equation the resistivity data is fit and the fitting parameters are summarized in 
+table ST2. The first term, ρo which represents the contribution from grain-boundary (GB) scattering is 
+0
+50
+100
+150
+200
+250
+300
+0
+2
+4
+6
+8
+0
+50
+100
+150
+200
+250
+300
+0
+30
+60
+90
+0
+50
+100
+150
+200
+250
+300
+0
+2
+4
+6
+8
+10
+12
+14
+16
+0
+50
+100
+150
+200
+250
+300
+0
+30
+60
+90
+120
+0
+50
+100
+150
+200
+250
+300
+0
+2
+4
+6
+8
+10
+12
+14
+0
+50
+100
+150
+200
+250
+300
+0
+5
+10
+15
+20
+25
+15
+.0
+
+
+
+T
+
+ Sample: G - A
+ Linear fit in 
+         FM-metallic region
+
+Temperature (K)
+TCRpeak %  5 %  
+
+TCRpeak %  21 %  
+74
+.5
+
+
+
+T
+
+ Sample: R - A
+ Linear fit in 
+         FM-metallic region
+
+Temperature (K)
+(b)
+TCRpeak %  4 %  
+0.28
+
+
+
+T
+
+
+Temperature (K)
+ Sample: G - B
+ Linear fit in 
+        FM-metallic region
+(c)
+TCRpeak %  14 %  
+76
+.6
+
+
+
+T
+
+
+Temperature (K)
+ Sample: R - B
+ Linear fit in 
+         FM-metallic region
+(d)
+TCRpeak %  8 %  
+0.35
+
+
+
+T
+
+ Sample: G - C
+ Linear fit in 
+         FM-metallic region
+
+Temperature (K)
+(e)
+TCRpeak %  18 %  
+20
+.1
+
+
+
+T
+
+
+Temperature (K)
+ Sample: R - C
+ Linear fit in 
+         FM-metallic region
+(a)
+(f)
+
+27 
+ 
+found to be higher by more than one-order in NS-G as compared to NS-R. This is expected as NS-G 
+has a granular morphology and increased contribution from GB scattering affects the transport 
+mechanism even at low-temperatures.   Additionally, ρo’s value is higher by orders of magnitude as 
+compared to the other coefficients. This shows that GB scattering effects dominate the transport 
+mechanism compared to other contributions to the electronic transport.  
+ 
+ 
+Figure S3: Low-temperature resistive up-turn is observed in the NSMO thin films NS-G and 
+NS-R. The temperature regime from 4 K up to 60K is fit using the low-temperature transport 
+equation.  
+  
+ 
+ 
+0
+10
+20
+30
+40
+50
+0.07
+0.08
+0.09
+0.10
+0
+10
+20
+30
+40
+50
+0.0020
+0.0025
+0.0030
+0.0035
+0.0040
+ Resistivity - NS-G
+ Low-temp uptrun
+TIMT  147 K
+(cm)
+Temperature (K)
+TIMT  135 K
+ Resistivity - NS-R
+ Low-temp uptrun
+(cm)
+Temperature (K)
+Sample 
+𝝆𝒐 
+𝝆𝟐 
+𝝆𝟒.𝟓 
+𝝆𝑷 
+𝝆𝟎.𝟓 
+R2 (%) 
+NS-G 
+0.09272 
+1.16E-5 
+-1.00E-9 
+1.33E-10 
+-0.0038 
+99.99 
+NS-R 
+0.00305 
+3.53E-7 
+-2.90E-11 
+4.90E-12 
+-8.38E-5 
+99.99 
+Table ST2: The table illustrates the values of coefficients of low-temperature transport after 
+fitting. 
+

9tE0T4oBgHgl3EQffwDm/content/tmp_files/2301.02410v1.pdf.txt

@@ -0,0 +1,1798 @@
+Codepod: A Namespace-Aware, Hierarchical Jupyter
+for Interactive Development at Scale
+Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian
+Dept. of Computer Science, Iowa State University
+Ames, Iowa, USA
+{hebi,qxiao,jtian}@iastate.edu,[email protected]
+ABSTRACT
+Jupyter is a browser-based interactive development environment
+that has been popular recently. Jupyter models programs in code
+blocks, and makes it easy to develop code blocks interactively by
+running the code blocks and attaching rich media output. How-
+ever, Jupyter provides no support for module systems and names-
+paces. Code blocks are linear and live in the global namespace;
+therefore, it is hard to develop large projects that require modular-
+ization in Jupyter. As a result, large-code projects are still devel-
+oped in traditional text files, and Jupyter is only used as a surface
+presentation. We present Codepod, a namespace-aware Jupyter
+that is suitable for interactive development at scale. Instead of
+linear code blocks, Codepod models code blocks as hierarchical
+code pods, and provides a simple yet powerful module system for
+namespace-aware incremental evaluation. Codepod is open source
+at https://github.com/codepod-io/codepod.
+ACM Reference Format:
+Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian. 2023. Codepod: A Namespace-
+Aware, Hierarchical Jupyter for Interactive Development at Scale. In Pro-
+ceedings of (Conference’23). ACM, New York, NY, USA, 10 pages. https:
+//doi.org/10.1145/nnnnnnn.nnnnnnn
+1
+INTRODUCTION
+Traditional software development is typically closely tied with file
+systems. Developers write code into a set of files in the file-system
+hierarchy. For example, developers write functions in files using
+a text editor and invoke a compiler or an interpreter to run or
+evaluate the code in the files. Modern Integrated Development
+Environments (IDEs) provide a file system browser and integrate
+debuggers to help run and debug over the files.
+Jupyter notebook [6] is a browser-based interactive development
+environment that has been widely adopted by many different com-
+munities, both in science and industry. Jupyter notebooks support
+literate programming that combines code, text, and execution re-
+sults with rich media visualizations. Juyter models the code as a
+sequence of "code cells". This provides a clean separation between
+code blocks, whereas text editors do not partition code in the same
+text file but instead relying on developers and editor plugins to do
+so. Code cells can be interactively (re)-run and display results in rich
+media such as data visualization right beside the cell, providing de-
+velopers an interactive Read-Eval-Print-Loop (REPL) development
+experience. Jupyter has been popular recently in software devel-
+opment [10–12, 15], proving such interactive cycle is beneficial to
+software development.
+Conference’23, Jan, 2023, Ames, IA, USA
+2023. ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+However, Jupyter falls short for module systems and namespaces.
+Code blocks in Jupyter notebooks are linear and live in the global
+namespace, making it non-scalable for large software projects of
+hundreds of function definitions with potential naming conflicts.
+As a result, large code projects are still developed in traditional
+text files, and Jupyter is primarily used as a surface presentation
+of the projects, consisting of only a fraction of the entire codebase.
+Our case study in Section 4.3 found that Jupyter notebook shares
+less than 5% of the code of real-world open-source projects. All
+functions defined in the Jupyter notebook are only accessed in the
+same notebook. There are calls from Jupyter to the code in the text
+files, but no calls from text files to Jupyter code.
+The Jupyter-file hybrid development model has several disadvan-
+tages. Changes in files are not in sync with the Jupyter runtime. This
+effectively breaks the REPL interactive development functionality.
+The hybrid model still relies on text editors, external debuggers,
+and IDEs and thus still suffers from the drawbacks of file-based
+software development, which we will detail below.
+Although computers store information into files, organizing code
+into text files where information is linearly presented is counter-
+productive. Complex software requires proper abstraction and seg-
+mentation of code, typically by defining functions and hierarchical
+modules. For simplicity, in this paper, we assume functions are the
+building blocks of software projects and refer to the functions when
+we talk about “code blocks”. File-based approaches force developers
+to maintain the correspondence between code and files, which differ
+significantly in granularity: code blocks are small in size, but large
+in amount, while files are typically long but few. The unbalance
+in granularity poses dilemmas to developers: including too many
+code blocks into one file makes the hierarchy hard to maintain,
+while including few code blocks into one file creates many small
+files and deep directories that are also hard to work with. Besides,
+programming languages typically design module systems around
+file systems, e.g., a file is a module. It becomes tedious to reference
+and import from different modules scattered over multiple files
+and levels of directories. This is the case in the real world. Among
+highly regarded open source projects, each project contains tens to
+hundreds of files, possibly with levels of different directories. For
+a file containing tens of functions, about half of the functions are
+internal to the file and are not called in other files.
+To overcome the above disadvantages of both Jupyter and text-
+file-based development, we propose Codepod, a namespace-aware
+Jupyter for interactive software development at scale. Codepod
+models a program as hierarchical code blocks and represents it
+accordingly. Developers write each function as a code pod and
+place it at an appropriate hierarchy. In Codepod, the code blocks are
+organized into a tree of code pods and decks. A code pod resembles
+arXiv:2301.02410v1  [cs.SE]  6 Jan 2023
+
+Conference’23, Jan, 2023, Ames, IA, USA
+Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian
+a cell in Jupyter. The partition of pods is maintained by grouping
+them into decks. A deck can also contain child decks. All code of
+the entire project can be developed without needing files.
+In addition, Codepod features a simple yet powerful module sys-
+tem that abstracts over the native module system of programming
+languages to provide a consistent and straightforward evaluation
+model for different languages. Codepod’s module system consists
+of five namespace rules, inspired by the hierarchical nature of code
+blocks and the access pattern among them. (1) namespace separa-
+tion by default: in Codepod, each deck is a namespace, and the root
+deck is the global namespace. Pods in different decks are defined in
+separate namespaces and cannot see each other; (2) public pods: a
+pod can be marked as "public" and is made available to its parent
+deck; (3) utility pods: A pod or deck in Codepod can be marked
+as a “utility pod/deck”. Such a pod is visible to the parent deck
+node’s sub-tree; (4) testing pods: a testing pod or deck can access
+its parent deck’s namespace; and (5) explicit path: a pod is always
+accessible by specifying the full path within the tree, providing the
+compatibility for arbitrary imports. The detailed rationale of the
+rules is discussed in Section 2.
+Last but not least, Codepod provides a namespace-aware incre-
+mental evaluation model. In Codepod, every pod can be executed,
+and the evaluation happens in the appropriate namespace. Similar
+to Jupyter notebooks, the results are displayed right beside the
+code pod for easy debugging and intuitive interactive development.
+When a pod is changed, the pod can be re-evaluated, and the up-
+dated definition is applied incrementally in that scope in the active
+runtime, and the new definition is visible to all other pods using it
+in the entire codebase without restarting the current runtime.
+We have implemented a fully working Codepod as a Web ap-
+plication and currently have implemented full namespace-aware
+runtime support for four language kernels: Python, JavaScript, Julia,
+and Scheme/Racket. New kernels can be easily developed based
+on existing Jupyter notebook kernels. Codepod is open-sourced at
+https://example.com
+In summary, we make the following contributions in this work:
+• we propose Codepod, a novel namespace-aware interactive
+development environment
+• we propose a simple yet powerful module system abstraction
+for Codepod
+• we provide a fully working Codepod implementation with
+namespace-awareness and incremental runtime support for
+four programming languages, and make it open source
+• we conduct case studies of real-world open-source projects
+to statistically show that our Codepod model will be useful
+for real-world development.
+2
+HIERARCHICAL PODS
+In this section, we introduce the Codepod model and its namespace
+rules. In the next section, we describe the incremental evaluation
+runtime and algorithms.
+2.1
+Codepod Interface
+In Codepod, code blocks are organized into a tree. In the tree, non-
+leaf nodes are decks, and leaf nodes are pods. Thus a deck may
+contain a list of pods and a list of sub-decks. A pod is a text editor
+containing the real code, and a deck is the container of the pods.
+We will use “node” to refer to a node in the tree, which can be either
+a deck or a pod.
+An overview demo of the Codepod interface is shown in Figure
+1, implementing a simplified Python regular expression compiler.
+The code is organized into a tree, which starts from the leftmost
+ROOT node, and grows to the right. The background level of grey
+of the deck indicates the level of the deck in the tree.
+In order to define the interactions between code pods in the tree,
+Codepod provides simple yet powerful namespace rules abstracting
+different languages’ native module systems and providing a consis-
+tent module system for all languages. In the following sections, we
+introduce the rules in detail. We will revisit this overview exam-
+ple in Section 2.7 for the meaning of different kinds of pods after
+introducing the namespace rules.
+A typical workflow using Codepod starts from an empty tree
+of a single ROOT deck. Developers can create pods as a child of
+the deck and start to develop in the global namespace. To develop
+hierarchical modules, developers can create a deck under the ROOT
+deck and create pods under the new deck. Pods and decks can be
+moved from one node to another node in different levels to group
+the pods and re-order the code hierarchy. Pods and decks can be
+folded so that only the pods of interest are displayed during the
+development. A pod can be evaluated, and the possibly rich media
+result will be displayed under the pod.
+2.2
+NS Rule 1: Namespace Separation
+In Codepod, the code blocks are organized into a tree of decks and
+pods. Each deck can contain multiple pods and child decks. A pod
+contains the actual code, and a deck declares a namespace. A pod
+belongs to the namespace of its parent deck. The first rule is the
+basic namespace separation: pods in the same namespace are visible
+to each other, but pods in different namespaces are not. For example,
+in Fig. 2, there are 5 decks, and thus 5 namespaces. In Deck-2, there
+are two pods defining functions a and b. Functions a and b can call
+each other without a problem because they are in the same deck
+and thus the same namespace. In all other four decks, the reference
+to either a or b will throw errors because they belong to different
+namespaces.
+2.3
+NS Rule 2: Public Interface to Parent
+In order to build up the software, we have to establish connections
+between the definitions of code pods in different namespaces. Soft-
+ware programs are often highly hierarchical: lower-level functions
+are composed together to build higher-level functions. This is a
+natural fit to the Codepod model, where code blocks are ordered
+hierarchically. Thus in this rule, we allow public interfaces to be
+exposed from child decks to parent decks. More specifically, each
+pod can be marked as “public”. Such public pods are visible in the
+parent deck of the current deck. For example, in Fig. 3, there are
+3 decks, thus 3 namespaces. The three namespaces are composed
+hierarchically; Deck-A is the parent of Deck-B, which is the parent
+of Deck-C. In Deck-C, there are four pods, defining four functions
+c1, c2, c3, and c4. Those functions can see each other because they
+are in the same namespace. The pods for c1, c2, and c4 are marked
+public (indicated by highlight), while c3 is not. In its parent deck
+
+Codepod: A Namespace-Aware, Hierarchical Jupyter
+for Interactive Development at Scale
+Conference’23, Jan, 2023, Ames, IA, USA
+Figure 1: Codepod overview example for Regular Expression code.
+Figure 2: NS Rule 1: separate namespace by default
+Figure 3: NS Rule 2: export to parent namespace. Yellow
+highlights indicate pods to be exported/exposed to parent
+decks.
+(Deck-B), the call of c1, c2, c4 is allowed, meaning that they are
+available in this parent namespace. However, the usage of c3 will
+raise an error because it is not exposed.
+The public functions are exported only to the parent deck but
+not to the child decks. For example, function b1 is defined in Deck-
+B, and the pod is marked public. This function b1 is visible to its
+parent deck, Deck-A, but not to its child deck, Deck-C.
+Lastly, the public interface is only exposed to one level up the
+hierarchy. If the names are desired to be visible further up, the
+names can be further exposed up to the root deck. For example,
+although c4 is marked as public, it is only visible to its immediate
+parent deck, Deck-B. Calling c4 in the for pod for a2 in Deck-A will
+raise an error as c4 is not visible in Deck-A. In the middle deck,
+Deck-B, the functions c1 and c2 are re-exported to the parent deck,
+and thus c1 and c2 are available in the top deck, Deck-A.
+In summary, this “up-rule” allows users to mark a pod public
+and expose it to one-level deck up, and can be re-exported to upper
+levels explicitly until the root pod. This namespace rule closely
+resembles the hierarchical nature of software and is natural to use
+this to build up complex functionalities from the ground up.
+2.4
+NS Rule 3: Utility Pods
+Although exposing pods from child decks to parent decks is natural
+for building software, it cannot cover all use-cases. One particular
+access pattern is utility functions that are supposed to be called
+in many other pods at different levels. This is commonly used in
+real-world software. For example, many software projects will have
+a utils folder that implements utility functions such as string
+manipulation, general parsing, logging functions. Such utility func-
+tions are used by other functions at different hierarchy levels. In
+the Codepod hierarchy, pods for such functions need to be children
+for all other pods calling the utility functions; thus, the model will
+no longer be a tree but a graph. However, modeling code blocks as
+graphs is not as scalable as trees, and too many utility pods will
+
+CPiwTe3yqmmC
+sre parse
+Pattern
+parse
+class Pattern():
+1 def parse():
+def closegroup(): pass
+2
+_parse_sub()
+3
+def opengroup(): pass
+3
+isstring()
+sre_compile
+4
+Tokenizer.match()
+SubPattern
+class SubPattern():
+compile
+_parse_sub
+2
+def closegroup(): pass
+1 def compile():
+1
+def
+_parse_sub():
+m
+def append(): pass
+re
+2
+_code()
+2
+_parse()
+4
+def getwidth(): pass
+3
+parse()
+3
+Tokenizer.match()
+compile
+4
+isstring()
+Tokenizer
+def re.compile():
+_parse
+1 class Tokenizer():
+compile()
+_code
+def _parse():
+2
+def get(): pass
+1
+def _code():
+2
+_parse_sub()
+3
+def match(): pass
+match
+2
+_compile()
+3
+_escape()
+def match():
+3
+compile_info()
+4
+Pattern.opengroup()
+_compile().search()
+CPQW6M4jqdqG
++Test
+5
+SubPattern.getwidth()
+_compile_info
+Tokenizer.get()
+1 p = Pattern()
+search
+1 def _compile_info():
+2 p.opengroup("a")
+def search():
+2
+SubPattern.getwidth()
+ROOT
+escape
+3 print(p)
+2
+_compile().match()
+1 def _escape():
+_compile
+1 _parse(p,"hello abc")
+Pattern.checkgroup()
+_compile
+1 def _compile():
+1 _escape(p,"a")
+def
+_compileO:
+2
+_simple()
+2
+sre_compile.compile()
+3
+SubPattern
+CPbxCz8GETTx
+AUtility
+_simple
+1 def _simple():
+isstring
+2
+SubPattern.getwidth()
+1 def isstring(obj):
+CPGGBRbcaBGb
++Test
+2
+return isinstance(
+1 print("Testing ..")
+3
+obj,(str,bytes))
+2 isstring("abc")
+isnumber
+1 isstring(1)
+def isnumber(obj):
+2
+return isinstance(
+3
+obj,(int, float))Deck-2
+Deck-4
+a
+# ERROR not defined
+1
+def a():
+2 a()
+2
+# ok
+3
+return b()
+Deck-1
+Deck-5
+b
+1 # ERROR not defined
+1
+def b():
+1 # ERROR not defined
+2 a()
+2
+# ok
+2 a()
+3
+return a()
+Deck-3
+1 # ERROR not defined
+2 a()Deck-B
+Deck-C
+Deck-A
+b1
+def b1():
+c1
+1
+al
+2
+# ok, cl exported
+def cl():
+1
+def al():
+3
+return cl()
+2
+return c1() + c3()
+2
+# ok
+3
+b1()
+cl,c2
+c2
+4
+# ok
+1 # dummy pod
+def c2():
+5
+c1()
+2 # re-export cl,c2
+2
+c3()
+a2
+b3
+c3
+1
+def a2():
+1
+def b3():
+1
+def c3():
+2
+# ERROR not
+defined
+2
+# 
+ERROR
+R not defined
+2
+pass
+3
+b4()
+3
+c3()
+c4
+4
+# ERROR not
+defined
+c4()
+b4
+5
+1
+def c4():
+1
+def b4():
+2
+pass
+c4()Conference’23, Jan, 2023, Ames, IA, USA
+Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian
+Figure 4: NS Rule 3: Utility pods/decks (indicated in green
+icon Utility)
+make it impossible to layout the pod hierarchy cleanly in a 2D space
+without many intersections.
+Thus, we design a “utility rule”: a deck/pod can be marked as
+a utility deck/pod. Such a utility pod is meant to provide utility
+functions to the parent deck’s sub-tree, and thus all the public
+functions in the utility deck are visible in the parent deck’s whole
+sub-tree. The utility pods are also namespace-aware: it is only
+visible to the parent deck, but not the grand-parent deck and above.
+Thus the utility decks can also be hierarchically ordered to build
+utility functions at different abstraction levels. As a special case, a
+utility deck under the root deck defines global utility functions that
+can be accessed throughout the entire codebase.
+For example, in Fig. 4, there are three regular decks and two
+utility decks. The public functions utils_b1 and utils_b2 defined
+in utility deck B are visible in its parent deck B’s sub-tree, including
+decks B, C. Another utility deck, A, is defined in the upper level
+and has a greater visible scope.
+2.5
+NS Rule 4: Testing Pods
+Figure 5: NS Rule 4: Testing pods/decks (indicated in green
+icon Test)
+Another essential pattern in interactive software development
+is to test whether the functions work as expected by writing some
+testing code and observing results. Such testing code must access
+the functions being tested, thus having to be in the same namespace
+or the parent namespace. However, either option has problems.
+On the one hand, testing code might create variables, introduce
+helper functions, and produce side effects. Thus they should be
+in a separate namespace to avoid polluting the namespace of the
+functions under testing. On the other hand, placing the testing deck
+as the parent deck of the function under testing is not logically
+natural because it does not provide upper-level functions.
+Therefore, we allow a deck/pod to be marked as a testing deck/-
+pod. A testing deck is placed as a child deck in the same namespace
+of the functions being tested. Although the testing deck/pod is
+a child-namespace, it can access the definitions visible within its
+parent deck, thus is able to call and test the function of interest.
+The testing pods are also namespace-aware: it can only access the
+function definitions in its parent deck, but not the grand-parent
+deck or siblings. Thus the testing decks can also be hierarchically
+ordered to build testing functions at different abstraction levels.
+For example, in Fig. 5, there are three regular decks and two
+testing decks, and one testing pod inside the regular deck A. The
+code pods in the testing deck are visible within the same testing
+deck, allowing for a testing setup like defining variables x and y
+and using them in other pods in the same testing deck. The pods in
+the testing deck run in a separate namespace, thus will not pollute
+other namespaces. A testing deck can access functions defined in its
+parent deck and can thus call and test whether the function yields
+expected results. A testing pod is similar to a testing deck, running
+in a separate namespace, and has access to the function definitions
+in the deck it belongs to.
+2.6
+NS Rule 5: Explicit Access via Full Path
+Figure 6: NS Rule 5: explicit imports by full path
+Finally, the 5th rule is the “brute-force rule”: a pod can always
+be accessible by specifying the full path within the tree. In other
+words, all pods are accessible via an explicit full path. This provides
+compatibility for arbitrary imports. This is considered the last resort
+and is ideally not needed but can be helpful in some cases. The path
+can be either a relative path connecting two pods or the absolute
+path from the root deck. In order to specify the path, the decks have
+to be named. In Codepod, an unnamed deck receives a UUID as the
+
+Deck-C
+1
+def c1():
+2
+util_bl()
+3
+Deck-B
+1
+def c2():
+1
+def b():
+2
+util_b2()
+2
+# OK
+3
+3
+util_al()
+Deck-A
+4
+util_a2()
+5
+# OK
+UtilityDeck-B
+AUtility
+1
+def al():
+6
+util_bl()
+2
+util_al()
+7
+util_b2()
+util_bl
+3
+util_a2()
+1 def util_b(): pass
+L
+def a() :
+2
+util_b2
+# ERROR not defined
+3
+util_bi()
+1 def util_b2(): pass
+UtilityDeck-A
+AUtility
+util_al
+1 def util_al(): pass
+util_a2
+1 def util a2(): passDeck-C
+c1
+def c1():
+Deck-B
+2
+pass
+b1
+c2
+def b1():
+1
+def c2():
+Deck-A
+2
+return c()
+2
+pass
+a
+b2
+1 
+def b2():
+1
+TestDeck-B
++Test
+2
+def a() :
+2
+1
+print("Testing b ..")
+3
+2
+assert bl() == 1
+4
+assert b2() == 2
+5
+4
+assert cl() == 3
+1
+print("Testing a
++ Test
+assert a() == 3
+TestDeck-A
++Test
+1 # test context setup
+2x=2
+3y=3
+1 # x and y are available
+2 print("Testing a ..")
+3 assert a(x) == yC
+C
+1 # explicit relative import
+2
+import d from ../D
+B
+3
+def c() :
+4
+()p
+1
+2
+A
+D
+1
+d
+2
+def d():
+2
+CPrFHVDyXTrC
+ # explicit absolute import
+2
+import c from /A/B/C
+3
+def f() :
+4
+c()Codepod: A Namespace-Aware, Hierarchical Jupyter
+for Interactive Development at Scale
+Conference’23, Jan, 2023, Ames, IA, USA
+name. Most of the time, developers do not need to specify names
+to the decks, as the first four rules will make the modules system
+usable without specifying names. Named decks are also helpful as
+a document for naming important module hierarchies.
+For example, in Fig. 6, there are 5 decks in the codebase. In Deck-
+D, a function d is defined. In Deck-C, the function d is not accessible.
+However, it is still able to be imported by a relative path ../D. As
+another example, in the bottom deck, a full absolute path /A/B/C
+is used to access the function c defined in Deck-C.
+2.7
+Discussion
+In summary, based on the hierarchical pod model, Codepod pro-
+vides a simple yet powerful module system including five rules:
+namespace separation, public pods, utility pods, testing pods, and
+full-path explicit access. These rules are highly hierarchical, and
+therefore are well suited for building hierarchical software projects
+from the ground up. This module system abstracts over different
+programming languages’ native module systems and provides a con-
+sistent module system across languages. The following section will
+discuss the runtime system and algorithms to support the Codepod
+module system.
+Let us revisit the Codepod example in Fig. 1, and see how these
+namespace rules are useful in real-world applications. This ex-
+ample implements a simplified Python regular expression com-
+piler. The functions are ordered into the decks re, sre_compile
+and sre_parse decks. sre_parse is the basic buiding block. It
+contains a child deck that defines three internal classes, Pattern,
+SubPattern, Tokenizer. These classes are only used in sre_parse
+module and not exposed to the upper level. The sre_parse mod-
+ule defines several helper functions including _parse_sub, _parse,
+_escape, and they are used to build a parse function which is ex-
+posed to parent module sre_compile. Similarly, module sre_compile
+defines some internal helper functions that are composed to pro-
+vide compile to the parent re module to build the top level API
+compile, match and search. The general functions isstring and
+isnumber are defined in a utility deck and are accessed through the
+modules sre_compile and sre_parse. Finally, the testing pods at
+different hierarchy make it easy to test and debug the functions at
+different levels.
+Ordering code blocks in Codepod is natural in building the dif-
+ferent levels of abstractions, and the hierarchy of code is close to
+the call graph, and is more cleanly maintained compared to file
+editors. The Codepod implementation maintains a clear code hi-
+erarchy and makes it easy to develop the project interactively. In
+comparison, the file-based implementation using VSCode would
+spread the functions into files, and within the file, the hierarchy of
+the functions is not clearly maintained. Writing all the functions
+into a Jupyter notebook is challenging due to the lack of namespace
+support, and the code hierarchy cannot be maintained within a
+single global namespace.
+2.8
+Version Control
+Implementing a version control system requires tremendous effort.
+There already exists mature file-based version control systems such
+as git and svn. We re-use the git version control system to apply
+version control to Codepod. Specifically, the code pods are first
+CodePod 
+Front-end
+Runtime 
+Kernel
+RunCode
+Request Complete
+EvalInNS(code, ns)
+AddImport(from, to, name)
+DeleteImport(from, name)
+Jupyter Protocol
+CodePod added protocol
+Figure 7: Kernel Communication Protocols
+exported to files. Then git version control is applied to the generated
+files. For front-end rendering, we query the git version control
+system for the diff between two versions, e.g., the current changes
+or specific git commit changes. Then the diff results are parsed to
+show pod-level diffs.
+3
+INCREMENTAL RUNTIME
+Codepod develops an intuitive, effective, consistent cross-language
+scope-aware incremental runtime abstraction. In Codepod, the run-
+time loads the code pods in the hierarchy. When a pod in some scope
+is changed, the updated definition can be applied incrementally in
+that scope in the active runtime without restarting the runtime,
+and the new definition is visible for other pods depending on it.
+3.1
+Runtime Kernel Communication
+We build the Codepod Kernel communication based on the Jupyter
+Message Queue protocol. The Jupyter kernel protocols and our
+added protocols are shown in Fig. 7. In its simplest form, Jupyter
+kernel messaging supports eval and complete. We add the fol-
+lowing protocol to the messaging queue: EvalInNS, AddImport,
+DeleteImport. The protocol EvalInNS instructs the langauge ker-
+nel to evaluate code in a specific namespace. The AddImport proto-
+col makes a function “name” defined in “from” namespace available
+in the “to” namespace, and DeleteImport undo the change. The
+algorithm for the language kernel is given in next section.
+3.2
+Language Kernel Algorithm
+The language kernel needs to support four functions to work with
+Codepod: GetModule, EvalInNS, AddImport, DeleteImport. Most
+languages do not natively support all these operations. We imple-
+ment a thin wrapper of these functions as shown in Algorithm 1.
+GetModule is a function that gets the module instance for a spe-
+cific namespace. Since we need to refer to the same module given
+the same name, we need to maintain a mapping from namespace
+names to the module instance. In line 1, the nsmap object is created
+as a global variable, initialized with an empty dictionary. The Get-
+Module function will query this map, and if the module already
+exists, return the module instance (lines 3-4). If the module is not
+found, create a new module by calling the language’s createMod
+API, record the module in the nsmap, and return the module (lines
+6-8).
+
+Conference’23, Jan, 2023, Ames, IA, USA
+Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian
+Algorithm 1 Namespace-aware Runtime: Language Kernel
+1: 𝑛𝑠𝑚𝑎𝑝 ← 𝐸𝑚𝑝𝑡𝑦𝐷𝑖𝑐𝑡 ()
+2: function GetModule(ns)
+3:
+if 𝑛𝑠 ∈ 𝑛𝑠𝑚𝑎𝑝 then
+4:
+return 𝑛𝑠𝑚𝑎𝑝 [𝑛𝑠]
+5:
+else
+6:
+𝑚𝑜𝑑 ← 𝑐𝑟𝑒𝑎𝑡𝑒𝑀𝑜𝑑 (𝑛𝑠)
+7:
+𝑛𝑠𝑚𝑎𝑝 [𝑛𝑠] ← 𝑚𝑜𝑑
+8:
+return 𝑚𝑜𝑑
+9:
+end if
+10: end function
+11: function EvalInNS(ns, code)
+12:
+𝑚𝑜𝑑 ← GetModule(ns)
+13:
+𝑖𝑠𝐸𝑥𝑝𝑟 ← IsExpr(code)
+14:
+if isExpr then
+15:
+return eval(code, mod)
+16:
+else
+17:
+exec(code, mod)
+18:
+return NULL
+19:
+end if
+20: end function
+21: function AddImport(from, to, name)
+22:
+𝑠 ← "$name=eval($name,from)"
+23:
+EvalInNS(to, s)
+24: end function
+25: function DeleteImport(from, name)
+26:
+𝑠 ← "del $name"
+27:
+EvalInNS(from, s)
+28: end function
+The EvalInNS function is responsible for loading the module
+specified by the namespace and evaluating code with that mod-
+ule instance active. It first loads or creates the module instance
+by the GetModule function. Many languages treat expression and
+statement separately, where expressions return values, while state-
+ments do not. Thus the algorithm first checks whether this code
+is an expression or statement. If it is an expression, the language’s
+EVAL API is called with the code and module instance and returns
+the expression results for display in the front-end. Otherwise, the
+language’s EXEC API is called to evaluate the code in the module
+instance for side-effect only and return NULL to the user.
+Finally, AddImport and DeleteImport work by meta-programming:
+a new program string is constructed to add or delete names in the
+target namespace. The function AddImport receives the name to
+import and the "from" and "to" namespace. The goal is to import the
+function "name" from the "from" namespace and make it available in
+the "to" namespace. A string is constructed to assign a variable with
+the same name "name" with the evaluation of the name in the "from"
+namespace (lines 22-23). The DeleteImport works by constructing
+a "delete $name" code and evaluating the target namespace (lines
+26-27).
+We note that the EVAL and EXEC API with module awareness are
+different across languages. Some languages provide native support,
+while others might need to perform reflection-level operations,
+such as manually passing in Python symbol maps. The code strings
+for AddImport and DeleteImport are generally different across
+languages too. We supply the core code of the four kernels we have
+implemented in Figure 8, 9, 10, and 11.
+3.3
+Pod Hierarchy Algorithm
+This section formally describes the key algorithms in Algorithm 2
+that implement Codepod’s module system, the namespace rules, and
+how the evaluation of pods/decks is handled in the pod hierarchy.
+A pod is evaluated with the RunPod function, which calls EvalInNS
+with the pod’s code content and namespace string. A deck is eval-
+uated with the RunDeck function, which evaluates the subtree of
+the deck. The function first evaluates all the child decks in a DFS
+(depth-first search) manner, then evaluates all the pods in this deck.
+1
+d = {}
+2
+3
+def getmod(ns):
+4
+if ns not in d:
+5
+d[ns] = types.ModuleType(ns)
+6
+d[ns]. __dict__["
+CODEPOD_GETMOD"] = getmod
+7
+return d[ns]
+8
+9
+def add_import(src , dst , name):
+10
+return eval_func("""
+11
+{name}= getmod ({src}).
+__dict__ ["{ name }"]
+12
+""", dst)
+13
+14
+def delete_import(ns, name):
+15
+eval_func("""del {name}""", ns)
+1
+def eval_func(code , ns):
+2
+mod = getmod(ns)
+3
+[stmt , expr] = code2parts(code)
+4
+if stmt:
+5
+exec(stmt , mod.__dict__)
+6
+if expr:
+7
+return eval(expr , mod.
+__dict__)
+Figure 8: Python Kernel Namespace Implementation
+1
+var NSMAP = NSMAP || {};
+2
+function eval(code , ns, names) {
+3
+if (!NSMAP[ns]) {
+4
+NSMAP[ns] = {};
+5
+}
+6
+for (let k of keys(NSMAP[ns])) {
+7
+eval(
+8
+`var ${k}=NSMAP["${ns}"].${k}`
+9
+);
+10
+}
+11
+let res = eval(code);
+12
+for (let name of names) {
+13
+eval(
+14
+`NSMAP["${ns}"].${name}=${name}`
+15
+);
+16
+}
+17
+return res;
+18
+}
+1
+function addImport(from , to, name) {
+2
+if (!NSMAP[from]) {
+3
+NSMAP[from] = {};
+4
+}
+5
+if (!NSMAP[to]) {
+6
+NSMAP[to] = {};
+7
+}
+8
+eval(`
+9
+NSMAP["${to}"].${name}=
+10
+NSMAP["${from}"].${name}`);
+11
+}
+12
+function deleteImport(ns, name) {
+13
+if (!NSMAP[ns]) {
+14
+NSMAP[ns] = {};
+15
+}
+16
+eval(
+17
+`delete NSMAP["${ns}"].${name}`)
+;
+18
+}
+Figure 9: Javascript Kernel Namespace Implementation
+1
+function isModuleDefined(names)
+2
+mod = :(Main)
+3
+for name in names
+4
+name = Symbol(name)
+5
+if !isdefined(eval(mod), name)
+6
+return false
+7
+end
+8
+mod = :($mod.$name)
+9
+end
+10
+return true
+11
+end
+12
+function ensureModule(namespace)
+13
+names = split(namespace , "/",
+14
+keepempty=false)
+15
+mod = :(Main)
+16
+for name in names
+17
+name = Symbol(name)
+18
+if !isdefined(eval(mod), name)
+19
+include_string(eval(mod),
+20
+"module $name end")
+21
+end
+22
+mod = :($mod.$name)
+23
+end
+24
+return mod , eval(mod)
+25
+end
+1
+function eval(code , ns)
+2
+_, mod = ensureModule(ns)
+3
+include_string(mod , code)
+4
+end
+5
+function addImport(from , to, name)
+6
+from_name , _ = ensureModule(from)
+7
+_, to_mod = ensureModule(to)
+8
+code = """
+9
+using $from_name: $name as CP$name
+10
+$name=CP$name
+11
+$name
+12
+"""
+13
+include_string(to_mod , code)
+14
+end
+15
+function deleteImport(ns, name)
+16
+_, mod = ensureModule(ns)
+17
+include_string(mod , "$name=nothing")
+18
+end
+Figure 10: Julia Kernel Namespace Implementation
+The reason to evaluate child decks before child pods is that the child
+decks might define public functions exposed to this deck; thus, the
+pods must have those definitions in place before execution. The
+reason to use DFS is to make sure the lowest-level pods are first
+evaluated before moving up the hierarchy. Child pods are evaluated
+sequentially. When running the child pods, the algorithm examines
+
+Codepod: A Namespace-Aware, Hierarchical Jupyter
+for Interactive Development at Scale
+Conference’23, Jan, 2023, Ames, IA, USA
+1
+(compile-enforce-module-constants #f)
+2
+3
+(define (ns- >submod ns)
+4
+(let ([names (string-split ns "/")])
+5
+(when (not (empty? names))
+6
+(let ([one (string- >symbol
+7
+(first names))]
+8
+[two (map string- >symbol
+9
+(rest names))])
+10
+‘(submod ',one
+11
+,@two)))))
+12
+(define (ns- >enter ns)
+13
+(let ([mod (ns- >submod ns)])
+14
+(if
+(void? mod) '(void)
+15
+‘(dynamic-enter! ',mod))))
+16
+17
+(define (ns- >ensure-module ns)
+18
+(let loop
+19
+([names (string-split ns "/")])
+20
+(if (empty? names)
+21
+'(void)
+22
+‘(module
+23
+,(string- >symbol
+24
+(first names))
+25
+racket/base
+26
+,(loop (rest names))))))
+1
+(define (add-import from to name)
+2
+(let ([name (string- >symbol name)])
+3
+(eval (ns- >enter to))
+4
+(eval
+5
+‘(define ,name
+6
+(dynamic-require/expose
+7
+',(ns- >submod from)
+8
+',name)))))
+9
+10
+(define (delete-import ns name)
+11
+(eval (ns- >enter ns))
+12
+(namespace-undefine-variable!
+13
+(string- >symbol name)))
+14
+15
+(define (string- >sexp s)
+16
+(call-with-input-string
+17
+s
+18
+(lambda (in)
+19
+(read in))))
+20
+21
+(define (codepod-eval code ns)
+22
+(eval (ns- >ensure-module ns))
+23
+(eval (ns- >enter ns))
+24
+(begin0
+25
+(eval
+26
+(string- >sexp
+27
+(~a "(begin " code ")")))
+28
+(enter! #f)))
+Figure 11: Racket Kernel Namespace Implementation
+Algorithm 2 Namespace-aware Runtime: Pod Hierarchy
+1: function RunPod(pod)
+2:
+EvalInNS(pod.ns, pod.code)
+3: end function
+4: function RunTest(ns, code)
+5:
+for 𝑛𝑎𝑚𝑒 ← 𝑝𝑜𝑑.𝑝𝑎𝑟𝑒𝑛𝑡.𝑛𝑎𝑚𝑒𝑠 do
+6:
+AddImport(pod.parent.ns, pod.ns, name)
+7:
+end for
+8:
+EvalInNS(pod.ns, pod.code)
+9: end function
+10: function RunUtility(from, name)
+11:
+EvalInNS(pod.ns, pod.code)
+12:
+function dfs(parent)
+13:
+for 𝑐ℎ𝑖𝑙𝑑 ← 𝑝𝑎𝑟𝑒𝑛𝑡.𝑐ℎ𝑖𝑙𝑑_𝑑𝑒𝑐𝑘𝑠 do
+14:
+for 𝑛𝑎𝑚𝑒 ← 𝑝𝑜𝑑.𝑝𝑎𝑟𝑒𝑛𝑡.𝑛𝑎𝑚𝑒𝑠 do
+15:
+AddImport(pod.parent.ns, pod.ns, name)
+16:
+end for
+17:
+dfs(child)
+18:
+end for
+19:
+end function
+20:
+dfs(pod.parent)
+21: end function
+22: function RunTree(root)
+23:
+function dfs(parent)
+24:
+for 𝑐ℎ𝑖𝑙𝑑 ← 𝑝𝑎𝑟𝑒𝑛𝑡.𝑐ℎ𝑖𝑙𝑑_𝑑𝑒𝑐𝑘𝑠 do
+25:
+RunTree(child)
+26:
+end for
+27:
+end function
+28:
+dfs(pod)
+29:
+for 𝑝𝑜𝑑 ← 𝑟𝑜𝑜𝑡.𝑐ℎ𝑖𝑙𝑑_𝑝𝑜𝑑𝑠 do
+30:
+if pod.type is "Pod" then
+31:
+RunPod(pod)
+32:
+else if pod.type is "Test" then
+33:
+RunTest(pod)
+34:
+else if pod.type is "Utility" then
+35:
+RunUtility(pod)
+36:
+end if
+37:
+end for
+38: end function
+the type of the pods and calls the corresponding functions RunPod,
+RunUtility, and RunTest for different types accordingly.
+Hierarchical
+Layout
+Commnication
+Protocol
+Kernel
+Runtime
+CodeServer
+API
+Total
+LOC
+4.1k
+1.8k
+1k
+1k
+7.9k
+Table 1: Codepod Implemenatation Statistics
+The RunUtility function will first evaluate the pods in the names-
+pace. Then, the public names are exported to the parent’s subtree
+by traversing the parent’s subtree in a DFS manner and calling
+AddImport for each of the namespaces in the sub-tree during tra-
+versal. In this way, the name of the utility function is available in
+all the decks of the parent’s sub-tree. The RunTest function will
+loop through the parent deck’s public names and run AddImport
+for all the names from the parent’s namespace to the testing pod’s
+namespace, and then evaluate the test pods in the namespace where
+the parent’s function definitions are available.
+3.4
+Fallback Execution
+Codepod requires the programming languages to support interpret-
+ing in order to run and evaluate code interactively. With interactive
+development becoming popular, there have been many interpreters
+implementations for even compiled languages. For example, C++
+has a highly regarded interpreter, Cling. Another requirement is
+that the language needs to support namespace and provide a way
+to evaluate code blocks within the namespace.
+Suppose the support of namespace-aware interactive develop-
+ment is not fully supported due to the limitation of the language
+interpreter. In that case, Codepod provides a fallback option: export-
+ing code to files and invoking the language interpreter/compiler to
+run the program as a whole. The downside of this approach is that
+the variables are not persisted in memory across runs because each
+invocation starts a new process.
+4
+CASE STUDIES
+4.1
+Implementation
+We have implemented a fully working Codepod as a Web applica-
+tion (front-end in React and backend server in Node.js) and cur-
+rently implemented full namespace-aware runtime support for
+four language kernels, including Python, JavaScript, Julia, and
+Scheme/Racket. New kernels can be easily developed upon exist-
+ing Jupyter notebook kernels. Codepod is open-sourced at https:
+//example.com.
+Codepod implementation contains 4 major parts, the LOC sta-
+tistics is shown in Table 1. The hierarchical layout implements the
+front-end hierarchical pods and tools. The communication proto-
+col implements the RunTree/RunPod logic and how the front-end
+communicates with the backend API server and language runtime
+kernels. Kernel runtime implements the kernels and WebSocket
+protocol message handling. Finally, CodeServer API implements the
+API for retrieving and updating pods hierarchy from the front-end
+and talking to the database.
+
+Conference’23, Jan, 2023, Ames, IA, USA
+Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian
+4.2
+Python and Julia Projects Statistics
+In this section, we study several highly regarded open-source projects
+and visually show how the representation of Codepod can poten-
+tially help develop code projects. This case study aims to see how
+functions are distributed in the files and directories and the calling
+relations of the functions. This helps us evaluate how the Codepod
+model can help with real-world open-source software projects.
+The projects are obtained from GitHub’s top-rated Python and
+Julia projects; the information about the projects is shown in Ta-
+ble 2. We analyze the projects with the help of tree-sitter [1] parser
+framework. We count only the source directory of the projects,
+ignoring testing code and documents. Our study focuses on func-
+tions as the code block granularity. Python projects might contain
+classes, and we treat each method as a function.
+In Table 2, we can see that software projects often contain a large
+number of functions, and those functions are distributed in a large
+number of files, possibly in a deep directory hierarchy of tens of
+directories. For example, the you-get project contains 444 functions
+and is distributed into 133 files in 8 directories. The LightGraph.jl
+project contains 242 functions, distributed in 110 files within 27
+directories. It can be pretty challenging to grasp the hierarchy of
+the functions by browsing through the files and directories.
+We also count the number of internal and external functions of
+each file. A file’s internal functions are defined as those only called
+by the other functions in the same file. These internal functions are
+helper functions that are used to implement the external functions
+that act as the public interface of the file, called by other files. In
+Table 2, we see that approximately half of the files are internal
+and are used as building blocks of other functions. However, this
+dependency information is not clear in a file because the functions
+are considered linear within a file, and no clear hierarchy can be
+effectively maintained. Thus potentially, Codepod can help to apply
+hierarchical relations to the functions inside each file.
+Figure 12: Statistics for top open source Python Projects
+To understand the distributions of functions in each file and call-
+ing relationships, we plot the in-degree,out-degree,LOC,#functions-
+per-file of the Python and Julia Projects in Fig. 12 and Fig. 13, re-
+spectively. The in/out-degrees of a function is defined by how many
+calls to the functions and how many calls this function makes to
+other functions. Both in/out-degrees are computed based on the call
+Figure 13: Statistics for top open source Julia Projects
+graphs over the functions defined in the project; thus, the statistics
+do not include language or library API calls.
+From the plot Fig. 12.(a) and Fig. 13.(a), we see that most func-
+tions are being called no more than two times. This shows that
+most functions are “local”, and only used to implement higher-level
+functions. Also, in Fig. 12.(b) and Fig. 13.(b), we observe that the
+out-degree is more than in-degree, and most functions have less
+than five function calls to other functions. This is also consistent
+with Codepod’s tree-based model: a deck in a tree node might have
+multiple sub-decks. A pod defined in a deck might use the functions
+defined in the sub-decks to implement higher-level functionality.
+From the function per file plot Fig. 12.(d) and Fig. 13.(d), we can
+see that the distribution of functions into files are not even: a large
+number of files contain only 1 or 2 functions, while there can also be
+a few very large file containing tens of functions. This data shows
+that in order to maintain a cleaner hierarchy, developers might use
+a separate file for each function, resulting in too many small files
+and deep directories, which are relatively hard to maintain, edit
+and reference using file browsers and file editors. Also, within the
+large files with tens of functions, the hierarchy of those functions
+cannot be cleanly maintained within a file, and Codepod could help
+build a finer-granular hierarchy for the functions.
+4.3
+Jupyter Project Statistics
+In this study, we investigate how Jupyter notebooks are used in real-
+world open projects. This study aims to see how code is distributed
+among Jupyter notebooks and text files and how the Jupyter note-
+books interact with the functions defined in text files regarding
+calling relationships. We query GitHub APIs to find top Python
+projects whose primary language is Python and whose secondary
+language is Jupyter Notebook. The projects and statistics are shown
+in Table 3.
+In Table 3, we can see that there are typically more text files
+than Jupyter notebooks. This is also the case for the total LOC
+and number of functions in Jupyter notebooks vs. text files. In
+fact, the LOC of the Jupyter notebook consists only 3.7% of the
+codebase for these projects. This means that the majority of the
+code is implemented in the text files.
+To understand the calling relationships of the Jupyter notebooks
+and text files, we calculate the percentage of internal functions
+
+ab-no
+DOE
+cookiecutter
+locust
+150
+unoo
+requests
+200
+100
+100
+50
+5
+10
+5
+1D
+15
+(afuncbion indegree
+bj functinoutdegree
+60 
+60
+40
+40
+no
+20
+20
+0
+0
+0
+25
+50
+75
+100
+0
+5
+10
+15
+20
+(ciavglocperfunction
+dyfunctionsperfileJuliaDB.jl
+100
+HTTP.jI
+100
+Flux.jl
+un
+LightGraphs.jl
+50
+50 
+5
+10
+0
+5
+10
+15
+al function indegree
+(b) functin outdegree
+30 
+60
+ 20
+40
+10 
+20
+0
+0
+0
+25
+50
+75
+100
+5
+10
+15
+20
+(c) avg loc perfunction
+(d) functionsper fileCodepod: A Namespace-Aware, Hierarchical Jupyter
+for Interactive Development at Scale
+Conference’23, Jan, 2023, Ames, IA, USA
+#stars
+#dirs
+#files
+#loc
+#funcs
+#internal funcs
+description
+soimort/you-get
+41.6k
+8
+133
+14707
+444
+273
+CMD-line utility to download media from Web
+cookiecutter/cookiecutter
+15.2k
+0
+18
+2139
+51
+30
+CMD-line utility to create projects from templates
+locustio/locust
+17k
+10
+59
+18671
+529
+144
+performance testing tool
+psf/requests
+45.9k
+0
+18
+5183
+135
+73
+HTTP library
+JuliaData/JuliaDB.jl
+706
+0
+20
+3113
+106
+82
+Parallel analytical database
+JuliaWeb/HTTP.jl
+439
+0
+38
+7513
+65
+36
+HTTP client and server
+FluxML/Flux.jl
+3.2k
+5
+33
+6408
+84
+41
+Machine Learning Framework
+JuliaGraphs/LightGraphs.jl
+675
+27
+110
+16963
+242
+101
+Network and graph analysis.
+Table 2: Function Statistics in Open Source Projects (Python and Julia)
+#stars
+#file
+(ipynb/files)
+#loc
+(ipynb/files)
+#call
+fs-to-nb
+#call
+nb-to-fs
+#func
+(ipynb/files)
+#%internal
+(ipynb/files)
+description
+blei-lab/edward
+4.6k
+14/42
+1340/5449
+0
+11
+10/102
+100%/93%
+A probabilistic programming language
+tqdm/tqdm
+19.3k
+2/30
+284/2257
+0
+9
+0/83
+NA/80%
+A Fast, Extensible Progress Bar
+google/jax
+14.1k
+3/196
+104/42879
+0
+8
+0/2418
+NA/38%
+Autograd and Optimizing Compiler for ML
+google-research/bert
+29k
+1/13
+322/4547
+0
+8
+7/92
+100%/88%
+State-of-the-art NLP language model
+quantopian/zipline
+14.4k
+2/183
+115/30087
+0
+3
+3/851
+100%/61%
+Algorithmic Trading Library
+Table 3: Jupyter Notebooks statistics in Open Source Projects
+for files and Jupyter notebooks. The internal function is defined
+the same as above, i.e., the functions that are only called from the
+same file or Jupyter notebook. From the result, two projects contain
+no function definitions in the Jupyter notebooks, and the other
+three projects’ functions are 100% internal to the notebook files. In
+contrast, the text files contain 38% to 93% internal functions. Also,
+we calculate the number of calls from the Jupyter notebook to text
+files and vice versa and show that there are no calls from source
+text files to notebooks, but only calls from the Jupyter notebook to
+the text files. This means that the Jupyter notebook is not used to
+develop functions. The projects are implemented in text files, and
+Jupyter notebooks are only used to call those files’ APIs and are
+most likely for presentation and tutorial purposes.
+5
+RELATED WORK
+Running and debugging code is a major activity in software de-
+velopment. There have been many tools to help the debugging
+process. In the simplest form, developers write code in files using
+some editors such as VIM and Emacs [13], and compile and run
+files in the command line. The problem with this approach is that
+it is non-interactive, and the program always needs to restart from
+the beginning. An interactive development method is to launch
+a Read-Eval-Print-Loop (REPL) [7, 14], type, load, and evaluate
+code expressions in the REPL. Although REPL is interactive, the
+code being evaluated is not editable, and users have to type code
+into the REPL. It is also common to open a file editor and send a
+code region to the REPL for evaluation. Integrated Development
+Environments (IDEs) such as VSCode integrate file editors with
+a file browser, command line, plugins, and debuggers. There also
+exist editor plugins to navigate between the functions of a project.
+However, those plugins still do not give a within-file hierarchy
+to the code and depend on the programming language designs to
+support the within-file module system, which only a handful of
+languages support to various degrees. File-based approaches force
+developers to maintain the correspondence between code and files,
+which is tricky due to the significantly different granularity of code
+blocks and files. The unbalance in granularity poses dilemmas to
+developers: including too many code blocks into one file makes the
+hierarchy hard to maintain, while including few code blocks into
+one file creates many small files and deep directories that are also
+hard to work with.
+Figure 14: Interactive Development with Jupyter. Image from [11].
+A recent new paradigm is a web-based interactive notebook
+called Jupyter Notebook [6]. The interface of Jupyter is shown in
+Figure 14. The notebook consists of code cells. Each cell can be run
+in arbitrary order, and the results are displayed under the cell. The
+code output can be visualized, e.g., plotting a figure. Thus Jupyter
+notebooks support literate programming that combines code, text,
+
+Fibonnaci
+In [3]:
+def fib(x):
+if x <= 1:
+Markdown
+return x
+Cells
+return fib(x-1) + fib(x-2)
+fib(10)
+Out[3]:
+55
+Output 1
+Code
+Cells
+Let's plot the numbers
+In [8]:
+from matplotlib import pyplot
+%matplotlib inline
+x = range(15)
+y = [fib(n) for n in x]
+pyplot.plot(x, y);
+Execution
+350
+300
+Counter
+250
+200
+ Output 2
+150
+100
+50 
+0
+6
+10
+12
+14Conference’23, Jan, 2023, Ames, IA, USA
+Hebi Li, Forrest Sheng Bao, Qi Xiao, Jin Tian
+and execution results with rich media visualizations. However, the
+Jupyter notebook cells are linear, and all the code blocks live in the
+global namespace. Jupyter notebook lacks a module system that
+is crucial for complex software. This makes it hard to develop a
+large-scale software system. Thus Jupyter notebook is typically
+used only for surface demo and visualization purposes. The real
+code of the projects is still developed in text files with text editors
+and external runtime. In order to define hierarchical code, users
+have to write code into text files and import the module from text
+files into the notebook. Such a Jupyter-file hybrid model has several
+disadvantages. Changes in files are not in sync with the Jupyter
+runtime. When a function definition in a file is changed, the Jupyter
+runtime must be restarted, and the file must be reloaded for the
+change to take effect.
+Another programming paradigm is Visual Programming, e.g. Lab-
+View [2], and Google’s Blockly [9], and Microsoft’s MakeCode [5].
+Such a visual programming paradigm has distinguished advantages:
+it is visually clean, easy to learn, and impossible to make syntax er-
+rors. However, all these visual programming systems use such block
+style items down to each expression (e.g., a+b), which can be ver-
+bose. Also, visual programming blocks live in a global namespace,
+and there is no module system available for developing large-scale
+software.
+There exist code analyzers to generate a visual presentation of
+code, such as Unified Modeling Language (UML) [3, 8]. Tools such
+as call graph visualizers [4] are also developed to help to understand
+a codebase. However, those visual presentations are not editable,
+making them less useful during development.
+6
+CONCLUSION
+In this paper, we propose Codepod, a namespace-aware hierarchi-
+cal interactive development environment. Codepod uses a novel
+hierarchical code block model to represent code and abstracts away
+files. We propose namespace rules that make it easy to organize
+the pods and provide a consistent module system across languages.
+Codepod provides an incremental evaluation runtime that helps
+interactively develop a large-scale software project.
+We hope Codepod can provide a novel way to drive the software
+development process and inspire other research. In the future, we
+will push Codepod forward with the contribution from the com-
+munity, perform user studies to compare Codepod with VSCode
+and Jupyter. It is interesting to integrate program analysis tools
+into the Codepod model. We are also interested in integrating other
+programming paradigms into the Codepod framework; for exam-
+ple, it would be helpful to integrate graphical programming as a
+“graphical pod” and mix graphical programs with plain-text code.
+REFERENCES
+[1] Tree-sitter: An incremental parsing system for programming tools. https://tree-
+sitter.github.io/tree-sitter/. Accessed: 2021-07-30.
+[2] Rick Bitter, Taqi Mohiuddin, and Matt Nawrocki. LabVIEW: Advanced program-
+ming techniques. Crc Press, 2006.
+[3] Grady Booch, Ivar Jacobson, James Rumbaugh, et al. The unified modeling
+language. Unix Review, 14(13):5, 1996.
+[4] David Callahan, Alan Carle, Mary W. Hall, and Ken Kennedy. Constructing the
+procedure call multigraph. IEEE Transactions on Software Engineering, 16(4):483–
+487, 1990.
+[5] James Devine, Joe Finney, Peli de Halleux, Michał Moskal, Thomas Ball, and
+Steve Hodges. Makecode and codal: intuitive and efficient embedded systems
+programming for education. ACM SIGPLAN Notices, 53(6):19–30, 2018.
+[6] Thomas Kluyver, Benjamin Ragan-Kelley, Fernando Pérez, Brian Granger,
+Matthias Bussonnier, Jonathan Frederic, Kyle Kelley, Jessica Hamrick, Jason
+Grout, Sylvain Corlay, Paul Ivanov, Damián Avila, Safia Abdalla, and Carol Will-
+ing. Jupyter notebooks – a publishing format for reproducible computational
+workflows. In F. Loizides and B. Schmidt, editors, Positioning and Power in
+Academic Publishing: Players, Agents and Agendas, pages 87 – 90. IOS Press, 2016.
+[7] John McCarthy, Michael I Levin, Paul W Abrahams, Daniel J Edwards, and
+Timothy P Hart. LISP 1.5 programmer’s manual. MIT press, 1965.
+[8] Nenad Medvidovic, David S Rosenblum, David F Redmiles, and Jason E Rob-
+bins. Modeling software architectures in the unified modeling language. ACM
+Transactions on Software Engineering and Methodology (TOSEM), 11(1):2–57, 2002.
+[9] Erik Pasternak, Rachel Fenichel, and Andrew N Marshall. Tips for creating a
+block language with blockly. In 2017 IEEE Blocks and Beyond Workshop (B&B),
+pages 21–24. IEEE, 2017.
+[10] Jeffrey M Perkel. Why jupyter is data scientists’ computational notebook of
+choice. Nature, 563(7732):145–147, 2018.
+[11] João Felipe Pimentel, Leonardo Murta, Vanessa Braganholo, and Juliana Freire. A
+large-scale study about quality and reproducibility of jupyter notebooks. In 2019
+IEEE/ACM 16th International Conference on Mining Software Repositories (MSR),
+pages 507–517. IEEE, 2019.
+[12] Bernadette M Randles, Irene V Pasquetto, Milena S Golshan, and Christine L
+Borgman. Using the jupyter notebook as a tool for open science: An empirical
+study. In 2017 ACM/IEEE Joint Conference on Digital Libraries (JCDL), pages 1–2.
+IEEE, 2017.
+[13] Richard M Stallman. Emacs the extensible, customizable self-documenting dis-
+play editor. In Proceedings of the ACM SIGPLAN SIGOA symposium on Text
+manipulation, pages 147–156, 1981.
+[14] L Thomas van Binsbergen, Mauricio Verano Merino, Pierre Jeanjean, Tijs van der
+Storm, Benoit Combemale, and Olivier Barais. A principled approach to repl
+interpreters. In Proceedings of the 2020 ACM SIGPLAN International Symposium on
+New Ideas, New Paradigms, and Reflections on Programming and Software, pages
+84–100, 2020.
+[15] Jiawei Wang, Li Li, and Andreas Zeller. Better code, better sharing: on the need of
+analyzing jupyter notebooks. In Proceedings of the ACM/IEEE 42nd International
+Conference on Software Engineering: New Ideas and Emerging Results, pages 53–56,
+2020.
+

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+arXiv:2301.02895v1  [cond-mat.stat-mech]  7 Jan 2023
+RENEWAL EQUATIONS FOR SINGLE-PARTICLE DIFFUSION IN
+MULTI-LAYERED MEDIA
+PAUL C. BRESSLOFF∗
+Abstract. Diffusion in heterogeneous media partitioned by semi-permeable interfaces has a wide
+range of applications in the physical and life sciences, ranging from thermal conduction in composite
+media, gas permeation in soils, diffusion magnetic resonance imaging (dMRI), drug delivery, and
+intercellular gap junctions. Many of these systems involve three-dimensional (3D) diffusion in an
+array of parallel planes with homogeneity in the lateral directions, so that they can be reduced to
+effective one-dimensional (1D) models.
+In this paper we develop a probabilistic model of single-
+particle diffusion in 1D multi-layered media by constructing a multi-layered version of so-called
+snapping out Brownian motion (BM). The latter sews together successive rounds of reflected BM,
+each of which is restricted to a single layer. Each round of reflected BM is killed when the local time
+at one end of the layer exceeds an independent, exponentially distributed random variable. (The
+local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of
+a boundary.) The particle then immediately resumes reflected BM in the same layer or the layer
+on the other side of the boundary with equal probability, and the process is iterated We proceed
+by constructing a last renewal equation for multi-layered snapping out BM that relates the full
+probability density to the probability densities of partially reflected BM in each layer. We then show
+how transfer matrices can be used to solve the Laplace transformed renewal equation, and prove that
+the renewal equation and corresponding multi-layer diffusion equation are equivalent. We illustrate
+the theory by analyzing the first passage time (FPT) problem for escape at the exterior boundaries
+of the domain. Finally, we use the renewal approach to incorporate a generalization of snapping out
+BM based on the encounter-based method for surface absorption; each round of reflected BM is now
+killed according to a non-exponential distribution for each local time threshold. This is achieved
+by considering a corresponding first renewal equation that relates the full probability density to
+the FPT densities for killing each round of reflected BM. We show that for certain configurations,
+non-exponential killing leads to an effective time-dependent permeability that is normalizable but
+heavy-tailed.
+1. Introduction. Diffusion in heterogeneous media partitioned by semi per-
+meable barriers has a wide range of applications in natural and artificial systems.
+Examples include multilayer electrodes and semi-conductors [27, 18, 24, 34], thermal
+conduction in composite media [3, 33, 17, 44], waste disposal and gas permeation
+in soils [55, 43, 52], diffusion magnetic resonance imaging (dMRI) [53, 13, 16], drug
+delivery [49, 54], and intercellular gap junctions [20, 15, 26]. Many of these systems
+involve three-dimensional (3D) diffusion in an array of parallel planes with homo-
+geneity in the lateral directions, which means that they can be reduced to effective
+one-dimensional (1D) models. Consequently, there have been a variety of analytical
+and numerical studies of 1D multilayer diffusion that incorporate methods such as
+spectral decompositions, Greens functions, and Laplace transforms [11, 51, 50, 19, 36,
+37, 29, 35, 30, 14, 6, 46].
+Almost all studies of multilayer diffusion have focused on macroscopic models
+in which the relevant field is the concentration of diffusing particles. Many of the
+analytical challenges concern the derivation of time-dependent solutions that charac-
+terize short-time transients or threshold crossing events. This requires either carrying
+out a non-trivial spectral decomposition of the solution and/or inverting a highly
+complicated Laplace transform. In general, it is necessary to develop some form of
+approximation scheme or to supplement a semi-analytical solution with numerical
+computations. As far as we are aware, single-particle diffusion or Brownian motion
+∗Department
+of
+Mathematics,
+University
+of
+Utah,
+Salt
+Lake
+City,
+UT
+84112
+USA
+([email protected])
+1
+
+(BM) in multilayer media has not been investigated to anything like the same ex-
+tent, with the possible exception of spatially discrete random walks [40, 47, 39, 2].
+On the other hand, a rigorous probabilistic formulation of 1D diffusion through a
+single semi-permeable barrier has recently been introduced by Lejay [41] in terms of
+so-called snapping out BM, see also Refs. [1, 42, 12]. Snapping out BM sews together
+successive rounds of reflected BM that are restricted to either x < 0 or x > 0 with a
+semi-permeable barrier at x = 0. Each round of reflected BM is killed when its local
+time at x = 0± exceeds an exponentially distributed random variable with constant
+rate κ0. (Roughly speaking, the local time at x = 0+ (x = 0−) specifies the amount
+of time a positively (negatively) reflected Brownian particle spends in a neighborhood
+of the right-hand (left-hand) side of the barrier [38].) It then immediately resumes
+either negatively or positively reflected BM with equal probability, and so on.
+We recently reformulated 1D snapping out BM in terms of a renewal equation
+that relates the full probability density to the probability densities of partially re-
+flected BMs on either side of the barrier [9]. (The theory of semigroups and resolvent
+operators were used in Ref. [41] to derive a corresponding backward equation.) We
+established the equivalence of the renewal equation with the corresponding single-
+particle diffusion equation, and showed how to solve the former using a combination
+of Laplace transforms and Green’s function methods. We subsequently extended the
+theory to bounded domains and higher spatial dimensions [10]. Formulating interfa-
+cial diffusion in terms of snapping out BM has at least two useful features. First, it
+provides a more general probabilistic framework for modeling semi-permeable mem-
+branes. For example, each round of partially reflected BM on either side of an in-
+terface could be killed according to a non-Markovian process, along analogous lines
+to encounter-based models of surface absorption [31, 32, 7, 8].
+That is, partially
+reflected BM is terminated when its local time at the interface exceeds a random
+threshold that is not exponentially distributed. As we have shown elsewhere, this
+leads to a time-dependent permeability that tends to be heavy-tailed [9, 10]. Sec-
+ond, numerical simulations of snapping out BM generate sample paths that can be
+used to obtain approximate solutions of boundary value problems in the presence of
+semi-permeable interfaces [41].1
+In this paper we develop a multi-layered version of snapping out BM and its as-
+sociated renewal equations for both exponential and non-Markovian killing processes.
+In particular, we consider a single particle diffusing in a finite interval [0, L] that is
+partitioned into m subintervals (or layers) (aj, aj+1), j = 0, . . . , m − 1, with a0 = 0,
+am = L, see Fig. 1.1. The interior interfaces at x = a1, . . . , am−1 are taken to be
+semi-permeable barriers with constant permeabilities κj, j = 1, . . . , m − 1, whereas
+partially reflecting or Robin boundary conditions are imposed at the ends x = 0, L
+with absorption rates 2κ0 and 2κl, respectively.
+(The factors of 2 are convenient
+when formulating snapping out BM.) The diffusion coefficient is also heterogeneous
+with D(x) = Dj for all x ∈ (aj−1, aj). We begin in section 2 by writing down the
+multi-layered diffusion equation, which we formally solve using Laplace transforms
+and an iterative method based on transfer matrices, following along analogous lines
+to Refs. [51, 46]. In section 3, we construct the multi-layered version of snapping out
+BM and write down the corresponding last renewal equation, which relates the full
+1An efficient computational schemes for finding solutions to the single-particle diffusion equation
+in the presence of one or more semi-permeable interfaces has also been developed in terms of under-
+damped Langevin equations [21, 22]. However, this is distinct from snapping out BM, which is an
+exact single-particle realization of diffusion through an interface in the overdamped limit.
+2
+
+x = a1
+x = 0
+x = L
+x = a m-2
+x = a m-1
+x = a 2
+layer 1
+layer 2
+layer m-1
+layer m
+Fig. 1.1. A 1D layered medium consisting of m layers x ∈ (aj, aj+1), j = 0, 1, . . . m − 1, with
+a0 = 0 and am = L. The interior interfaces at x = aj, j = 1, . . . m − 1 act as semi-permeable
+membranes, whereas partially absorbing boundary conditions are imposed on the exterior boundaries
+at x = 0, L.
+probability density to the probability densities of partially reflected BM in each layer.
+We then show how transfer matrices can be used to solve the Laplace transformed
+renewal equation, although the details differ significantly from the iterative solution
+of the diffusion equation. We also prove that the renewal equation and diffusion equa-
+tion are equivalent. This exploits a subtle feature of partially reflected BM, namely,
+the Robin boundary condition is modified when the initial position of the particle is
+on the boundary itself. In section 4 we illustrate the theory by analyzing the first
+passage time (FPT) problem for the particle to escape from one of the ends of the
+domain. The FPT statistics can be analyzed in terms of the small-s behavior of the
+Laplace transformed probability fluxes at the ends x = 0, L, where s is the Laplace
+variable. This means that it is sufficient to solve the multi-layer renewal equation in
+Laplace space, without having to invert the Laplace transformed solution using some
+form of spectral decomposition, for example. Finally, in section 5, we use the renewal
+approach to incorporate a generalization of snapping out BM based on the encounter-
+based method for surface absorption. This is achieved by considering a corresponding
+first renewal equation that relates the full probability density to the FPT densities
+for killing each round of reflected BM.
+2. Single-particle diffusion equation in a 1D layered medium. Before
+developing the more general renewal approach for single-particle diffusion in the multi-
+layer domain of Fig. 1.1, it is useful to briefly consider the classical formulation in
+terms of the diffusion equation with constant permeabilities. Let ρj(x, t) denote the
+probability density of the particle position in the j-th layer. For concreteness, we
+assume that the particle starts in the first layer, that is, x0 ∈ [0, a1], although it
+is straightforward to adapt the analysis to include more general initial conditions,
+see section 3. (For notational convenience, we drop the explicit dependence of ρj on
+x0.) Single-particle diffusion can be represented by the following piecewise system of
+partial differential equations (PDEs):
+∂ρj
+∂t = Dj
+∂2ρj
+∂x2 ,
+x ∈ (aj−1, aj), j = 1, . . . m,
+(2.1a)
+Dj
+∂ρj(x, t)
+∂x
+����
+x=a−
+j
+= Dj+1
+∂ρj+1(x, t)
+∂x
+����
+x=a+
+j
+= κj[ρj+1(a+
+j , t) − ρj(a−
+j , t)],
+j = 1, . . . m − 1,
+(2.1b)
+D1
+∂ρ1(x, t)
+∂x
+����
+x=0
+= 2κ0ρ1(0, t),
+Dm
+∂ρm(x, t)
+∂x
+����
+x=L
+= −2κmρm(L, t),
+(2.1c)
+together with the initial condition ρj(x, t) = δ(x − x0)δj,1. Finally, we denote the
+composite solution on the domain G = ∪m
+j=1[a+
+j−1, a−
+j ] by ρ(x, t). Laplace transforming
+3
+
+equations (2.1a)–(2.1c) gives
+Dj
+∂2�ρj
+∂x2 − s�ρj = −δ(x − x0)δj,1,
+x ∈ (aj−1, aj), j = 1, . . . m,
+(2.2a)
+Dj
+∂�ρj(x, s)
+∂x
+����
+x=a−
+j
+= Dj+1
+∂�ρj+1(x, s)
+∂x
+����
+x=a+
+j
+= κj[�ρj+1(a+
+j , s) − �ρj(a−
+j , s)],
+j = 1, . . . m − 1,
+(2.2b)
+D1
+∂�ρ1(x, s)
+∂x
+����
+x=0
+= 2κ0�ρ1(0, s),
+Dm
+∂�ρm(x, s)
+∂x
+����
+x=L
+= −2κm�ρm(L, s).
+(2.2c)
+Equations (2.2a)–(2.2b) can be solved using transfer matrices along similar lines to
+Refs. [51, 46]. We sketch the steps here.
+First, note that for all 1 ≤ j ≤ m, equation (2.2a) has the general solution
+�ρj(x, s) = Al
+j(s) cosh(
+�
+s/Dj[x − aj−1]) + Bl
+j(s) sinh(
+�
+s/Dj[x − aj−1])
+(2.3)
+or, equivalently
+�ρj(x, s) = Ar
+j(s) cosh(
+�
+s/Dj[x − aj]) + Br
+j (s) sinh(
+�
+s/Dj[x − aj]).
+(2.4)
+For 1 < j ≤ m, the coefficients Al
+j, Bl
+j are related to Ar
+j, Br
+j according to
+� Ar
+j
+Br
+j
+�
+= Uj(s)
+� Al
+j
+Bl
+j
+�
+,
+Uj(s) =
+�
+cosh(
+�
+s/DjLj)
+sinh(
+�
+s/DjLj)
+sinh(
+�
+s/DjLj)
+cosh(
+�
+s/DjLj)
+�
+,
+(2.5)
+where Lj = aj − aj−1 is the length of the j-th layer. The presence of the Dirac delta
+function for j = 1 means that the relationship between the coefficients (Ar
+1(s), Br
+1(s))
+and (Al
+1(s), Bl
+1(s)) is determined by imposing the continuity condition �ρ1(x+
+0 , s) =
+�ρ1(x−
+0 , s) and the flux discontinuity condition ∂x�ρ1(x+
+0 , s) − ∂x�ρ1(x−
+0 , s) = −1/D1.
+This yields the result
+� Ar
+1
+Br
+1
+�
+= U1(s)
+� Al
+1
+Bl
+1
+�
++
+1
+√sD1
+�
+sinh(
+�
+s/D1[x0 − a1])
+− cosh(
+�
+s/D1[x0 − a1])
+�
+.
+(2.6)
+Given the relationships �ρj(aj, s) = Ar
+j(s), �ρj(aj−1, s) = Al
+j(s), Dj∂x�ρj(aj, s) =
+�sDjBr
+j (s) and Dj∂x�ρj(aj−1, s) = �sDjBl
+j(s), the boundary conditions (2.2b) can
+be written in the form
+�
+sDjBr
+j (s) =
+�
+sDj+1Bl
+j+1(s) = κj[Al
+j+1(s) − Ar
+j(s)].
+(2.7)
+That is, for 1 ≤ j < m,
+� Al
+j+1
+Bl
+j+1
+�
+= Vj(s)
+� Ar
+j
+Br
+j
+�
+,
+Vj(s) =
+
+
+1
+�
+sDj/κj
+0
+�
+Dj/Dj+1
+
+ .
+(2.8)
+Iterating equations (2.5) and (2.8) for m ≥ 2, we have
+� Ar
+m
+Br
+m
+�
+= Mm(s)
+� Ar
+1
+Br
+1
+�
+,
+(2.9)
+4
+
+with
+M2(s) = U2(s)V1(s),
+Mm(s) = Um(s)
+
+
+m−1
+�
+j=2
+Vj(s)Uj(s)
+
+ V1(s) for m ≥ 3.
+(2.10)
+Hence, we have shown how the solution in any layer can be expressed in terms of
+the two unknown coefficients Al
+1(s) and Bl
+1(s). The latter are then determined by
+imposing the Robin boundary conditions at x = 0, L:
+�
+sD1Bl
+1(s) = 2κ0Al
+1(s),
+�
+sDmBr
+m(s) = −2κmAr
+m(s).
+(2.11)
+3. Snapping out BM in a 1D layered medium. We now develop an al-
+ternative formulation of multi-layer diffusion, which is based on a generalization of
+1D snapping out BM for a single semi-permeable interface [41, 7]. In particular, we
+construct a renewal equation that relates ρ(x, t) on G to the probability densities of
+partially reflected BM in each of the layers [aj−1, aj], j = 1, . . . , m.
+3.1. Single layer with partially reflecting boundaries. Consider BM in the
+interval [aj−1, aj] with both ends totally reflecting. Let X(t) ∈ [aj−1, aj] denote the
+position of the Brownian particle at time t and introduce the pair of Brownian local
+times
+ℓ+
+j−1(t) = lim
+h→0
+Dj
+h
+ˆ t
+0
+H(aj−1 + h − X(τ))dτ,
+(3.1a)
+ℓ−
+j (t) = lim
+h→0
+Dj
+h
+ˆ t
+0
+H(aj − h − X(τ))dτ,
+(3.1b)
+where H is the Heaviside function. Note that ℓ+
+j−1(t) determines the amount of time
+that the Brownian particle spends in a neighborhood to the right of x = aj−1 over the
+interval [0, t]. Similarly, ℓ−
+j (t) determines the amount of time spent in a neighborhood
+to the left of x = aj. (The inclusion of the factor Dj means that the local times have
+units of length.) It can be shown that the local times exist and are nondecreasing,
+continuous function of t [38]. The corresponding stochastic differential equation (SDE)
+for X(t) is given by the Skorokhod equation
+dX(t) =
+�
+2DjdW(t) + dℓ+
+j−1(t) − dℓ−
+j (t).
+(3.2)
+Roughly speaking, each time the particle hits one of the ends it is given an impul-
+sive kick back into the bulk domain. It can be proven that the probability density
+for particle position evolves according to the single-particle diffusion equation with
+Neumann boundary conditions at both ends.
+Partially reflected BM can now be defined by introducing a pair of exponentially
+distributed independent random local time thresholds �ℓ+
+j−1 and �ℓ−
+j such that
+P[�ℓ+
+j−1 > ℓ] = e−2κj−1ℓ/Dj,
+P[�ℓ−
+j > ℓ] = e−2κjℓ/Dj.
+(3.3)
+The stochastic process is then killed as soon as one of the local times exceeds its
+corresponding threshold, which occurs at the stopping time Tj = min{τ −
+j , τ +
+j } with
+τ +
+j = inf{t > 0 : ℓ+
+j−1(t) > �ℓ+
+j−1},
+τ −
+j = inf{t > 0 : ℓ−
+j (t) > �ℓ−
+j }.
+(3.4)
+5
+
+local time lj-1
+Tj
+x
+time t
+x0
+interface
+aj-1
+aj
+totally reflecting
+*
+Fig. 3.1. Sketch of a course-grained trajectory of a Brownian particle in the interval [aj−1, aj]
+with a partially reflecting boundary at x = aj−1 and a totally reflecting boundary at x = aj. The
+particle is absorbed as soon as the time ℓj−1(t) spent in a boundary layer around x = aj−1 exceeds
+an exponentially distribution threshold �ℓj−1, which occurs at the stopping time Tj.
+In Fig. 3.1 we illustrate the basic construction using a simplified version of partially
+reflected BM in which x = aj−1 is partially reflecting (0 < κj−1 < ∞) but x = aj is
+totally reflecting (κj = 0).
+It can be shown that the probability density for particle position prior to absorp-
+tion at one of the ends (see also section 5),
+pj(x, t|x0)dx = P[x ≤ X(t) < x + dx, t < Tj|X0 = x0], x ∈ [aj−1, aj],
+(3.5)
+satisfies the single-particle diffusion equation (Fokker-Planck equation) with Robin
+boundary conditions at x = aj−1, aj [25, 48, 45, 5, 28]:
+∂pj(x, t|x0)
+∂t
+= Dj
+∂2pj(x, t|x0)
+∂x2
+,
+aj−1 < x0, x < aj,
+(3.6a)
+Dj∂xpj(aj−1, t|x0) = 2κj−1p(aj−1, t|x0),
+(3.6b)
+Dj∂xpj(aj, t|x0) = −2κjp(aj, t|x0),
+(3.6c)
+and pj(x, 0|x0) = δ(x − x0).
+It is convenient to Laplace transform with respect to t, which gives
+Dj
+∂2�pj(x, s|x0)
+∂x2
+− s�pj(x, s|x0) = −δ(x − x0),
+aj−1 < x0, x < aj
+(3.7a)
+Dj∂x�pj(aj−1, s|x0) = 2κj−1�pj(aj−1, s|x0),
+(3.7b)
+Dj∂x�p(aj, s|x0) = −2κj�pj(aj, s|x0).
+(3.7c)
+We can identify �pj(x, s|x0) as the Green’s function of the modified Helmholtz equation
+with Robin boundary conditions at x = aj−1, aj:
+�pj(x, s|x0) =
+
+
+
+AjFj(x, s)F j(x0, s),
+aj−1 ≤ x ≤ x0
+AjFj(x0, s)Fj(x, s),
+x0 ≤ x ≤ aj
+(3.8)
+6
+
+where
+Fj(x, s) =
+�
+sDj cosh(
+�
+s/Dj[x − aj−1]) + 2κj−1 sinh(
+�
+s/Dj[x − aj−1]),
+(3.9a)
+Fj(x, s) =
+�
+sDj cosh(
+�
+s/Dj[aj − x]) + 2κj sinh(
+�
+s/Dj[aj − x]),
+(3.9b)
+Aj =
+1
+�sDj
+1
+2(κj−1 + κj)
+�
+sDj cosh(
+�
+s/DjLj) + [sDj + 4κj−1κj] sinh(
+�
+s/DjLj)
+,
+(3.9c)
+and Lj = aj − aj−1 is the width of the layer. It can be checked that the Robin
+boundary conditions are satisfied at x = aj−1, aj for all aj−1 < x0 < aj. However, for
+x0 = aj−1, aj, we have
+Dj∂x�pj(aj−1, s|aj−1) = 2κj−1�p(aj−1, s|aj−1) − 1,
+(3.10a)
+Dj∂x�pj(aj, s|aj) = −2κj�pj(aj, s|aj) + 1.
+(3.10b)
+In other words,
+lim
+ǫ→0
+�
+∂
+∂x
+����
+x=aj
+�pj(x, s|aj − ǫ)
+�
+̸=
+∂
+∂x
+����
+x=aj
+�
+lim
+ǫ→0 �pj(x, s|aj − ǫ)
+�
+(3.11)
+etc. The modification of the Robin boundary condition when the particle starts at
+the barrier plays a significant role in establishing the equivalence of snapping out BM
+with single particle diffusion in a multi-layered medium (see section 3.3).
+3.2. Last renewal equation. We now construct snapping out BM in the multi-
+layered domain shown in Fig. 1.1 by sewing together multiple rounds of reflected BM.
+For the moment, assume that the exterior boundaries are totally reflecting. For each
+interface we introduce a pair of local time ℓ±
+j and a corresponding pair of independent
+exponentially distributed thresholds �ℓ±
+j with rates 2κj, j = 1, . . . , m − 1. Suppose
+that the particle starts at x = x0 in the first layer. It realizes positively reflected
+BM until its local time ℓ−
+1 (t) at x = a1 exceeds the random threshold �ℓ−
+1 with rate
+2κ1. The process immediately restarts as a new reflected BM with probability 1/2 in
+either [0, a1] or [a1, a2]. If the particle is in layer 2, then the reflected BM is stopped
+as soon as one of the local times (ℓ+
+1 (t), ℓ−
+2 (t)) exceeds its corresponding threshold.
+Each time the BM is restarted all local times are reset to zero. Finally, taking the
+exterior boundaries to be partially reflecting, we introduce an additional pair of local
+times, ℓ0(t), ℓm(t) for the external boundaries at x = 0, L, and a corresponding pair of
+exponentially distributed random thresholds �ℓ0, �ℓm with rates 2κ0, 2κm, respectively.
+The stochastic process is then permanently terminated at the stopping time
+T = min{T0, Tm},
+Tk = inf{t > 0 : ℓk(t) > �ℓk}, k = 0, m.
+(3.12)
+We illustrate the basic construction in Fig.
+3.2 in the simplified case of a single
+semi-permeable interface at x = aj and totally reflecting boundaries x = aj−1 and
+x = aj+1. The statistics of diffusion across the interface can be captured by sewing
+together successive rounds of partially reflected BM in the intervals [aj−1, a−
+j ] and
+[a+
+j , aj+1] with each round killed according to an exponentially distributed local time
+threshold, and the new domain selected with probability 1/2.
+7
+
+x = aj
+x = aj-1
+reflecting
+x0
+(a)
+Robin
+x = aj+1
+Robin
+reflecting
+reflecting
+reflecting
+x = aj
+x = aj-1
+x = aj+1
+x0
+reflecting
+reflecting
+x = aj
+x = aj-1
+x = aj+1
+(b)
+(a)
+(c)
+Fig. 3.2. Decomposition of snapping out BM on the interval [aj−1, aj+1] with reflecting bound-
+ary conditions at the ends and a semi-permeable barrier at x = aj. (a) Diffusion across the interface.
+(b) Partially reflected BM in [a+
+j , aj+1]. (c) Partially reflected BM in [aj−1, a−
+j ].
+Consider a general initial probability density φ(x0) with x0 ∈ G and set
+ρj(x, t) =
+ˆ
+G
+ρj(x, t|x0)φ(x0)dx0,
+pj(x, t) =
+ˆ
+G
+pj(x, t|x0)φ(x0)dx0.
+(3.13)
+Following our previous work on snapping out BM for single semi-permeable interfaces
+[9, 10], the renewal equation for the j-th interior layer, j = 2, . . . , m − 1, takes the
+form
+ρj(x, t) = pj(x, t) + κj−1
+ˆ t
+0
+pj(x, τ|aj−1)[ρj−1(a−
+j−1, t − τ) + ρj(a+
+j−1, t − τ)]dτ
++ κj
+ˆ t
+0
+pj(x, τ|aj)[ρj(a−
+j , t − τ) + ρj+1(a+
+j , t − τ)]dτ
+(3.14a)
+for all x ∈ (a+
+j−1, a−
+j ), with the probability density pj(x, τ|y) given by the solution
+to equations (3.6). The first term pj(x, t) on the right-hand side of equation (3.14a)
+represents all trajectories that reach x at time t without ever being absorbed by the
+interfaces at x = a+
+j−1, a−
+j . The first integral on the right-hand side sums over all
+trajectories that were last absorbed (stopped) at time t − τ by hitting the interface
+at x = aj−1 from either the left-hand or right-hand side and then switching with
+probability 1/2 to BM in the j-th layer such that it is at position x ∈ (a+
+j−1, a−
+j ) at
+time t. Since the particle is not absorbed over the interval (t − τ, t], the probability of
+reaching x is pj(x, τ|aj−1). In addition, the probability that the last stopping event
+occurred in the interval (t−τ, t−τ+dτ) irrespective of previous events is 2κj−1dτ. (We
+see that the inclusion of the factor 2 in the definition of the permeability cancels the
+probability factor of 1/2.) The second integral has the corresponding interpretation
+for trajectories that were last stopped by hitting the interface at x = aj. In the case
+of the end layers, we have
+ρ1(x, t) = p1(x, t) + κ1
+ˆ t
+0
+p1(x, τ|a1)[ρ1(a−
+1 , t − τ) + ρ2(a+
+1 , t − τ)]dτ,
+(3.14b)
+ρm(x, t) = pm(x, t)
+(3.14c)
++κm−1
+ˆ t
+0
+pm(x, τ|am−1)[ρm−1(a−
+m−1, t − τ) + ρm(a+
+m−1, t − τ)]dτ.
+8
+
+Note that there is only a single integral contribution in the end layers since only one
+of the boundaries is semi-permeable. One interesting difference between the renewal
+equation formulation and the PDE analyzed in section 2 is that the exterior boundary
+conditions are already incorporated into the solutions p1(x, t|x0) and pm(x, t|x0), so
+that they do not have to be imposed separately.
+Given the fact that the renewal equations (3.14a)–(3.14c) are convolutions in time,
+it is convenient to Laplace transform them by setting �ρj(x, s) =
+´ ∞
+0
+e−stρj(x, t)dt etc.
+This gives
+�ρ1(x, s) = �p1(x, s) + κ1�p1(x, s|a1)Σ1(s), x ∈ [0+, a−
+1 ],
+(3.15a)
+�ρj(x, s) = �pj(x, s) + κj−1�pj(x, s|aj−1)Σj−1(s) + κj �pj(x, s|aj)Σj(s), x ∈ [a+
+j−1, a−
+j ],
+1 < j < m,
+(3.15b)
+�ρm(x, s) = �pm(x, s) + κm−1�pm(x, s|am−1)Σm−1(s), x ∈ [a+
+m−1, L−],
+(3.15c)
+where
+Σj(s) = �ρj(a−
+j , s) + �ρj+1(a+
+j , s).
+(3.16)
+The functions Σj(s) can be determined self-consistently by setting x = a±
+k for k =
+1, . . . , m−1 and performing various summations. More specifically, substituting equa-
+tion (3.15b) into the right-hand side of (3.16) for 1 < j < m gives
+Σj(s) = Σp
+j(s) + κj−1�pj(aj, s|aj−1)Σj−1(s) + κj �pj(aj, s|aj)Σj(s)
++ κj �pj+1(aj, s|aj)Σj(s) + κj+1�pj+1(aj, s|aj+1)Σj+1(s)
+(3.17a)
+for 1 < j < m − 1 and Σp
+j(s) ≡ �pj(aj, s) + �pj+1(aj, s). On the other hand, equations
+(3.15b) and (3.15a) for j = 2 implies that
+Σ1(s) = Σp
+1(s) + κ1�p1(a1, s|a1)Σ1(s)
++ κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s),
+(3.17b)
+while equations (3.15c) and (3.15a) for j = m − 1 yields
+Σm−1(s) = Σp
+m−1(s) + κm−1�pm(am−1, s|am−1)Σm−1(s)
+(3.17c)
++ κm−2�pm−1(am−1, s|am−2)Σm−2(s) + κm−1�pm−1(am−1, s|am−1)Σm−1(s).
+Equations (3.17a)–(3.17c) can be rewritten in the more compact matrix form
+m−1
+�
+k=1
+Θjk(s)Σk(s) = −Σp
+j(s),
+(3.18)
+where Θ(s) is a tridiagonal matrix with non-zero elements
+Θj,j(s) = dj(s) ≡ κj[�pj+1(aj, s|aj) + �pj(aj, s|aj)] − 1, j = 1, . . . m − 1,
+(3.19a)
+Θj,j−1(s) = cj(s) ≡ κj−1�pj(aj, s|aj−1),
+j = 2, . . . m − 1,
+(3.19b)
+Θj,j−1(s) = bj(s) ≡ κj+1�pj+1(aj, s|aj+1),
+j = 1, . . . , m − 2.
+(3.19c)
+Assuming that the matrix Θ(s) is invertible, we obtain the formal solution
+Σj(s) = −
+m−1
+�
+k=1
+Θ−1
+jk (s)Σp
+k(s).
+(3.20)
+9
+
+Substituting into equations (3.15a)–(3.15c) gives
+�ρj(x, s) = �pj(x, s) −
+m−1
+�
+k=1
+�
+κj−1�pj(x, s|aj−1)Θ−1
+j−1,k(s) + κj �pj(x, s|aj)Θ−1
+jk (s)
+�
+× [�pj(aj, s) + �pj+1(aj+1, s)].
+(3.21)
+An alternative way to solve for Σj(s) is to use transfer matrices analogous to the
+analysis of the PDE in section 2. For simplicity, suppose that the particle starts in
+the first layer at a point x0 ∈ [0, a1] so that �pj(x, s) = �p1(x, s|x0)δj,1. It follows that
+equations (3.17a)–(3.17c) can be rewritten in the iterative form
+�
+Σj
+Σj+1
+�
+= Wj(s)
+� Σj−1
+Σj
+�
+,
+Wj(s) =
+
+
+0
+1
+−cj(s)
+bj(s)
+−dj(s)
+bj(s)
+
+
+(3.22)
+for 1 < j < m − 1. In particular,
+� Σm−2
+Σm−1
+�
+= N(s)
+� Σ1
+Σ2
+�
+,
+N(s) =
+m−2
+�
+k=2
+Wk(s),
+(3.23)
+with, see equation (3.17b),
+Σ2(s) = −
+1
+b1(s) (�p1(a1, s|x0) + d1(s)Σ1(s)) .
+(3.24)
+Finally, having determined Σ2, . . . , Σm−1 in terms of Σ1, we can calculate Σ1 by
+imposing equation (3.17c), after rewriting it in the more compact form
+Σm−2(s) = −dm−1(s)
+cm−2(s) Σm−1(s).
+(3.25)
+We thus obtain the following self-consistency condition for Σ1:
+�
+1, dm−1(s)
+cm−2(s)
+�
+N(s)
+�
+Σ1(s)
+−
+1
+b1(s) (�p1(a1, s|x0) + d1(s)Σ1(s))
+�
+= 0.
+(3.26)
+3.3. Equivalence of the renewal and diffusion equations. We now have
+two alternative methods of solution in Laplace space, one based on the diffusion
+equations (2.2a)–(2.2c) and the other based on the renewal equations (3.15a)–(3.15c).
+Both methods involve transfer matrices that can be iterated to express the solution in
+the final layer in terms of the solution in the first layer. It is useful to check that the
+renewal equations (3.15a)–(3.15c) are indeed equivalent to the Laplace transformed
+diffusion equations (2.1a)–(2.1c).
+(This is simpler than showing that the iterative
+solutions are equivalent.) Clearly, the composite density �ρ(x, s) satisfies the diffusion
+equation in the bulk and the exterior boundary conditions, so we only have to check
+the boundary conditions across the interior interfaces. First, differentiating equations
+(3.15a) and (3.15b) for j = 2 with respect to x and setting x = a±
+1 gives
+∂x�ρ1(a−
+1 , s) = ∂x�p1(a1, s|x0) + κ1∂x�p1(a1, s|a1)Σ1(s),
+(3.27a)
+∂x�ρ2(a+
+1 , s) = κ1∂x�p2(a1, s|a1)Σ1(s) + κ2∂x�p2(a1, s|a2)Σ2(s).
+(3.27b)
+10
+
+Imposing the Robin boundary condition (3.6) implies that
+D1∂x�p1(a1, s|x0) = −2κ1�p(a1, s|x0),
+D2∂x�p2(a1, s|a2) = 2κ1�p(a1, s|a2).
+On the other hand, equations (3.10a) and (3.10b) yield
+D1∂x�p1(a1, s|a1) = −2κ1�p(a1, s|a1) + 1,
+D2∂x�p2(a1, s|a1) = 2κ1�p2(a1, s|a1) − 1.
+Substituting into equations (3.27a) and (3.27b), we have
+D1∂x�ρ1(a−
+1 , s) = −2κ1�p1(a1, s|x0) − κ1[2κ1�p1(a1, s|a1) − 1]Σ1(s),
+(3.28a)
+D2∂x�ρ2(a+
+1 , s) = κ1[2κ1�p2(a1, s|a1) − 1]Σ1(s) + 2κ2κ1�p2(a1, s|a2)Σ2(s).
+(3.28b)
+Subtracting equations (3.28a) and (3.28b), and using equation (3.17b) implies that
+D2∂x�ρ2(a+
+1 , s) − D1∂x�ρ1(a−
+1 , s) = 2κ1
+�
+κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s)
++ �p1(a1, s|x0) + κ1�p1(a1, s|a1)Σ1(s) − Σ1(s)
+�
+= 0.
+(3.29)
+Similarly, adding equations (3.28a) and (3.28b) gives
+D2∂x�ρ2(a+
+1 , s) + D1∂x�ρ1(a−
+1 , s)] = 2κ1
+�
+κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s)
+− �p1(a1, s|x0) − κ1�p1(a1, s|a1)Σ1(s)
+�
+.
+(3.30)
+On the other hand setting x = a±
+1 in equations (3.15a) and (3.15b) for j = 2 shows
+that
+�ρ1(a−
+1 , s) = �p1(a1, s|x0) + κ1�p1(a1, s|a1)Σ1(s),
+(3.31a)
+�ρ2(a+
+1 , s) = κ1�p2(a1, s|a1)Σ1(s) + κ2�p2(a1, s|a2)Σ2(s).
+(3.31b)
+Hence, we obtain the expected semi-permeable boundary conditions at x = a1,
+D2∂x�ρ2(a+
+1 , s) = D1∂x�ρ1(a−
+1 , s) = κ1[�ρ2(a+
+1 , s) − �ρ1(a−
+1 , s)].
+(3.32)
+A similar analysis can be carried out at the other interfaces.
+We have thus established the equivalence of the renewal equations (3.14a)–(3.14c)
+and the Laplace transformed diffusion equations (2.2a)–(2.2c). Hence, snapping out
+BM X(t) on G is the single-particle realization of the stochastic process whose prob-
+ability density evolves according to the multi-layer diffusion equation.
+4. First-passage time problem. One of the useful features of working in
+Laplace space is that one can solve various first passage time problems without having
+to calculate any inverse Laplace transforms. We will illustrate this by considering the
+escape of the Brownian particle from one of the ends at x = 0, L. For simplicity,
+we again assume that the particle starts in the first layer. Let Q(x0, t) denote the
+survival probability that a particle starting at x0 ∈ (0, a1) has not been absorbed at
+either end over the interval [0, t). It follows that
+Q(x0, t) =
+ˆ L
+0
+ρ(x, t)dx =
+m−1
+�
+j=0
+ˆ aj+1
+aj
+ρj(x, t)dx.
+(4.1)
+11
+
+(We drop the explicit dependence of ρ and ρj on the initial position x0 for notational
+convenience.) Differentiating both sides of equation (4.1) with respect to t and using
+equations (2.1a)–(2.1c) shows that
+dQ(x0, t)
+dt
+=
+m
+�
+j=1
+ˆ aj
+aj−1
+∂ρj(x, t)
+∂t
+dx =
+m
+�
+j=1
+ˆ aj
+aj−1
+Dj
+∂2ρj(x, t)
+∂x2
+dx
+=
+m
+�
+j=1
+Dj
+�∂ρj(aj, t)
+∂x
+− ∂ρj(aj−1, t)
+∂x
+�
+= Dm
+∂ρm(am, t)
+∂x
+− D1
+∂ρ1(a0, t)
+∂t
+≡ −Jm(x0, t) − J0(x0, t).
+(4.2)
+We have used flux continuity across each interior interface so that the survival proba-
+bility decreases at a rate equal to the sum of the outward fluxes at the ends x = 0, L,
+which are denoted by J0 and JL respectively. Laplace transforming equation (4.2)
+and imposing the initial condition Q(x0, 0) = 1 gives
+s �Q(x0, s) − 1 = − �J0(x0, s) − �JL(x0, s).
+(4.3)
+Assuming that κ0+κm > 0, the particle is eventually absorbed at one of the ends with
+probability one, which means that limt→∞ Q(x0, t) = lims→0 s �Q(x0, s) = 0. Hence,
+�J0(x0, 0)+ �Jm(x0, 0) = 1. Let π0(x0) and πL(x0) denote the splitting probabilities for
+absorption at x = 0 and x = L, respectively, and denote the corresponding conditional
+MFPTs by T0(x0) and TL(x0). It can then be shown that
+π0(x0) = �J0(x0, 0),
+πL(x0) = �JL(x0, 0),
+(4.4)
+and
+π0(x0)T0(x0) = − ∂
+∂s
+�J0(x0, s)
+����
+s=0
+,
+πL(x0)TL(x0) = − ∂
+∂s
+�JL(x0, s)
+����
+s=0
+.
+(4.5)
+Hence, analyzing the statistics of escape from the domain [0, L] reduces to determining
+the small-s behavior of the solutions ∂x�ρ1(0, s) and ∂x�ρm(L, s). We will proceed using
+the renewal equation approach of section 3.
+4.1. Identical layers. A considerable simplification of the iterative equation
+(3.22) occurs in the case of identical layers with Dj = D, κj = κ and aj = ja for all
+j = 1, . . . , m. The solution (3.8) for partially reflected BM is now the same in each
+layer. That is, �pj(x, s|x0) = �p(x − (j − 1)a, s|x0 − (j − 1)a) for x, x0 ∈ [aj−1, aj] with
+�p(x, s|x0) =
+
+
+
+AF(x, s)F(x0, s),
+a ≤ x ≤ x0
+AF(x0, s)F(x, s),
+x0 ≤ x ≤ a
+,
+(4.6)
+F(x, s) =
+√
+sD cosh(
+�
+s/D[x − a]) + 2κ sinh(
+�
+s/D[x − a]),
+(4.7a)
+F(x, s) =
+√
+sD cosh(
+�
+s/D[a − x]) + 2κ sinh(
+�
+s/D[a − x]),
+(4.7b)
+A =
+1
+√
+sD
+1
+4κ
+√
+sD cosh(
+�
+s/Da) + [sD + 4κ2] sinh(
+�
+s/Da)
+.
+(4.7c)
+12
+
+In addition equations (3.22)–(3.26) for identical layers imply that
+N(s) = W(s)m−3,
+W(s) =
+�
+0
+1
+−1
+−g(a, s)
+�
+,
+(4.8)
+with
+g(y, s) ≡ 2κ�p(a, s|y) − 1
+κ�p(a, s|0)
+= 2g0(y, s) − g1(s),
+(4.9)
+where
+g0(y, s) ≡ �p(a, s|y)
+�p(a, s|0) =
+√
+sD cosh(
+�
+s/Dy) + 2κ sinh(
+�
+s/Dy)
+√
+sD
+,
+(4.10a)
+g1(s) ≡
+1
+κ�p(a, s|0) = 4κ
+√
+sD cosh(
+�
+s/Da) + [sD + 4κ2] sinh(
+�
+s/Da)
+κ
+√
+sD
+.
+(4.10b)
+The matrix W(s) can be diagonalized according to
+W(s) = UWd(s)U†,
+Wd(s) = diag(λ+(s), λ−(s)),
+(4.11)
+with
+λ±(s) = −g(a, s) ±
+�
+g(a, s)2 − 4
+2
+,
+λ+ + λ− = −g,
+λ+λ− = 1,
+(4.12)
+and
+U =
+�
+1
+1
+λ+
+λ−
+�
+,
+U† =
+�
+1
+1−λ2
++
+−
+λ+
+1−λ2
++
+1
+1−λ2
+−
+−
+λ−
+1−λ2
+−
+�
+,
+U†U = UU† =
+� 1
+0
+0
+1
+�
+.
+(4.13)
+Substituting (4.11) into (3.23) and (3.26) gives
+�
+1, g(a, s)
+�
+U(s)Wd(s)m−3U†(s)
+�
+Σ1(s)
+Σ2(s)
+�
+= 0,
+(4.14)
+and
+Σm−1(s) =
+�
+0, 1
+�
+U(s)Wd(s)m−3U†(s)
+� Σ1(s)
+Σ2(s)
+�
+,
+(4.15)
+with
+Σ2(s) = −g0(x0, s)
+κ
+− g(a, s)Σ1(s).
+(4.16)
+In addition, from equations (3.15a) and (3.15c) we have
+�J0(x0, s) = f(x0, s) + κf(a, s)Σ1(s),
+�JL(x0, s) = κf(a, s)Σm−1(s),
+(4.17)
+where
+D∂x�p(0, s|y) = f(y, s) ≡ 2κ[
+√
+sD cosh(
+�
+s/D[a − y]) + 2κ sinh(
+�
+s/D[a − y])]
+4κ
+√
+sD cosh(
+�
+s/Da) + [sD + 4κ2] sinh(
+�
+s/Da)
+,
+(4.18)
+13
+
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+κ = 100
+κ = 10
+κ = 1
+κ = 0.1
+splitting probability
+initial position x0
+0
+0.2
+0.5
+0.8
+1
+0.3
+0.7
+0.1
+0.4
+0.6
+0.9
+Fig. 4.1. Splitting probabilities for escape from a three-layer, homogeneous medium. Plots of
+π0(x0) and πL(x0) as a function of x0 for various rates κ. Other parameters are D = 1 and a = 1.
+and D∂x�p(L, s|L − a) = −f(a, s).
+For the sake of illustration, consider three layers (m = 3). Equation (4.14) implies
+that for κ > 0
+Σ1(s) = 1
+κ
+g(a, s)g0(x0, s)
+1 − g(a, s)2
+,
+Σ2(s) = − 1
+κ
+g0(x0, s)
+1 − g(a, s)2 .
+(4.19)
+Using the limits
+lim
+s→0 g0(y, s) = 1 + 2κy/D,
+lim
+s→0 g1(s) = 4(1 + κa/D),
+(4.20)
+lim
+s→0 g(y, s) = −2(1 + 2κ[a − y]/D),
+lim
+s→0 f(y, s) = (1 + 2κ[a − y]/D)
+2(1 + κa/D)
+,
+(4.21)
+we can thus determine the splitting probabilities π0(x0) and πL(x0). Example plots
+of π0(x0) and πL(x0) as a function of x0 ∈ [0, a] are shown in Fig. 3.2 for a = D = 1.
+It can be checked that π0(x0) + πL(x0) = 1 for all x0. Moreover, in the limit κ → ∞,
+we see that π0(0) → 1 and πL(0) → 0 as expected. Also note that for x0 < 1/2
+(x0 > 1/2), π0(x0) is an increasing (a decreasing) function of κ.
+4.2. Large number of layers (m → ∞). For a large number of layers (m ≫ 1)
+we have
+Wm−3
+d
+=
+�
+λm−3
++
+0
+0
+λm−3
+−
+�
+= λm−3
+−
+�
+ǫ
+0
+0
+1
+�
+,
+ǫ =
+�λ+
+λ−
+�m−3
+(4.22)
+with |ǫ| ≪ 1 since |λ−| > |λ+|. It follows that
+N(s) = U(s)Wd(s)m−3U†(s) = λ−(s)m−3{M0(s) + ǫM1(s)},
+(4.23)
+where
+M0 =
+1
+1 − λ2
+−
+�
+1
+−λ−
+λ−
+−λ2
+−
+�
+,
+M1 =
+1
+1 − λ2
++
+�
+1
+−λ+
+λ+
+−λ2
++
+�
+.
+(4.24)
+14
+
+The next step is to introduce the series expansions
+Σj(s) = Σ(0)
+j (s) + ǫΣ(1)
+j (s) + O(ǫ2),
+j = 1, 2,
+(4.25)
+with
+Σ(0)
+2 (s) = −g0(x0, s)
+κ
+− g(a, s)Σ(0)
+1 (s),
+Σ(n)
+2 (s) = −g(a, s)Σ(n)
+1 (s) for n ≥ 1. (4.26)
+Substituting equations (4.23) and (4.25) into (4.14) and collecting terms in powers of
+ǫ gives the O(1) and O(ǫ) equations
+�
+1, g(a, s)
+�
+M0(s)
+�
+Σ(0)
+1 (s)
+Σ(0)
+2 (s)
+�
+= 0,
+(4.27a)
+�
+1, g(a, s)
+� �
+M0(s)
+�
+Σ(1)
+1 (s)
+Σ(1)
+2 (s)
+�
++ M1(s)
+�
+Σ(0)
+1 (s)
+Σ(0)
+2 (s)
+��
+= 0.
+(4.27b)
+Equation (4.27a) has the solution
+Σ(0)
+1 (s) = −
+λ−(s)g0(x0, s)
+κ(1 + g(a, s)λ−(s)) = g0(x0, s)
+κλ−(s) ,
+(4.28)
+so that
+Σ(1)
+1 (s) = λ+(s)4
+λ−(s)4
+1 − λ+(s)2
+1 − λ−(s)2
+�
+Σ(0)
+1
+− g0(x0, s)
+κλ+(s)
+�
+.
+(4.29)
+Finally,
+Σm−1(s) = (0, 1)N(s)
+� Σ1(s)
+Σ2(s)
+�
+(4.30)
+= λ−(s)m−3(0, 1){M0(s) + ǫM1(s)}
+�
+Σ(0)
+1 (s) + ǫΣ(1)
+1 (s) + O(ǫ2)
+Σ(0)
+2 (s) + ǫΣ(1)
+1 (s) + O(ǫ2)
+�
+= λ+(s)m−3(0, 1)
+�
+M0(s)
+�
+Σ(1)
+1 (s)
+Σ(1)
+2 (s)
+�
++ M1(s)
+�
+Σ(0)
+1 (s)
+Σ(0)
+2 (s)
+�
++ O(ǫ)
+�
+.
+We have used the fact the O(1) solution (Σ(0)
+1 , Σ(0)
+2 )⊤ is actually a null-vector of the
+matrix M0 so the leading contribution to Σm−1(s) is proportional to ǫλ−(s)m−3 =
+λ+(s)m−3. Hence, Σm−1(s) → 0 as m → ∞ due to the fact that |λ+(s)| < 1 for all s.
+Equations (4.4) and (4.17) then imply that πm(x0) → 0 as m → ∞, with the rate of
+decay determined by λ+(0)m−3.
+5. Generalized model of multi-layer diffusion. The analysis of the FPT
+problem in section 4 could also have been carried out using the solution of the diffusion
+equation constructed in section 2. However, one advantage of the renewal approach
+is that it is based on snapping out BM, which can be used to generate sample paths
+of single-particle diffusion in a multi-layer medium. Rather than exploring numerical
+aspects here, we consider another advantage of the renewal approach, namely, it
+supports a more general model of semi-permeable membranes. This is based on an
+extension of snapping out BM that modifies the rule for killing each round of reflected
+BM within a layer. We proceed by applying the encounter-based model of absorption
+[31, 32, 7, 8] to reflected BM in each of the layers separately.
+15
+
+5.1. Local time propagator for a single layer. As we mentioned in sec-
+tion 3.1, partially reflected BM in an interval can be implemented by introducing
+exponentially distributed local time thresholds at either end of the interval, which
+then determine when reflected BM is killed. Here we generalize the killing mecha-
+nism. Given the local times (3.1a) and (3.1b) of the j-th layer with totally reflecting
+boundaries, the local time propagator is defined according to [31]
+Pj(x, ℓ, ℓ′, t|x0)dx dℓ ℓ′
+= P[x < X(t) < x + dx, ℓ < ℓ+
+j−1 < ℓ + dℓ, ℓ′ < ℓ−
+j < ℓ′ + dℓ′|X(0) = x0].
+(5.1)
+Next, for each interface we introduce a pair of independent identically distributed
+random local time thresholds �ℓ±
+j such that P[�ℓ±
+j > ℓ] ≡ Ψ±
+j (ℓ). The special case of
+exponential distributions is given by equations (3.3). The stochastic process in the
+j-th layer is then killed as soon as one of the local times ℓ+
+j−1 and ℓ−
+j exceeds its
+corresponding threshold, which occurs at the FPT time Tj = min{τ +
+j , τ −
+j }, see equa-
+tion (3.4). Since the corresponding local time thresholds �ℓ+
+j−1 and �ℓ−
+j are statistically
+independent, the relationship between the resulting probability density pj(x, t|x0) for
+partially reflected BM in the j-th layer and Pj(x, ℓ1, ℓ2, t|x0) can be established as
+follows:
+pj(x, t|x0)dx = P[X(t) ∈ (x, x + dx), t < Tj|X0 = x0]
+= P[X(t) ∈ (x, x + dx), ℓ+
+j−1(t) < �ℓ+
+j−1, ℓ−
+j (t) < �ℓ−
+j |X0 = x0]
+=
+ˆ ∞
+0
+dℓψ+
+j−1(ℓ)
+ˆ ∞
+0
+dℓ′ψ−
+j (ℓ′)P[X(t) ∈ (x, x + dx), ℓ+
+j−1 < ℓ, ℓ−
+j < ℓ′|X0 = x0]
+=
+ˆ ∞
+0
+dℓψ+
+j−1(ℓ)
+ˆ ∞
+0
+dℓ′ψ−
+j (ℓ′)
+ˆ ℓ
+0
+dˆℓ
+ˆ ℓ′
+0
+dˆℓ′[Pj(x, ˆℓ, ˆℓ′, t|x0)dx].
+We have also introduced the probability densities ψ±
+j (ℓ) = −∂ℓΨ±
+j (ℓ). Reversing the
+orders of integration yields the result
+pj(x, t|x0) =
+ˆ ∞
+0
+dℓΨ+
+j−1(ℓ)
+ˆ ∞
+0
+dℓ′Ψ−
+j (ℓ′)Pj(x, ℓ, ℓ′, t|x0).
+(5.2)
+An evolution equation for the local time propagator can be derived as follows
+[7, 8]. Since the local times only change at the boundaries x = aj−1, aj, the propagator
+satisfies the diffusion equation in the bulk of the domain
+∂Pj
+∂t = Dj
+∂2Pj
+∂x2 , x ∈ (aj−1, aj).
+(5.3)
+The nontrivial step is determining the boundary conditions at x = aj−1, aj. Here we
+give a heuristic derivation based on a boundary layer construction. For concreteness,
+consider the left-hand boundary layer [aj−1, aj−1 + h] and define
+ℓh
+j−1(t) = Dj
+h
+ˆ t
+0
+�ˆ h
+0
+δ(Xt′ − x)dx
+�
+dt′.
+(5.4)
+By definition, hℓh
+j−1(t)/Dj is the residence or occupation time of the process X(t)
+in the boundary layer up to time t. Although the width h and the residence time
+16
+
+in the boundary layer vanish in the limit h → 0, the rescaling by 1/h ensures that
+limh→0 ℓh
+j−1(t) = ℓ+
+j−1(t). Moreover, from conservation of probability, the flux into
+the boundary layer over the residence time hδℓ/Dj generates a corresponding shift in
+the probability Pj within the boundary layer from ℓ → ℓ + δℓ. That is, for ℓ > 0,
+−Jj(aj−1 + h, ℓ, ℓ′, t|x0)hδℓ = [Pj(aj−1, ℓ + δℓ, ℓ′, t|x0) − Pj(aj−1, ℓ, ℓ′, t|x0)]h,
+where Jj(x, ℓ, ℓ′, t|x0) = −D∂xPj(x, ℓ, ℓ′, t|x0). Dividing through by hδℓ and taking
+the limits h → 0 and δℓ → 0 yields
+−Jj(aj−1, ℓ, ℓ′, t|x0) = ∂ℓPj(aj−1, ℓ, ℓ′, t|x0), ℓ > 0.
+Moreover, when ℓ = 0 the probability flux Jj(aj−1, 0, ℓ′, t|x0) is identical to that
+of a Brownian particle with a totally absorbing boundary at x = aj−1, which we
+denote by Jj,∞(aj−1, ℓ′, t|x0). In addition, it can be shown that Pj(aj−1, 0, ℓ′, t|x0) =
+−Jj,∞(aj−1, ℓ′, t|x0). Applying a similar argument at the end x = aj, we obtain the
+pair of boundary conditions
+D∂xPj(aj−1, ℓ, ℓ′, t|x0) = −Pj(aj−1, 0, ℓ′, t|x0)δ(ℓ) + ∂Pj(aj−1, ℓ, ℓ′, t|x0)
+∂ℓ
+,
+(5.5a)
+−D∂xPj(aj, ℓ, ℓ′, t|x0) = −Pj(aj, ℓ, 0, t|x0)δ(ℓ′) + ∂Pj(aj, ℓ, ℓ′, t|x0)
+∂ℓ′
+.
+(5.5b)
+The crucial step in the encounter-based approach is to note that for exponentially
+distributed local time thresholds, see equation (3.3), the right-hand side of equation
+(5.2) reduces to a double Laplace transform of the local time propagator:
+pj(x, t|x0) = Pj(x, z+
+j−1, z−
+j , t|x0),
+z+
+j−1 = 2κj−1
+Dj
+,
+z−
+j = 2κj
+Dj
+,
+(5.6)
+with
+Pj(x, z, z′, t|x0) ≡
+ˆ ∞
+0
+dℓe−zℓ
+ˆ ∞
+0
+dℓ′e−z′ℓ′Pj(x, ℓ, ℓ′, t|x0).
+(5.7)
+Laplace transforming the propagator boundary conditions (5.5a) and (5.5b) then
+shows that the probability density pj of equation (5.6) is the solution to the Robin
+BVP given by equations (3.6a) and (3.6b). Hence, the probability density of partially
+reflected BM in the j-th layer is equivalent to the doubly Laplace transformed local
+time propagator with the pair of Laplace variables z+
+j−1 and z−
+j . Assuming that the
+Laplace transforms can be inverted, we can then incorporate non-exponential proba-
+bility distributions Ψ+
+j−1(ℓ) and Ψ−
+j (ℓ′) such that the corresponding marginal density
+is now
+pj(x, t|x0) =
+ˆ ∞
+0
+dℓΨ+
+j−1(ℓ)
+ˆ ∞
+0
+dℓ′Ψ−
+j (ℓ′)L−1
+ℓ L−1
+ℓ′ Pj(x, z, z′, t|x0),
+(5.8)
+where L−1 denotes the inverse Laplace transform. One major difference from the
+exponential case is that the stochastic process X(t) is no longer Markovian.
+5.2. Killing time densities. In order to sew together successive rounds of re-
+flected BM in the case of general distributions Ψj we will need the conditional FPT
+densities f +
+j−1(x0, t) and f −
+j (x0, t) for partially reflected BM in the j-th layer to be
+killed at the ends x = aj−1 and x = aj, respectively. The corresponding conditional
+17
+
+killing times were defined in equation (3.4).
+The FPT densities are given by the
+outward probability fluxes at the two ends:
+f +
+j−1(x0, t) = Dj∂xpj(aj−1, t|x0),
+f −
+j (x0, t) = −Dj∂xpj(aj, t|x0).
+(5.9)
+As in previous sections, it is convenient to Laplace transform with respect to t. Laplace
+transforming equation (5.8) and using the Green’s function (3.8) gives
+�pj(x, s|x0) =
+ˆ ∞
+0
+dℓΨ+
+j−1(ℓ)
+ˆ ∞
+0
+dℓ′Ψ−
+j (ℓ′)L−1
+ℓ L−1
+ℓ′ �Pj(x, z, z′, s|x0),
+(5.10)
+where
+�Pj(x, z, z′, s|x0) =
+
+
+
+Aj(z, z′, s)Fj(x, z, s)Fj(x0, z′, s),
+aj−1 ≤ x ≤ x0,
+Aj(z, z′, s)Fj(x0, z, s)Fj(x, z′, s),
+x0 ≤ x ≤ aj,
+(5.11)
+with
+Fj(x, z, s) =
+�
+s/Dj cosh(
+�
+s/Dj[x − aj−1]) + z sinh(
+�
+s/Dj[x − aj−1]),
+(5.12a)
+Fj(x, z′, s) =
+�
+s/Dj cosh(
+�
+s/Dj[aj − x]) + z′ sinh(
+�
+s/Dj[aj − x]),
+(5.12b)
+Aj =
+1
+�
+sDj
+1
+(z + z′)
+�
+s/Dj cosh(
+�
+s/DjLj) + [s/Dj + zz′] sinh(
+�
+s/DjLj)
+.
+(5.12c)
+Since �Pj(x, z, z, s|x0) satisfies the Robin boundary conditions
+Dj∂x �Pj(aj−1, z, z′, s|x0) = Djz �Pj(aj−1, z, z′, s|x0),
+Dj∂x �Pj(aj, z, z′, s|x0) = −Djz′ �Pj(aj, z, z′, s|x0),
+it follows that
+�f +
+j−1(x0, s) ≡ Dj∂x�pj(aj−1, s|x0)
+= Dj
+ˆ ∞
+0
+dℓΨ+
+j−1(ℓ)
+ˆ ∞
+0
+dℓ′Ψ−
+j (ℓ′)
+�
+∂ℓ �Pj(aj−1, ℓ, ℓ′, s|x0) + �Pj(aj−1, 0, ℓ′, s|x0)
+�
+= Dj
+ˆ ∞
+0
+dℓψ+
+j−1(ℓ)
+ˆ ∞
+0
+dℓ′Ψ−
+j (ℓ′) �Pj(aj−1, ℓ, ℓ′, s|x0).
+(5.13)
+Similarly,
+�f −
+j (x0, s) ≡ −Dj∂x�pj(aj, s|x0) = Dj
+ˆ ∞
+0
+dℓΨ+
+j−1(ℓ)
+ˆ ∞
+0
+dℓ′ψ−
+j (ℓ′) �Pj(aj, ℓ, ℓ′, s|x0).
+(5.14)
+Evaluation of the FPT densities reduces to the problem of calculating the prop-
+agator �Pj(ak, ℓ, ℓ′, s|x0) by inverting the double Laplace transform �Pj(ak, z, z′, s|x0)
+with respect to z and z′, k = j − 1, j, and then evaluating the double integrals in
+equations (5.13) and (5.14). In general, this is a non-trivial calculation. However, a
+18
+
+major simplification occurs if we take one of the densities Ψ+
+j−1 or Ψ−
+j to be an expo-
+nential. First suppose that Ψ+
+j−1(ℓ) = e−2κj−1ℓ/Dj. We then have a Robin boundary
+condition at x = aj−1,
+�f +
+j−1(x0, s) = 2κj−1�pj(aj−1, s|x0),
+(5.15)
+whereas
+�f −
+j (x0, s) = Dj
+ˆ ∞
+0
+dℓ′ψ−
+j (ℓ′) �Pj(aj, z+
+j−1, ℓ′, s|x0).
+(5.16)
+From equation (5.10) we find that
+�Pj(aj, zj, z′, s|x0) = 1
+Dj
+Λj(x0, s)
+z′ + hj(s),
+(5.17)
+where
+Λj(x0, s) =
+�
+s/Dj cosh(
+�
+s/Dj[x0 − aj−1]) + z+
+j−1 sinh(
+�
+s/Dj[x0 − aj−1])
+�
+s/Dj cosh(
+�
+s/DjLj) + z+
+j−1 sinh(
+�
+s/DjLj
+,
+(5.18)
+and
+hj(s) =
+�
+s/Dj
+�
+s/Dj tanh(
+�
+s/DjLj) + z+
+j−1
+�
+s/Dj + z+
+j−1 tanh(
+�
+s/DjLj)
+.
+(5.19)
+Inverting the Laplace transform with respect to z′ then gives
+�Pj(aj, z+
+j−1, ℓ′, s|x0) = D−1
+j Λj(x0, s)e−hj(s)ℓ′
+(5.20)
+and, hence,
+�f −
+j (x0, s) = Λj(x0, s) �ψ−
+j (hj(s)).
+(5.21)
+On the other hand,
+�pj(aj, s|x0) = D−1
+j Λj(x0, s)�Ψ−
+j (hj(s)).
+(5.22)
+We thus obtain the following boundary condition at x = aj:
+�f −
+j (x0, s) = �K−
+j (s)�pj(aj, s|x0),
+�K−
+j (s) =
+Dj �ψ−
+j (hj(s))
+�Ψ−
+j (hj(s))
+.
+(5.23)
+Finally, using the convolution theorem, the boundary condition at x = aj in the time
+domain takes the form
+Dj∂xpj(aj, t|x0) = −
+ˆ t
+0
+K−
+j (τ)pj(aj, t − τ|x0)dτ.
+(5.24)
+That is, in the case of a non-Markovian density for killing partially reflected BM at
+one end of an interval, the corresponding boundary condition involves an effective
+time-dependent absorption rate K−
+j (t), which acts as a memory kernel.
+19
+
+Now suppose that Ψ−
+j (ℓ) = e−2κjℓ/Dj so that
+�f −
+j (x0, s) = 2κj�pj(aj, s|x0), �f +
+j−1(x0, s) = Dj
+ˆ ∞
+0
+dℓψ+
+j−1(ℓ) �Pj(aj−1, ℓ, z−
+j , s|x0).
+(5.25)
+From equation (5.10) we have
+�Pj(aj, z, z−
+j , s|x0) = 1
+Dj
+Λj(x0, s)
+z + hj(s),
+(5.26)
+where
+Λj(x0, s) =
+�
+s/Dj cosh(
+�
+s/Dj[aj − x0]) + z−
+j sinh(
+�
+s/Dj[aj − x0])
+�
+s/Dj cosh(
+�
+s/DjLj) + z−
+j sinh(
+�
+s/DjLj
+,
+(5.27)
+and
+hj(s) =
+�
+s/Dj
+�
+s/Dj tanh(
+�
+s/DjLj) + z−
+j
+�
+s/Dj + z−
+j tanh(
+�
+s/DjLj)
+.
+(5.28)
+Using identical arguments to the previous case we find that the boundary condition
+at x = aj−1 is
+�f +
+j−1(x0, s) = �K+
+j−1(s)�pj(aj−1, s|x0),
+�K+
+j−1(s) =
+Dj �ψ+
+j−1(hj(s))
+�Ψ+
+j−1(hj(s))
+.
+(5.29)
+5.3. Generalized snapping out BM and the first renewal equation. We
+now define a generalized snapping out BM by sewing together successive rounds of
+reflected BM along identical lines to section 3.2, except that now each round is killed
+according to the general process introduced in section 5.1. (For simplicity, we assume
+that the exterior boundaries at x = 0, L are totally reflecting.) Although each round
+of partially reflected Brownian motion is non-Markovian, all history is lost following
+absorption and restart so that we can construct a renewal equation. However, it is
+now more convenient to use a first rather than a last renewal equation. Again we
+consider a general probability density φ(x0) of initial conditions x0 ∈ G.
+Let f +
+j−1(t) and f −
+j (t) denote the conditional FPT densities for partially reflected
+BM in the j-th layer to be killed at the end x = aj−1 and x = aj, respectively, in the
+case of a general initial distribution φ(x0). It follows that
+f +
+j−1(t) =
+ˆ aj
+aj−1
+f +
+j−1(x0, t)φ(x0)dx0 = Dj
+ˆ aj
+aj−1
+∂xpj(aj−1, t|x0)φ(x0)dx0,
+(5.30a)
+f −
+j (t) =
+ˆ aj
+aj−1
+f −
+j (x0, t)φ(x0)dx0 = −Dj
+ˆ aj
+aj−1
+∂xpj(aj, t|x0)φ(x0)dx0.
+(5.30b)
+with f +
+j−1(x0, t) and f −
+j (x0, t) defined in equations (5.9). We also set f +
+1 (t) ≡ 0 and
+f −
+m(t) ≡ 0. Generalizing previous work [9, 10], the first renewal equation in the j-th
+layer, 1 ≤ j ≤ m, takes the form
+ρj(x, t) ≡
+ˆ
+G
+ρj(x, t|x0)φ(x0)dx0
+(5.31)
+= pj(x, t) + 1
+2
+m−1
+�
+k=1
+ˆ t
+0
+[ρj(x, t − τ|a−
+k ) + ρj(x, t − τ|a+
+k )][f −
+k (τ) + f +
+k (τ)]dτ
+20
+
+for x ∈ (aj−1, aj) and
+pj(x, t) =
+ˆ aj
+aj−1
+pj(x, t|x0)φ(x0)dx0.
+(5.32)
+The first term on the right-hand side of equation (5.31) represents all sample trajecto-
+ries that start in the k-th layer and have not been absorbed at the ends x = ak−1, ak
+up to time t. The integral term represents all trajectories that were first absorbed
+(stopped) at a semi-permeable interface at time τ and then switched to either posi-
+tively or negatively reflected BM state with probability 1/2, after which an arbitrary
+number of switches can occur before reaching x ∈ (aj−1, aj) at time t. The probability
+that the first stopping event occurred at the k-th interface in the interval (τ, τ + dτ)
+is [f +
+k (τ) + f −
+k (τ)]dτ. Laplace transforming the renewal equation (5.31) with respect
+to time t gives
+�ρj(x, s) = �pj(x, s) + 1
+2
+m−1
+�
+k=1
+[�ρj(x, s|a−
+k ) + �ρj(x, s|a+
+k )][ �f −
+k (s) + �f +
+k (s)].
+(5.33)
+In order to determine the factors
+Σjk(x, s) = �ρj(x, s|a−
+k ) + �ρj(x, s|a+
+k ),
+1 ≤ k < m,
+(5.34)
+we substitute into equation (5.33) the initial density φ(x0) = 1
+2[δ(x0−a−
+k )+δ(x0−a+
+k )].
+This gives
+Σjk(x, s) = �pj(x, s|ak)[δj,k + δj,k+1] + 1
+2Σjk(x, s)[ �f −
+k (a−
+k , s) + �f +
+k (a+
+k , s)]
++ 1
+2Σj,k−1(x, s) �f +
+k−1(a−
+k , s) + 1
+2Σj,k+1(x, s) �f −
+k+1(a+
+k , s)].
+(5.35)
+Comparison with equations (3.17a)–(3.17c) implies that the above equation can
+be rewritten in the matrix form
+m−1
+�
+l=1
+Θkl(s)Σjl(s) = −�pj(x, s|ak)[δj,k + δj,k+1],
+(5.36)
+where Θ(s) is a tridiagonal matrix with non-zero elements
+Θk,k(s) = dk(s) ≡ �f −
+k (a−
+k , s) + �f +
+k (a+
+k , s) − 1, k = 1, . . . m − 1,
+(5.37a)
+Θk,k−1(s) = ck(s) ≡ �f +
+k−1(a−
+k , s),
+k = 2, . . . m − 1,
+(5.37b)
+Θk,k+1(s) = bk(s) ≡ �f −
+k+1(a+
+k , s),
+k = 1, . . . , m − 2.
+(5.37c)
+Assuming that the matrix Θ(s) is invertible, we obtain the formal solution
+Σjk(x, s) = −Θ
+−1
+kj (s)�pj(x, s|aj) − Θ
+−1
+k,j−1(s)�pj(x, s|aj−1).
+(5.38)
+Substituting into equation (5.33) yields the result
+�ρj(x, s) = �pj(x, s) + 1
+2
+m−1
+�
+k=1
+[Θ
+−1
+kj (s)�pj(x, s|aj) + Θ
+−1
+k,j−1(s)�pj(x, s|aj−1)]
+(5.39)
+× [ �f −
+k (s) + �f +
+k (s)].
+21
+
+Equivalence of first and last renewal equations for exponential killing. An im-
+portant check of our analysis is to show that the solution (5.39) of the first renewal
+equation is equivalent to the solution (3.21) of the last renewal equation when each
+round of reflecting BM is killed according to an independent exponential distribu-
+tion for each local time threshold. Since �pj(x, s|x0) then satisfies Robin boundary
+conditions at x = aj−1, aj we find that
+Θk,k(s) = κk[�pk+1(ak, s|ak) + �pk(ak, s|ak)] − 1 = Θkk(s),
+(5.40a)
+Θk,k−1(s) = κk−1�pk(ak−1, s|ak) = κk−1�pk(ak, s|ak−1) = Θk,k−1(s),
+(5.40b)
+Θk,k+1(s) = κk+1�pk+1(ak+1, s|ak) = κk+1�pk+1(ak, s|ak+1) = Θk,k+1(s).
+(5.40c)
+We have used two important properties of partially reflected BM:
+i) Symmetry of the Green’s function �p(x, s|x0) = �p(x0, s|x).
+ii) The solution for the functions Σjk(x, s) is obtained by introducing the initial con-
+ditions (5.34). The FPT densities are thus evaluated at the initial points a±
+k . This
+means that when we impose the Robin boundary conditions we do not pick up the
+additional constant term in equations (3.10a) and (3.10b).
+It follows that the solution (5.39) reduces to the form
+�ρj(x, s) = �pj(x, s) +
+m−1
+�
+k=1
+�
+�pj(x, s|aj−1)Θ−1
+k,j−1(s) + �pj(x, s|aj)Θ−1
+kj (s)
+�
+× κk[�pk(ak, s) + �pk+1(ak, s)].
+(5.41)
+Finally, using the fact that κkΘ−1
+kj (s) = κjΘ−1
+jk (s), we recover the solution (3.21).
+Non-exponential killing. The above analysis shows that the same solution struc-
+ture holds for both exponential and non-exponential killing, provided that we ex-
+press the tridiagonal matrix Θij in terms of the conditional FPT densities �f ±
+k (a±
+k , s),
+�f +
+k−1(a−
+k , s) and �f −
+k+1(a+
+k , s). The latter are themselves determined from equations
+(5.13) and (5.14). One configuration that is analytically tractable is a 1D domain
+with a sequence of semi-permeable barriers whose distributions Ψ±
+j
+alternate be-
+tween exponential and non-exponential. For example, suppose Ψ−
+j (ℓ) = e−2κj/Dj and
+Ψ+
+j (ℓ) = e−2κj/Dj+1 for all odd layers j = 1, 3, . . ., whereas Ψ±
+j (ℓ) are non-exponential
+for even layers j = 2, 4, . . .. Combining the analysis of the FPT densities in section
+5.2 with the analysis of the first renewal equation and its relationship with the last
+renewal equation, we obtain the following generalization of the interfacial boundary
+conditions (2.2b):
+Dj∂xρj(a−
+j , s) = Dj+1∂xρj+1(a+
+j , s) = 1
+2[ �K+
+j (s)ρj+1(a+
+j , s) − �K−
+j (s)ρj(a−
+j , s)],
+(5.42)
+with �K±
+j (s) = 2κj for odd j and
+�K−
+j (s) =
+Dj �ψ−
+j (hj(s))
+�Ψ−
+j (hj(s))
+,
+�K+
+j (s) =
+Dj+1 �ψ+
+j (hj+1(s))
+�Ψ+
+j (hj+1(s))
+(5.43)
+for even j, with hj(s) and hj+1(s) given by equations (5.19) and (5.28), respectively.
+We thus have the setup shown in Fig. 5.1. Note, in particular, that the time-dependent
+permeability kernels of the even interfaces are asymmetric.
+22
+
+x = a1
+x = 0
+x = a4
+x = a2
+x = a3
+K2-(t)
+κ1
+κ1
+K2+(t)
+K4-(t)
+κ2
+κ2
+K4+(t)
+Fig. 5.1. A 1D layered medium partitioned by a sequence of semi-permeable interfaces that
+alternate between symmetric constant permeabilities κj, j = 1, 3, . . . and asymmetric time-dependent
+permeabilities K±
+j (t), j = 2, 4, . . ..
+Permeability kernels for the gamma distribution. For the sake of illustration, sup-
+pose that ψ±
+j (ℓ) for even j are given by the gamma distributions
+ψ±
+j (ℓ) =
+z±
+j (z±
+j ℓ)µ−1e−z±
+j ℓ
+Γ(µ)
+, µ > 0,
+(5.44)
+where Γ(µ) is the gamma function. The corresponding Laplace transforms are
+�ψ±
+j (z) =
+�
+z±
+j
+z±
+j + z
+�µ
+,
+�Ψ±
+j (z) = 1 − �ψ±
+j (z)
+z
+.
+(5.45)
+If µ = 1 then ψ±
+j reduce to the exponential distributions with constant reactivity κj.
+The parameter µ thus characterizes the deviation of ψ±
+j (ℓ) from the exponential case.
+If µ < 1 (µ > 1) then ψ±
+j (ℓ) decreases more rapidly (slowly) as a function of the local
+time ℓ. Substituting the gamma distributions into equations (5.43) yields
+�K−
+j (s) =
+Djhj(s)(z−
+j )µ
+[z−
+j + hj(s)]µ − (z−
+j )µ , �K+
+j (s) =
+Dj+1hj+1(s)(z+
+j )µ
+[z+
+j + hj+1(s)]µ − (z+
+j )µ .
+(5.46)
+If µ = 1 then �K±
+j (s) = 2κj as expected. On the other hand if µ = 2, say, then
+�K−
+j (s) =
+2κj
+2 + Djhj(s)/2κj
+,
+�K+
+j (s) =
+2κj
+2 + Dj+1hj+1(s)/2κj
+.
+(5.47)
+The corresponding time-dependent kernels K±
+j (t) are normalizable since
+ˆ ∞
+0
+K−
+j (t)dt = �K−
+j (0) =
+2κj
+2 + κ−1
+j κj−1/[1 + 2κj−1Lj/Dj),
+(5.48a)
+ˆ ∞
+0
+K+
+j (t)dt = �K+
+j (0) =
+2κj
+2 + κ−1
+j κj+1/[1 + 2κj+1Lj/Dj+1).
+(5.48b)
+However, the kernels are heavy-tailed with infinite moments. For example,
+⟨t⟩− ≡
+1
+�K−
+j (0)
+ˆ ∞
+0
+tK−
+j (t)dt = −
+1
+�K−
+j (0)
+lim
+s→0 ∂s �K−
+j (s)
+=
+1
+�K−
+j (0)
+lim
+s→0
+Djh′
+j(s)/2
+[2 + Djhj(s)/2κj]2 = Dj
+4κj
+�K−
+j (0)
+2κj
+lim
+s→0 h′
+j(s) = ∞.
+(5.49)
+That is, all moments are infinite since all derivatives of hj(s) are singular at s = 0.
+An analogous result was previously found for a single interface in 1D and 3D [9, 10].
+23
+
+6. Discussion. In this paper we developed a probabilistic framework for analyz-
+ing single-particle diffusion in heterogeneous multi-layered media. Our approach was
+based on a multi-layered version of snapping out BM. We showed that the distribution
+of sample trajectories satisfied a last renewal equation that related the full probabil-
+ity density to the probability densities of partially reflected BM in each layer. The
+renewal equation was solved using a combination of Laplace transforms and transfer
+matrices.
+We also proved the equivalence of the renewal equation and the corre-
+sponding multi-layered diffusion equation in the case of constant permeabilities. We
+then used the renewal approach to incorporate a more general probabilistic model of
+semipermeable interfaces. This involved killing each round of partially reflected BM
+according to a non-Markovian encounter-based model of absorption at an interface.
+We constructed a corresponding first renewal equation that related the full probability
+density to the FPT densities for killing each round of reflected BM. In particular, we
+showed that non-Markovian models of absorption can generate asymmetric, heavy-
+tailed time-dependent permeabilities.
+In developing the basic mathematical framework, we focused on relatively simple
+examples such as identical layers with constant permeabilities or alternating Marko-
+vian and non-Markovian interfaces. We also restricted our analysis to the Laplace
+domain rather than the time domain. However, it is clear that in order to apply
+the theory more widely, it will be necessary to develop efficient numerical schemes
+for solving the last or first renewal equations in Laplace space, and then inverting
+the Laplace transformed probability density to obtain the solution in the time do-
+main. In the case of non-Markovian models of absorption at both ends of a layer, it
+will also be necessary to compute the double inverse Laplace transform of the local
+time propagator and evaluate the resulting double integral in equation (5.8). Another
+computational issue is developing an efficient numerical scheme for simulating sample
+trajectories of snapping out BM in heterogeneous multi-layer media.
+Finally, from a modeling perspective, it would be interesting to identify plausible
+biophysical mechanisms underlying non-Markovian models of semi-permeable mem-
+branes. As previously highlighted within the context of encounter-based models of
+absorption [31, 32, 7, 8], various surface-based reactions are better modeled in terms
+of a reactivity that is a function of the local time. For example, the surface may
+become progressively activated by repeated encounters with a diffusing particle, or an
+initially highly reactive surface may become less active due to multiple interactions
+with the particle (passivation) [4, 23].
+REFERENCES
+[1] V. Aho, K. Mattila, T. K¨uhn, P. Kek¨al¨ainen, O. Pulkkine, R. B. Minussi, M. Vihinen-
+Ranta and J. Timonen Diffusion through thin membranes: Modeling across scales. Phy.
+Rev. E 93 (2016) 043309
+[2] I. Alemany, J. N. Rose, J. Garnier-Brun, A. D. Scott and D. J. Doorly Random walk dif-
+fusion simulations in semi-permeable layered media with varying diffusivity Science Reports
+12 (2022) 10759
+[3] S. Barbaro, C. Giaconia and A. Orioli A computer oriented method for the analysis of non
+steady state thermal behaviour of buildings. Build. Environ. 23 (1988) 19-24
+[4] C. H. Bartholomew. Mechanisms of catalyst deactivation. Appl. Catal. A: Gen. 212 (2001)
+17-60.
+[5] A. N. Borodin and P. Salminen. Handbook of Brownian Motion:
+Facts and Formulae
+Birkhauser Verlag, Basel-Boston-Berlin (1996).
+[6] P.C. Bressloff Diffusion in cells with stochastically-gated gap junctions. SIAM J. Appl. Math.
+76 (2016) 1658-1682
+24
+
+[7] P.C. Bressloff Diffusion-mediated absorption by partially reactive targets: Brownian function-
+als and generalized propagators. J. Phys. A. 55 (2022) 205001
+[8] P.C. Bressloff Spectral theory of diffusion in partially absorbing media. Proc. R. Soc. A 478
+(2022) 20220319
+[9] P.C. Bressloff A probabilistic model of diffusion through a semipermeable barrier. Proc. Roy.
+Soc. A 478 (2022) 20220615.
+[10] P.C. Bressloff Renewal equation for single-particle diffusion through a semipermeable inter-
+face. Phys. Rev. E. In press (2023)
+[11] P. R. Brink and S. V. Ramanan A model for the diffusion of fluorescent probes in the sep-
+tate giant axon of earthworm: axoplasmic diffusion and junctional membrane permeability.
+Biophys. J. 48 (1985) 299-309
+[12] A. Bobrowski. Semigroup-theoretic approach to diffusion in thin layers separated by semi-
+permeable membranes. J. Evol. Equ. 21 (2021) 1019-1057
+[13] P. T. Callaghan, A. Coy, T. P. J. Halpin, D. MacGowan, K. J. Packer and F. O. Zelaya.
+Diffusion in porous systems and the influence of pore morphology in pulsed gradient spin-
+echo nuclear magnetic resonance studies. J. Chem. Phys. 97 (1992) 651-662
+[14] E. Carr and I. Turner A semi-analytical solution for multilayer diffusion in a composite
+medium consisting of a large number of layers. Appl. Math. Model. 40 (2016) 7034-7050
+[15] B. W. Connors and M. A. Long Electrical synapses in the mammalian brain. Ann. Rev.
+Neurosci. 27 (2004) 393-418
+[16] A. Coy and P. T. Callaghan. Pulsed gradient spin echo nuclear magnetic resonance for
+molecules diffusing between partially reflecting rectangular barriers. J. Chem. Phys. 101
+(1994) 4599-4609.
+[17] F. deMonte. Transient heat conduction in one-dimensional composites lab. A natural analytic
+approach. Int. J. Heat Mass Transf. 43 (2000) 3607-3619
+[18] J.-P. Diard, N. Glandut, C. Montella and J.-Y. Sanchez. One layer, two layers, etc. An
+introduction to the EIS study of multilayer electrodes. Part 1: Theory. J. Electroanal. Chem.
+578 (2005) 247-257
+[19] O. K. Dudko, A. M. Berezhkovskii and G. H. Weiss. Diffusion in the presence of periodically
+spaced permeable membranes. J. Chem. Phys. 121 (2004) 11283
+[20] W. J. Evans and P. E. Martin Gap junctions: structure and function. Mol. Membr. Biol. 19
+(2002) 121-136
+[21] S. Regev and O. Farago. Application of underdamped Langevin dynamics simulations for the
+study of diffusion from a drug-eluting stent. Phys. A, Stat. Mech. Appl. 507 (2018) 231-239
+[22] O. Farago Algorithms for Brownian dynamics across discontinuities. J. Comput. Phys. 423
+(2020) 109802.
+[23] M. Filoche, D. S. Grebenkov, J. S. Andrade and B. Sapoval. Passivation of irregular
+surfaces accessed by diffusion. Proc. Natl. Acad. Sci. 105 (2008) 7636-7640.
+[24] V. Freger. Diffusion impedance and equivalent circuit of a multilayer film. Electrochem. Com-
+mun. 7 (2005) 957-961
+[25] M. Freidlin. Functional Integration and Partial Differential Equations Annals of Mathematics
+Studies, Princeton University Press, Princeton (1985) New Jersey
+[26] D. A. Goodenough and D. L. Paul Gap junctions. Cold Spring Harb Perspect Biol 1 (2009)
+a002576
+[27] G. L. Graff, R. E. Williford and P. E. Burrows. Mechanisms of vapor permeation through
+multilayer barrier films: lag time versus equilibrium permeation. J. Appl. Phys. 96 (2004)
+1840-1849
+[28] D. S. Grebenkov Partially Reflected Brownian Motion: A Stochastic Approach to Transport
+Phenomena, in “Focus on Probability Theory”, Ed. Velle LR pp. 135-169. Hauppauge: Nova
+Science Publishers (2006)
+[29] D. S. Grebenkov Pulsed-gradient spin-echo monitoring of restricted diffusion in multilayered
+structures. J. Magn. Reson. 205 (2010) 181-195
+[30] D. S. Grebenkov, D. V. Nguyen and J.-R. Li Exploring diffusion across permeable barriers
+at high gradients. I. Narrow pulse approximation. J. Magn. Reson. 248 (2014) 153-163.
+[31] D. S. Grebenkov Paradigm shift in diffusion-mediated surface phenomena. Phys. Rev. Lett.
+(2020) 125, 078102.
+[32] D. S. Grebenkov An encounter-based approach for restricted diffusion with a gradient drift.
+J. Phys. A. (2022) 55 045203.
+[33] P. Grossel and F. Depasse. Alternating heat diffusion in thermophysical depth profiles: mul-
+tilayer and continuous descriptions. J. Phys. D: Appl. Phys. 31 (1998) 216.
+[34] Y. Gurevich, I. Lashkevich and C. G. delaCruz. Effective thermal parameters of layered
+films:an application to pulsed photothermal techniques. Int. J. Heat Mass Transf. 52 (2009)
+25
+
+4302-4307.
+[35] D. W. Hahn and M. N. Ozisik One-Dimensional Composite Medium Ch. 10 pp. 393-432.
+Wiley, Hoboken (2012)
+[36] R. Hickson, S. Barry and G. Mercer. Critical times in multilayer diffusion. Part 1: Exact
+solutions. Int. J. Heat Mass Transf. 52 (2009) 5776-5783.
+[37] R. Hickson, S. Barry and G. Mercer. Critical times in multilayer. diffusion. Part. 2: Ap-
+proximate solutions. Int. J. Heat Mass Transf. 52 (2009) 5784-5791.
+[38] K. Ito and H. P. McKean. Diffusion Processes and Their Sample Paths Springer-Verlag,
+Berlin (1965)
+[39] T. Kay and Giuggioli. Diffusion through permeable interfaces: Fundamental equations and
+their application to first-passage and local time statistics. Phys. Rev. Res. 4 (2022) L032039
+[40] V. M. Kenkre, L. Giuggiol and Z. Kalay. Molecular motion in cell membranes: analytic
+study of fence-hindered random walks. Phys. Rev. E 77 (2008) 051907
+[41] A. Lejay The snapping out Brownian motion. The Annals of Applied Probability 26 (2016)
+1727-1742.
+[42] A. Lejay Monte Carlo estimation of the mean residence time in cells surrounded by thin layers.
+Mathematics and Computers in Simulation 143 (2018) 65-77
+[43] G. Liu, L. Barbour and B. C. Si. Unified multilayer diffusion model and application to diffu-
+sion experiment in porous media by method of chambers. Environ. Sci. Technol. 43 (2009)
+2412-2416
+[44] X. Lu and P. Tervola. Transient heat conduction in the composites lab-analytical method. J.
+Phys. A: Math. Gen. 38 (2005) 81
+[45] G. N. Milshtein. The solving of boundary value problems by numerical integration of stochastic
+equations. Math. Comp. Sim. 38(1995) 77-85
+[46] N. Moutal and D. S. Grebenkov Diffusion across semi-permeable barriers: spectral proper-
+ties, efficient computation, and applications J. Sci. Comput. 81 (2019) 1630-1654
+[47] D. Novikov, E. Fieremans, J. Jensen and J. A. Helpern. Random walks with barriers. Nat.
+Phys. 7 (2011) 508-514
+[48] V. G. Papanicolaou. The probabilistic solution of the third boundary value problem for second
+order elliptic equations Probab. Th. Rel. Fields 87 (1990) 27-77
+[49] G. Pontrelli and F. de Monte. Mass diffusion through two-layer porous media: an applica-
+tion to the drug-eluting stent. Int. J. Heat Mass Transf. 50 (2007) 3658-3669.
+[50] J. G. Powles, M. Mallett, G. Rickayzen and W. Evans. Exact analytic solutions for dif-
+fusion impeded by an infinite array of partially permeable barriers. Proc. R. Soc. Lond. A
+436 (1992) 391
+[51] S. V. Ramanan and P. R. Brink. Exact solution of a model of diffusion in an infinite chain
+or monlolayer of cells coupled by gap junctions. Biophys. J. 58 (1990) 631-639
+[52] C. D. Shackelford and S. M. Moore. Fickian diffusion of radio nuclides for engineered
+containment barriers: diffusion coefficients, porosities, and complicating issues. Eng. Geol.
+152 (2013) 133-147. 123
+[53] J. E. Tanner. Transient diffusion in a system partitioned by permeable barriers. application to
+NMR measurements with a pulsed field gradient. J. Chem. Phys. 69 (1978) 1748.
+[54] H. Todo, T. Oshizaka, W. R. Kadhum and K. Sugibayashi. Mathematical model to predict
+skin concentration after topical application of drugs. Pharmaceutics 5 (2013) 634-651.
+[55] S. R. Yates, S. K. Papiernik, F. Gao and J. Gan. Analytical solutions for the transport of
+volatile organic chemicals in unsaturated layered systems. Water Resour. Res. 36 (2000)
+1993-2000.
+26
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+Analysis of Label-Flip Poisoning Attack
+on Machine Learning Based Malware Detector
+Kshitiz Aryal
+Department of Computer Science
+Tennessee Technological University
+Cookeville, TN, USA
+[email protected]
+Maanak Gupta
+Department of Computer Science
+Tennessee Technological University
+Cookeville, TN, USA
+[email protected]
+Mahmoud Abdelsalam
+Department of Computer Science
+North Carolina A&T State University
+Greensboro, NC, USA
+[email protected]
+Abstract—With the increase in machine learning (ML) applica-
+tions in different domains, incentives for deceiving these models
+have reached more than ever. As data is the core backbone of ML
+algorithms, attackers shifted their interest towards polluting the
+training data itself. Data credibility is at even higher risk with
+the rise of state-of-art research topics like open design principles,
+federated learning, and crowd-sourcing. Since the machine learn-
+ing model depends on different stakeholders for obtaining data,
+there are no existing reliable automated mechanisms to verify
+the veracity of data from each source.
+Malware detection is arduous due to its malicious nature with
+the addition of metamorphic and polymorphic ability in the
+evolving samples. ML has proven to solve the zero-day malware
+detection problem, which is unresolved by traditional signature-
+based approaches. The poisoning of malware training data can
+allow the malware files to go undetected by the ML-based
+malware detectors, helping the attackers to fulfill their malicious
+goals. A feasibility analysis of the data poisoning threat in the
+malware detection domain is still lacking. Our work will focus on
+two major sections: training ML-based malware detectors and
+poisoning the training data using the label-poisoning approach.
+We will analyze the robustness of different machine learning
+models against data poisoning with varying volumes of poisoning
+data.
+Index
+Terms—Cybersecurity,
+Poisoning
+Attacks,
+Machine
+Learning, Malware Detectors, Adversarial Malware Analysis
+I. INTRODUCTION
+Machine Learning (ML) techniques have been emerging
+rapidly, providing computational intelligence to various ap-
+plications. The ability of machine learning to generalize to
+unseen data has paved its way from labs to the real world. It
+has already gained unprecedented success in many fields like
+image processing [1], [2], natural language processing [3], [4],
+recommendation systems used by Google, YouTube and Face-
+book, cybersecurity [5], [6], robotics [7], drug research [8], [9],
+and many other domains. ML-based systems are achieving
+unparalleled performance through modern deep neural net-
+works bringing revolutions in AI-based services. Recent works
+have shown significant achievements in fields like self-driving
+cars and voice-controlled systems used by tech giants like
+autopilot in Tesla, Apple Siri, Amazon Alexa, and Microsoft
+Cortana. With machine learning being applied to such critical
+applications, continuous security threats are never a bombshell.
+In addition to traditional security threats like malware at-
+tack [10], phishing [11], man-in-the-middle attack [12], denial-
+of-service [13], SQL injection [14], adversaries are finding
+novel ways to sneak into ML models [15].
+Data poisoning and evasion attacks [16]–[20] are the latest
+menaces against the security of machine learning models.
+Poisoning attacks enable attackers to control the model’s
+behavior by manipulating a model’s data, algorithms, or hyper-
+parameters during the model training phase. On the other hand,
+an evasion attack is carried out during the test time by manip-
+ulating the test sample. Adversaries can craft legitimate inputs
+imperceptible to humans but force models to make wrong
+predictions. Szegedy et al. [21] discovered the vulnerability
+of deep learning architecture against adversarial attacks, and
+ever since, there have been several major successful adversar-
+ial attacks against machine learning architectures [22], [23].
+Sophisticated attackers are motivated by very high incentives
+to manipulate the result of the machine learning models. With
+the current data scale with which machine learning models are
+trained, it is impossible to verify each data point individually.
+In most scenarios, it is unlikely that an attacker gets access
+to training data. However, with many systems adopting online
+learning [24], crowd-sourcing [25] for training data, open
+design principles, and federated learning, poisoning attacks
+already pose a serious threat to ML models [26]. There have
+been instances [27] when big companies have been compro-
+mised by a data poisoning attack. Malware public databases
+like VirusTotal1, which rely on crowdsourced malware files
+for training its algorithm, can be poisoned by attackers while
+Google’s mail spam filter can be thrown out of track by wrong
+reporting of spam emails.
+Data poisoning relates to adding training data that either
+leaves a backdoor on the model or negatively impacts the
+model’s performance. Figure 1 shows the architecture of the
+poisoning attack. In the given figure, the addition of poisoned
+data in the training bag forces the model to learn and predict
+so that attackers benefit from it. This type of poisoning is
+not limited to particular domains but has extended across
+all ML applications. Label flipping attack is carried out to
+flip the prediction of machine learning detectors. Among
+all the existing approaches, we chose one of the simplest
+poisoning techniques called label poisoning. We swap the
+1https://www.virustotal.com/
+arXiv:2301.01044v1  [cs.CR]  3 Jan 2023
+
+Fig. 1. General architecture for Poisoning Machine Learning Models
+existing training data labels in label poisoning to check the
+ML models’ robustness.
+In this work, we perform a comparative analysis of different
+machine learning-based malware detectors’ robustness against
+label-flipping data poisoning attacks. Unlike the existing ap-
+proaches, we are demonstrating the impact of simple label-
+switching data poisoning in different malware detectors. We
+will first train eight different ML models widely used to detect
+malware, namely Stochastic Gradient Descent (SGD), Random
+Forest (RF), Logistic Regression (LR), K-Nearest Neighbor
+Classifier (KNN), Linear Support Vector Machine (SVM),
+Decision Tree (DT), Perceptron, and Multi-Layer Perceptron
+(MLP). This will be followed by poisoning 10% and 20% of
+training data by flipping the label of data samples. All of the
+models are retrained after data poisoning, and the performance
+of each model is evaluated. The major contributions of this
+paper are as follows.
+• We taxonomize the existing data poisoning attacks on ma-
+chine learning models in terms of domains, approaches, and
+targets.
+• We provide threat modeling for adversarial poisoning attacks
+against malware detectors. The threat is modeled in terms of
+the attack surface, the attacker’s knowledge, the attacker’s
+capability, and adversarial goals.
+• We train eight different machine learning-based malware
+detectors from malware data obtained from VirusTotal and
+VirusShare2. We compare the performance of these malware
+detectors with training and testing data in terms of accuracy,
+precision, and recall.
+• Finally, we show a simple label-switching approach to
+poison the data without any knowledge of training models.
+2https://virusshare.com/
+Fig. 2. Taxonomy of poisoning attack on attack domain, approach and target
+The performance of malware detectors is analyzed while
+poisoning 10% and 20% of the total training data.
+The rest of the paper is organized as follows. The existing
+literature for data poisoning attacks in different domains,
+including malware, is discussed in Section II. Section III
+provides the threat modeling for data poisoning attacks. An
+overview of ML algorithms that are used to train the malware
+detector in this paper is discussed in Section IV. Section V dis-
+cusses experimental methodology elaborating on the algorithm
+and the testbed used for the experiment. The evaluation and
+discussion on the performed experiments are given in Section
+VI. Finally, Section VII concludes this work.
+II. LITERATURE REVIEW
+Data poisoning attacks have been used against the machine
+learning domain for a long time. The existing literature on
+data poisoning attacks can be taxonomized in terms of attack
+domains, approach, and the target (victim), as illustrated in
+Figure 2. The recently trending technologies like crowd-
+sourcing and federated learning are always vulnerable as the
+veracity of individual data can never be verified. The recent
+victims of poisoning attacks have spread in security, network,
+and speech recognition domains. We also classified the major
+approaches that are taken to produce or optimize the poisoning
+attacks in Figure 2. The existing data poisoning approaches
+have targeted almost all the machine learning algorithms
+ranging from traditional algorithms like regression to modern
+deep neural network architectures.
+Table I summarizes the existing literature on poisoning
+attacks. Biggio et al. [43] attacked a support vector machine
+using gradient ascent. To make poisoning attacks closer to
+the real world, Yang et al. [44] used a generative adversarial
+network with an autoencoder to poison deep neural nets.
+Gongalez et al. [45] extended poisoning from binary learning
+to multi-class problems. Shafahi et al. [28] proposed a targeted
+clean label poisoning attack on neural networks using an
+optimization-based crafting method. Shen et al. [31] performed
+an imperceptible poisoning attack on a deep neural network
+by clogging the back-propagation from gradient tensors during
+training while also minimizing the gradient norm. Jiang et
+
+Machine learning
+Machine learning with
+without Data Poisoning
+Data Poisoning
+Data
+Data
+Collection
+Collection
+Poisoned
+直盲自
+自自自
+Data
+18168
+0100
+1818
+0100
+Preprocessing
+Preprocessing
+Data
+110
+0110
+Data
+Training
+Training Data with
+Data
+Poisoned dataset
+Training
+Training
+Poisoned Mode
+Trained
+Prediction
+Poisoned
+Model
+Prediction ModelData Poisoning
+Y
+Domains
+Approach
+Target
+> Image
+Gradient
+Neural
+Network
+Crowd Sourcing
+Reinforcement learning
+Support Vector
+Machine
+> Graph
+Label Flipping
+Regression
+Federated Learning
+Generative Adversarial
+Network
+Truth-Finder
+Dawid-Skene
+Security
+Empirical Inverstigations
+Graph
+Recommendation
+FakeUsers Insertion
+Embedding
+Spectrum
+Online learning
+Network
+SpeechRecognitionTABLE I
+DATA POISONING ATTACKS
+Domains
+Approach
+Target
+Publications
+Image
+Crowd
+Sourcing
+Graph
+Federated
+Learning
+Security
+Online
+Learning
+Gradient
+Reinforcement
+Learning
+Label
+Flipping
+GAN
+Others
+Neural
+Network
+SVM
+Regression
+Graph
+Embedding
+Customized
+Shafahi et al. [28]
+√
+√
+√
+Liu et al. [29]
+√
+√
+√
+√
+Cao et al. [30]
+√
+√
+√
+Shen et al. [31]
+√
+√
+√
+Zhang et al. [32]
+√
+√
+√
+Jiang et al. [33]
+√
+√
+√
+Kwon et al. [34]
+√
+√
+√
+Zhang et al. [35]
+√
+√
+√
+Bagdasaryal et al. [36]
+√
+√
+√
+√
+Li et al. [37]
+√
+√
+√
+Sasaki et al. [38]
+√
+√
+√
+Zhang et al. [39]
+√
+√
+√
+√
+Lovisotto et al. [40]
+√
+√
+√
+Li et al. [41]
+√
+√
+√
+√
+Kravchik et al. [42]
+√
+√
+√
+This Work
+√
+√
+√
+√
+√
+Domains:Poisoning domain for crafted attack, Approach: Approach to poison the training data, Target: Target of poisoning attack
+al. [33] performed a flexible poisoning attack against linear and
+logistic regression. Kwon et al. [34] could selectively poison
+particular classes against deep neural networks. Cao et al. [30]
+proposed a distributed label-flipping poisoning approach to
+poison the DL model in federated architecture. Miao et al. [46]
+poisoned Dawid-Skene [47] model by exploiting the reliability
+degree of workers. Fang et al. [48] proposed a poisoning attack
+against a graph-based recommendation system by maximizing
+the hit ratio of target items using fake users.
+In the given Table I, we can observe that only a handful
+of works have been carried out in the security domain.
+Sasaki et al. [38] proposed an attack framework for backdoor
+embedding, which prevented the detection of specific types of
+malware. They generated poisoning samples by solving an op-
+timization problem and tested it against a logistic regression-
+based malware detector. To poison the Android malware de-
+tectors, Lie et al. [41] experimented backdoor poisoning attack
+against Drebin [49], DroidCat [50], MamaDroid [51] and
+DroidAPIMiner [52]. Kravchik et al. [42] attacked the cyber
+attack detectors deployed in the industrial control system. The
+back gradient optimization techniques used to pollute the train-
+ing data successfully poison the neural network-based model.
+These works have focused their approach on some algorithm
+testing against some defense mechanism. However, none of
+the works compared the feebleness of multiple algorithms
+against data poisoning attacks. In this work, we demonstrate
+the effectiveness of label switch poisoning of the training
+data against eight machine learning algorithms widely used
+in malware detectors.
+III. THREAT MODEL: KNOW THE ADVERSARY
+All security threats are defined in terms of their goals and
+attack capabilities. Modeling the threat allows for identifying
+and better understanding the risk arriving with a threat. A
+poisoning attack is performed by manipulating the training
+data either at the initial learning or incremental learning
+Fig. 3. Threat model for poisoning attack
+period. The threat model of a poisoning attack reflects the
+attacker’s knowledge, goal, capabilities, and attack surface, as
+shown in Figure 3.
+Attack Surface: Attack surface denotes how the adversary
+attacks the model under analysis. Machine learning algorithms
+require data to pass through different stages in the pipeline,
+and each stage offers some kind of vulnerability. In this work,
+we are only concerned about poisoning attacks which make
+the training data an attack surface.
+Attacker’s Knowledge: The attacker’s knowledge is the
+amount of information about the model under attack that
+an attacker has. Based on the amount of knowledge of the
+attacker, the poisoning approach is determined. Attacker’s
+knowledge can be broadly classified into the two following
+categories:
+• White box model: In the white box model, an attacker has
+complete information about the underlying target model,
+such as the algorithm used, training data, hyper-parameters,
+and gradient information. It’s easier to carry out a white box
+attack due to the information available that helps the attacker
+to create a worst-case scenario for the target model.
+• Black box model: In the black-box model, an attacker only
+
+Threat Model
+Attack Surface
+Attacker's
+Attacker's
+Capability
+Attacker's Goal
+Knowledge
+Training
+White Box
+Data
+Untargeted
+data
+model
+Injection
+Misclassifcation
+Black Box
+Data
+Targeted
+model
+Modification
+Misclassification
+Logic
+Confidence
+Corruption
+Reductionhas information about the model’s input and output. An
+attacker has no information about the internal structure of
+the model. Black-box models can also be divided further
+into complete black-box models and gray-box models. In
+the gray box model, the model’s performance for each input
+the attacker provides can be known. As such, the gray box
+attack is considered to be relatively easier than the complete
+black box model.
+In this paper, we perform a black box attack on different
+malware detection models. Our experiments will prove the vul-
+nerability of these models to random label poisoning attacks
+without having any information about the models.
+Attacker’s Capability: The attacker’s capability represents
+the ability of an adversary to manipulate the data and model in
+different stages of the ML pipeline. It defines the sections that
+can be manipulated, the mechanism used for manipulation, and
+constraints to the attacker. Poisoning can be carried out in a
+well-controlled environment if the attacker has complete infor-
+mation about the underlying model and training data. Attacker
+capabilities can be classified into the following categories:
+• Data Injection: It is the ability to insert new data into the
+training dataset, leading machine learning models to learn
+on contaminated data.
+• Data Modification: It is the ability to access and modify
+the training data as well as the data labels. Label flipping
+is a well-known approach carried out in poisoning attack
+domains.
+• Logic Corruption: It is the ability to manipulate the logic of
+ML models. This ability is out of scope for data poisoning
+and is considered a model poisoning approach.
+Adversarial Goals: The attacker’s objective is to deceive the
+ML model by injecting poisoned data. However, poisoning
+training data might differ depending on the goals of an
+attacker. Attacker goals can be categorized as:
+• Untargeted Misclassification:
+An attacker tries to change
+the model’s output to a value different than the original
+prediction. Untargeted misclassification is a relatively easier
+goal for attackers.
+• Targeted Misclassification:
+An attacker’s goal is to add a
+certain backdoor in the models so that particular samples
+are classified to a chosen class.
+• Confidence Reduction: An attacker can also poison training
+data to reduce the confidence of the machine learning model
+for a particular prediction. In this approach, changing the
+classification label is unnecessary, but reducing the confi-
+dence score is enough to meet the attacker’s goal.
+Our paper aims to cause the malware detector models to
+misclassify. However, since we are dealing with binary clas-
+sification, it can be considered either targeted or untargeted
+misclassification.
+IV. OVERVIEW OF MACHINE LEARNING ALGORITHMS
+Almost all of the ML architectures have already been
+victimized by data poisoning attacks. In this section, we will
+brief some ML architectures in which we performed data
+poisoning attacks later in this paper.
+Stochastic Gradient Descent:
+Stochastic gradient descent
+(SGD) is derived from the gradient descent algorithm, which
+is a popular ML optimization technique. A gradient gives the
+slope of the function and measures the degree of change of
+a variable in response to the changes of another variable.
+Starting from an initial value, gradient descent runs iteratively
+to find the optimal values of the parameters, which are the
+minimal possible value of the given cost function. In Stochastic
+Gradient Descent, a few samples are randomly selected in
+place of the whole dataset for each iteration. The term batch
+determines the number of samples to calculate each iteration’s
+gradient. In normal gradient descent optimization, a batch is
+taken to be the whole dataset leading to the problem when the
+dataset gets big. Stochastic gradient descent considers a small
+batch in each iteration to lower the computing cost of the
+gradient descent approach while working with a large dataset.
+Random Forest:
+A random forest is a supervised ML
+algorithm that is constructed from an ensemble of decision tree
+algorithms. Its ensemble nature helps to provide a solution to
+complex problems. The random forest is made up of a large
+number of decision trees that have been trained via bagging
+or bootstrap aggregation. The average mean of the output
+of constituent decision trees is the random forest’s ultimate
+forecast. The precision of the output improves as the number
+of decision trees used grows. A random forest overcomes the
+decision tree algorithm’s limitations by eliminating over-fitting
+and enhancing precision.
+Logistic Regression: The probability for classification prob-
+lems is modeled using logistic regression, which divides
+them into two possible outcomes. For classification, logistic
+regression is an extension of the linear regression model. For
+regression tasks, linear regression works well; however, it fails
+to replicate for classification. The linear model considers the
+class a number and finds the optimum hyperplane that mini-
+mizes the distances between the points and the hyperplane. As
+it interpolates between the points, it cannot be interpreted as
+probabilities. Because there is no relevant threshold for class
+separation, logistic regression is applied. It is a widely used
+classification algorithm due to its ease of implementation and
+strong performance in linearly separable classes.
+K-Nearest Neighbors (KNN) Classifier: The KNN algorithm
+relies on the assumption that similar things exist in close
+proximity. It is a non-parametric and lazy learning algorithm.
+KNN does not carry any assumption for underlying data
+distribution. It does not require training data points for model
+generation, as all the training data are used in a testing phase.
+This results in faster training and a slower testing process. The
+costly testing phase will consume more time and memory. In
+KNN, K is the number of nearest neighbors and is generally
+considered odd. KNN, however, suffers from the curse of
+dimensionality. With increased feature dimension, it requires
+more data and becomes prone to overfitting.
+Support Vector Machine (SVM): A support vector machine
+
+Algorithm 1: Data Poisoning Algorithm
+Input: Non-poisoned feature set
+Output: Poisoned feature set
+Data: Static features obtained from malware and
+benign training set
+1 for all the samples do
+2
+Train the machine learning models and measure
+the performance
+3
+for 10% each of Malware and Benign data do
+4
+if Training label is not flipped then
+5
+label=Get training label of given data
+6
+if label==0 then
+7
+Flip the label to 1
+8
+else if label==1 then
+9
+Flip the label to 0
+10
+Train all the models and measure the performance
+11
+for 20% each of Malware and Benign data do
+12
+if Training label is not flipped then
+13
+label=Get training label of given data
+14
+if label==0 then
+15
+Flip the label to 1
+16
+else if label==1 then
+17
+Flip the label to 0
+18
+Train all the models and measure the performance
+is a popular supervised ML algorithm applied in both classi-
+fication and regression tasks. SVM aims to find a hyperplane
+that classifies the data points. In SVM, there are several pos-
+sible hyperplanes, and we need to determine the optimal hy-
+perplane that maximizes the margin between the two classes.
+Hyperplanes are the decision boundary for SVM, where data
+points near to hyperplane are the support vectors. Due to its
+effectiveness in high dimensional spaces and memory-efficient
+properties, it is widely adopted in different domains.
+Multi-Layer Perceptron:
+The term ’Perceptron’ is derived
+from the ability to perceive, see, and recognize images in a
+human-like manner. A perceptron machine is based on the
+neuron, a basic unit of computation, with a cell receiving a
+series of pairs of inputs and weights. Although the perceptron
+was originally thought to represent any circuit and logic, non-
+linear data cannot be represented by a perceptron with only one
+neuron. Multi-Layer Perceptron was developed to overcome
+this limitation. In multi-layer perceptron, the mapping between
+input and output is non-linear. It has input and output layers
+and several hidden layers stacked with numerous neurons.
+Because the inputs are merged with the initial weights in
+a weighted sum and applied to the activation function, the
+multi-layer perceptron falls under the category of feedforward
+algorithms. Each linear combination is propagated to the
+following layer, unlike with a perceptron.
+V. EXPERIMENTAL METHODOLOGY
+In this paper, we are using the label-flipping approach to
+poison the training data. With source class CS and a target
+class CT from a set of classes C, the dataset DI is poisoned.
+The detailed poisoning performed in the paper is shown in
+Algorithm 1. We perform a label poisoning attack of differ-
+ent volumes to training data without guiding the poisoning
+mechanism through machine learning architecture or the loss
+function. It is an efficient way to showcase the ability of
+random poisoning to hamper the model’s performance. We are
+training all eight malware detector models three times in total.
+As illustrated in Algorithm 1, we begin the model training
+with clean data without adding any noise. After recording the
+model’s performance on clean data, we proceed towards the
+first stage of poisoning our data. We take 10% of shuffled
+training data belonging to each malware and benign class, and
+we change their labels. We retrain all the models and again
+measure the performance of the models. We repeat the same
+operation with 20% of shuffled training data. The percentage
+of poisoned data is taken randomly for this experimental
+purpose, as the goal is to show the impact on the models.
+The algorithm we followed in carrying out this experiment is
+not a novel approach but a generic approach to poison the
+data.
+A. Experimental Environment and Dataset
+All the experiments are performed in Google-Colab us-
+ing Google’s GPU. All the implementation will be worked
+around using python libraries and Scikit-Learn. The training
+dataset [53] is obtained from the Kaggle repository, where
+data are collected from VirusTotal and VirusShare. The dataset
+comprises windows PE malware and benign files processed
+through static executable analysis. The dataset comprises
+216,352 files (75,503 benign files and 140,849 malware files)
+with 54 features.
+VI. EVALUATION RESULTS AND ANALYSIS
+A. Data Pre-processing and Transformation
+We begin our experiment by loading data from Kaggle
+dataset [53]. To clean the data, we followed two different
+approaches. First, we ignored rows that are missing more than
+50% of data, whereas we replaced the null values with the
+arithmetic mean value of the column for rows with less than
+50% missing values. Second, we normalized the data by scal-
+ing the values from 0 to 1. Afterward, 85% of data were used
+for training purposes while the remaining 15% were used for
+testing purposes. We trained selected eight machine learning
+models with standard hyper-parameters for each model. We
+didn’t tweak many machine learning parameters to fine-tune
+the detection accuracy, resulting in significant overfitting in a
+few models.
+B. Performance Indicators
+We evaluated the malware detectors’ performance using the
+following metrics:
+
+TABLE II
+MALWARE DETECTION TRAINING RESULT
+Algorithm
+Clean Data
+Training Data
+Testing Data
+Accuracy
+Precision
+Recall
+F1
+Accuracy
+Precision
+Recall
+F1
+Stochastic Gradient Descent
+93.41
+92.49
+88.29
+90.34
+72.98
+58.6
+78.77
+67.20
+Decision Tree
+99.96
+99.98
+99.91
+99.94
+59.65
+44.5
+59.85
+51.05
+Random Forest
+99.97
+99.92
+99.97
+99.94
+83.65
+98.82
+54.12
+69.94
+Logistic Regression
+93.2
+92.21
+87.94
+90.02
+92.33
+92.24
+85.36
+88.67
+KNN Classifier
+98.38
+97.33
+98.05
+97.69
+97.42
+96.38
+96.25
+96.31
+Support Vector Machine
+93.15
+92.44
+87.51
+89.91
+92.03
+90.89
+85.94
+88.34
+Perceptron
+90.93
+88.6
+84.91
+86.72
+75.39
+60.28
+87.86
+71.50
+Multi-Layer Perceptron
+91.28
+91.07
+83.16
+86.94
+71.93
+57.45
+77.66
+66.04
+TABLE III
+MALWARE DETECTION PERFORMANCE WITH 10% POISONING DATA
+Algorithm
+10% Poisoned Data
+Training Data
+Testing Data
+Accuracy
+Precision
+Recall
+F1
+Accuracy
+Precision
+Recall
+F1
+Stochastic Gradient Descent
+85.12
+82.49
+77.14
+79.73
+72.39
+64.23
+61.38
+62.77
+Decision Tree
+96.77
+99.44
+92.01
+95.58
+51.92
+38.33
+43.98
+40.96
+Random Forest
+96.77
+98.92
+92.51
+95.61
+80.13
+82.68
+60.22
+69.68
+Logistic Regression
+84.51
+82.29
+75.39
+78.69
+83.26
+81.06
+72.91
+76.77
+KNN Classifier
+89.49
+85.47
+87.1
+86.28
+86.59
+83.1
+81.15
+82.11
+Support Vector Machine
+84.75
+82.84
+75.42
+78.96
+66.99
+63.14
+31.16
+41.73
+Perceptron
+77.94
+67.78
+79.69
+73.25
+40.16
+25.89
+31
+73.25
+Multi-Layer Perceptron
+83.85
+82.72
+72.58
+77.32
+83.33
+82.81
+70.74
+76.30
+Accuracy =
+TP + TN
+TP + TN + FP + FN
+Precision =
+TP
+TP + FP , Recall =
+TP
+TP + FN
+F1-score = 2 ∗ (Precision ∗ Recall)
+Precision + Recall
+A positive outcome corresponds to a malware sample, while
+a negative result corresponds to a benign example. TP, TN,
+FP, and FN are true positives, true negatives, false positives,
+and false negatives, respectively. Accuracy is the percentage
+of correct predictions on the given data. Precision measures
+the ratio between true positives and all the positives. Recall
+provides the ability of our model to predict true positives
+correctly. The F1 score is the harmonic mean, the combination
+of a classifier’s precision and recall.
+C. Results and Discussion
+Table II shows the accuracy, precision, and recall for train-
+ing and testing data. Stochastic Gradient Descent, Decision
+Trees, Random Forest, and Perceptron looked overfitted to
+training data compared to other models. Since the data volume
+is a little bit high, decision tree-based classifiers are prone to
+overfitting problems. We used shallow layer neural networks
+leading perceptron to overfit in the data. However, classifiers
+like logistic regression, KNN classifier, and Support Vector
+Machine have shown the best performance in all three metrics.
+We have compared the performance of both the training and
+testing sets as we have only poisoned the training data while
+preserving the test data from attack.
+We flipped the labels of 10% training data as a poisoning
+attack. On poisoning 10% of total data, the performance metric
+for each detector is displayed in Table III. The results show the
+robustness of decision trees and random forest-based malware
+detectors compared to other malware detectors. We further
+poisoned 20% of total training data to see the impact of
+increased poisoned data in each model, whose results are
+shown in Table IV. The left-most confusion matrix in each of
+the figures from Figure 4 to Figure 11 shows the number of
+TP, TN, FP, and FN for each classifier on clean data, whereas
+the middle and right one shows results with 10% and 20%
+poisoning, respectively. In the confusion matrix, label ’0’ is
+for malware, and label ’1’ is for benign samples. The top-left
+corner in the confusion matrix gives True Positive, the top-
+right corner gives False Positive, the bottom-left gives False
+Negative, and the bottom-right corner gives True Negative
+samples.
+D. Analysis and Observations
+The goal of this work is to show the vulnerability of popular
+machine-learning models that are used for malware detection.
+Results in Tables II, III and IV reflect the limitations of
+all the experimented machine learning models even with the
+basic label poisoning attack. Figure 12 shows the ROC curve,
+comparing the models’ performance on the clean data, 10%
+and 20% poisoned data. In the ROC curve, the blue curve
+corresponds to the performance of clean data, the orange
+curve corresponds to 10% poisoned data, and the green curve
+corresponds to the 20% poisoned data. The curve closest to
+the top-left corner is the one performing best. We can infer
+from the graph that logistic regression, K-Nearest Neighbors,
+
+TABLE IV
+MALWARE DETECTION PERFORMANCE WITH 20% POISONING DATA
+Algorithm
+20% Poisoned data
+Training Data
+Testing Data
+Accuracy
+Precision
+Recall
+F1
+Accuracy
+Precision
+Recall
+F1
+Stochastic Gradient Descent
+78.56
+75.65
+70.21
+72.83
+62.69
+54.86
+50.72
+52.71
+Decision Tree
+96.54
+93.54
+98.34
+95.88
+40.26
+34.25
+49.67
+40.54
+Random Forest
+96.54
+93.04
+98.94
+95.90
+72.8
+68.77
+61.66
+65.02
+Logistic Regression
+78.38
+74.3
+72.13
+73.20
+77.58
+75.1
+76.78
+75.93
+KNN Classifier
+87.41
+82.48
+87.94
+85.12
+82.15
+76.16
+82.2
+79.06
+Support Vector Machine
+78.58
+74.45
+72.6
+73.51
+75.39
+74.74
+60.37
+66.79
+Perceptron
+75.16
+68.58
+72.57
+72.57
+49.37
+38.28
+38.28
+38.28
+Multi Layer Perceptron
+77.6
+75.45
+67.1
+71.03
+76.85
+74.81
+65.66
+69.94
+Fig. 4. Confusion Matrix for Stochastic Gradient Descent Based Malware Detector
+Fig. 5. Confusion Matrix for Decision Tree Based Malware Detector
+Fig. 6. Confusion Matrix for Random Forest-Based Malware Detector
+Fig. 7. Confusion Matrix for Logistic Regression Based Malware Detector
+Support Vector Machine, and Multi-Layer Perceptron are the
+best models on the clean data. However, the distance between
+the three curves represents the robustness of the model toward
+the poisoning attack. If the separation between the curves of
+clean data and poisoning data is low, it infers that the poisoning
+attack has a minimal impact on the model’s performance. In
+the ROC graph, we can observe that Random Forest, Logistic
+Regression, K-Nearest Neighbors, and Multi-Layer Perceptron
+
+ConfusionMatrixforSGD
+Predicted Class
+12000
+12148
+1884
+10000
+Actual Class
+8000
+6000
+1522
+6082
+4000
+2000
+0
+1Confusion Matrix for SGD with 10% Poisoning
+PredictedClass
+10000
+IClass
+0
+11778
+1652
+8000
+Actual
+6000
+2495
+5711
+4000
+2000
+0
+1Confusion Matrix for SGD with 20% Poisoning
+PredictedClass
+10000
+IClass
+0
+10830
+1936
+8000
+Actual
+6000
+3294
+5576
+4000
+2000
+0
+1ConfusionMatrixforDT
+Predicted Class
+7000
+7593
+6439
+Actual Class
+6000
+5000
+2677
+4927
+4000
+-3000
+0
+1Confusion Matrix for DT with 10% Poisoning
+PredictedClass
+8000
+Class
+0
+4915
+8515
+7000
+Actual
+6000
+5000
+4763
+3443
+4000
+0
+1Confusion Matrix for DT with 20% Poisoning
+PredictedClass
+7000
+IClass
+0
+5440
+7326
+6000
+Actual
+5000
+4945
+3925
+4000
+0
+1ConfusionMatrixforRF
+Predicted Class
+12500
+14000
+32
+10000
+Actual Class
+7500
+5000
+4405
+3199
+-2500
+0
+-Confusion Matrix for RF with 10% Poisoning
+PredictedClass
+12000
+12145
+1285
+10000
+ActualClass
+8000
+6000
+3293
+4913
+4000
+2000
+0
+1Confusion Matrix for RF with 20% Poisoning
+PredictedClass
+10000
+ActualClass
+0
+10036
+2730
+8000
+6000
+3154
+5716
+4000
+0
+1ConfusionMatrixfor LR
+PredictedClass
+12500
+13492
+540
+10000
+Actual Class
+7500
+-5000
+1119
+6485
+2500
+0
+1Confusion Matrix for LR with 10% Poisoning
+PredictedClass
+12000
+10000
+Actual Class
+0
+12032
+1398
+8000
+6000
+2223
+5983
+4000
+2000
+0
+1Confusion Matrix for LR with 20% Poisoning
+PredictedClass
+10000
+IClass
+0
+10774
+1992
+8000
+Actual
+6000
+2859
+6011
+4000
+2000
+0
+1Fig. 8. Confusion Matrix for KNN Based Malware Detector
+Fig. 9. Confusion Matrix for Support Vector Machine-Based Malware Detector
+Fig. 10. Confusion Matrix for Perceptron Based Malware Detector
+Fig. 11. Confusion Matrix for Multi-Layer Perceptron Based Malware Detector
+have their graphs close to each other, proving their robustness
+against poisoned data. Random Forest’s robustness can be
+attributed to its ensemble nature which helps it to capture
+better insights about the data. The robustness of logistic
+regression and K-Nearest Neighbors can be due to the low
+dimensionality of our training data. Further, we can observe
+the performance of models, like SVM and perceptron, doing
+better with the 20% poisoned data than with 10% poisoned
+data. The gain in the performance of these models is due
+to unrestricted data poisoning. Since we are not guiding our
+poisoning approach according to the models, further adding
+poisoning data after some threshold point slightly improves the
+models’ performance. In the end, even the least sophisticated
+attacks, like label poisoning, are causing the performance
+decay of the models to a large extent. This further alerts us
+toward the catastrophic consequences of more sophisticated
+attacks like gradients and reinforcement learning.
+VII. CONCLUSION
+In this work, we perform a feasibility analysis of label-
+flip poisoning attacks on ML-based malware detectors. We
+evaluated eight different ML models that are widely used in
+malware detection. Spotting the lack of poisoning attacks work
+in the malware domain, this paper analyses the robustness
+of ML-based malware detectors against different volumes of
+poisoned data. We observed the decay in performance of all
+the models while poisoning 10% and 20% of total training
+data. The significant decrease in the performance of the models
+shows the severe vulnerability of malware detectors to guided
+poisoning approaches. We also observed differences in the
+effect of poisoning attacks across the different models. Our
+work is carried out within the limited scope of one generic
+poisoning algorithm and a single malware dataset. There are
+few future research directions that are clearly visible. The
+malware detectors can be tested against many advanced poi-
+soning approaches using numerous datasets from the industry.
+
+ConfusionMatrixforPerceptron
+Predicted Class
+7000
+7831
+6201
+Actual Class
+6000
+5000
+4000
+2139
+5465
+3000
+1ConfusionMatrixforPerceptronwith10%Poisoning
+PredictedClass
+7000
+IClass
+0
+7734
+5696
+6000
+Actual
+5000
+4897
+3309
+4000
+0
+1Confusion Matrix for Perceptron with 20% Poisoning
+PredictedClass
+10000
+ActualClass
+0
+1393
+11373
+8000
+6000
+4507
+4363
+4000
+2000
+0
+1ConfusionMatrixforMLE
+Predicted Class
+12500
+13412
+620
+10000
+Actual Class
+7500
+5000
+1277
+6327
+2500
+0
+1Confusion Matrix for MLP with 10% Poisoning
+PredictedClass
+12000
+1205
+10000
+IClass
+0
+12225
+8000
+Actual
+6000
+2401
+5805
+4000
+2000
+0
+1Confusion Matrix for MLP with 20% Poisoning
+PredictedClass
+10000
+IClass
+0
+10805
+1961
+8000
+Actual
+6000
+3046
+5824
+4000
+2000
+0
+1ConfusionMatrixforKNN
+Predicted Class
+12500
+13767
+265
+10000
+Actual Class
+7500
+5000
+261
+7343
+2500
+0
+1Confusion Matrix for KNN with 10% Poisoning
+PredictedClass
+10000
+IClass
+0
+11860
+1570
+8000
+Actual
+6000
+1531
+6675
+4000
+2000
+0
+1Confusion Matrix for KNN with 20% Poisoning
+PredictedClass
+10000
+IClass
+0
+10484
+2282
+8000
+Actual
+6000
+1578
+7292
+4000
+2000
+0
+1ConfusionMatrixforSVM
+Predicted Class
+12500
+13371
+661
+10000
+Actual Class
+7500
+5000
+1061
+6543
+2500
+0
+1Confusion Matrix for SVM with 10% Poisoning
+PredictedClass
+10000
+IClass
+0
+11989
+1441
+8000
+Actual
+6000
+5718
+2488
+-4000
+2000
+0
+1Confusion Matrix for SVM with 20% Poisoning
+PredictedClass
+10000
+IClass
+0
+10955
+1811
+8000
+Actual
+6000
+3510
+5360
+-4000
+2000
+0
+1Fig. 12. ROC Curve for Malware detectors under Poisoning Environments
+The poisoning can be tested in a more real environment by
+poisoning the executable files. The research community still
+lacks exhaustive studies on the vulnerabilities of malware
+detectors and how to make detectors more robust against these
+poisoning attacks.
+REFERENCES
+[1] D. Ciregan, U. Meier, and J. Schmidhuber, “Multi-column deep neural
+networks for image classification,” in 2012 IEEE Conference on Com-
+puter Vision and Pattern Recognition, 2012, pp. 3642–3649.
+[2] J. Schmidhuber, U. Meier, and D. Ciresan, “Multi-column deep neural
+networks for image classification,” in 2012 IEEE Conference on Com-
+puter Vision and Pattern Recognition.
+IEEE Computer Society, 2012.
+[3] K. Chowdhary, “Natural Language Processing,” Fundamentals of Arti-
+ficial Intelligence, pp. 603–649, 2020.
+[4] J. Hirschberg and C. D. Manning, “Advances in Natural Language
+Processing,” Science, vol. 349, no. 6245, pp. 261–266, 2015.
+[5] C.-F. Tsai, Y.-F. Hsu, C.-Y. Lin, and W.-Y. Lin, “Intrusion detection by
+machine learning: A review,” Expert Systems with Applications, vol. 36,
+no. 10, pp. 11 994–12 000, 2009.
+[6] N. Peiravian and X. Zhu, “Machine learning for android malware detec-
+
+KNNCleanData,auc=0.971460832443106
+KNN 10% Poison, auc=0.8552929416737186
+KNN 20% Poison, auc=0.8216704426092349SVM Clean Data, auc=0.9073593040810904
+SVM 10% Poison, auc=0.6002161213967441
+SVM 20% Poison, auc=0.73096877256933DT Clean Data, auc=0.5872804184858597
+DT 10% Poison, auc=0.5037422175699491
+DT 20% Poison, auc=0.4169380476360457MLp CleanData,auc=0.8939386759774156
+MLP 10% Poison, auc=0.8088423576886244
+MLP 20% Poison, auc=0.7514920551542543Perceptron Clean Data, auc=0.7824525005443335
+Perceptron 10% Poison, auc=0.3838246137390344
+Perceptron 20% Poison, auc=0.476954003915064SGD Clean Data, auc=0.7482652186900373
+SGD 10% Poison, auc=0.7025164286923702
+SGD 20% Poison, auc=0.6086523161420353RF Clean Data.auC=0.7456458394939469
+RF 10% Poison, auc=0.7625879961069476
+RF 20% Poison, auc=0.711092219132663LR Clean Data, auc=0.9073593040810904
+LR 10% Poison, auc=0.8125026745227009
+LR 20% Poison, auc=0.7608190424784267tion using permission and api calls,” in 2013 IEEE 25th International
+Conference on Tools with Artificial Intelligence, 2013, pp. 300–305.
+[7] J. Kober and J. Peters, “Learning motor primitives for robotics,” in 2009
+IEEE International Conference on Robotics and Automation, 2009.
+[8] R. Manicavasaga, P. B. Lamichhane, P. Kandel, and D. A. Talbert, “Drug
+repurposing for rare orphan diseases using machine learning techniques,”
+in The International FLAIRS Conference Proceedings, vol. 35, 2022.
+[9] A. Dhakal, C. McKay, J. J. Tanner, and J. Cheng, “Artificial intelligence
+in the prediction of protein–ligand interactions: recent advances and
+future directions,” Briefings in Bioinformatics, vol. 23, no. 1, p. bbab476,
+2022.
+[10] M. H. R. Khouzani, S. Sarkar, and E. Altman, “Maximum Damage
+Malware Attack in Mobile Wireless Networks,” IEEE/ACM Transactions
+on Networking, vol. 20, no. 5, pp. 1347–1360, 2012.
+[11] S. Gupta, A. Singhal, and A. Kapoor, “A literature survey on social
+engineering attacks: Phishing attack,” in 2016 International Conference
+on Computing, Communication and Automation (ICCCA), 2016.
+[12] F. Callegati, W. Cerroni, and M. Ramilli, “Man-in-the-Middle Attack to
+the HTTPS Protocol,” IEEE Security Privacy, vol. 7, no. 1, 2009.
+[13] C. Schuba, I. Krsul, M. Kuhn, E. Spafford, A. Sundaram, and D. Zam-
+boni, “Analysis of a denial of service attack on TCP,” in Proceedings.
+1997 IEEE Symposium on Security and Privacy (Cat. No.97CB36097),
+1997.
+[14] W. G. Halfond, J. Viegas, A. Orso et al., “A classification of sql-injection
+attacks and countermeasures,” in Proceedings of the IEEE International
+Symposium on Secure Software Engineering, vol. 1.
+IEEE, 2006.
+[15] I. Yilmaz and R. Masum, “Expansion of cyber attack data from
+unbalanced
+datasets
+using
+generative
+techniques,”
+arXiv
+preprint
+arXiv:1912.04549, 2019.
+[16] B. Kolosnjaji, A. Demontis, B. Biggio, D. Maiorca, G. Giacinto,
+C. Eckert, and F. Roli, “Adversarial malware binaries: Evading deep
+learning for malware detection in executables,” in IEEE European Signal
+Processing Conference, 2018, pp. 533–537.
+[17] F. Kreuk, A. Barak, S. Aviv-Reuven, M. Baruch, B. Pinkas, and
+J. Keshet, “Adversarial examples on discrete sequences for beating
+whole-binary malware detection,” arXiv preprint :1802.04528, 2018.
+[18] L. Demetrio, B. Biggio, G. Lagorio, F. Roli, and A. Armando, “Explain-
+ing Vulnerabilities of Deep Learning to Adversarial Malware Binaries,”
+arXiv preprint arXiv:1901.03583, 2019.
+[19] O. Suciu, S. E. Coull, and J. Johns, “Exploring adversarial examples
+in malware detection,” in 2019 IEEE Security and Privacy Workshops,
+2019.
+[20] K. Aryal, M. Gupta, and M. Abdelsalam, “A Survey on Adversarial
+Attacks for Malware Analysis,” arXiv preprint arXiv:2111.08223, 2021.
+[21] C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow,
+and R. Fergus, “Intriguing properties of neural networks,” arXiv preprint
+arXiv:1312.6199, 2013.
+[22] S. M. P. Dinakarrao, S. Amberkar, S. Bhat, A. Dhavlle, H. Sayadi,
+A. Sasan, H. Homayoun, and S. Rafatirad, “Adversarial attack on
+microarchitectural events based malware detectors,” in Proceedings of
+the 56th Annual Design Automation Conference 2019, 2019, pp. 1–6.
+[23] W. Hu and Y. Tan, “Generating adversarial malware examples for black-
+box attacks based on GAN,” arXiv preprint arXiv:1702.05983, 2017.
+[24] S. Shalev-Shwartz et al., “Online learning and online convex optimiza-
+tion,” Foundations and Trends® in Machine Learning, vol. 4, 2012.
+[25] A. Rai, K. K. Chintalapudi, V. N. Padmanabhan, and R. Sen, “Zee:
+Zero-effort Crowdsourcing for Indoor Localization,” in Proceedings of
+the 18th Annual International Conference on Mobile Computing and
+Networking, 2012, pp. 293–304.
+[26] K. Bonawitz, H. Eichner, W. Grieskamp, D. Huba, A. Ingerman,
+V. Ivanov, C. Kiddon, J. Koneˇcn`y, S. Mazzocchi, B. McMahan et al.,
+“Towards federated learning at scale: System design,” Proceedings of
+Machine Learning and Systems, vol. 1, pp. 374–388, 2019.
+[27] “Tay, Microsoft’s AI chatbot, gets a crash course in racism from
+Twitter,”
+https://www.theguardian.com/technology/2016/mar/24/
+tay-microsofts-ai-chatbot-gets-a-crash-course-in-racism-from-twitter,
+2016.
+[28] A. Shafahi, W. R. Huang, M. Najibi, O. Suciu, C. Studer, T. Dumitras,
+and T. Goldstein, “Poison Frogs! Targeted Clean-Label Poisoning At-
+tacks on Neural Networks,” Advances in Neural Information Processing
+Systems, vol. 31, 2018.
+[29] X. Liu, S. Si, X. Zhu, Y. Li, and C.-J. Hsieh, “A unified framework for
+data poisoning attack to graph-based semi-supervised learning,” arXiv
+preprint arXiv:1910.14147, 2019.
+[30] D. Cao, S. Chang, Z. Lin, G. Liu, and D. Sun, “Understanding distributed
+poisoning attack in federated learning,” in 2019 IEEE 25th International
+Conference on Parallel and Distributed Systems (ICPADS). IEEE, 2019.
+[31] J. Shen, X. Zhu, and D. Ma, “TensorClog: An imperceptible poisoning
+attack on deep neural network applications,” IEEE Access, vol. 7, 2019.
+[32] J. Zhang, J. Chen, D. Wu, B. Chen, and S. Yu, “Poisoning Attack
+in Federated Learning using Generative Adversarial Nets,” in 2019
+18th IEEE International Conference On Trust, Security And Privacy In
+Computing And Communications/13th IEEE International Conference
+On Big Data Science And Engineering (TrustCom/BigDataSE).
+IEEE,
+2019.
+[33] W. Jiang, H. Li, S. Liu, Y. Ren, and M. He, “A Flexible Poisoning Attack
+Against Machine Learning,” in ICC 2019-2019 IEEE International
+Conference on Communications (ICC).
+IEEE, 2019, pp. 1–6.
+[34] H. Kwon, H. Yoon, and K.-W. Park, “Selective poisoning attack on deep
+neural network to induce fine-grained recognition error,” in IEEE Sec-
+ond International Conference on Artificial Intelligence and Knowledge
+Engineering, 2019, pp. 136–139.
+[35] H. Zhang, T. Zheng, J. Gao, C. Miao, L. Su, Y. Li, and K. Ren, “Data
+poisoning attack against knowledge graph embedding,” in Proceedings
+of the Twenty-Eighth International Joint Conference on Artificial
+Intelligence, IJCAI-19.
+International Joint Conferences on Artificial
+Intelligence Organization, 7 2019, pp. 4853–4859. [Online]. Available:
+https://doi.org/10.24963/ijcai.2019/674
+[36] E. Bagdasaryan, A. Veit, Y. Hua, D. Estrin, and V. Shmatikov, “How to
+backdoor federated learning,” in International Conference on Artificial
+Intelligence and Statistics.
+PMLR, 2020, pp. 2938–2948.
+[37] M. Li, Y. Sun, H. Lu, S. Maharjan, and Z. Tian, “Deep reinforcement
+learning for partially observable data poisoning attack in crowdsensing
+systems,” IEEE Internet of Things Journal, vol. 7, 2020.
+[38] S. Sasaki, S. Hidano, T. Uchibayashi, T. Suganuma, M. Hiji, and S. Kiy-
+omoto, “On embedding backdoor in malware detectors using machine
+learning,” in IEEE International Conference on Privacy, Security and
+Trust, 2019, pp. 1–5.
+[39] X. Zhang, X. Zhu, and L. Lessard, “Online Data Poisoning Attack,” in
+Learning for Dynamics and Control.
+PMLR, 2020, pp. 201–210.
+[40] G. Lovisotto, S. Eberz, and I. Martinovic, “Biometric backdoors: A
+poisoning attack against unsupervised template updating,” in 2020 IEEE
+European Symposium on Security and Privacy (EuroS&P). IEEE, 2020.
+[41] C. Li, X. Chen, D. Wang, S. Wen, M. E. Ahmed, S. Camtepe, and
+Y. Xiang, “Backdoor attack on machine learning based android malware
+detectors,” IEEE Trans. on Dependable and Secure Computing, 2021.
+[42] M. Kravchik, B. Biggio, and A. Shabtai, “Poisoning attacks on cyber
+attack detectors for industrial control systems,” in Proceedings of the
+36th Annual ACM Symposium on Applied Computing, 2021.
+[43] B. Biggio, B. Nelson, and P. Laskov, “Poisoning attacks against support
+vector machines,” arXiv preprint arXiv:1206.6389, 2012.
+[44] C. Yang, Q. Wu, H. Li, and Y. Chen, “Generative Poisoning Attack
+Method Against Neural Networks,” preprint arXiv:1703.01340, 2017.
+[45] L. Mu˜noz-Gonz´alez, B. Biggio, A. Demontis, A. Paudice, V. Wongras-
+samee, E. C. Lupu, and F. Roli, “Towards Poisoning of Deep Learning
+Algorithms with Back-gradient Optimization,” in Proceedings of the
+10th ACM workshop on Artificial Intelligence and Security, 2017.
+[46] C. Miao, Q. Li, L. Su, M. Huai, W. Jiang, and J. Gao, “Attack under
+Disguise: An Intelligent Data Poisoning Attack Mechanism in Crowd-
+sourcing,” in Proceedings of the 2018 World Wide Web Conference,
+2018.
+[47] A. P. Dawid and A. M. Skene, “Maximum Likelihood Estimation of
+Observer Error-Rates Using the EM Algorithm,” Journal of the Royal
+Statistical Society: Series C (Applied Statistics), vol. 28, no. 1, 1979.
+[48] M. Fang, G. Yang, N. Z. Gong, and J. Liu, “Poisoning attacks to
+graph-based recommender systems,” in Proceedings of the 34th Annual
+Computer Security Applications Conference, 2018, pp. 381–392.
+[49] D. Arp, M. Spreitzenbarth, M. Hubner, H. Gascon, K. Rieck, and
+C. Siemens, “Drebin: Effective and explainable detection of android
+malware in your pocket.” in NDSS, vol. 14, 2014, pp. 23–26.
+[50] H. Cai, N. Meng, B. Ryder, and D. Yao, “Droidcat: Effective android
+malware detection and categorization via app-level profiling,” IEEE
+Transactions on Information Forensics and Security, vol. 14, no. 6, 2018.
+[51] E. Mariconti, L. Onwuzurike, P. Andriotis, E. De Cristofaro, G. Ross,
+and G. Stringhini, “Mamadroid: Detecting android malware by building
+markov chains of behavioral models,” preprint arXiv:1612.04433, 2016.
+
+[52] Y. Aafer, W. Du, and H. Yin, “Droidapiminer: Mining api-level features
+for robust malware detection in android,” in International Conference
+on Security and Privacy in Communication Systems.
+Springer, 2013.
+[53] “Malware
+detection,”
+https://www.kaggle.com/competitions/
+malware-detection/data.
+

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+arXiv:2301.02908v1  [math.FA]  7 Jan 2023
+DYNAMICAL PROPERTIES AND SOME CLASSES OF
+NON-POROUS SUBSETS OF LEBESGUE SPACES
+STEFAN IVKOVI´C, SERAP ¨OZTOP, AND SEYYED MOHAMMAD TABATABAIE∗
+Abstract. In this paper, we introduce several classes of non-σ-porous
+subsets of a general Lebesgue space. Also, we study some linear dynam-
+ics of operators and show that the set of all non-hypercyclic vectors of a
+sequences of weighted translation operators on Lp-spaces is not σ-porous.
+1. Introduction
+σ-porous sets, as a collection of very thin subsets of metric spaces, were
+introduced and studied first time in [8] through a research on boundary be-
+havior of functions, and then were applied in differentiation and Banach spaces
+theories in [3, 14]. The concepts related to porosity have been active topics in
+recent decades because they can be adapted for many known notions in several
+kind of metric spaces; see the monograph [21]. σ-porous subsets of R are null
+and of first category, while in every complete metric space without any isolated
+points these two categories are different [20]. On the other hand, linear dy-
+namics including hypercyclicity in operator theory received attention during
+the last years; see books [2, 11] and for instance [6, 16, 17]. Recently, F. Bayart
+in [1] through study of hypercyclic shifts (which was previously studied in [15];
+see also [10]) proved that the set of non-hypercyclic vectors of some classes of
+weighted shift operators on ℓ2(Z) is a non-σ-porous set. This would be a new
+example of a first category set which is not σ-porous. In this work, by some
+idea from the proof of [1, Theorem1] first we introduce a class of non-σ-porous
+subsets of general Lebesgue spaces, and then we develop the main result of [1]
+to sequences of weighted translation operators on general Lebesgue spaces in
+the context of discrete groups and hypergroups. In particular, we prove that
+if p ≥ 1, K is a discrete hypergroup, (an) is a sequence with distinct terms in
+K, and w : K → (0, ∞) is a bounded measurable function such that
+�
+n∈N
+1
+w(a0)w(a1) . . . w(an)χ{an+1} ∈ Lp(K),
+then the set of all non-hypercyclic vectors of the sequence (Λn)n is not σ-
+porous, where the operators Λn are given in Definition 3.8. Also, we study
+2010 Mathematics Subject Classification. 47A16, 28A05, 43A15, 43A62.
+∗Corresponding author.
+Key words and phrases. non-σ-porous sets, Lebesgue spaces, σ-porous operators, locally
+compact groups, locally compact hypergroups, hypercyclic vectors.
+1
+
+2
+S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE
+non-σ-porosity of non-hypercyclic vectors of weighted composition operators
+on L∞(Ω) for a general measure space Ω equipped with a nonnegative Radon
+measure and on Lp(R, τ), where τ is the Lebesgue measure on R. We show
+that if G is a locally compact group, µ is a left Haar measure on G, a ∈ G,
+and w : G → (0, ∞) be a weight such that
+�
+1
+w(a)w(a2) . . . w(an)
+�
+n ∈ L∞(G, µ),
+then the set of all non-hypercyclic vectors of the weighted translation operator
+Ta,w,∞ on L∞(G, µ) is not σ-porous.
+2. Non-σ-porous subsets of Lebesgue spaces
+In this section, we will introduce some classes of non-σ-porous subsets of
+Lebesgue spaces related to a fixed function. First, we recall the definition of
+the main notion of this paper.
+Definition 2.1. Let 0 < λ < 1. A subset E of a metric space X is called
+λ-porous at x ∈ E if for each δ > 0 there is an element y ∈ B(x; δ) \ {x} such
+that
+B(y; λ d(x, y)) ∩ E = ∅.
+E is called λ-porous if it is λ-porous at every element of E. Also, E is called
+σ-λ-porous if it is a countable union of λ-porous subsets of X.
+The following lemma plays a key role in the proof of main results of this
+section. This fact is a special case of [19, Lemma2]; see also [1, Lemma2].
+Lemma 2.2. Let F be a non-empty family of non-empty closed subsets of a
+complete metric space X such that for each F ∈ F and each x ∈ X and r > 0
+with B(x; r) ∩ F ̸= ∅, there exists an element J ∈ F such that
+∅ ̸= J ∩ B(x; r) ⊆ F ∩ B(x; r)
+and F ∩B(x; r) is not λ-porous at all elements of J ∩B(x; r). Then, every set
+in F is not σ-λ-porous.
+The next result is a development of of [1, Theorem1]. Same as [1], the proof
+of this theorem is based on Lemma 2.2.
+Theorem 2.3. Let p ≥ 1, Ω be a locally compact Hausdorff space, µ be a
+nonnegative Radon measure on Ω, and A ⊆ Ω be a Borel set such that
+|f|χA ≤ ∥f∥p a.e.
+(f ∈ Lp(Ω, µ)).
+(2.1)
+Then, for each measurable function g on Ω with gχA ∈ Lp(Ω, µ), the set
+Γg :=
+�
+f ∈ Lp(Ω, µ) : |f| ≥ |g|χA a.e.
+�
+is not σ-porous in Lp(Ω, µ).
+
+DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS
+3
+Proof. Fix an arbitrary number 0 < λ ≤ 1
+2, and pick 0 < β < λ. Denote
+F :=
+�
+Γg : gχA ∈ Lp(Ω, µ)
+�
+.
+We will show that the collection F satisfies the conditions of Lemma 2.2.
+Let g ∈ Lp(Ω, µ). Without lossing the generality, we can assume that g is a
+nonnegative function. Trivially, Γg ̸= ∅. Let (fn) be a sequence in Γg and
+fn → f in Lp(Ω, µ). Then, by (2.1), |f| ≥ gχA a.e., and so f ∈ Γg. Therefore,
+every element of the collection F is a closed subset of Lp(Ω, µ). Now, assume
+that f ∈ Lp(Ω, µ) and r > 0 with B(f; r) ∩ Γg ̸= ∅. We find a measurable
+function h with 0 ≤ hχA ∈ Lp(Ω, µ) such that
+∅ ̸= B(f; r) ∩ Γh ⊆ B(f; r) ∩ Γg,
+and B(f; r) ∩ Γg is not λ-porous at elements of B(f; r) ∩ Γh.
+Since
+�
+|f| + β−1gχA
+�p ∈ L1(Ω, µ) and µ is a Radon measure, the mapping
+ν defined by
+ν(B) :=
+�
+B
+�
+|f| + β−1gχA
+�p dµ
+(for every Borel set B ⊆ Ω)
+is a Radon measure [9]. Hence, there are some 0 < ǫ < 1, a function k ∈
+B(f; r) ∩ Γg and a compact subset D of Ω with µ(D) > 0 such that
+∥k − f∥p < ǫ1/p r.
+and
+�
+Dc
+�
+|f| + β−1gχA
+�p dµ < (1 − ǫ) rp.
+(2.2)
+Pick some α with
+∥k − f∥p < α < ǫ1/p r,
+and denote
+δ := ǫ1/p r − α
+2µ(D)
+1
+p
+.
+Now, we define two functions h, ξ : Ω → C by
+h := (gχA + δ)χD + β−1gχA χΩ\D
+and
+ξ := (|k| + δ)η χD + hχΩ\D,
+where
+η(x) :=
+
+
+
+k(x)
+|k(x)|,
+if k(x) ̸= 0
+1,
+if k(x) = 0
+for all x ∈ Ω. Since D is compact, we have hχA ∈ Lp(Ω, µ). Also, for each
+x ∈ D,
+|k(x) − ξ(x)| =
+��k(x) −
+�
+|k(x)| + δ
+�
+η(x)
+��
+=
+��k(x) − k(x) − δ η(x)
+��
+= δ,
+
+4
+S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE
+and therefore
+∥(ξ − k) χD∥p = δ µ(D)
+1
+p = ǫ1/p r − α
+2
+.
+This implies that
+∥(ξ − f) χD∥p ≤ ∥(ξ − k) χD∥p + ∥(k − f) χD∥p
+≤ ǫ1/p r − α
+2
++ α < ǫ1/p r.
+Hence,
+∥ξ − f∥p
+p =
+�
+D
+|ξ − f|p dµ +
+�
+Ω\D
+|ξ − f|p dµ
+< ǫ rp +
+�
+Ω\D
+|β−1gχA − f|p dµ
+≤ ǫ rp +
+�
+Ω\D
+(β−1gχA + |f|)p dµ
+< ǫ rp + (1 − ǫ) rp = rp,
+and so, ξ ∈ B(f; r). Moreover,
+|ξ(x)| = |k(x)| + δ ≥ g(x) + δ = h(x)
+a.e. on D ∩ A,
+and for each x ∈ (Ω \ D) ∩ A we have |ξ(x)| = h(x). This shows that ξ ∈ Γh,
+and so
+∅ ̸= B(f; r) ∩ Γh ⊆ B(f; r) ∩ Γg
+because h ≥ g. Now, let u ∈ B(f; r)∩Γh and put r′ := min{δ, λ (r−∥f −u∥p)}.
+Let v ∈ B(u; r′). We define the function γ : Ω → C by
+γ(x) :=
+
+
+
+
+
+v(x),
+if x ∈ D
+�
+|v(x)| + β|u(x) − v(x)|
+�
+θ(x),
+if x ∈ Ω \ D
+where
+θ(x) :=
+
+
+
+v(x)
+|v(x)|,
+if v(x) ̸= 0
+1,
+if v(x) = 0.
+Therefore, for each x ∈ Ω \ D we have
+|γ(x) − v(x)| = β |u(x) − v(x)|
+and
+|γ(x)| ≥ β |u(x)|.
+Easily,
+∥γ − v∥p
+p = ∥(γ − v) χD∥p
+p + ∥(γ − v) χΩ\D∥p
+p
+= ∥(γ − v) χΩ\D∥p
+p
+= βp ∥(u − v) χΩ\D∥p
+p
+≤ βp ∥u − v∥p
+p < λp ∥u − v∥p
+p,
+
+DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS
+5
+and hence,
+γ ∈ B
+�
+v; λ ∥u − v∥p
+�
+⊆ B(f; r).
+In addition,
+|γ(x)| ≥ β |u(x)| ≥ β h(x) = g(x)
+for a.e. x ∈ (Ω \ D) ∩ A
+and
+|γ(x)| = |v(x)| ≥ |u(x)| − δ ≥ g(x)
+for a.e. x ∈ D ∩ A,
+because ∥u − v∥p ≤ δ and also |u| ≥ h. Therefore,
+B
+�
+v; λ ∥u − v∥p
+�
+∩ B(f; r) ∩ Γg ̸= ∅.
+and this competes the proof.
+□
+Remark 2.4. Note that, in general, the condition (2.1) in the statement of
+Theorem 2.3 does not implies that Ω is a discrete space. In particular, if in
+the condition (2.1) we set A := Ω, then it implies that Lp(Ω, µ) ⊆ L∞(Ω, µ),
+and this inclusion is equivalent to
+α := inf{µ(E) : µ(E) > 0} > 0,
+(2.3)
+and equivalently, for each q > p, Lp(Ω, µ) ⊆ Lq(Ω, µ); see [18]. If in addition,
+suppµ = Ω, then the condition (2.3) implies that for each x ∈ Ω,
+µ({x}) = inf{µ(F) : F is a compact neighborhood of x} > 0.
+Specially, if Ω is a locally compact group (or hypergroup) and µ is a left Haar
+measure of it, then the condition (2.1) implies that Ω is a discrete topological
+space.
+The next result is a direct conclusion of Theorem 2.3.
+Corollary 2.5. Let Ω be a discrete topological space and ϕ := (ϕj)j∈Ω ⊆
+[1, ∞) such that for each j, ϕj ≥ 1. Put µϕ := �
+j∈Ω ϕj δj, where δj is the
+point-mass measure at j. Then, for each g ∈ Lp(Ω, µϕ), the set
+Γg :=
+�
+f ∈ Lp(Ω, µϕ) : |f| ≥ |g|
+�
+is not σ-porous in Lp(Ω, µϕ).
+Proof. Just note that for each k ∈ Ω and f ∈ Lp(Ω, µϕ),
+∥f∥p
+p =
+�
+j∈Ω
+|f(j)|p µϕ({j}) ≥ |f(k)| ϕk ≥ |f(k)|p.
+□
+In particular, if a set is endowed with the counting measure, we get the fact.
+Corollary 2.6. Let p ≥ 1 and A be a non-empty set. Then, for each g ∈
+ℓp(A), the set
+Γg :=
+�
+f ∈ ℓp(A) : |f| ≥ |g|
+�
+is not σ-porous in ℓp(A).
+The situation for L∞-spaces is different.
+
+6
+S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE
+Theorem 2.7. Let Ω be a locally compact Hausdorff space and µ be a non-
+negative Radon measure on Ω. Then, for each g ∈ L∞(Ω, µ), the set
+Γg :=
+�
+f ∈ L∞(Ω, µ) : |f| ≥ |g| a.e.
+�
+is not σ-porous in L∞(Ω, µ).
+Proof. Same as the proof of Theorem 2.3 fix 0 < λ ≤ 1
+2, and set
+F :=
+�
+Γg : g ∈ L∞(Ω, µ)
+�
+.
+This collection satisfies the conditions of Lemma 2.2. Trivially, Γg is a closed
+subset of L∞(Ω, µ) for all g ∈ L∞(Ω, µ). Let Assume that 0 ≤ g ∈ L∞(Ω, µ),
+and let f ∈ L∞(Ω, µ) and r > 0.
+If B(f; r) ∩ Γg ̸= ∅, we choose some
+k ∈ B(f; r) ∩ Γg and we find some ε ∈ (0, 1) such that ∥k − f∥∞ < εr. Pick
+some δ ∈ (0, (1 − ε)r), and set
+h := g + δ
+and
+ξ := (|k| + δ)η,
+where η is as in the proof of Theorem 2.3. Then, we get
+∥ξ − f∥∞ ≤ ∥ξ − k∥∞ + ∥k − f∥∞ ≤ δ + εr < (1 − ε)r + εr = r,
+so ξ ∈ B(f; r), and ξ ∈ Γh since |k| ≥ g. Next, let u ∈ B(f; r) ∩ Γh, and set
+r′ := min{δ, λ (r − ∥f − u∥∞)}. Pick some v ∈ B(u; r′). Then, ∥u − v∥∞ <
+r′ ≤ δ, so |v| ≥ |u| − δ ≥ h − δ = g a.e., so v ∈ Γg. Thus,
+v ∈ B(v; λ∥u − v∥∞) ∩ B(f; r) ∩ Γg.
+This completes the proof.
+□
+Remark 2.8. The main Theorem 2.3 is valid also for the sequence space c0,
+because the sequences with finitely many non-zero coefficients approximate
+sequences in c0.
+At the end of this section, we give a class of non-σ-porous subsets of the
+Lp-space on real line. In the proof of this result, which is also based on Lemma
+2.2, we apply some functions defined in the proof of Theorem 2.3.
+Theorem 2.9. Let p ≥ 1, and τ be the Lebesgue measure on R. For each
+g ∈ Lp(R, τ) put
+Θg :=
+�
+f ∈ Lp(R, τ) : ∥fχ[m,m+1]∥p ≥ ∥gχ[m,m+1]∥p for all m ∈ Z
+�
+.
+Then, Θg is not σ-porous in Lp(R, τ).
+Proof. Let 0 < λ ≤ 1
+2, and 0 < β < λ. Denote
+F :=
+�
+Θg : g ∈ Lp(R, τ)
+�
+.
+We prove that the collection F satisfies the conditions of Lemma 2.2. Let
+0 ≤ g ∈ Lp(R, τ). Then, easily Θg ̸= ∅ and it is closed in Lp(R, τ). Now,
+assume that f ∈ Lp(R, τ) and r > 0 with B(f; r)∩Θg ̸= ∅. Then, there exists
+
+DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS
+7
+a large enough number N ∈ N, some 0 < ǫ < 1 and a function k ∈ B(f; r)∩Θg
+such that
+∥k − f∥p < ǫ
+1
+p r
+and
+�
+[−N,N]c
+�
+|f| + β−1g
+�p dτ < (1 − ǫ) rp.
+Pick some α with ∥k − f∥p < α < ǫ
+1
+p r, and denote δ := ǫ
+1
+p r−α
+2(2N)
+1
+p . Put
+A1 := {m ∈ [N] : g = 0 a.e. on [m, m + 1]},
+A2 := [N] \ A1
+and
+B1 := {m ∈ [N] : k = 0 a.e. on [m, m + 1]},
+B2 := [N] \ B1,
+where [N] := {−N, . . . , N − 1}, and then define
+ρ :=
+�
+m∈A1
+χ[m,m+1] +
+�
+m∈A2
+gχ[m,m+1]
+∥gχ[m,m+1]∥p
+,
+and
+η :=
+�
+m∈B1
+χ[m,m+1] +
+�
+m∈B2
+kχ[m,m+1]
+∥kχ[m,m+1]∥p
+.
+Now, we define h, ξ : R → C by
+h := gχ[−N,N] + δρ + β−1g χ[−N,N]c
+and
+ξ := |k| χ[−N,N] + δη + hχ[−N,N]c.
+Clearly, h ∈ Lp(R, τ). For each x ∈ [−N, N] we have |k(x) − ξ(x)| = δ |η(x)|,
+and so
+∥(k − ξ)χ[−N,N]∥p
+p = δp ∥ηχ[−N,N]∥p
+p
+= δp
+�
+m∈[N]
+∥ηχ[m,m+1]∥p
+p
+= δp 2N.
+Hence, ∥(k − ξ)χ[−N,N]∥p = δ (2N)
+1
+p . Now, similar to the proof of Theorem
+2.3 we have ξ ∈ B(f; r). Moreover,
+∥ξχ[m,m+1]∥p = ∥kχ[m,m+1]∥p + δ ≥ ∥gχ[m,m+1]∥p + δ = ∥hχ[m,m+1]∥p
+for all m ∈ [N]. And also for each m /∈ [N],
+∥ξχ[m,m+1]∥p = ∥hχ[m,m+1]∥p ≥ ∥gχ[m,m+1]∥p.
+So,
+ξ ∈ B(f; r) ∩ Θh ⊆ B(f; r) ∩ Θg.
+
+8
+S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE
+Now, let u ∈ B(f; r) ∩ Θh and put r′ := min{δ, λ (r − ∥f − u∥p)}. Let v ∈
+B(u; r′). We define the function γ : R → C by
+γ(x) :=
+
+
+
+
+
+v(x),
+if x ∈ [−N, N]
+�
+|v(x)| + β|u(x) − v(x)|
+�
+θ(x),
+if x ∈ [−N, N]c
+where
+θ(x) :=
+
+
+
+v(x)
+|v(x)|,
+if v(x) ̸= 0
+1,
+if v(x) = 0.
+Similar to the proof of Theorem 2.3, we have γ ∈ B
+�
+v; λ ∥u − v∥p
+�
+. Now, for
+each m /∈ [N],
+|γ|χ(m,m+1) = (|v| + β|u − v|)χ(m,m+1) ≥ β|u|χ(m,m+1).
+Hence,
+∥γχ[m,m+1]∥p ≥ β ∥uχ[m,m+1]∥p ≥ β ∥hχ[m,m+1]∥p
+since u ∈ B(f; r) ∩ Θh. However, in this case we have (m, m + 1) ∈ [−N, N]c,
+so hχ(m,m+1) = β−1gχ(m,m+1). Thus, β∥hχ[m,m+1]∥p = ∥gχ[m,m+1]∥p. If m ∈
+[N], we have γχ[m,m+1] = vχ[m,m+1] because γχ[−N,N] = vχ[−N,N] and [m, m+
+1] ⊆ [−N, N]. We get
+�� ∥uχ[m,m+1]∥p − ∥vχ[m,m+1]∥p
+�� ≤ ||(u − v)χ[m,m+1]∥p ≤ ∥u − v∥p < δ
+because v ∈ B(u; r′), hence
+∥γχ[m,m+1]∥p = ∥vχ[m,m+1]∥p
+≥ ∥uχ[m,m+1]∥p − δ
+≥ ∥hχ[m,m+1]∥p − δ
+= ∥gχ[m,m+1]∥p.
+Therefore,
+γ ∈ B
+�
+v; λ ∥u − v∥p
+�
+∩ B(f; r) ∩ Θg,
+and the proof is complete.
+□
+3. Applications
+In this section, we will apply the results of the previous section, to prove that
+the set of all non-hypercyclic vectors of some sequences of weighted translation
+operators is non-σ-porous.
+Definition 3.1. Let X be a Banach space. A sequence (Tn)n∈N0 of operators
+in B(X) is called hypercyclic if there is an element x ∈ X (called hypercyclic
+vector) such that the orbit {Tn(x) : n ∈ N0} is dense in X. The set of all
+hypercyclic vectors of a sequence (Tn)n∈N0 is denoted by HC((Tn)n∈N0). An
+operator T ∈ B(X) is called hypercyclic if the sequence (T n)n∈N0 is hyper-
+cyclic.
+
+DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS
+9
+Let G be a locally compact group and a ∈ G.
+Then, for each function
+f : G → C we define Laf : G → C by Laf(x) := f(a−1x) for all x ∈ G. Note
+that if p ≥ 1, then the left translation operator
+La : Lp(G) → Lp(G),
+f �→ Laf
+is not hypercyclic because ∥La∥ ≤ 1. Hypercyclicity of weigted translation
+operators on Lp(G) and regarding an aperiodic element a was studied in [5]
+(an element a ∈ G is called aperiodic if the closed subgroup of G generated by
+a is not compact).
+Definition 3.2. Let G be a locally compact group with a left Haar measure
+µ. Fix p ≥ 1. We denote Lp(G) := Lp(G, µ). Assume that w : G → (0, ∞)
+is a bounded measurable function (called a weight) and a ∈ G. Then, the
+weighted translation operator Ta,w,p : Lp(G) → Lp(G) is defined by
+Ta,w,p(f) := w Laf,
+(f ∈ Lp(G)).
+For each n ∈ N we denote ϕn := w Law . . . Lan−1w, where a0 := e, the
+identity element of G.
+Theorem 3.3. Let p ≥ 1, G be a discrete group and a ∈ G. Let µ be a left
+Haar measure on G with µ({e}) ≥ 1 and (γn)n be an unbounded sequence of
+non-negative integers. Let w : G → (0, ∞) be a bounded function such that for
+some finite nonempty set F ⊆ G and some N > 0 we have
+aγnF ∩ F = ∅
+(n ≥ N),
+and
+β := inf
+� γn
+�
+k=1
+w(akt) : n ≥ N, t ∈ F
+�
+> 0.
+Then, the set
+Λ :=
+�
+f ∈ Lp(G, µ) : ∥T γn
+a,w,pf − χF ∥p ≥ µ(F)
+1
+p for all n ≥ N
+�
+is non-σ-porous.
+
+10
+S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE
+Proof. Let Γ := {f ∈ Lp(G, µ) : |f| ≥
+1
+βχF }. Then, Γ is not σ-porous in
+Lp(G, µ) thanks to Theorem 2.7. Also, for each f ∈ Γ and n ≥ N we have
+∥T γn
+a,w,pf − χF∥p
+p =
+�
+G
+|
+n
+�
+k=1
+w(a−γn+kx) f(a−γnx) − χF (x)|p dµ(x)
+=
+�
+G
+|
+γn
+�
+k=1
+w(akx) f(x) − χF(aγnx)|p dµ(x)
+=
+�
+G
+|
+γn
+�
+k=1
+w(akx) f(x) − χa−γnF (x)|p dµ(x)
+≥
+�
+F
+|
+γn
+�
+k=1
+w(akx) f(x) − χa−γnF (x)|p dµ(x)
+=
+�
+F
+|
+γn
+�
+k=1
+w(akx) f(x)|p dµ(x)
+≥
+�
+F
+|β 1
+β |p dµ(x)
+= µ(F).
+This completes the proof.
+□
+Example 3.4. Let G be the additive group Z with the counting measure.
+Let F be a finite non-empty subset of Z. Put N := max{|j| : j ∈ F}. If
+w := (wn)n∈Z ⊆ (0, ∞) is a bounded sequence with wn ≥ 1 for all n ≥ N.
+Then the required conditions in the previous theorem hold with respect to F
+and a := 1.
+The following fact is a direct conclusion of the previous theorem.
+Corollary 3.5. Let p ≥ 1, G be a discrete group and a ∈ G with infinite order.
+Let µ be the counting measure on G and (γn)n be an unbounded sequence of
+non-negative integers. Let w : G → (0, ∞) be a bounded function such that for
+some t ∈ G,
+inf
+� γn
+�
+k=1
+w(akt) : n ∈ N
+�
+> 0.
+Then, the set
+�
+f ∈ Lp(G, µ) : ∥T γn
+a,wf − χ{t}∥p ≥ 1 for all n
+�
+is non-σ-porous.
+Theorem 3.6. Let p ≥ 1, G be a discrete group, and a ∈ G. Let µ be a left
+Haar measure on G with µ({e}) ≥ 1. Let (γn)n be an unbounded sequence of
+non-negative integers and let w : G → (0, ∞) be a bounded function such that
+inf
+n∈N
+γn
+�
+k=1
+w(ak) > 0.
+
+DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS
+11
+Then, the set
+Γ :=
+�
+f ∈ Lp(G, µ) : |f(e)| inf
+n∈N
+γn
+�
+k=1
+w(ak) ≥ 1
+�
+is non-σ-porous. In particular, setting Tn := T γn
+a,w,p for all n, the set of all
+non-hypercyclic vectors of the sequence (Tn)n is not σ-porous in Lp(G, µ).
+Proof. Since µ({e}) ≥ 1, applying Theorem 2.3 the set Γ is non-σ-porous,
+because
+[ inf
+n∈N
+γn
+�
+k=1
+w(ak)]−1 χ{e} ∈ Lp(G, µ).
+Let f ∈ Γ. If n is a nonnegative integer, then for every x in G we have
+∥Tnf∥p ≥
+��ϕγn(x) Laγn f(x)
+��,
+and so setting x = aγn we have
+∥Tnf∥p ≥
+��ϕn(aγn) Laγn f(aγn)
+��
+=
+� γn
+�
+k=1
+w(ak)
+�
+|f(e)|
+≥ |f(e)| inf
+m∈N
+γm
+�
+k=1
+w(ak) ≥ 1.
+This implies that the set {Tnf : n ∈ N} is not dense in Lp(G, µ), and so Γ
+is a subset of the set of all non-hypercyclic vectors of T. This completes the
+proof.
+□
+Now, we recall the definition of hypergroups which are generalizations of
+locally compact groups; see the monograph [4] and the basic paper [12] for
+more details. In locally compact hypergroups the convolution of two Dirac
+measures is not necessarily a Dirac measure.
+Let K be a locally compact
+Hausdorff space. We denote by M(K) the space of all regular complex Borel
+measures on K, and by δx the Dirac measure at the point x. The support of
+a measure µ ∈ M(K) is denoted by supp(µ).
+Definition 3.7. Suppose that K is a locally compact Hausdorff space, (µ, ν) �→
+µ∗ν is a bilinear positive-continuous mapping from M(K)×M(K) into M(K)
+(called convolution), and x �→ x− is an involutive homeomorphism on K (called
+involution) with the following properties:
+(i)
+M(K) with ∗ is a complex associative algebra;
+(ii)
+if x, y ∈ K, then δx∗δy is a probability measure with compact support;
+(iii)
+the mapping (x, y) �→ supp(δx ∗ δy) from K × K into C(K) is contin-
+uous, where C(K) is the set of all non-empty compact subsets of K
+equipped with Michael topology;
+(iv) there exists a (necessarily unique) element e ∈ K (called identity) such
+that for all x ∈ K, δx ∗ δe = δe ∗ δx = δx;
+
+12
+S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE
+(v) for all x, y ∈ K, e ∈ supp(δx ∗ δy) if and only if x = y−;
+Then, K ≡ (K, ∗,− , e) is called a locally compact hypergroup.
+A nonzero nonnegative regular Borel measure m on K is called the (left)
+Haar measure if for each x ∈ K, δx∗m = m. For each x, y ∈ K and measurable
+function f : K → C we denote
+f(x ∗ y) :=
+�
+K
+f d(δx ∗ δy),
+while this integral exists.
+Definition 3.8. Suppose that a := (an)n∈N0 is a sequence in a hypergroup
+K, and w is a weight function on K. For each n ∈ N0 we define the bounded
+linear operator Λn+1 on Lp(K) by
+Λn+1f(x) := w(a0 ∗ x) w(a1 ∗ x) . . . w(an ∗ x) f(an+1 ∗ x)
+(f ∈ Lp(K))
+for all x ∈ K. Also, we assume that Λ0 is the identity operator on Lp(K).
+Some linear dynamical properties of this sequence of operators were studied
+in [13]. The sequence {Λn}n is a generalization of the usual powers of a single
+weighted translation operator on Lp(G), where G is a locally compact group.
+In fact, any locally compact group G with the mapping
+µ ∗ ν �→
+�
+G
+�
+G
+δxydµ(x)dν(y)
+(µ, ν ∈ M(G))
+as convolution, and x �→ x−1 from G onto G as involution is a locally compact
+hypergroup. Let η := (an)n∈N0 be a sequence in G, and w be a weight on G.
+Then for each f ∈ Lp(G), n ∈ N0 and x ∈ G, we have
+Λn+1f(x) = w(a0x) w(a1x) . . . w(anx) f(an+1x).
+In particular, let a ∈ G and for each n ∈ N0, put an := a−n. Then, Λn = T n
+a,w,p
+for all n ∈ N. In this case, the operator Ta,w,p is hypercyclic if and only if the
+sequence (Λn)n is hypercyclic.
+Let K be a discrete hypergroup with the convolution ∗ between Radon
+measures of K and the involution ·− : K → K. Then, by [12, Theorem7.1A],
+the measure µ on K given by
+µ({x}) :=
+1
+δx ∗ δx−({e}),
+(x ∈ K)
+(3.1)
+is a left Haar measure on K.
+Proposition 3.9. Let K be a discrete hypergroup, µ be the Haar measure
+(3.1), and p ≥ 1. Then for each g ∈ Lp(K, µ), the set
+�
+f ∈ Lp(K, µ) : |f| ≥ |g|
+�
+is not σ-porous in Lp(K, µ).
+
+DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS
+13
+Proof. Just note that for each x ∈ K we have µ({x}) ≥ 1 because
+1 = δx ∗ δx−(K) ≥ δx ∗ δx−({e}).
+Hence, the measure space (K, µ) satisfies the condition of Corollary 2.5.
+□
+Let a := (an)n∈N be a sequence in a discrete hypergroup K such that
+an ̸= am for each m ̸= n, and let w : K → (0, ∞) be bounded. We define
+ha,w : K → C by
+ha,w :=
+�
+n∈N0
+1
+w(a0)w(a1) . . . w(an)χ{an+1}.
+Theorem 3.10. Let p ≥ 1, and K be a discrete hypergroup endowed with the
+left Haar measure (3.1). Let a := (an)n∈N0 ⊆ K with distinct terms, and w be
+a weight on K such that ha,w ∈ Lp(K). Then, the set of all non-hypercyclic
+vectors of the sequence (Λn)n is not σ-porous.
+Proof. First, thanks to Proposition 3.9, the set
+E :=
+�
+f ∈ Lp(K) : |f(an+1)| ≥
+1
+w(a0)w(a1) . . . w(an) for all n
+�
+is not σ-porous because it equals to the set
+�
+f ∈ Lp(K) : |f| ≥ ha,w
+�
+. Now,
+for each f ∈ E,
+∥Λn+1f∥p ≥ sup
+x∈K
+w(a0 ∗ x) w(a1 ∗ x) . . . w(an ∗ x) |f(an+1 ∗ x)|
+≥ w(a0) w(a1) . . . w(an) |f(an+1)| ≥ 1
+for all n ∈ N0. This implies that 0 does not belong to the closure of {Λnf :
+n ∈ N} in Lp(K), and so E ⊆ [HC((Λn)n)]c. This completes the proof.
+□
+Since any group is a hypergroup, we can give the fact below.
+Corollary 3.11. Let p ≥ 1, and G be a discrete group. Let a ∈ G be of
+infinite order, (γn)n∈N0 ⊆ N be with distinct terms and w : G → (0, ∞) be a
+weight such that
+�
+1
+w(aγ0)w(aγ1) . . . w(aγn)
+�
+n ∈ ℓp(G).
+Then, the set of all non-hypercyclic vectors of the sequence (T γn
+a,w,p)n is not
+σ-porous in ℓp(G).
+Now, we can write the next corollary which is a generalization of [1, Theorem
+1].
+Corollary 3.12. Let p ≥ 1, (γn)n ⊆ N be strictly increasing and (wn)n∈Z be
+a bounded sequence in (0, ∞) such that
+�
+1
+wγ0wγ1wγ2 . . . wγn
+�
+n ∈ ℓp(Z).
+
+14
+S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE
+Then, the set of all non-hypercyclic vectors of the sequence (Tn)n is not σ-
+porous, where
+(Tn+1a)k := wγ0wγ1wγ2 . . . wγnak+γn+1
+(k ∈ N0)
+for all a := (aj)j ∈ ℓp(Z).
+Applying Theorem 2.7 we can speak regarding some more general situation
+in the case of p = ∞. Let Ω be a locally compact Hausdorff space endowed with
+a nonnegative Radon measure µ. Let w : Ω → (0, ∞) be a bounded measurable
+function, and α : Ω → Ω be a bi-measurable mapping such that ∥f ◦ α±1∥∞ =
+∥f∥∞ for all f ∈ L∞(Ω, µ). Then, we define Tα,w,∞ : L∞(Ω, µ) → L∞(Ω, µ)
+by
+Tα,w,∞(f) := w (f ◦ α)
+(f ∈ L∞(Ω, µ)).
+If Ω be a locally compact group and a ∈ Ω, setting αa(x) := ax for all x ∈ Ω,
+we denote Ta,w,∞ := Tαa,w,∞. Note that α−1 means the inverse function of α,
+and for each k ∈ N, α−k := (α−1)k.
+Theorem 3.13. Let Tα,w,∞ be the weighted composition operator defined as
+above and let {γn}n ⊆ N be a fixed unbounded sequence. Suppose that there
+exists a sequence {An}n of disjoint subsets of Ω with µ(An) > 0 for all n such
+that
+jα,w :=
+�
+n∈N
+1
+(w ◦ α−γn) (w ◦ α−γn+1) . . . (w ◦ α−1)χAn ∈ L∞(Ω, µ).
+Then, the set {f ∈ L∞(Ω, µ) : ∥T γn
+α,w,∞(f)∥∞ ≥ 1 for all n} is not σ-porous.
+In particular, the set of all non-hypercyclic vectors of the sequence {T γn
+α,w,∞}n
+is not σ-porous.
+Proof. Let E := {f ∈ L∞(Ω, µ) : |f| ≥ jα,w}. Then, E is not σ-porous thanks
+to Theorem 2.7. For each f ∈ E and n ∈ N we have
+∥T γn
+α,w,∞(f)∥∞ = ∥
+γn
+�
+k=1
+(w ◦ αγn−k) (f ◦ αγn)∥∞
+= ∥
+γn
+�
+k=1
+(w ◦ α−k) f∥∞
+≥ ∥
+γn
+�
+k=1
+(w ◦ α−k) χAn f∥∞
+≥ ∥
+γn
+�
+k=1
+(w ◦ α−k) χAn jα,w∥∞
+= 1.
+This completes the proof.
+□
+
+DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS
+15
+Corollary 3.14. Let G be a locally compact group and µ be a left Haar mea-
+sure on G. Let a ∈ G and w : G → (0, ∞) be a bounded measurable function.
+Then, if
+�
+1
+w(a)w(a2) . . . w(an)
+�
+n ∈ L∞(G, µ),
+then the set of all non-hypercyclic vectors of the operator Ta,w,∞ on L∞(G, µ)
+is not σ-porous.
+Corollary 3.15. If (wn)n∈Z is a bounded sequence such that
+�
+1
+w1 . . . wn
+�
+n ∈ ℓ∞,
+then the set of all non-hypercyclic vectors of the sequence (Tγn,w)n is not σ-
+porous in ℓ∞.
+Theorem 3.16. Let Tα,w,∞ be the weighted composition operator on L∞(Ω, µ)
+and let F ⊆ Ω be a Borel set with 0 < µ(F) < ∞. Let there exists a constant
+N > 0 such that for all n ≥ N,
+αn(F) ∩ F = ∅,
+(3.2)
+and
+β := inf{
+n
+�
+k=1
+(w ◦ α−k)(t) : n ≥ N, t ∈ F} ̸= 0.
+Then, the set
+{f ∈ L∞(Ω, µ) : ∥T n
+α,w,∞f − χF∥∞ ≥ 1 for all n ≥ N}
+is not σ-porous in L∞(Ω, µ).
+Proof. Let Γ := {f ∈ L∞(Ω, µ) : |f| ≥ 1
+βχF }. Then by Theorem 2.7, Γ is not
+σ-porous in L∞(Ω, µ). Also, for each f ∈ Γ we have
+∥T n
+α,w,∞f − χF ∥∞ = ∥
+n
+�
+k=1
+(w ◦ αn−k) (f ◦ αn) − χF∥∞
+= ∥
+n
+�
+k=1
+(w ◦ α−k) f − χF ◦ αn∥∞
+= ∥
+n
+�
+k=1
+(w ◦ α−k) f − χαn(F )∥∞
+≥ ∥
+n
+�
+k=1
+(w ◦ α−k) fχF − χαn(F )χF∥∞
+= ∥
+n
+�
+k=1
+(w ◦ α−k) fχF∥∞
+≥ β∥fχF∥∞ ≥ β 1
+β ∥χF∥∞ = 1.
+
+16
+S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE
+This completes the proof.
+□
+Example 3.17. Let Ω := R and µ be the Lebesgue measure. Put α(t) := t−1
+for all t ∈ R and F := [0, 1]. If w ∈ Cb(R) such that |w(t)| ≥ 1 for all t ≥ k > 0
+and inf{|w(t)| : t ∈ [0, 1]} > 0, then the required conditions in the previous
+theorem hold with respect to F.
+With some similar proof, one can prove the next fact without the condition
+(3.2).
+Theorem 3.18. Let Tα,w,∞ be the weighted composition operator on L∞(Ω, µ)
+and let F ⊆ Ω be a Borel set with 0 < µ(F) < ∞ such that
+inf{
+n
+�
+k=1
+(w ◦ α−k)(t) : n ≥ N, t ∈ F} ̸= 0.
+Then, the set
+{f ∈ L∞(Ω, µ) : ∥T n
+α,w,∞f∥∞ ≥ 1 for all n ≥ N}
+is not σ-porous in L∞(Ω, µ). In particular, the set of all non-hypercyclic vec-
+tors of the operator Tα,w,∞ is not σ-porous.
+In sequel, we find some application for Theorem 2.9 regarding hypercyclicity
+of shift operators on Lp(R, τ).
+Theorem 3.19. Consider the weighted translation operator Tα,w on Lp(R, τ)
+given by Tα,wf := w · (f ◦α), where 0 < w, w−1 ∈ Cb(R) and α(t) = t + 1. For
+each n ∈ N put An := [n, n + 1] = αn([0, 1]). Set
+yα,w :=
+�
+n∈N
+1
+inft∈An
+�n
+k=1(w ◦ α−k)(t)χAn
+and assume that yα,w ∈ Lp(R, τ) (in particular inft∈An
+�n
+k=1(w ◦ α−k)(t) > 0
+for all n ∈ N). Then, the set
+{f ∈ Lp(R, τ) : ∥T n
+α,w(f)∥p ≥ 1 for all n ∈ N}
+is not σ-porous.
+Proof. By Theorem 2.9, the set
+E := {f ∈ Lp(R, τ) : ∥fχAn∥p ≥ ∥yα,wχAn∥p for all n ∈ N}
+is not σ-porous, because it equals to
+{f ∈ Lp(R, τ) : ∥fχ[m,m+1]∥p ≥ ∥yα,wχ[m,m+1]∥p for all m ∈ Z},
+
+DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS
+17
+as yα,wχ[m,m+1] = 0 for all m ∈ Z with m ≤ 0. Now, note that for each f ∈ E
+and n ∈ N,
+∥T n
+α,w(f)∥p
+p =
+�
+R
+� n
+�
+k=1
+(w ◦ αn−k)(t)
+�p
+|(f ◦ αn)(t)|p dτ
+=
+�
+R
+� n
+�
+k=1
+(w ◦ α−k)(t)
+�p
+|f(t)|p dτ
+≥
+�
+An
+� n
+�
+k=1
+(w ◦ α−k)(t)
+�p
+|f(t)|p dτ
+≥ inf
+t∈An
+� n
+�
+k=1
+(w ◦ α−k)(t)
+�p
+∥yα,wχAn∥p
+p
+= inf
+t∈An
+� n
+�
+k=1
+(w ◦ α−k)(t)
+�p
+1
+inft∈An [�n
+k=1(w ◦ α−k)(t)]p τ(An) = 1.
+□
+Assume now that there exists some l ∈ Z such that
+β := inf{
+n
+�
+k=1
+(w ◦ α−k)(t) : t ∈ [l, l + 1], n ∈ N} > 0.
+Put
+F := {f ∈ Lp(R, τ) : ∥fχ[m,m+1]∥p ≥ ∥ 1
+β χ[l,l+1]χ[m,m+1]∥p for all m ∈ Z}.
+So by Theorem 2.9, F is not σ-porous. For every f ∈ F we have
+∥T n
+α,w(f)∥p
+p =
+�
+R
+� n
+�
+k=1
+(w ◦ αn−k)(t)
+�p
+|(f ◦ αn)(t)|p dτ
+=
+�
+R
+� n
+�
+k=1
+(w ◦ α−k)(t)
+�p
+|f(t)|p dτ
+≥
+�
+[l,l+1]
+� n
+�
+k=1
+(w ◦ α−k)(t)
+�p
+|f(t)|p dτ
+≥ 1.
+Hence, the set
+{f ∈ Lp(R, τ) : ∥T n
+α,w(f)∥p ≥ 1 for all n ∈ N}
+is not σ-porous.
+Next, suppose that α is an aperiodic function on R (this means that for each
+compact set C ⊂ R, there exists a constant N > 0 such that αn(C) ∩ C = ∅
+for all n ≥ N) and β > 0, where β is as above. Then, the set
+{f ∈ Lp(R, τ) : ∥T n
+α,w(f) − χ[l,l+1]∥p ≥ 1 for all n ≥ N}
+
+18
+S. IVKOVI´C, S. ¨OZTOP, AND S.M. TABATABAIE
+is not σ-porous. Indeed, for all f ∈ F, and n ≥ N we have
+∥T n
+α,w(f) − χ[l,l+1]∥p ≥ ∥
+n
+�
+k=1
+(w ◦ α−k) fχ[l,l+1]∥p
+by the similar calculations as in the proof of Theorem 3.16. However,
+∥
+n
+�
+k=1
+(w ◦ α−k) fχ[l,l+1]∥p ≥ β ∥fχ[l,l+1]∥p ≥ 1.
+References
+1. F. Bayart, Porosity and hypercyclic operators, Proc. Amer. Math. Soc. 133(11) (2005)
+3309-3316.
+2. F. Bayart and ´E. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Math.
+179, Cambridge University Press, Cambridge, 2009.
+3. C. L. Belna, M. J. Evans and P.D.Humke, Symmetric and ordinary differentiation, Proc.
+Amer. Math. Soc. 72(2) (1978) 261-267.
+4. W.R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups,
+De Kruyter, Berlin, 1995.
+5. C-C. Chen and C-H. Chu, Hypercyclic weighted translations on groups, Proc. Amer.
+Math. Soc. 139 (2011) 2839-2846.
+6. C-C. Chen, S. ¨Oztop and S. M. Tabatabaie, Disjoint dynamics on weighted Orlicz spaces,
+Disjoint dynamics on weighted Orlicz spaces, Complex Anal. Oper. Theory, 14(72)
+(2020). https://doi.org/10.1007/s11785-020-01034-x
+7. C.-C. Chen and S. M. Tabatabaie, Chaotic operators on hypergroups, Oper. Mat. 12(1)
+(2018) 143-156.
+8. E. P. Dolˇzenko, Boundary properties of arbitrary functions, Izv. Akad. Nauk SSSR Ser.
+Mat. 31 (1967) 3-14.
+9. G.B. Folland, Real Analysis, Modern Techniques and Their Applications; Second Edition,
+John Wiley and Sons, Inc. New York (1999).
+10. K-G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math. 139 (2000)
+47-68.
+11. K-G. Grosse-Erdmann and A. Peris, Linear Chaos, Universitext, Springer, 2011.
+12. R. I. Jewett, Spaces with an abstract convolution of measures, Adv. Math., 18 (1975)
+1-101.
+13. V. Kumar and S. M. Tabatabaie, Hypercyclic sequences of weighted translations on hy-
+pergroups, Semigroup Forum, 103 (2021) 916–934.
+14. D. Preiss and L. Zaj´ıˇcek, Fr´echet differentiation of convex functions in a Banach space
+with a separable dual, Proc. Amer. Math. Soc. 91(2) (1984) 202–204.
+15. H. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995) 993-1004.
+16. Y. Sawano, S. M. Tabatabaie and F. Shahhoseini, Disjoint Dynamics of Weighted
+Translations
+on
+Solid
+Spaces,
+Topology
+Appl.
+298,
+107709,
+14
+pp.
+(2021)
+DOI:10.1016/J.TOPOL.2021.107709
+17. S. M. Tabatabaie and S. Ivkovic, Linear dynamics of cosine operator functions on solid
+Banach function spaces, Positivity, 25 (2021) 1437–1448.
+18. A. Villani, Another note on the inclusion Lp(µ) ⊂ Lq(µ), Amer. Math. Monthly, 92
+(1985) 485–487.
+19. L. Z´ajiˇcek, Porosity and σ-porous, Real Anal. Exchange 13 (1987/1988) 314-350.
+20. L. Zaj´ıˇcek, Small non-σ-porous sets in topologically complete metric spaces, Colloq.
+Math. 77(2) (1998) 293-304.
+21. L. Z´ajiˇcek, On σ-porous sets in abstract spaces, Abstr. Appl. Anal. 5 (2005) 509-534.
+
+DYNAMICAL PROPERTIES AND SOME CLASSES OF NON-POROUS SUBSETS
+19
+Mathematical Institute of the Serbian Academy of Sciences and Arts, p.p.
+367, Kneza Mihaila 36, 11000 Beograd, Serbia.
+Email address: [email protected]
+Department of Mathematics, Faculty of Science, Istanbul University, Istan-
+bul, Turkey
+Email address: [email protected]
+Department of Mathematics, University of Qom, Qom, Iran.
+Email address: [email protected]
+

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@@ -0,0 +1,2070 @@
+Distributed Sparse Linear Regression
+under Communication Constraints ∗
+Rodney Fonseca and Boaz Nadler
+Department of Computer Science and Applied Mathematics,
+Weizmann Institute of Science, Rehovot, Israel
+e-mail: [email protected]; [email protected]
+Abstract: In multiple domains, statistical tasks are performed in dis-
+tributed settings, with data split among several end machines that are con-
+nected to a fusion center. In various applications, the end machines have
+limited bandwidth and power, and thus a tight communication budget.
+In this work we focus on distributed learning of a sparse linear regression
+model, under severe communication constraints. We propose several two
+round distributed schemes, whose communication per machine is sublinear
+in the data dimension. In our schemes, individual machines compute debi-
+ased lasso estimators, but send to the fusion center only very few values. On
+the theoretical front, we analyze one of these schemes and prove that with
+high probability it achieves exact support recovery at low signal to noise
+ratios, where individual machines fail to recover the support. We show in
+simulations that our scheme works as well as, and in some cases better,
+than more communication intensive approaches.
+MSC2020 subject classifications: Primary 62J07, 62J05; secondary
+68W15.
+Keywords and phrases: Divide and conquer, communication-efficient,
+debiasing, high-dimensional.
+1. Introduction
+In various applications, datasets are stored in a distributed manner among sev-
+eral sites or machines (Fan et al., 2020, chap. 1.2). Often, due to communication
+constraints as well as privacy restrictions, the raw data cannot be shared be-
+tween the various machines. Such settings have motivated the development of
+methods and supporting theory for distributed learning and inference. See, e.g.,
+the reviews by Huo and Cao (2019), Gao et al. (2022) and references therein.
+∗This research was supported by a grant from the Council for Higher Education Compet-
+itive Program for Data Science Research Centers. RF acknowledges support provided by the
+Mor´a Miriam Rozen Gerber Fellowship for Brazilian postdocs.
+1
+arXiv:2301.04022v1  [cs.LG]  9 Jan 2023
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+2
+In this paper we consider distributed learning of a sparse linear regression
+model. Specifically, we assume that the response y ∈ R and the vector X ∈ Rd
+of explanatory variables are linearly related via
+y = X⊤θ∗ + w,
+(1)
+where w ∼ N(0, σ2), σ > 0 is the noise level, and θ∗ ∈ Rd is an unknown
+vector of coefficients. We further assume X ∈ Rd is random with mean zero
+and covariance matrix Σ. We focus on a high-dimensional setting d ≫ 1, and
+assume that θ∗ is sparse with only K ≪ d nonzero coefficients. The support set
+of θ∗ ∈ Rd is denoted by S = {i ∈ [d] | |θ∗
+i | > 0}.
+Given N samples {(Xi, yi)}N
+i=1 from the model (1), common tasks are to
+estimate the vector θ∗ and its support set S. Motivated by contemporary ap-
+plications, we consider these tasks in a distributed setting where the data are
+randomly split among M machines. Specifically, we consider a star topology
+network, whereby the end machines communicate only with a fusion center.
+As reviewed in Section 2, estimating θ∗ and its support in the above or simi-
+lar distributed settings were studied by several authors, see for example Mateos,
+Bazerque and Giannakis (2010); Chen and Xie (2014); Lee et al. (2017); Battey
+et al. (2018); Chen et al. (2020); Liu et al. (2021); Barghi, Najafi and Mota-
+hari (2021) and references therein. Most prior works on distributed regression
+required communication of at least O(d) bits per machine, as in their schemes
+each machine sends to the fusion center its full d-dimensional estimate of the
+unknown vector θ∗. Some works in the literature denote this as communication
+efficient, in the sense that for a machine holding n samples, an O(d) communi-
+cation is still significantly less than the size O(n · d) of its data.
+The design and analysis of communication efficient distributed schemes is
+important, as in various distributed settings the communication channel is the
+critical bottleneck. Moreover, in some practical cases, such as mobile devices and
+sensor networks, the end machines may have very limited bandwidth. Thus, in
+high dimensional settings with d ≫ 1, it may not even be feasible for each ma-
+chine to send messages of length O(d). In this work, we study such a restricted
+communication setting, assuming that each machine is allowed to send to the fu-
+sion center only a limited number of bits, significantly lower than the dimension
+d. Our goals are to develop low communication distributed schemes to estimate
+θ∗ and its support and to theoretically analyze their performance.
+We make the following contributions. On the methodology side, in Section 4
+we present several two round distributed schemes. The schemes vary slightly by
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+3
+the messages sent, but in all of them, the fusion center estimates the support set
+of θ∗ in the first round and the vector θ∗ in the second round. In our schemes,
+each machine computes its own debiased lasso estimate. However, it sends to
+the center only the indices of its top few largest values, possibly along with their
+signs. Hence, the communication per machine is significantly less than d bits.
+In the simplest variant, the fusion center estimates the support of θ∗ by voting,
+selecting the few indices that were sent by the largest number of machines.
+Next, on the theoretical side, we prove in Section 5 that under suitable condi-
+tions, with high probability the first round of our scheme achieves exact support
+recovery with communication per machine sublinear in d. Specifically, we present
+support guarantees under two different parameter regimes. Theorem 2 consid-
+ers a case with a relatively large number of machines. Here, each machine sends
+a short message of O(K ln d) bits. Next, Theorem 3 considers a setting with
+relatively few machines, M = O(ln d). Here, to achieve exact support recovery
+each machine sends a much longer message, of length O(dα) for some suitable
+α < 1. This is still sublinear in d, and much less than the communication re-
+quired if a machine were to send its full d-dimensional estimated vector. The
+proofs of our theorems rely on recent results regarding the distribution of debi-
+ased lasso estimators, combined with sharp bounds on tails of binomial random
+variables. Exact support recovery follows by showing that with high probability,
+all non-support indices receive fewer votes than support indices.
+In Section 6 we present simulations comparing our schemes to previously
+proposed methods. These illustrate that with our algorithms, the fusion center
+correctly detects the support of θ∗ and consequently accurately estimates θ∗,
+even at low signal to noise ratios where each machine is unable to do so. Fur-
+thermore, this is achieved with very little communication per machine compared
+to the dimension d. One insight from both the simulations and our theoretical
+analysis is that for the fusion center to detect the correct support, it is not
+necessary to require M/2 votes as suggested in Barghi, Najafi and Motahari
+(2021) and Chen and Xie (2014). Instead, as few as O(ln d) votes suffice to dis-
+tinguish support from non-support indices. Interestingly, under a broad range
+of parameter values, our schemes work as well as, and in some cases better than
+more communication intensive approaches. Our simulations also highlight the
+importance and advantages of a second round of communication. Specifically,
+even though a single-round scheme based on averaging debiased lasso estimates,
+as proposed by Lee et al. (2017), is minimax rate optimal and finds the correct
+support, it nonetheless may output an estimate with a larger mean squared er-
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+4
+ror than that of our scheme. We conclude with a summary and discussion in
+Section 7. Proofs appear in the Appendix.
+Notation
+For an integer k ≥ 1, we denote [k] = {1, 2, . . . , k}. The indicator
+function is denoted as I(A), which equals one if condition A holds and zero
+otherwise. The ℓq norm of a vector Y ∈ Rn for q ≥ 1 is ∥Y ∥q = (�n
+i=1 |Yi|q)1/q,
+whereas ∥Y ∥0 = �n
+i=1 I(Yi ̸= 0) is its number of nonzero entries. We denote
+by |Y | the vector whose entries are (|Y1|, |Y2|, . . . , |Yn|). For a d × d matrix
+A = {aij}d
+i,j=1, we denote ∥A∥∞ = max1≤i≤d
+�d
+j=1 |aij|. We further denote by
+σmin(A) and σmax(A) its smallest and largest singular values, respectively. For
+a subset J ⊂ [d], AJ is the d×|J| matrix whose columns are those in the subset
+J. Similarly, AJ,J is the |J|×|J| submatrix whose rows and columns correspond
+to the indices in J. The cumulative distribution function (CDF) of a standard
+Gaussian is denoted by Φ(·) whereas Φc(·) = 1−Φ(·). We write an ≳ bn for two
+sequences {an}n≥1 and {bn}n≥1 if there are positive constants C and n0 such
+that an ≥ Cbn for all n > n0.
+2. Previous works
+Distributed linear regression schemes under various settings, not necessarily
+involving sparsity, have been proposed and theoretically studied in multiple
+fields, including sensor networks, statistics and machine learning, see for example
+(Guestrin et al., 2004; Predd, Kulkarni and Poor, 2006; Boyd et al., 2011; Zhang,
+Duchi and Wainwright, 2013; Heinze et al., 2014; Rosenblatt and Nadler, 2016;
+Jordan, Lee and Yang, 2019; Chen et al., 2020; Dobriban and Sheng, 2020; Zhu,
+Li and Wang, 2021; Dobriban and Sheng, 2021).
+Mateos, Bazerque and Giannakis (2010) were among the first to study dis-
+tributed sparse linear regression in a general setting without a fusion center,
+where machines are connected and communicate with each other. They devised
+a multi-round scheme whereby all the machines reach a consensus and jointly
+approximate the centralized solution, that would have been computed if all data
+were available at a single machine. Several later works focused on the setting
+which we also consider in this paper, where machines are connected in a star
+topology to a fusion center, and only one or two communication rounds are
+made. In a broader context of generalized sparse linear models, Chen and Xie
+(2014) proposed a divide-and-conquer approach where each machine estimates
+θ∗ by minimizing a penalized objective with a sparsity inducing penalty, such
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+5
+as ∥θ∥1. Each machine sends its sparse estimate to the fusion center, which es-
+timates the support by voting over the indices of the individual estimates of
+the M machines. Finally, the center estimates θ∗ by a weighted average of these
+M estimates. For sparse linear regression, with each machine computing a lasso
+estimate of θ∗, their method suffers from the well known bias of the lasso, which
+is not reduced by averaging.
+To overcome the bias of the lasso, in recent years several debiased lasso es-
+timators were derived and theoretically studied, see Zhang and Zhang (2014);
+van de Geer et al. (2014); Javanmard and Montanari (2018). For distributed
+learning, debiased estimators have been applied in various settings, including
+hypothesis testing, quantile regression and more, see for example Lee et al.
+(2017); Battey et al. (2018); Liu et al. (2021); Lv and Lian (2022).
+In particular, Lee et al. (2017) proposed a single round scheme whereby each
+machine computes its own debiased lasso estimator, and sends it to the fusion
+center. The center averages these debiased estimators and thresholds the result
+to estimate θ∗ and recover its support. Lee et al. (2017) proved that the re-
+sulting estimator achieves the same error rate as the centralized solution, and
+is minimax rate optimal. However, their scheme requires a communication of
+O(d) bits per machine and is thus not applicable in the restricted communica-
+tion setting considered in this manuscript. Moreover, as we demonstrate in the
+simulation section, unless the signal strength is very low, our two round scheme
+in fact achieves a smaller mean squared error, with a much lower communica-
+tion. This highlights the potential sub-optimality of lasso and debiased lasso in
+sparse regression problems with sufficiently strong signals.
+Most related to our paper is the recent work by Barghi, Najafi and Motahari
+(2021). In their method, each machine computes a debiased lasso estimator
+ˆθ, but sends to the fusion center only the indices i for which |ˆθi| is above a
+certain threshold. The support set estimated by the fusion center consists of all
+indices that were sent by at least half of the machines, i.e., indices that received
+at least M/2 votes. Focusing on the consistency of feature selection, Barghi,
+Najafi and Motahari (2021) derive bounds on the type-I and type-II errors
+of the estimated support set. Their results, however, are given as rates with
+unspecified multiplicative constants. As we show in this work, both theoretically
+and empirically, consistent support estimation is possible with a much lower
+voting threshold. Furthermore, requiring at least M/2 votes implies that their
+scheme achieves exact support recovery only for much stronger signals.
+We remark that voting is a natural approach for distributed support esti-
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+6
+mation under communication constraints. Amiraz, Krauthgamer and Nadler
+(2022) analyzed voting-based distributed schemes in the context of a simpler
+problem of sparse Gaussian mean estimation. They proved that even at low
+signal strengths, their schemes achieve exact support recovery with high prob-
+ability using communication sublinear in the dimension. Their setting can be
+viewed as a particular case of sparse linear regression but with a unitary design
+matrix. Their proofs, which rely on this property, do not extend to our setting.
+3. The lasso and debiased lasso estimators
+For our paper to be self contained, we briefly review the lasso and debiased lasso
+and some of their theoretical properties. The lasso (Tibshirani, 1996) is perhaps
+the most popular method to fit high-dimensional sparse linear models. Given
+a regularization parameter λ > 0 and n samples (Xi, yi), stacked in a design
+matrix X ∈ Rn×d and a response vector Y ∈ Rn, the lasso estimator is given by
+˜θ = ˜θ(X, Y, λ) = arg min
+θ∈Rd
+� 1
+2n∥Y − Xθ∥2
+2 + λ∥θ∥1
+�
+.
+(2)
+The lasso has two desirable properties. First, computationally Eq. (2) is a convex
+problem for which there are fast solvers. Second, from a theoretical standpoint,
+it enjoys strong recovery guarantees, assuming the data follows the model (1)
+with an exact or approximately sparse θ∗, see for example (Candes and Tao,
+2005; Bunea, Tsybakov and Wegkamp, 2007; van de Geer and B¨uhlmann, 2009;
+Hastie, Tibshirani and Wainwright, 2015). However, the lasso has two major
+drawbacks: it may output significantly biased estimates and it does not have
+a simple asymptotic distribution. The latter is needed for confidence intervals
+and hypothesis testing. To overcome these limitations, and in particular derive
+confidence intervals for high-dimensional sparse linear models, several authors
+developed debiased lasso estimators (Zhang and Zhang, 2014; van de Geer et al.,
+2014; Javanmard and Montanari, 2014a,b, 2018).
+For random X with a known population covariance matrix Σ, Javanmard and
+Montanari (2014a) proposed 1
+nΣ−1X⊤(Y − X˜θ) as a debiasing term. . As Σ is
+often unknown, both van de Geer et al. (2014) and Javanmard and Montanari
+(2014b) developed methods to estimate its inverse Ω = Σ−1. In our work, we
+estimate Ω using the approach of van de Geer et al. (2014), who assume that Ω
+is sparse. In their method, presented in Algorithm 1, ˆΩ is constructed by fitting
+a lasso regression with regularization λΩ > 0 to each column of X against all
+the other columns. Hence, it requires solving d separate lasso problems.
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+7
+Given the lasso estimate ˜θ of Eq. (2) and the matrix ˆΩ, the debiased lasso is
+ˆθ = ˆθ(Y, X, λ, λΩ) = ˜θ + 1
+n
+ˆΩX⊤(Y − X˜θ).
+(3)
+An appealing property of ˆθ is that, under some conditions, it is asymptotically
+unbiased with a Gaussian distribution. For our analysis, we shall use the follow-
+ing result (Javanmard and Montanari, 2018, Theorem 3.13).
+Theorem 1. Consider the linear model Y = Xθ∗+W, where W ∼ N(0, σ2In×n)
+and X ∈ Rn×d has independent Gaussian rows with zero mean and covariance
+matrix Σ ∈ Rd×d. Suppose that Σ satisfies the following conditions:
+i. For all i ∈ [d], Σii ≤ 1.
+ii. For some constants Cmax, Cmin > 0,
+0 < Cmin < σmin(Σ) ≤ σmax(Σ) < Cmax.
+(4)
+iii. For C0 = (32Cmax/Cmin) + 1, and a constant ρ > 0,
+max
+J⊆[d], |J|≤C0K ∥Σ−1
+J,J∥∞ ≤ ρ.
+Let KΩ be the maximum row-wise sparsity of Ω = Σ−1, that is,
+KΩ = max
+i∈[d] |{j ∈ [d]; Ωij ̸= 0, j ̸= i}| .
+Let ˜θ be the lasso estimator computed using λ = κσ
+�
+(ln d)/n for κ ∈ [8, κmax],
+and let ˆθ be the debiased lasso estimator in Eq. (3) with ˆΩ computed by Algorithm
+1 with λΩ = κΩ
+�
+(ln d)/n for some suitable large κΩ > 0. Let ˆΣ = X⊤X/n de-
+note the empirical covariance matrix. Then there exist constants c, c∗, C depend-
+ing solely on Cmin, Cmax, κmax and κΩ such that, for n ≥ c max{K, KΩ} ln d,
+the following holds:
+√n(ˆθ − θ∗) = Z + R,
+Z|X ∼ N(0, σ2 ˆΩˆΣˆΩ⊤),
+(5)
+where Z = n−1/2 ˆΩX⊤W and R = √n
+�
+ˆΩˆΣ − I
+�
+(θ∗ − ˜θ), and with probability
+at least 1 − 2de−c∗n/K − de−cn − 6d−2,
+∥R∥∞ ≤ Cσ ln d
+√n
+�
+ρ
+√
+K + min{K, KΩ}
+�
+.
+(6)
+Assumptions (i) and (ii) in this theorem are common in the literature. As-
+sumption (iii) is satisfied, for example, by circulant matrices Σij = ς|i−j|,
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+8
+Algorithm 1 Computation of a precision matrix estimate ˆΩ
+Input: design matrix X ∈ Rn×d, regularization parameter λΩ > 0.
+Output: precision matrix estimate ˆΩ ∈ Rd×d.
+xi ∈ Rn denotes the i-th column of X.
+X−i ∈ Rn×(d−1) denotes the design matrix with the i-th column removed.
+1: for i = 1, . . . , d do
+2:
+Fit a lasso with response xi, design matrix X−i and regularization parameter λΩ.
+3:
+Let ˜γi = {˜γi,j}d
+j=1,j̸=i ∈ Rd−1 be the estimated regression coefficients of step 2.
+4:
+Compute ˜τ 2
+i = (2n)−1∥xi − X−i˜γi∥2
+2 + λΩ∥˜γi∥1, i ∈ [d].
+5: end for
+6: Construct d × d matrix
+˜C =
+�
+����
+1
+−˜γ1,2
+· · ·
+−˜γ1,d
+−˜γ2,1
+1
+· · ·
+−˜γ2,d
+...
+...
+...
+...
+−˜γd,1
+−˜γd,2
+· · ·
+1
+�
+���� .
+7: return ˆΩ = diag{˜τ −2
+1
+, . . . , ˜τ −2
+d
+} ˜C.
+ς ∈ (0, 1). The quantity R in Eq. (5) can be viewed as a bias term. By Theorem
+1, this bias is small if the sample size and dimension are suitably large, which
+in turn implies that ˆθi is approximately Gaussian. The following lemma, proven
+in the Appendix, bounds the error of this approximation. It will be used in
+analyzing the probability of exact support recovery of our distributed scheme.
+Lemma 1. Under the assumptions of Theorem 1, for any τ > 0,
+���Pr
+� √n(ˆθi−θ∗
+i )
+σ√cii
+≤ τ
+�
+− Φ (τ)
+��� ≤
+δR
+σ√cii
+φ (τ) + 2de−c∗n/K + de−cn + 6
+d2 ,
+(7)
+where φ(·) denotes the Gaussian density function, cii = (ˆΩˆΣˆΩ⊤)ii, and δR is
+the upper bound on the bias term in Eq. (6), namely
+δR = Cσ ln d
+√n
+�
+ρ
+√
+K + min{K, KΩ}
+�
+.
+(8)
+4. Distributed sparse regression with restricted communication
+As described in Section 1, we consider a distributed setting with M machines
+connected in a star topology to a fusion center. For simplicity, we assume that
+each machine m has a sample (Xm, Y m) of n = N/M i.i.d. observations from
+the model (1), where Y m ∈ Rn and Xm ∈ Rn×d. In describing our schemes,
+we further assume that the noise level σ is known. If σ is unknown, it may
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+9
+Algorithm 2 Distributed voting based scheme for support estimation
+Input: Data (Xm, Y m) ∈ Rn×(d+1), threshold τ and regularization parameters λΩ and λ.
+Output: Support estimate ˆS.
+At each local machine m = 1, . . . , M
+1: Compute a lasso estimator ˜θm via Eq. (2) with regularization parameter λ.
+2: Compute a precision matrix estimate ˆΩm ∈ Rd×d by Algorithm 1 with Xm and λΩ.
+3: Compute a debiased lasso estimate ˆθm ∈ Rd, Eq. (3), with data (Xm, Y m), λ and ˆΩm.
+4: Calculate the empirical covariance matrix ˆΣm = n−1(Xm)⊤Xm.
+5: Use ˆΩm and ˆΣm to compute the standardized estimator ˆξm ∈ Rd, Eq. (9).
+6: Set Sm = {i; |ˆξm
+i | > τ} and send it to the fusion center.
+At the fusion center
+7: For each i ∈ [d], compute Vi = �M
+m=1 I(i ∈ Sm).
+8: Sort Vj1 ≥ Vj2 ≥ · · · ≥ Vjd.
+9: return ˆS = {j1, . . . , jK}.
+be consistently estimated, for example, by the scaled lasso of Sun and Zhang
+(2012), see also (Javanmard and Montanari, 2018, Corollary 3.10).
+We present several two round distributed schemes to estimate the sparse
+vector θ∗ of Eq. (1) under the constraint of limited communication between the
+M machines and the fusion center. Here we present the simplest scheme and
+discuss other variants in section 4.1. In all variants, the fusion center estimates
+the support of θ∗ in the first round, and θ∗ itself in the second round.
+The first round of our scheme is described in Algorithm 2, whereas the full two
+round scheme is outlined in Algorithm 3. In the first round, each machine m ∈
+[M] computes the following quantities using its own data (Xm, Y m): (i) a lasso
+estimate ˜θm by Eq. (2); (ii) a matrix ˆΩm by Algorithm 1; and (iii) a debiased
+lasso ˆθm by Eq. (3). Up to this point, this is identical to Lee et al. (2017). The
+main difference is that in their scheme, each machine sends to the center its
+debiased lasso estimate ˆθm ∈ Rd, incurring O(d) bits of communication.
+In contrast, in our scheme each machine sends only a few indices. Towards
+this end and in light of Eq. (7) of Lemma 1, each machine computes a normalized
+vector ˆξm whose coordinates are given by
+ˆξm
+k =
+√nˆθm
+k
+σ(ˆΩm ˆΣm(ˆΩm)⊤)1/2
+kk
+,
+∀k ∈ [d].
+(9)
+In the simplest variant, each machine sends to the center only indices k such
+that |ˆξm
+k | > τ for some suitable threshold τ > 0.
+Given the messages sent by the M machines, the fusion center counts the
+number of votes received by each index. If the sparsity level K is known, its
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+10
+Algorithm 3 Two round distributed scheme to estimate θ∗
+Input: Data (Xm, Y m) ∈ Rn×(d+1), sparsity K, threshold τ and regularizations λΩ and λ.
+Output: A two-round estimate ˆθ ∈ Rd of θ∗.
+First round
+1: The fusion center estimates ˆS with Algorithm 2.
+Second round
+2: The fusion center sends ˆS to all M machines.
+At each local machine m = 1, . . . , M
+3: Let Xm
+ˆ
+S ∈ Rn×K be the K columns of Xm corresponding to indices in ˆS.
+4: Compute ˆβm = arg minβ ∥Xm
+ˆ
+S β − Y m∥2
+2.
+5: Send ˆβm to the fusion center.
+At the fusion center
+6: Given ˆβ1, . . . , ˆβM, compute the estimate ˆθ according to Eq. (11).
+7: return ˆθ.
+estimated support set ˆS consists of the K indices with the largest number of
+votes. Otherwise, as discussed in Remark 4.5 below, the center may estimate the
+support set by the indices whose number of votes exceed a suitable threshold.
+Next, we describe the second round. At its start, the fusion center sends
+the estimated support ˆS to all M machines. Next, each machine computes the
+standard least squares regression solution, restricted to the set ˆS, namely
+ˆβm = arg min
+β ∥Xm
+ˆ
+S β − Y m∥2
+2
+(10)
+where Xm
+ˆ
+S ∈ Rn×| ˆ
+S| consists of the columns of Xm corresponding to the indices
+in ˆS. Each machine then sends its vector ˆβm to the fusion center. Finally, the
+fusion center estimates θ∗ by averaging these M vectors,
+ˆθi =
+�
+1
+M
+�M
+m=1 ˆβm
+i
+i ∈ ˆS
+0
+otherwise
+(11)
+In the next section we present several variants of this basic two round scheme.
+Before that we make a few remarks and observations.
+Remark 4.1. The communication of the first round (Algorithm 2) depends on
+the threshold τ. A high threshold leads to only few sent indices. However, at
+low signal strengths, the signal coordinates may not have the highest values |ˆξm
+k |
+and thus may not be sent. Hence, for successful support recovery by the fusion
+center, a lower threshold leading to many more sent coordinates is required. Since
+the maxima of d standard Gaussian variables scales as
+√
+2 ln d, to comply with
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+11
+the communication constraints, the threshold τ should also scale as O(
+√
+ln d). In
+Section 5, we present suitable thresholds and sufficient conditions on the number
+of machines and on the signal strength, which guarantee support recovery by
+Algorithm 2, with high probability and little communication per machine.
+Remark 4.2. With known K, the communication per machine of the second
+round is O(K ln d) bits. For suitable choices of the threshold τ in the first round,
+this is negligible or at most comparable to the communication of the first round.
+Remark 4.3. A two round scheme, whereby given an estimated set ˆS, the
+second round is identical to ours was discussed by Battey et al. (2018) in Eq.
+(A.2) of their supplementary. The difference is that in their first round, similar
+to Lee et al. (2017), each machine sends its full debiased lasso vector, with a
+communication of O(d) bits. Battey et al. (2018) showed that, under certain
+conditions, their two-round estimator attains an optimal rate. In Section 5, we
+prove that for a sufficiently high SNR, our method achieves the same rate, but
+using much less communication.
+Remark 4.4. With a higher communication per machine in the second round,
+it is possible for the fusion center to compute the exact centralized least squares
+solution corresponding to the set ˆS, denoted ˆθLS. Specifically, suppose that each
+machine sends to the center both the vector (Xm
+ˆ
+S )⊤Y m of length | ˆS|, and the
+| ˆS| × | ˆS| matrix (Xm
+ˆ
+S )⊤Xm
+ˆ
+S . The center may then compute ˆθLS as follows
+ˆθ
+LS =
+� M
+�
+m=1
+(Xm
+ˆ
+S )⊤Xm
+ˆ
+S
+�−1
+M
+�
+m=1
+(Xm
+ˆ
+S )⊤Y m.
+(12)
+With K known and | ˆS| = K, such a second round has a communication of
+O(K2) bits. If the sparsity K is non-negligible, this is much higher than the
+O(K) bits of our original scheme. In particular, if K = O(d1/2), the resulting
+communication is comparable to that of sending the full debiased lasso vector.
+Remark 4.5. In practice, the sparsity K is often unknown. Instead of step 9
+in Algorithm 2, one alternative is to estimate S by thresholding the number of
+votes. For some threshold τvotes > 0, ˆS could be set as all indices i such that
+Vi > τvotes. Lemma 3 in Appendix A shows that, under suitable conditions,
+non-support indices have a small probability of receiving more than 2 ln d votes.
+Hence, τvotes = 2 ln d is a reasonable choice for such a threshold value.
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+12
+4.1. Variations of Algorithm 2
+Various adaptations of Algorithm 2 are possible and may offer better perfor-
+mance. One example is a top L algorithm where each machine sends to the cen-
+ter the indices of the L largest entries of |ˆξm|, for some parameter K ≤ L ≪ d.
+A similar approach was proposed in Amiraz, Krauthgamer and Nadler (2022)
+for the simpler problem of sparse normal means estimation. One advantage of
+this variant is that its communication per machine is fixed and known a priori
+O(L ln d). This is in contrast to the above thresholding based scheme, whose
+communication per machine is random.
+A different variant is to use sums of signs to estimate the support. Here
+machines send both the indices corresponding to the largest entries in |ˆξm| and
+their signs. Hence, in step 5 of Algorithm 2 the message sent by machine m is
+Sm =
+��
+i, sign(ˆξm
+i )
+�
+; |ˆξm
+i | > τ
+�
+.
+Next, the fusion center computes for each index i ∈ [d] its corresponding sum
+of received signs, i.e.,
+V sign
+i
+=
+M
+�
+m=1
+sign(ˆξm
+i )I
+��
+i, sign(ˆξm
+i )
+�
+∈ Sm�
+.
+(13)
+For known K, the estimated support set are the K indices with largest values
+of |V sign
+i
+|. This algorithm uses a few more bits than a voting scheme. How-
+ever, sums of signs are expected to better distinguish between support and
+non-support coefficients when the number of machines is large. The reason is
+that at non-support indices j ̸∈ S, the random variable V sign
+j
+has approximately
+zero mean, unlike sums of votes Vj, whereas at support indices |V sign
+i
+| ≈ Vi since
+support indices are unlikely to be sent to the fusion center with the opposite
+sign of θ∗
+i . In the simulation section we illustrate the improved performance of
+a sign-based over a votes-based distributed scheme.
+5. Theoretical results
+In this section, we present a theoretical analysis for one of our schemes. Specif-
+ically, both Theorems 2 and 3 show that under suitable conditions, with high
+probability Algorithm 2 achieves exact support recovery with little communi-
+cation per machine. In Theorem 2, the number of machines is relatively large,
+and the communication per machine is linear in the sparsity K. In Theorem 3
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+13
+the number of machines M is logarithmic in d, in which case the communica-
+tion per machine is much higher, though still sublinear in d. Both theorems are
+based on the Gaussian approximation in Lemma 1 and on probability bounds
+for binomial random variables. Their proofs appear in Appendix A.
+To put our theorems in context, let us briefly review previous results on exact
+support recovery in the simpler (non-distributed) sparse linear regression set-
+ting. A key quantity characterizing the ability of exact support recovery is the
+signal strength, defined as θmin = mini∈S |θ∗
+i |. As proven by Wainwright (2009),
+under suitable conditions on the design matrix, the lasso estimator based on
+n samples and an appropriately chosen regularization parameter λn, achieves
+exact support recovery with high probability, provided that θmin ≳
+�
+(ln d)/n.
+The same rate θmin ≳
+�
+(ln d)/n is also sufficient for support recovery using
+a debiased lasso estimator (see, e.g., section 2.2 of Javanmard and Montanari
+(2014b)). In a distributed setting, Lee et al. (2017) proved that with high proba-
+bility, their scheme achieves exact support recovery when θmin ≳
+�
+(ln d)/(nM).
+While this result matches the centralized setting, their scheme requires each ma-
+chine to send to the center its d-dimensional debiased lasso estimate, incurring
+O(d) communication per machine. Hence, an interesting range for the signal
+strength, for the study of support recovery under communication constraints,
+is
+�
+ln d
+nM ≲ θmin ≲
+�
+ln d
+n . In this range, individual machines may be unable to
+exactly recover the support using the lasso or debiased lasso estimators.
+To derive support recovery guarantees, we assume the smallest nonzero co-
+efficient of θ∗ is sufficiently large, namely |θ∗
+i | ≥ θmin for all i ∈ S and some
+suitable θmin > 0. For our analysis below, conditional on the design matrices
+X1, . . . , XM at the M machines, it will be convenient to make the following
+change of variables from θmin to the (data-dependent) SNR parameter r,
+θmin = θmin(d, σ, r, n, cΩ) = σ
+�
+2cΩ
+n r ln d,
+(14)
+where cΩ is defined as
+cΩ =
+max
+i∈[d],m∈[M]
+�
+ˆΩm ˆΣm(ˆΩm)⊤�
+ii .
+(15)
+Recall from Eq. (9) that by Theorem 1, σ2 �
+n−1 ˆΩm ˆΣm(ˆΩm)⊤�
+ii is the asymp-
+totic variance of ˆθm
+i . Hence, σ2cΩ/n is the largest variance of all d coordinates
+of the M debiased estimators computed by the M machines. In terms of the
+SNR parameter r, the range of interest is thus
+1
+M < r < 1.
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+14
+Recall that our scheme is based on thresholding the normalized debiased lasso
+estimators ˆξm
+k of Eq. (9). We denote the corresponding normalized signal by
+ϑm
+k =
+√nθ∗
+k
+σ
+�
+ˆΩm ˆΣm(ˆΩm)⊤
+�1/2
+kk
+,
+∀k ∈ [d].
+(16)
+Lemma 1 states that under suitable conditions, ˆξm
+k − ϑm
+k has approximately a
+standard Gaussian distribution. This property plays an important role in our
+theoretical results. Eq. (7) of Lemma 1 provides a bound on the error between
+the CDF of ˆξm
+k −ϑm
+k and that of a standard Gaussian. For a threshold τ, let ϵ(τ)
+be the largest of these error bounds over all d coordinates in all M machines,
+ϵ(τ) =
+max
+k∈[d],m∈[M]
+�
+�
+�
+�
+�
+δRφ (τ − ϑm
+k )
+σ
+�
+ˆΩm ˆΣm(ˆΩm)⊤
+�1/2
+kk
++ 2de−c∗n/K + de−cn + 6
+d2
+�
+�
+�
+�
+�
+. (17)
+Recall that δR, defined in Eq. (8), is an upper bound on the bias ˆθm
+k − θ∗
+k.
+By Lemma 5.4 of van de Geer et al. (2014), if the row sparsity of Ω satisfies
+KΩ = o(n/ ln d) and ˆΩm is computed with regularization λΩ ∝
+�
+(ln d)/n, then
+�
+ˆΩm ˆΣm(ˆΩm)⊤�
+kk ≥ Ωkk + oP (1) ≥ C−1
+max + oP (1) when ln d
+n → 0. Hence, when
+n and d are large, all terms on the right hand side of Eq. (17) are small, and
+the Gaussian approximation is accurate.
+To prove that our scheme recovers S with high probability, we assume that:
+(C1) The n samples in each of the M machines are i.i.d. from the model (1)
+and the conditions of Theorem 1 all hold. Additionally, all machines use
+the same regularization parameters λ and λΩ to compute the lasso (2) and
+debiased lasso (3) estimators, respectively.
+(C2) |θ∗
+i | ≥ θmin(d, σ, r, n, cΩ) for all i ∈ S, where θmin and cΩ are defined in
+Eqs. (14) and (15), respectively.
+The following theorem provides a recovery guarantee for Algorithm 2, where
+the sparsity K is assumed to be known to the fusion center.
+Theorem 2. Suppose Algorithm 2 is run with threshold τ =
+√
+2 ln d. Assume
+that d is sufficiently large and that the SNR in Eq. (14) satisfies
+1
+4
+ln2(48√π ln3/2 d)
+ln2(d)
+< r < 1.
+Additionally, assume conditions C1 and C2 hold and the approximation error
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+15
+in Eq. (17) satisfies ϵ(τ) ≤ 1/d. Then, if the number of machines satisfies
+8 ln d
+�
+2(1−√r)
+2 ln d
+√
+2π
+�
+2(1−√r)
+2 ln d+1
+� − ϵ(τ)d(1−√r)
+2 d(1−√r)
+2
+≤ M ≤ d
+3,
+(18)
+with probability at least 1 − K+1
+d , Algorithm 2 achieves exact support recovery.
+Additionally, the expected communication per machine is O (K ln d) bits.
+Let us make a few remarks regarding this theorem. The upper bound M <
+d/3 is rather artificial and stems from the fact that in our proof we assume
+M < d. It is possible to derive support guarantees also for the case M > d,
+though this setting seems to be unlikely in practice. The lower bound on the
+number of machines is required to guarantee that with high probability, all
+support indices receive more votes than any non-support coordinate. The lower
+bound on the SNR r ensures that the lower bound on the number of machines
+in Eq. (18) is indeed smaller than d/3, so the range of possible values for M is
+not empty. A similar lower bound on r appeared in Amiraz, Krauthgamer and
+Nadler (2022) after their Theorem 1.B.
+Another important remark is that the threshold τ =
+√
+2 ln d in Theorem 2
+is relatively high, so each machine sends only few indices to the center. How-
+ever, to guarantee support recovery, this requires a relatively large number of
+machines M = polylog(d) · d(1−√r)2. In Theorem 3, we give sufficient conditions
+to still achieve a high probability of exact support recovery when the number
+of machines is much smaller, of order only logarithmic in d. The price to pay is
+a higher communication per machine, which nonetheless is still sub-linear in d,
+namely much lower than the communication required to send the whole debi-
+ased lasso vector. For the next theorem, we assume that a lower bound on the
+SNR is known to all machines, which set a threshold that depends on it.
+Theorem 3. Suppose Algorithm 2 is run with threshold τ =
+√
+2r ln d. Assume
+that d is sufficiently large and that the SNR in Eq. (14) satisfies
+ln(16 ln d)
+ln d
+< r < 1.
+Additionally, assume conditions C1 and C2 hold and the approximation error
+in Eq. (17) satisfies ϵ(τ) < 1/(4dr). If the number of machines satisfies
+16 ln d
+1 − 2ϵ(τ) ≤ M ≤ dr,
+(19)
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+16
+then with probability at least 1 − K+1
+d , Algorithm 2 achieves exact support re-
+covery, with expected communication per machine O
+�
+d1−r ln d
+�
+bits.
+Beyond support recovery, another quantity of interest is the accuracy of the
+distributed estimator ˆθ of Eq. (11). The following corollary, proven in Appendix
+A, shows that once S is precisely recovered, ˆθ is close to the oracle least squares
+estimator ˆθLS computed with all data in a single machine and with knowledge
+of the true support. Consequently, ˆθ is also close to the true vector θ∗.
+Corollary 1. Assume the conditions of Theorem 2 hold. Let N = nM denote
+the total sample size in all M machines. If M = O
+�
+NK
+(max{K,ln N})2
+�
+, then
+∥ˆθ − ˆθ
+LS∥2 = OP
+�√
+M max{K, ln N}
+N
+�
+and
+∥ˆθ − θ∗∥2 = OP
+��
+K
+N
+�
+,
+as d, N → ∞ and
+ln d
+N/M → 0, where ˆθ is defined in Eq. (11) and ˆθLS is the
+least squares solution using all N samples and with a known S, as in Eq. (12),
+appended by zeros at all coordinates j /∈ S.
+Corollary 1 shows that in a high dimensional sparse setting, for a sufficiently
+strong signal, Algorithm 3 with a threshold τ =
+√
+2 ln d achieves the same error
+rate as the oracle estimator. Let us put this result in a broader context. If the
+support S were known, then each machine could have computed its least squares
+solution restricted to S and send it to the center for averaging. As discussed
+in Rosenblatt and Nadler (2016), in a general setting of M-estimators, if the
+number of machines is not too large, averaging is optimal and to leading order
+coincides with the centralized solution. Yet, while being rate optimal, we note
+that averaging does lead to a loss of accuracy and is not as efficient as the oracle
+estimator, see Dobriban and Sheng (2021).
+As mentioned in Remark 4.3, Battey et al. (2018) also proposed a two-round
+estimator that attains the optimal rate in Corollary 1, but requires each machine
+to send at least d values to the fusion center. In contrast, ˆθ is computed using
+a much lower communication cost. Similar results can also be established for
+Algorithm 3 under the conditions of Theorem 3.
+5.1. Comparison to other works
+Theorems 2 and 3 can be viewed as analogous to Theorems 2.A and 2.B of
+Amiraz, Krauthgamer and Nadler (2022), who studied distributed estimation
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+17
+of a sparse vector under communication constraints. A key difference is that
+in their setting, the d coordinates of the estimated vector at each machine are
+all independent and unbiased. This allowed them to analyze a top-L scheme
+since the probability of sending any non-support index was the same and could
+be easily bounded by L/d. As such, their proofs do not directly apply to our
+setting. In our case, the debiased lasso ˆθm still has a small bias term and its
+d coordinates are in general correlated. This implies that the probabilities of
+sending non-support indices are not all identical. In our analysis, we bypass
+this issue by analyzing instead a thresholding approach at step 5 of Algorithm
+2, where each ˆθm
+i
+for i ∈ [d] is compared separately to a fixed threshold. This
+way, we do not need to account for the complex dependence among different
+coordinates of the debiased lasso.
+Barghi, Najafi and Motahari (2021) considered a similar distributed scheme
+to estimate the support of θ∗. They did not normalize the debiased lasso estimate
+at each machine, and more importantly the estimated support set consists only
+of those indices that received at least M/2 votes. The authors performed a
+theoretical analysis of their scheme, though various quantities are described only
+up to unspecified multiplicative constants. We remark that both theoretically as
+well as empirically, the SNR must be quite high for support indices to obtain at
+least M/2 votes. In our work, we present explicit expressions for the minimum
+SNR in Eq. (14), sufficient for exact support recovery, requiring far fewer votes
+at the fusion center. The simulations in the next section illustrate the advantages
+of our scheme as compared to that of Barghi, Najafi and Motahari (2021).
+6. Simulations
+We present simulations that illustrate the performance of our proposed methods
+in comparison to other distributed schemes. We focus on methods based on
+debiased lasso estimates, and specifically consider the following five distributed
+schemes to estimate θ∗ and its support.
+• thresh-votes: Algorithm 2 with a threshold of τ =
+√
+2 ln d.
+• top-L-votes: The top L algorithm presented in section 4.1. Each machine
+sends the indices of its top L values of |ˆξm
+i | to the fusion center.
+• top-L-signs: Each machine sends both the indices and signs of the top
+L values of |ˆξm
+i |. The center forms ˆS using sums of signs as in Eq. (13).
+• BNM21: The algorithm proposed by Barghi, Najafi and Motahari (2021)
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+18
+with a threshold τ =
+√
+2 ln d. It is similar to Algorithm 2, the difference
+is that ˆS consists of the indices i ∈ [d] with Vi ≥ M/2.
+• AvgDebLasso: Based on Lee et al. (2017), each machine sends its debiased
+lasso estimate ˆθm. The center computes ˆθavg =
+1
+M
+�M
+m=1 ˆθm and estimates
+the support as the indices with the K largest values |ˆθavg
+i
+|, i ∈ [d].
+In all these algorithms, each machine computes a debiased lasso estimator using
+its own data. The methods differ by the content and length of the messages sent
+to the fusion center and hence by the manner in which the fusion center estimates
+S and θ∗. For a fair comparison, we run all methods with the same regularization
+parameters λΩ = 2
+�
+(ln d)/n and λ = 8
+�
+(ln d)/n in each machine to compute
+the precision matrix ˆΩm and the lasso estimator ˜θm, respectively.
+We performed simulations with both known sparsity, where the methods
+were run as described above, as well as unknown sparsity. In the latter case,
+for thresh-votes, top-L-votes and top-L-signs, the center computes ˆS as
+the indices i such that Vi or |V sign
+i
+| is larger than 2 ln d, see Remark 4.5 in
+Section 4; For AvgDebLasso, we set ˆS as the indices i such that |ˆθavg
+i
+| > 11 ln d
+n .
+This scaling is motivated by Theorem 16 of Lee et al. (2017), whereby in our
+simulation setting this term is larger than the term O
+��
+ln d/(nM)
+�
+in their
+bound for ∥ˆθavg − θ∗∥∞. The factor 11 was manually tuned for good results.
+BNM21 is unchanged, as it does not require knowledge of K.
+We evaluate the accuracy of an estimated support set ˆS by the F-measure,
+F-measure = 2 · precision · recall
+precision + recall ,
+where precision = |S ∩ ˆS|/| ˆS| and recall = |S ∩ ˆS|/K. An F-measure equal to
+one indicates that exact support recovery was achieved.
+Given a support estimate ˆS, the vector θ∗ is estimated as follows. For all
+four methods excluding AvgDebLasso, we perform a second round and compute
+the estimator ˆθ given by Eq. (11). AvgDebLasso is a single round scheme. Its
+estimate of θ∗ consists of ˆθavg restricted to the indices i ∈ ˆS. The error of
+an estimate ˆθ is measured by its ℓ2-norm ∥ˆθ − θ∗∥2. As a benchmark for the
+achievable accuracy, we also computed the oracle centralized estimator ˆθLS that
+knows the support S and estimates θ∗ by least squares on the whole data.
+We generated data as follows: The design matrix Xm ∈ Rn×d in machine
+m has n rows i.i.d. N(0, Σ), with Σi,j = 0.5|i−j|. We then computed for each
+machine its matrix ˆΩm, and the quantity cΩ in Eq. (15). Next, we generated a K
+sparse vector θ∗ ∈ Rd, whose nonzero indices are sampled uniformly at random
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+19
+(a)
+(b)
+Fig 1: Results for known sparsity, averaged over 500 realizations, as a function
+of SNR in the range r ∈ [1/M, 1]. (a) F-measure; (b) ℓ2 error, on a log scale.
+from [d]. Its nonzero coefficients have random ±1 signs and their magnitudes
+are chosen from K equally spaced values {θmin, . . . , 2θmin}, where
+θmin =
+�
+2cΩ ln d
+n
+.
+These matrices and the vector θ∗ are then kept fixed. For a simulation with
+SNR parameter r, we set σ = 1/√r. Finally, in each realization we generated
+the response Y m ∈ Rn according to the model (1).
+Our first simulation compared the performance of various schemes as a func-
+tion of the SNR, with a known sparsity K = 5. We fixed the dimension d = 5000,
+the sample size in each machine n = 250, and the number of machines M = 100.
+The top L method was run with L = K (see Sec. 4.1). Figure 1a displays the
+F-measure of each method, averaged over 500 realizations. As expected, at low
+SNR values, AvgDebLasso achieved the best performance in terms of support
+recovery. However, for stronger signals with r > 0.4, both top-K-votes and
+thresh-votes achieved an F-measure of one, in accordance with our theoret-
+ical results regarding exact recovery. In particular, at sufficiently high SNR,
+our methods estimate the support as accurately as AvgDebLasso, but with 2-3
+orders of magnitude less communication. The scheme of BNM21 achieves good
+
+n=250.d=5000.K=5.M=100
+1.0
+0.8
+measure
+0.6
+0.4
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+r
+AvgDebLasso
+top-K-votes
+BNM21
+oracle
+thresh-votesn=250,d=5000,K=5,M=100
+0.00
+-0.25
+error
+-0.50
+-0.75
+)
+1.00
+-1.25
+1.50
+-1.75
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+r
+AvgDebLasso
+top-K-votes
+BNM21
+oracle
+thresh-votesR. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+20
+(a)
+(b)
+Fig 2: Results for unknown sparsity, averaged over 500 realizations, as a function
+of SNR in the range r ∈ [1/M, 1]. (a) F-measure; (b) ℓ2 error on a log scale.
+performance only at higher SNR.
+Figure 1b shows the errors ∥ˆθ − θ∗∥2, averaged over 500 realizations. At low
+SNR, 1/M < r < 0.1, AvgDebLasso has the smallest error. However, for r > 0.4,
+thresh-votes and top-K-votes yield more accurate estimates. Thus, Figure
+1b shows the benefits of an accurate support estimate followed by a distributed
+least squares in a second round. Indeed, at these SNR levels, our methods exactly
+recover the support. Consequently, the second round reduces to a distributed
+ordinary least squares restricted to the correct support set S. In accordance
+with Corollary 1, Algorithm 2 then has the same error rate as the oracle.
+Next we present simulation results for unknown sparsity, as a function of the
+SNR in the range r ∈ [ 1
+M , 1]. As seen in Figure 2, throughout this SNR range,
+AvgDebLasso with a threshold of 11(ln d)/n achieves an F-measure close to one
+and ℓ2 errors close to those of the oracle. In contrast, thresh-votes achieves
+accurate estimates only for r > 0.6. These results illustrate that even when
+sparsity is unknown, our schemes can accurately estimate the vector θ∗ and its
+support, albeit with a higher SNR as compared to the case of known sparsity.
+Finally, we present simulation results as a function of number of samples
+n ∈ [nmin, nmax] = [100, 400] with M = 100 machines, and as a function of
+number of machines M ∈ [Mmin, Mmax] = [40, 160] with n = 250 samples
+
+n=250,d=5000,K=5,M=100
+1.0
+0.8
+measure
+0.6
+0.4
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+r
+AvgDebLasso
+thresh-votes
+BNM21
+oraclen=250,d=5000,K=5,M=100
+0.00
+-0.25
+error)
+-0.50
+-0.75
+log1o(l2
+-1.00
+-1.25
+-1.50
+-1.75
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+r
+AvgDebLasso
+thresh-votes
+BNM21
+oracleR. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+21
+(a)
+(b)
+Fig 3: ℓ2 error vs. sample size n (a) and vs. number of machines M (b), both on
+a log-log scale, at an SNR of r = 0.4. Values are averaged over 1000 realizations.
+per machine. Initially, for each machine m ∈ [Mmax], we generated its full data
+matrix Xm with nmax number of samples. We then computed the corresponding
+matrix ˆΩm and the quantity cΩ of Eq. (15). We then generated the sparse
+vector θ∗ as described above. To save on run-time, in simulations with a smaller
+number of samples n < nmax we nonetheless used the decorrelation matrices
+Ωm that correspond to nmax samples. This can be viewed as a semi-supervised
+setting, as also mentioned in Javanmard and Montanari (2018). We fixed r = 0.4
+and d = 5000, and the sparsity K = 5 was known to the center. Figure 3
+shows the resulting ℓ2 errors averaged over 1000 simulations. In this simulation,
+top-K-votes and thresh-votes are close to the centralized least squares oracle
+since their support estimates are accurate, as can be seen in Figure 1 for r = 0.4.
+Both plots in Figure 3 show a linear dependence on a log-log scale with a slope of
+approximately −1/2, namely that the resulting errors decay as 1/√n and 1/
+√
+M,
+respectively. This is in agreement with our theoretical result in Corollary 1.
+6.1. The advantages of sending signs
+We now illustrate the advantages of using sums of signs instead of sums of votes,
+in terms of both support recovery and parameter estimation. In this simulation,
+we fixed n = 250, d = 5000, M = 100 and K = 25. The results in Figure
+4 show that using sums of signs is more accurate than sums of votes for low
+
+d=5000,K=5,r=0.40,M=100
+-1.0
+log1o(l2 error)
+-1.2
+-1.4
+-1.6
+1.8
+2.0
+2.2
+2.4
+2.6
+log1o(n)
+AvgDebLasso
+top-K-votes
+thresh-votes
+oracled=5000,K=5,r=0.40,n=250
+-1.0
+log1o(l2 error)
+-1.2
+-1.4
+-1.6
+-1.8
+1.6
+1.8
+2.0
+2.2
+log1o(M)
+AvgDebLasso
+top-K-votes
+thresh-votes
+oracleR. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+22
+(a)
+(b)
+Fig 4: Results for schemes using sums of signs and sums of votes, averaged over
+500 realizations, as a function of r∈[ 1
+M , 0.1]. (a) F-measure; (b) log10 of ℓ2 error.
+SNR values in the range r ∈ [1/M, 0.1]. Figure 4 also shows that using L = 5K
+instead of L = K significantly improves the accuracy of the support estimator,
+at the expense of increasing the communication. This illustrates the potential
+trade-offs between the accuracy of support estimation and communication.
+7. Summary and Discussion
+The development and analysis of distributed statistical inference schemes having
+low communication are important contemporary problems. Given its simplicity
+and ubiquity, the sparse linear regression model has attracted significant atten-
+tion in the literature. Most previous inference schemes for this model require
+communication per machine of at least O(d) bits. In this work we proved the-
+oretically and showed via simulations that, under suitable conditions, accurate
+distributed inference for sparse linear regression is possible with a much lower
+communication per machine.
+Over the past years, several authors studied distributed statistical infer-
+ence under communication constraints. Specifically, for sparse linear regression,
+Braverman et al. (2016) proved that without a lower bound on the SNR, to ob-
+
+n=250.d=5000.K=25.M=100
+1.0
+0.8
+measure
+0.6
+0.4
+0.2
+0.0
+0.02
+0.04
+0.06
+0.08
+0.10
+r
+AvgDebLasso
+top-5K-votes
+thresh-votes
+top-5K-signs
+top-K-votes
+oracle
+top-K-signsn=250,d=5000,K=25,M=100
+0.4
+0.2
+error)
+0.0
+0.2
+)0
+0.4
+0.6
+-0.8
+-1.0
+0.02
+0.04
+0.06
+0.08
+0.10
+r
+AvgDebLasso
+top-5K-votes
+thresh-votes
+top-5K-signs
+top-K-votes
+oracle
+top-K-signsR. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+23
+tain a risk comparable to that of the minimax lower bound, a communication of
+at least Ω(M min(n, d)/ log d) bits is required. Acharya et al. (2019) proved that,
+under certain conditions, rate optimal estimates of a linear regression model can
+be computed using total communication sublinear in the dimension. However,
+as they mention in their appendix B.3, a precise characterization of the ability
+to recover the support with sublinear communication in d and its dependency on
+other parameters such as SNR and the number of machines is still an open prob-
+lem. In our theoretical results, we presented explicit expressions for the minimal
+SNR at which our scheme is guaranteed to achieve exact recovery with high
+probability and with sublinear communication. While we did not address the
+open problem of tight lower bounds, our results highlight the potential tradeoffs
+between SNR, communication and number of machines.
+We believe that using more refined techniques, our theoretical analysis can be
+extended and improved. For example, since the d coordinates of a debiased lasso
+estimator are correlated, sharp concentration bounds for dependent variables,
+like those of Lopes and Yao (2022), could improve our analysis and extend it to
+other schemes such as top-L. In our analysis, we focused on a setting where both
+the noise and covariates have Gaussian distribution. Lee et al. (2017) and Battey
+et al. (2018), for example, considered sub-Gaussian distributions for these terms.
+Our results can be adapted for this case, but a careful control of the various
+constants in probability bounds is needed to derive explicit expressions.
+Finally, our low-communication schemes could also be applied to other prob-
+lems, such as sparse M-estimators, sparse covariance estimation and distributed
+estimation of jointly sparse signals. We leave these for future research.
+Appendix A: Proofs
+A.1. Proof of Lemma 1
+Proof. Consider the debiased lasso estimator ˆθi given in Eq. (3). Making the
+change of variables t = σ√ciiτ/√n, gives that
+Pr
+�√n(ˆθi − θ∗
+i )
+σ√cii
+≤ τ
+�
+= Pr(ˆθi − θ∗
+i ≤ t).
+(20)
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+24
+It follows from Eq. (5) that Pr(ˆθi − θ∗
+i ≤ t) = Pr (Zi ≤ √nt − Ri). By the law
+of total probability,
+Pr
+�
+Zi ≤ √nt − Ri
+�
+=
+Pr
+�
+{Zi ≤ √nt − Ri} ∩ {|Ri| ≤ δR}
+�
++ Pr
+�
+{Zi ≤ √nt − Ri} ∩ {|Ri| > δR}
+�
+≤
+Pr
+�
+Zi ≤ √nt + δR
+�
++ Pr (|Ri| > δR) .
+(21)
+From Eq. (5) it follows that Pr (Zi ≤ √nt + δR) = Φ
+� √nt+δR
+σ√cii
+�
+. Hence, from
+Eqs. (20) and (21) we get that
+Pr
+�√n(ˆθi − θ∗
+i )
+σ√cii
+≤ τ
+�
+≤ Φ
+�√nt + δR
+σ√cii
+�
++ Pr (|Ri| > δR) .
+(22)
+The second term on the right-hand side of Eq. (22) can be bounded by Eq. (6).
+This gives the last three terms on the right hand side in Eq. (7).
+Let us analyze Φ
+� √nt+δR
+σ√cii
+�
+. For any fixed x and δ > 0, by the mean value
+theorem, |Φ(x + δ) − Φ(x)| ≤ δφ(x∗), where x∗ ∈ (x, x + δ). Since φ(x) is a
+decreasing function for x > 0, we have φ(x∗) ≤ φ(x). Thus,
+|Φ(x + δ) − Φ(x)| ≤ δφ(x).
+Applying this result with x =
+√nt
+σ√cii and δ =
+δR
+σ√cii , gives
+����Φ
+�√nt + δR
+σ√cii
+�
+− Φ
+� √nt
+σ√cii
+����� ≤
+δR
+σ√cii
+φ
+� √nt
+σ√cii
+�
+.
+Combining the above with Eq. (22), and replacing t = σ√ciiτ
+√n
+proves Eq. (7).
+A.2. Proofs of Theorem 2 and Theorem 3
+Let us first provide an overview of the proofs. Recall that we consider a dis-
+tributed setting with M machines each with its own data, and each machine
+sends an independent message containing a few indices to the fusion center. For
+any index i ∈ [d] and machine m, let pm
+i
+denote the probability that index i
+is sent by machine m, namely that |ˆξm
+i | > τ. Since data at different machines
+are statistically independent, the total number of votes Vi received at the fusion
+center for index i is distributed as Vi ∼ �M
+i=1 Ber(pm
+i ). Our proof strategy is as
+follows: we compute an upper bound for pm
+j for non-support indices j ̸∈ S, and
+a lower bound for pm
+i for support indices i ∈ S. Next, we employ tail bounds for
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+25
+binomial random variables. Combining these implies that under suitable condi-
+tions on the SNR and number of machines, with high probability, the number
+of votes Vi can perfectly distinguish between support and non-support indices.
+To carry out this proof outline, we now introduce some auxiliary lemmas and
+results. The following are standard Gaussian tail bounds,
+t
+√
+2π(t2 + 1)e−t2/2 ≤ 1 − Φ(t) ≤
+1
+√
+2πte−t2/2,
+∀t > 0.
+(23)
+We also use the following inequality for a binomial variable V ∼ Bin(M, p)
+(Boucheron, Lugosi and Massart, 2013, exercise 2.11). For any 0 < p ≤ a < 1,
+Pr (V > Ma) ≤
+��p
+a
+�a �1 − p
+1 − a
+�1−a�M
+= eMF (a,p)
+(24)
+where
+F(a, p) = a ln
+�p
+a
+�
++ (1 − a) ln
+�1 − p
+1 − a
+�
+.
+(25)
+The following result appeared in (Amiraz, Krauthgamer and Nadler, 2022,
+Lemma A.3). It is used in our proof to show that with high probability, support
+indices receive a relatively large number of votes.
+Lemma 2. Assume that mini∈S |θ∗
+i | is sufficiently large so that for some suitable
+pmin > 0, for all i ∈ S, and m ∈ [M], pm
+i ≥ pmin. If pmin ≥ 8 ln d
+M , then
+Pr
+�
+min
+i∈S Vi < 4 ln d
+�
+≤ K
+d .
+The next lemma shows that under suitable conditions, non-support indices
+receive relatively few total number of votes.
+Lemma 3. Assume that d ≥ 4 and M > 2 ln d. In addition, assume that
+pm
+j ≤
+1
+M for all non-support indices j ̸∈ S and all machines m ∈ [M]. Then
+Pr
+�
+max
+j̸∈S Vj > 2 ln d
+�
+≤ 1
+d.
+(26)
+Proof. Recall that the number of votes received by an index j ̸∈ S at the fusion
+center is distributed as Vj ∼ �M
+m=1 Ber(M, pm
+j ). Since pm
+j ≤
+1
+M for all j ̸∈ S,
+then Vj is stochastically dominated by
+V ∼ Bin(M, p),
+where
+p = 1/M.
+(27)
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+26
+Thus, by a union bound,
+Pr
+�
+max
+j̸∈S Vj > t
+�
+≤ (d − K) · Pr(V > t) ≤ d · Pr
+�
+V > M t
+M
+�
+.
+We now apply Eq. (24) with t ≥ 1, so that the value of a = t/M = tp indeed
+satisfies that a ≥ p. With F(a, p) defined in Eq. (25), this gives
+Pr
+�
+max
+j̸∈S Vj > t
+�
+≤ d · eMF (tp,p).
+(28)
+Next, we upper bound F(tp, p). Since ln(1 + x) ≤ x holds for all x ≥ 0,
+F(tp, p) = −tp ln(t) + (1 − tp) ln
+�
+1 + tp − p
+1 − tp
+�
+≤ −tp ln(t) + tp − p
+< −tp ln(t) + tp = −tp ln(t/e).
+Inserting this into Eq. (28) with t = 2 ln d, p = 1/M and M > 2 ln d gives
+Pr
+�
+max
+j̸∈S Vj > 2 ln d
+�
+≤ deMF (tp,p) ≤ de−2 ln(d) ln(2 ln(d)/e).
+For Eq. (26) to hold, we thus require that
+2 ln(d) ln
+�2 ln d
+e
+�
+≥ ln
+�
+d2�
+.
+(29)
+This holds for d ≥ exp{e/2} ≈ 3.89.
+Proof of Theorem 2. Our goal is to show that the event { ˆS = S} occurs with
+high probability. Recall that ˆS is determined at the fusion center as the K
+indices with the largest number of votes. Further recall that pm
+i
+denotes the
+probability that machine m sends index i. Our strategy is to show that these
+probabilities are sufficiently large for support indices and sufficiently small for
+non-support indices. This shall allow us to apply Lemmas 2 and 3 to prove the
+required result. To derive bounds on pm
+i
+we employ Lemma 1.
+First, we prove that the condition of Lemma 2 holds, i.e. that
+pm
+i ≥ 8 ln d
+M
+for all i ∈ S,
+(30)
+where pm
+i
+= Pr(|ˆξm
+i | > τ), and ˆξm
+i
+=
+√nˆθm
+i
+σ(ˆΩm ˆΣm(ˆΩm)⊤)1/2
+ii
+is the standardized
+debiased lasso estimator, defined in Eq. (9). Without loss of generality, assume
+that θ∗
+i > 0. Otherwise, we could do the same calculations for −ˆξm
+i . Clearly,
+pm
+i = Pr(|ˆξm
+i | > τ) ≥ Pr(ˆξm
+i
+> τ) = Pr
+�
+√n
+ˆθm
+i − θ∗
+i
+σ(ˆΩm ˆΣm(ˆΩm)⊤)1/2
+ii
+> τ − ϑm
+i
+�
+,
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+27
+where ϑm
+i
+is defined in Eq. (16). Since condition C1 holds, applying Eq. (7) of
+Lemma 1 gives that
+pm
+i ≥ Φc(τ − ϑm
+i ) − ϵ(τ),
+(31)
+where ϵ(τ) is the error defined in Eq. (17).
+Next, by condition C2, with the definition of cΩ in Eq. (15) and Eq. (14)
+ϑm
+i =
+√nθ∗
+i
+�
+ˆΩm ˆΣm(ˆΩm)⊤
+�1/2
+ii
+≥
+√nθmin
+√cΩ
+=
+√
+2r ln d.
+At a threshold τ =
+√
+2 ln d we thus obtain
+pm
+i ≥ Φc �
+(1 − √r)
+√
+2 ln d
+�
+− ϵ(τ).
+By the Gaussian tail bound (23),
+Φc
+��
+2(1 − √r)2 ln d
+�
+≥ C(r, d)d−(1−√r)2,
+where
+C(r, d) =
+�
+2(1 − √r)2 ln(d)
+√
+2π{2(1 − √r)2 ln(d) + 1}.
+Therefore, for the condition (30) to hold, it suffices that
+M ≥
+8 ln d
+C(r, d)d−(1−√r)2 − ϵ(τ) =
+8 ln d
+C(r, d) − ϵ(τ)d(1−√r)2 d(1−√r)2,
+which is precisely condition (18) of the theorem. Notice that the requirement
+ϵ(τ) < 1/d guarantees that the denominator in the fraction above is positive.
+The lower bound on r in the theorem, guarantees that the range of possible
+values for M is non-empty.
+Next, we prove that the conditions of Lemma 3 hold. The condition M >
+2 ln d is satisfied given the requirement of Eq. (18). The next condition to verify
+is pm
+j ≤ 1/M for all j ̸∈ S. Since pm
+j = Pr(|ˆξm
+j | > τ) and ϑm
+j = 0 for j ̸∈ S, then
+pm
+j − 2Φc(τ) =
+�
+Pr(ˆξm
+j > τ) − Φc(τ)
+�
++
+�
+Pr(ˆξm
+j < −τ) − Φc(τ)
+�
+.
+According to Eq. (5) of Theorem 1, apart from a bias term, ξm
+j
+and −ξm
+j
+have
+the same distribution because ϑm
+j = 0. Hence, applying Eq. (7) in Lemma 1 to
+each of the above bracketed terms separately gives that
+pm
+j ≤ 2Φc(τ) + 2ϵ(τ),
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+28
+where ϵ(τ) was defined in Eq. (17). By Eq. (23), for τ =
+√
+2 ln d we have 2Φc(τ) ≤
+2
+√
+2π
+√
+2 ln d
+1
+d. Since ϵ(τ) < 1/d, we obtain that for all j ̸∈ S
+pm
+j ≤
+1
+√
+π ln d
+1
+d + 2
+d ≤ 3
+d ≤ 1
+M ,
+(32)
+where the last inequality follows from the assumption that M ≤ d/3 in Eq. (18).
+Therefore, all conditions of Lemma 3 are satisfied.
+Since Lemmas 2 and 3 hold, we apply their results with a union bound to
+derive a lower bound on the probability of exact support recovery, as follows:
+Pr( ˆS = S) ≥ Pr
+��
+max
+j̸∈S Vj ≤ 4 ln d
+�
+∩
+�
+min
+i∈S Vi ≥ 4 ln d
+��
+≥ 1 − Pr
+�
+max
+j̸∈S Vj ≥ 4 ln d
+�
+− Pr
+�
+min
+i∈S Vi ≤ 4 ln d
+�
+≥ 1 − (K + 1)
+d
+.
+We remark that Lemma 3 provides a bound on Pr
+�
+maxj /∈S Vj > 2 ln d
+�
+. Clearly,
+the probability for a higher threshold 4 ln d above is much smaller.
+Finally, we analyze the communication per machine. Let Bm denote the num-
+ber of bits sent by machine m. Note that Bm is a sum of Bernoulli random
+variables Bm
+k ∼ Ber(pm
+k ) times some factor ∝ ln d corresponding to the num-
+ber of bits necessary to represent indices in [d]. The random variable Bm
+k is an
+indicator whether machine m sends index k to the center. Then
+E(Bm) = O
+� d
+�
+k=1
+E(Bm
+k ) ln d
+�
+= O
+�
+�
+�
+�
+�
+�
+i∈S
+pm
+i +
+�
+j̸∈S
+pm
+j
+�
+�
+� ln d
+�
+� .
+(33)
+Since pm
+j ≤ 3/d for all j ̸∈ S, then �
+j̸∈S pm
+j ≤ 3. Additionally, �
+i∈S pm
+i ≤ K.
+Therefore, E(Bm) = O (K ln d).
+Proof of Theorem 3. The proof is similar to that of Theorem 2. We first show
+that the conditions of Lemmas 2 and 3 hold. Then, we derive a lower bound
+for the probability of Algorithm 2 achieving exact support recovery. Recall that
+here the threshold is τ =
+√
+2r ln d, where r is the SNR introduced in Eq. (14).
+Let us start by proving that the condition of Lemma 2 holds. We need to
+show that pm
+i ≥ 8 ln d
+M
+for all i ∈ S. As in Eq. (31) in the proof of Theorem 2,
+pm
+i ≥ Φc(τ − ϑm
+i ) − ϵ(τ)
+where ϑm
+i
+is given by Eq. (16).
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+29
+By condition C2, ϑm
+i ≥ √nθmin/√cΩ =
+√
+2r ln d, where θmin and cΩ are given
+by Eqs. (14) and (15), respectively. Since τ =
+√
+2r ln d, then τ − ϑm
+i ≤ 0 and
+pm
+i ≥ Φc(0) − ϵ(τ) ≥ 1
+2 − ϵ(τ).
+The inequality pm
+i
+≥ 8 ln d
+M
+follows directly from the requirement on M in Eq.
+(19), i.e., M ≥
+8 ln d
+1/2−ϵ(τ).
+Next, we verify the conditions of Lemma 3. Its first condition that M > 4 ln d
+holds given Eq. (19). For the second condition, we need to show that pm
+j ≤ 1/M
+for all j ̸∈ S. As in the proof of Theorem 2,
+pm
+j ≤ 2Φc(τ) + 2ϵ(τ).
+Plugging the value τ =
+√
+2r ln d into the Gaussian tail bound (23) gives
+pm
+j ≤
+1
+√
+πr ln d
+1
+dr + 2ϵ(τ)
+(i)
+≤ 1
+dr ,
+(34)
+where inequality (i) follows from the assumptions that ϵ(τ) < 1/(4dr) and
+r > ln(16 ln d)/(ln d). This latter assumption implies that
+1
+√
+πr ln(d) ≤
+1
+2 for
+sufficienly large d. Since we assume that M ≤ dr in Eq. (19), then pm
+j ≤ 1/dr ≤
+1/M for all j ̸∈ S. Hence, both conditions of Lemma 3 are satisfied.
+Applying a union bound and the result of Lemmas 2 and 3, it follows that
+Pr( ˆS = S) ≥ Pr
+��
+max
+j̸∈S Vj ≤ 4 ln d
+�
+∩
+�
+min
+i∈S Vi ≥ 4 ln d
+��
+≥ 1 − (K + 1)
+d
+.
+Finally, let us analyze the average communication per machine. Let B de-
+note the number of bits sent by a single machine. Following the same steps
+used to compute Eq. (33), the expectation of B may be bounded as E(B) ≤
+O
+��
+K + d−K
+dr
+�
+ln d
+�
+, where the factor 1/dr is due to Eq. (34). Hence, the ex-
+pected communication of a single machine is O
+�
+d1−r ln d
+�
+bits.
+A.3. Proof of Corollary 1
+Proof. We proceed similar to Battey et al. (2018) in the proof of their Corollary
+A.3. By the law of total probability, for any constant C′ > 0,
+Pr
+�
+∥ˆθ − ˆθ
+LS∥2 > C′
+√
+M max{K, ln N}
+N
+�
+≤ Pr
+��
+∥ˆθ − ˆθ
+LS∥2 > C′
+√
+M max{K, ln N}
+N
+�
+∩ { ˆS = S}
+�
++ Pr( ˆS ̸= S).
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+30
+By Theorem 2, Pr( ˆS ̸= S) ≤ (K + 1)/d. In addition, when ˆS = S, then ˆθj =
+ˆθLS
+j
+= 0 for all j ̸∈ S. Consequently, ∥ˆθ − ˆθLS∥2 = ∥ˆθS − ˆθLS
+S ∥2. Furthermore, the
+first term on the right hand side above may be bounded by
+Pr
+�
+∥ˆθS − ˆθ
+LS
+S ∥2 > C′
+√
+M max{K, ln N}
+N
+�
+.
+The next step is to apply a result from the proof of Theorem A.1 of Battey
+et al. (2018), which appeared in the last line of their proof. For clarity we state
+it here as a lemma.
+Lemma 4. Consider the linear model in dimension K,
+y = X⊤β∗ + σw,
+where w ∼ N(0, 1), X ∼ N(0, Σ), and Σ ∈ RK×K satisfies 0 < Cmin ≤
+σmin(Σ) ≤ σmax(Σ) ≤ Cmax < ∞. Suppose N i.i.d. samples from this model
+are uniformly distributed to M machines, with n > K. Denote by ˆβm the least
+squares solution at the m-th machine and ˆβLS the centralized least squares solu-
+tion. If the number of machines satisfies M = O
+�
+NK
+(max{K,ln N})2
+�
+, then
+Pr
+���� 1
+M
+�
+m
+ˆβm − ˆβ
+LS���
+2 > C′
+√
+M max{K,ln N}
+N
+�
+≤ cMe− max{K ln N} + Me−c N
+M ,
+where c, C′ > 0 are constants that do not depend on K or N.
+Applying this lemma to our case gives
+Pr
+�
+∥ˆθS − ˆθ
+LS
+S ∥2 > C′
+√
+M max{K, ln N}
+N
+�
+≤ cMe− max{K ln N} + Me−c N
+M .
+Since M = O
+�
+NK
+(max{K,ln N})2
+�
+, it follows that ∥ˆθ − ˆθLS∥2 = OP
+��
+K
+N
+�
+. As the
+oracle estimator has rate ∥ˆθLS − θ∗∥2 = OP
+��
+K
+N
+�
+, by the triangle inequality
+∥ˆθ − θ∗∥2 = OP
+��
+K
+N
+�
+as well.
+References
+Acharya, J., De Sa, C., Foster, D. J. and Sridharan, K. (2019).
+Distributed
+Learning
+with
+Sublinear
+Communication.
+arXiv
+preprint
+arXiv:1902.11259.
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+31
+Amiraz, C., Krauthgamer, R. and Nadler, B. (2022). Distributed sparse
+normal means estimation with sublinear communication. Information and In-
+ference: A Journal of the IMA iaab030.
+Barghi, H., Najafi, A. and Motahari, S. A. (2021). Distributed sparse
+feature selection in communication-restricted networks. arXiv preprint
+arXiv:2111.02802.
+Battey, H., Fan, J., Liu, H., Lu, J. and Zhu, Z. (2018). Distributed testing
+and estimation under sparse high dimensional models. Annals of Statistics 46
+1352–1382.
+Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequal-
+ities: A Nonasymptotic Theory of Independence. Oxford University Press, Ox-
+ford.
+Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J. (2011).
+Distributed optimization and statistical learning via the alternating direction
+method of multipliers. Foundations and Trends in Machine learning 3 1–122.
+Braverman, M., Garg, A., Ma, T., Nguyen, H. L. and Woodruff, D. P.
+(2016). Communication lower bounds for statistical estimation problems via
+a distributed data processing inequality. In Proceedings of the forty-eighth
+annual ACM symposium on Theory of Computing 1011–1020.
+Bunea, F., Tsybakov, A. and Wegkamp, M. (2007). Sparsity oracle inequal-
+ities for the Lasso. Electronic Journal of Statistics 1 169–194.
+Candes, E. J. and Tao, T. (2005). Decoding by linear programming. IEEE
+Transactions on Information Theory 51 4203–4215.
+Chen, X. and Xie, M. (2014). A split-and-conquer approach for analysis of
+extraordinarily large data. Statistica Sinica 24 1655–1684.
+Chen, X., Liu, W., Mao, X. and Yang, Z. (2020). Distributed high-
+dimensional regression under a quantile loss function. Journal of Machine
+Learning Research 21 1–43.
+Dobriban, E. and Sheng, Y. (2020). WONDER: Weighted one-shot dis-
+tributed ridge regression in high dimensions. Journal of Machine Learning
+Research 21 1–52.
+Dobriban, E. and Sheng, Y. (2021). Distributed linear regression by averag-
+ing. Annals of Statistics 49 918–943.
+Fan, J., Li, R., Zhang, C.-H. and Zou, H. (2020). Statistical Foundations of
+Data Science. CRC Press, Boca Raton.
+Gao, Y., Liu, W., Wang, H., Wang, X., Yan, Y. and Zhang, R. (2022).
+A review of distributed statistical inference. Statistical Theory and Related
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+32
+Fields 6 89–99.
+Guestrin, C., Bodik, P., Thibaux, R., Paskin, M. and Madden, S. (2004).
+Distributed regression: an efficient framework for modeling sensor network
+data. In Third International Symposium on Information Processing in Sensor
+Networks 1–10.
+Hastie, T., Tibshirani, R. and Wainwright, M. (2015). Statistical Learning
+with Sparsity: The Lasso and Generalizations. CRC Press, Boca Raton.
+Heinze, C., McWilliams, B., Meinshausen, N. and Krummenacher, G.
+(2014). LOCO: Distributing ridge regression with random projections. arXiv
+preprint arXiv:1406.3469.
+Huo, X. and Cao, S. (2019). Aggregated inference. Wiley Interdisciplinary
+Reviews: Computational Statistics 11 e1451.
+Javanmard, A. and Montanari, A. (2014a). Hypothesis testing in high-
+dimensional regression under the Gaussian random design model: Asymptotic
+theory. IEEE Transactions on Information Theory 60 6522–6554.
+Javanmard, A. and Montanari, A. (2014b). Confidence intervals and hy-
+pothesis testing for high-dimensional regression. The Journal of Machine
+Learning Research 15 2869–2909.
+Javanmard, A. and Montanari, A. (2018). Debiasing the lasso: Optimal
+sample size for Gaussian designs. The Annals of Statistics 46 2593–2622.
+Jordan, M. I., Lee, J. D. and Yang, Y. (2019). Communication-efficient dis-
+tributed statistical inference. Journal of the American Statistical Association
+114 668–681.
+Lee, J. D., Liu, Q., Sun, Y. and Taylor, J. E. (2017). Communication-
+efficient sparse regression. The Journal of Machine Learning Research 18 1–
+30.
+Liu, M., Xia, Y., Cho, K. and Cai, T. (2021). Integrative high dimensional
+multiple testing with heterogeneity under data sharing constraints. Journal
+of Machine Learning Research 22 1–26.
+Lopes, M. E. and Yao, J. (2022). A sharp lower-tail bound for Gaussian
+maxima with application to bootstrap methods in high dimensions. Electronic
+Journal of Statistics 16 58–83.
+Lv, S. and Lian, H. (2022). Debiased distributed learning for sparse partial
+linear models in high dimensions. Journal of Machine Learning Research 23
+1–32.
+Mateos, G., Bazerque, J. A. and Giannakis, G. B. (2010). Distributed
+sparse linear regression. IEEE Transactions on Signal Processing 58 5262–
+
+R. Fonseca and B. Nadler/Distributed Sparse Linear Regression
+33
+5276.
+Predd, J. B., Kulkarni, S. B. and Poor, H. V. (2006). Distributed learning
+in wireless sensor networks. IEEE Signal Processing Magazine 23 56–69.
+Rosenblatt, J. D. and Nadler, B. (2016). On the optimality of averaging in
+distributed statistical learning. Information and Inference: A Journal of the
+IMA 5 379–404.
+Sun, T. and Zhang, C.-H. (2012). Scaled sparse linear regression. Biometrika
+99 879–898.
+Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso. Jour-
+nal of the Royal Statistical Society: Series B 58 267–288.
+van de Geer, S. A. and B¨uhlmann, P. (2009). On the conditions used to
+prove oracle results for the Lasso. Electronic Journal of Statistics 3 1360–
+1392.
+van de Geer, S., B¨uhlmann, P., Ritov, Y. and Dezeure, R. (2014).
+On asymptotically optimal confidence regions and tests for high-dimensional
+models. The Annals of Statistics 42 1166–1202.
+Wainwright, M. J. (2009). Sharp thresholds for high-dimensional and noisy
+sparsity recovery using ℓ1 - constrained quadratic programming (Lasso). IEEE
+Transactions on Information Theory 55 2183–2202.
+Zhang, Y., Duchi, J. C. and Wainwright, M. J. (2013). Communication-
+efficient algorithms for statistical optimization. Journal of Machine Learning
+Research 14 3321–3363.
+Zhang, C. H. and Zhang, S. S. (2014). Confidence intervals for low dimen-
+sional parameters in high dimensional linear models. Journal of the Royal
+Statistical Society: Series B 76 217–242.
+Zhu, X., Li, F. and Wang, H. (2021). Least-square approximation for a dis-
+tributed system. Journal of Computational and Graphical Statistics 30 1004–
+1018.
+

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@@ -0,0 +1,1950 @@
+arXiv:2301.02216v1  [gr-qc]  5 Jan 2023
+Logarithm Corrections and Thermodynamics for Horndeski gravity like Black Holes
+Riasat Ali,1, ∗ Zunaira Akhtar,2, † Rimsha Babar,3, ‡ G. Mustafa,4, § and Xia Tiecheng1, ¶
+1Department of Mathematics, Shanghai University,
+Shanghai-200444, Shanghai, People’s Republic of China
+2Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan
+3Division of Science and Technology, University of Education, Township, Lahore-54590, Pakistan
+4Department of Physics Zhejiang Normal University Jinhua 321004, People’s Republic of China
+In this paper, we compute the Hawking temperature by applying quantum tunneling approach for
+the Horndeski like black holes. We utilize the semi-classical phenomenon and WKB approximation
+to the Lagrangian field equation involving generalized uncertainty principle (GUP) and compute
+the tunneling rate as well as Hawking temperature.
+For the zero gravity parameter, we obtain
+results consistent without correction parameter or original tunneling.
+Moreover, we study the
+thermal fluctuations of the considered geometry and examine the stable state of the system by
+heat capacity technique.
+We also investigate the behaviour of thermodynamic quantities under
+the influence of thermal fluctuations. We observe from the graphical analysis, the corresponding
+system is thermodynamically stable with these correction terms.
+keywords:
+Horndeski like black holes;
+Quantum gravity;
+Tunneling radiation,
+Thermal
+fluctuations, Corrected entropy, Phase transition.
+I.
+INTRODUCTION
+Tunneling is the semi-classical mechanism in which particles have radiated from black hole (BH) outer horizon.
+Some analysis shows the keen interest in the Hawking temperature (TH) via tunneling method from different BHs.
+The main aspect to examine the TH is the imaginary part of classical action which leads to the tunneling radiation
+of boson particles appearing from the Horndeski like BHs.
+The quantum tunneling and TH of charged fermions in BHs has been observed [1]. In this paper, they examined that
+the tunneling and TH depend on charges of electric and magnetic, acceleration, rotation, mass and NUT parameter of
+the charged pair BHs. The tunneling strategy from Reissner Nordstom-de Sitter BH like solution in global monopole
+has been analyzed [2].
+In this article, the authors observed that the modified TH depends on the parameter of
+global monopole. The BH thermodynamics have been examined [3] with some parameters like acceleration, NUT
+and rotation. The researchers studied thermodynamical quantities like the area, entropy, surface gravity and TH.
+The tunneling spectrum of bosonic particles has been computed from the modified BHs horizon by utilizing the
+Proca field equation. Hawking evaluated tunneling probability from BH [4] by utilizing theoretical technique and
+later, it has been explained by Parikh and Wilczek [5, 6]. The important of this radiation represents that vacuum
+thermal fluctuation produce pairs of particle (particle and anti-particle) from the horizon. Hawking considered that
+the particle’s have ability to emit from the BH and the anti-particles have no ability to radiate from the horizon.
+Parikh and Wilczek explained a mathematical approach by utilizing WKB approximation. This phenomenon use
+geometrical optic approximation which is another view of eikonal approximation in wave clarification [7]. The set of
+all particles remain at the front boundary and with the emission of these particles, the BH mass reduces in the form
+of particles energy.
+In the Parikh-Wilczek method, a precisely tunneling was established and there were as still unanswered problems
+like information release, temperature unitary and divergence.
+Many authors have made efforts on the tunneling
+strategy and semi-classical phenomenon from the different BHs horizon; one of the important explanations can be
+checked in [8]-[31]. The radiate particles for many BHs have been analyzed and also computed the radiate particles
+with the influences of the geometry of BH with different parameters. It is possible to study modified thermodynamic
+properties of BH by considering generalized uncertainty principle (GUP) influences [32]. The GUP implies high energy
+∗Electronic address: [email protected]
+†Electronic address: [email protected]
+‡Electronic address: [email protected]
+§Electronic address: [email protected]
+¶Electronic address: [email protected]
+
+2
+result to thermodynamic of BH, by considering the quantum gravity theory with a minimal length. By considering
+the GUP influences, it is viable to examine the modified thermodynamic of BHs.
+It is a well known fact that thermal fluctuations are a result of statistical perturbations in the dense matter. With
+the emission of Hawking radiations from the BH, the size of BH reduces and consequently its temperature increases.
+Faizal and his colleague [33] have studied the thermodynamics and thermal fluctuations of generalized Schwarschild
+BH, (i.e., Reissner-Nordstrom, Kerr and charged AdS BHs) with the help first-order corrections and discussed the
+stability of BHs.
+The thermodynamics of rotating Kerr-AdS BH and its phase transition have been studied by
+Pourhassan and Faizal [34]. They concluded that the entropy corrections are very helpful to examine the geometry
+of small BHs. By applying the stability test of heat capacity and Hessian matrix, the phase transition as well as
+thermodynamics of non-minimal regular BHs in the presence of cosmological constant has been investigated [35] and
+the authors concluded that the local and global stability of the corresponding BHs increases for higher values of
+correction parameters. Zhang and Pradhan [36, 37] have investigated the corrected entropy and second order phase
+transition via thermal fluctuations on the charged accelerating BHs.
+Moreover, the thermodynamics and geometrical analysis of new Schwarzschild BHs have been studied [38, 39]. By
+using the tunneling approach under the influence of quantum gravity the Hawking temperture for different types
+of BH have been discussed [40]-[44]. Sharif and Zunaira [45, 46] have computed the thermodynamics, quasi-normal
+modeand thermal fluctuations of charged BHs with the help of Weyl corrections. The authors found that the system
+is unstable for the small radii of BHs under the influence of first order corrections and by using the heat capacity and
+Hessian matrix technique, they have also studied the stable conditions of the system. The authors in [47, 48] have
+investigated the thermodynamics, phase transition and local/global stability of NUT BH via charged, accelerating as
+well as rotating pairs. Ilyas et al.[49–52] discussed the energy conditions and calculated the new solutions for stellar
+structures by taking black hole geometry as exterior spacetime in the background of different modified theories of
+gravity. Recently, Ditta et al. [53] discussed the thermal stability and Joule–Thomson expansion of regular BTZ-like
+black hole.
+The main intention of this paper is to investigate the tunneling radiation without self-gravity and back-reaction and
+also explain the modified tunneling rate. The tunneling radiation is evaluated under the conditions of charge-energy
+conservation, Horndeski parameter and GUP parameter influences.
+The modified TH depends on the Horndeski
+parameter as well as GUP parameter and also investigated the behaviour of thermodynamic quantities via thermal
+fluctuations.
+This paper is based on the analysis of quantum tunneling, TH, stability and instability conditions for the Horndeski
+like BH. The paper is outlined as follows: in Sec. II, we study the tunneling radiation of bosonic particles for 4D
+Horndeski like BH and also calculate the effects of GUP parameters on tunneling and TH. In section III, we study
+the graphical presentation of tunneling radiation for this type of BH and analyze the stable and unstable conditions
+for Horndeski like BH. In section IV, we investigate the behaviour of thermodynamic quantities under the effects of
+thermal fluctuations. In section V, we express the discussion and conclusion of the whole analysis.
+II.
+HORNDESKI LIKE BLACK HOLES
+Hui and his coauthor Nicolis [58] argued that the no-hair theorems cannot be applied on a Galileon field, as it is
+coupled to gravity under the effect of peculiar derivative interactions. Further, they demonstrated that static and
+spherically symmetric spacetime defining the geometry of the black hole could not sustain nontrivial Galileon profiles.
+Babichev and Charmousis [59] examined the no-hair theorem in Ref. [58] by considering Horndeski theories and
+beyond. Furthermore, they provided the Lagrangian of Horndeski theory which can be expressed as a generalized
+Galileon Lagrangian, which is defined as
+S =
+� √−g
+�
+Q2(χ) + Q3(χ)□φ + Q4(χ)R + Q4,χ
+�
+(□φ)2��
+.
+− (∇ǫ∇εφ) (∇ǫ∇εφ)] + Q5(χ)Gǫε∇ǫ∇εφ
+− 1
+6Q5,χ
+�
+(□φ)3 − 3(□φ) (∇ǫ∇vφ) (∇ǫ∇εφ)
++2 (∇ǫ∇εφ) (∇v∇γφ) (∇γ∇ǫφ)]} d4x.
+(1)
+where Q2, Q3, Q4, and Q5 are the arbitrary functions of the scalar field φ and χ = −∂ǫφ∂ǫφ/2 represents the canonical
+kinetic term. Additionally, in the current analysis, fχ stands for ∂f(χ)/∂χ, Gǫε is the Einstein tensorR is the Ricci
+scalar, and other relations are defined as:
+(∇ǫ∇εφ)2 ≡ ∇ǫ∇εφ∇ε∇ǫφ
+(∇ǫ∇εφ)3 ≡ ∇ǫ∇εφ∇ε∇ρφ∇ρ∇ǫφ
+(2)
+
+3
+The scalar field admits the Galilean shift symmetry ∂ǫφ → ∂ǫφ + bǫ in flat spacetime for Q2 ∼ Q3 ∼ χ and Q4 ∼
+Q5 ∼ χ2, which resembles the Galilean symmetry [60]. In the current study, we investigate the tunneling radiation of
+spin-1 massive boson particles from Horndeski-like BH. For this purpose, we adopted the procedure, which is already
+reported in [57] for Horndeski spacetime. Finally, we have the following spacetime:
+ds2 = −
+�
+1 − 2rM(r)
+Σ
+�
+dt2 +
+1
+∇(r)Σdr2 + Σ2dθ2 − A
+Σ sin2 dφ2 − 4ar
+Σ M(r) sin2 θdtdφ,
+(3)
+with Σ2 = a2 cos2 θ + r2 and ∇(r) = a2 − 2rM(r)+ r2, M(r) = M − 1
+2QIn r
+r0 and A = (a2 + r2)2 − ∇a2 sin2 θ, while a,
+Q and M represent the rotation parameter, Horndeski parameter and mass of BH, respectively. If Q → 0, the metric
+(3) goes over to the Kerr BH [61] and if Q = a = 0 the metric (3) also goes over to the Schwarzschild metric. The
+line-element (3) can be re-written as
+ds2 = −f(r)dt2 + g−1(r)dr2 + I(r)dφ2 + h(r)dθ2 + 2R(r)dtdφ
+(4)
+where
+f(r) =
+�
+1 − 2rM(r)
+Σ
+�
+,
+g−1(r) = 1
+∇Σ,
+h(r) = Σ2,
+I(r) = −A
+Σ sin2,
+R(r) = −2ar
+Σ M(r) sin2 θ.
+We study the tunneling radiation of spin-1 particles from four-dimensional Horndeski like BHs.
+By utilizing the
+Hamilton-Jacobi ansatz and the WKB approximation to the modified field equation for the Horndeski space-time, the
+tunneling phenomenon is successfully applied. We study the modified filed equation on a four dimensional space-time
+with the background of rotation parameter, Horndeski parameter and evaluated for the radial function. As a result,
+we get the tunneling probability of the radiated particles and derive the modified TH of Horndeski like BHs. The
+modified filed equation is expressed by [27, 30]
+∂µ
+�√−gΨνµ�
++ √−g m2
+ℏ2 Ψν + √−g i
+ℏAµΨνµ + √−g i
+ℏeF νµΨµ + ℏ2β∂0∂0∂0
+�√−gg00Ψ0ν�
+−ℏ2β∂i∂i∂i
+�√−ggiiΨiν�
+= 0,
+(5)
+here Ψνµ, m and g present the anti-symmetric tensor, bosonic particle mass and determinant of coefficient matrix, so
+Ψνµ =
+�
+1 − ℏ2β∂2
+ν
+�
+∂νΨµ −
+�
+1 − ℏ2β∂2
+µ
+�
+∂µΨν +
+�
+1 − ℏ2β∂2
+ν
+� i
+ℏeAνΨµ
+−
+�
+1 − ℏ2β∂2
+ν
+� i
+ℏeAµΨν,
+and Fνµ = ∇νAµ − ∇µAν,
+with β, e , ∇µ and Aµ are the GUP parameter(quantum gravity), bosonic particle charge, covariant derivative and
+BH potential, respectively. The Ψνµ can be computed as
+Ψ0 = −IΨ0 + RΨ3
+fI + R2
+,
+Ψ1 =
+1
+g−1 Ψ1,
+Ψ2 = 1
+hΨ2,
+Ψ3 = RΨ0 + fΨ3
+fI + R2
+,
+Ψ01 =
+˜
+−DΨ01 + RΨ13
+(R2 + fI)g−1 ,
+Ψ02 =
+˜
+−DΨ02
+(R2 + fI)h,
+Ψ03 = (f 2 − fI)Ψ03
+(fI + R2)2 ,
+Ψ12 =
+1
+g−1hΨ12,
+Ψ13 =
+1
+g−1(fI + R2)Ψ13,
+Ψ23 = fΨ23 + RΨ02
+(fI + R2)h .
+In order to observe the bosonic tunneling, we have assumed Lagrangian gravity equation. Further, we utilized the
+WKB approximation to the Lagrangian gravity equation and computed set of equations.
+Furthermore, we have
+utilized the variable separation action to get required solutions. The approximation of WKB is defined [? ] as
+Ψν = ην exp
+� i
+ℏK0(t, r, φ, θ) + ΣℏnKn(t, r, φ, θ)
+�
+.
+(6)
+we get set of equations in Appendix A. Utilizing variable separation technique, we can take
+K0 = −(E − Lω)t + W(r) + Lφ + ν(θ),
+(7)
+
+4
+where E and L present the particle energy and particle angular, respectively, corresponding to angle φ.
+After considering Eq. (7) into Eqs. (22)-(25), we reach a matrix in the form
+U(η0, η1, η2, η3)T = 0,
+which express a 4 × 4 matrix presented as ”U”, whose elements are given as follows:
+U00 =
+−I
+g−1(fI + R2)
+�
+W 2
+1 + βW 4
+1
+�
+−
+I
+(fI + R2)h
+�
+L2 + βL4�
+−
+fI
+(fI + R2)2
+�
+ν2
+1 + βν4
+1
+�
+−
+m2I
+(fI + R2),
+U01 =
+−I
+g−1(fI + R2)
+�
+((E − Lω) + (E − Lω)3β + A0e + (E − Lω)2βeA0
+�
+W1 +
+R
+g−1(fI + R2) +
+�
+ν1 + βν3
+1
+�
+,
+U02 =
+−I
+h(fI + R2)
+�
+(E − Lω) + (E − Lω)3β − A0e − (E − Lω)2βeA0
+�
+L,
+U03 =
+−R
+g−1(fI + R2)
+�
+W 2
+1 + βW 4
+1
+�
+−
+fI
+h(fI + R2)2
+�
+(E − Lω)3β − (E − Lω)2βeA0 + (E − Lω) − eA0
+�
+ν1
++
+m2R
+(fI + R2)2 ,
+U12 =
+1
+g−1h
+�
+W1 + βW 3
+1
+�
+L,
+U11 =
+−I
+g−1(fI + R2)
+�
+β(E − Lω)4 − βeA0EW 2
+1 + (E − Lω)2 − eA0(E − Lω)
+�
++
+R
+(fI + R2)g−1
++
+�
+ν1 + βν3
+1
+�
+(E − Lω) −
+1
+g−1h
+�
+L2 + βL4�
+−
+1
+(fI + R2)g−1
+�
+ν1 + βν3
+1
+�
+− m2
+g−1 −
+eA0I
+(fI + R2)g−1
+×
+�
+(E − Lω) + (E − Lω)3β − A0e − (E − Lω)2βeA0
+�
++
+eA0R
+g−1(fI + R2)
+�
+ν1 + βν3
+1
+�
+,
+U13 =
+−R
+g−1(fI + R2)
+�
+W1 + βW 3
+1
+�
+(E − Lω) +
+1
+g−1(fI + R2)2
+�
+W1 + βW 3
+1
+�
+ν1 +
+ReA0
+g−1(fI + R2)
+�
+W1 + βW 3
+1
+�
+,
+U20 =
+I
+h(fI + R2)
+�
+(E − Lω)L + β(E − Lω)L3�
++
+R
+h(fI + R2)
+�
+(E − Lω) + β(E − Lω)3ν2
+1
+�
+−
+IeA0
+h(fI + R2)
+�
+L + βL3�
+,
+U22 =
+I
+h(R2 + fI)
+�
+βE4 − βeA0E + E2 − eA0(E − Lω)
+�
+−
+1
+g−1h +
+R
+h(R2 + fI)
+�
+(E − Lω)3β
++ −(E − Lω)2βeA0 − A0e + (E − Lω)
+�
+ν1 −
+f
+h(R2 + fI)
+�
+ν2
+1 + βν4
+1
+�
+− m2
+h −
+eA0I
+h(fI + R2)
+�
+(E − Lω) + (E − Lω)3β − A0e − (E − Lω)2βeA0
+�
+,
+U23 =
+f
+h(fI + R2)
+�
+L + βL3�
+ν1,
+U30 =
+(fI − f 2)
+(fI + R2)2
+�
+ν1 + βν3
+1
+�
+E +
+R
+h(fI + R2)
+�
+L2 + βL4�
+−
+m2R
+(fI + R2) − eA0(fI − f 2)
+(fI + R2)2
+�
+ν1 + βν3
+1
+�
+,
+U31 =
+1
+g−1(fI + R2)
+�
+ν1 + βν3
+1
+�
+W1,
+U32 =
+R
+h(R2 + fI)
+�
+L + βL3�
+E +
+f
+h(R2 + fI)
+�
+ν1 + βν3
+1
+�
+L,
+U33 =
+(fI − f 2)
+(fI + R2)
+�
+(E − Lω)2 − eA0(E − Lω) + β(E − Lω)4 − βeA0(E − Lω)3�
+−
+1
+g−1(R2 + fI)
+�
+W 2
+1 + βW 4
+1
+�
+−
+f
+(R2 + fI)h
+�
+L2 + βL4�
+−
+m2f
+(fI + R2) − eA0(fI − f 2)
+(fI + R2)
+×
+�
+(E − Lω) + β(E − Lω)3 − eA0(E − Lω)2�
+,
+
+5
+with ∂tK0 = (E − Lω),
+∂φK0 = L, W1 = ∂rK0 and ν1 = ∂θK0. For non-trivial solution, we get
+ImW ± = ±
+�
+�
+�
+�
+�
+�
+E − LΩ − eA0
+�2
++ Z1
+�
+1 + β Z2
+Z1
+�
+(fI + R2)gI−1
+dr,
+= ±iπ
+�
+R − LΩ − A0e
+�
++
+�
+1 + βA
+�
+2k(r+)
+,
+(8)
+where
+Z1 = (E − Lω)ν1
+g−1R
+fI + R2 +
+fg−1
+fI + R2 ν2
+1 − g−1m2,
+Z2 =
+g−1I
+fI + R2
+�
+(R − LΩ)4 + (eA0)2(R − LΩ)2 − 2A0e(R − LΩ)3�
++
+g−1R
+h(fI + R2)
+�
+(R − LΩ)3 − eA0(R − LΩ)2�
+ν1 −
+fg−1
+fI + R2 ν4
+1 − W 4
+1 .
+and A is a arbitrary parameter. In particular case, we take the radial component of the action of particle, for this aim
+we choose a components of matrix equals to zero. Since, we have found the tunneling radiation (related to Horndeski
+gravity and quantum gravity) for BH. This tunneling and TH quantities relate on the Horndeski gravity and quantum
+gravity of this particular physical object. Thus, we have found the corresponding TH which important as a component
+of leading metric with Horndeski gravity and quantum gravity. Such that, in this method we are not concerned in
+order of higher for Planck’s constant only obtained appropriate result. The generalized tunneling depends on the BHs
+metric and Horndeski gravity and GUP parameter. The generalized tunneling for Horndeski like BH can be written
+as
+T = Temission
+Tabsorption
+= exp
+�
+−2π(R − LΩ − A0e)
+k(r+)
+�
+[1 + βA] ,
+(9)
+with
+k(r+) = 4πr+
+�
+a2 + r2
++
+�
+Qr+ + −a2 + r2
++
+(10)
+In the presence of GUP terms, we calculate the TH of the Horndeski gravity BHs. by taking the Boltzmann factor
+TB = exp [(E − Lω − eA0)/TH] as
+TH = −a2 + Qr+ + r2
++
+4πr+
+�
+a2 + r2
++
+� [1 − βA] .
+(11)
+The above result shows that the TH depends on the Horndeski gravity, GUP parameter, rotation parameter, Horndeski
+gravity, arbitrary parameter A and radius (r+) of BH. When β = 0, we obtain the general TH in [57]. In the absence
+of charge i.e., Q = 0, the above temperature reduces into Kerr BH temperature [58, 59]. For β = 0 and a = 0,
+the temperature reduces into Reissner Nordstr¨om BH. Moreover, when Q = 0 = a, we recover the temperature of
+Schwarzschild BH [60]. The quantum corrections slow down the increase in TH throughout the radiation phenomenon.
+A.
+TH versus r+
+We observe the geometrical presentation of TH w.r.t r+ for the 4D Horndeski like metric. Moreover, we observe
+the physical significance of these graphs under Horndeski gravity and GUP parameter and study the stability and
+instability analysis of corresponding TH. For β equals to zero, the tunneling radiation will be independent of GUP
+parameter. In the left plot of Fig. 1, the TH increases with increasing β in small region of horizon 0 ≤ r+ ≤ 5, that
+indicates the stable state of BH till r+ → ∞. In the right plot of Fig. 1, the rotating parameter and β are fixed,
+then we take changing values of hairy parameter of Horndeski gravity and get the completely unstable form of BH
+with negative temperature.
+
+6
+β=10
+β=20
+β=30
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+6
+r+
+TH
+Q=0.5
+Q=1
+Q=1.5
+0
+1
+2
+3
+4
+5
+- 6
+- 4
+- 2
+0
+2
+r+
+TH
+Figure 1: TH w.r.t horizon r+ for a = 5, Q = 0.5 and Ξ = 1 and left β = 10 (black), β = 20 (blue), β = 30 (red).
+Right a = 0.5, Ξ = 1, β = 5, Q = 0.5 (black), a = 0.5, Ξ = 1, β = 5, Q = 1 (blue), a = 0.5, Ξ = 1, β = 5, Q = 1.5
+(red).
+III.
+THERMODYNAMICS AND EFFECTS OF FIRST ORDER CORRECTIONS
+Thermal fluctuations plays important role on the study of BH thermodynamics. With the concept of Euclidean
+quantum gravity, the temporal coordinates shifts towards complex plan. To check the effects of these correction in
+entropy, we find Hawking temperature and usual entropy of the given system with the help of first law of thermody-
+namics
+S = π
+�
+a2 + r2
++
+�
+,
+T = −a2 + Qr+ + r2
++
+4πr+
+�
+a2 + r2
++
+�
+(12)
+To check the corrected entropy along these thermal fluctuations, the partition function is Z(µ) in terms of density of
+states η(E) is given as [37]
+Z(µ) =
+� ∞
+0
+exp(−µE)η(E)dE,
+(13)
+where T+ = 1
+µ and E is the mean energy of thermal radiations. By using the Laplace inverse transform, the expression
+of density takes the form
+ρ(E) =
+1
+2πi
+� µ0+i∞
+µ0−i∞
+Z(µ) exp(µE)dµ =
+1
+2πi
+� µ0+i∞
+µ0−i∞
+exp( ˜S(µ))dµ,
+(14)
+where ˜S(µ) = µE + ln Z(µ) represents the modified entropy of the considered system that is dependent on Hawking
+temperature. Moreover, the expression of entropy gets modified with the help of steepest decent method,
+˜S(µ) = S + 1
+2(µ − µ0)2 ∂2 ˜S(µ)
+∂µ2
+���
+µ=µ0 + higher-order terms.
+(15)
+Using the conditions ∂ ˜S
+∂µ = 0 and ∂2 ˜S
+∂µ2 > 0, the corrected entropy relation under the first-order corrections modified.
+By neglecting higher order terms, the exact expression of entropy is expressed as
+˜S = S − δ ln(ST 2),
+(16)
+where δ is called correction parameter, the usual entropy of considered system is attained by fixing δ = 0 that is
+without influence of these corrections. Furthermore, inserting the Eq. (12) into (16), we have
+˜S = (a2 + r2
++) − δ log
+��
+a2 − r+ (Q + r+)
+�2
+16πr2
++
+�
+a2 + r2
++
+�
+�
+.
+(17)
+
+7
+δ=0
+δ=0.4
+δ=0.6
+δ=0.6
+0.0
+0.1
+0.2
+0.3
+0.4
+10
+20
+30
+40
+r+
+S
+˜
+Figure 2: Corrected entropy versus r+ for a=0.2, Q=0.4. .
+In the Fig. 2, the graph of corrected entropy is monotonically increasing throughout the considered domain. It is
+noted the graph (black) of usual entropy is increasing just for small value of horizon radius but corrected expression
+of energy is increasing smoothly. Thus, these corrections terms are more effective for small BHs. Now, using the
+expression of corrected entropy and check the other other thermodynamic quantities via thermal fluctuations. In this
+way, the the Helmholtz energy (F = − � ˜SdT ) leads to the form
+F =
+�
+a4 − r2
++
+�
+−4a2 + 2Qr+ + r2
++
+�� �
+δ log
+�
+(a2−r+(Q+r+))
+2
+{
+r2
++
+�
+a2 + r2
++
+�
+}
+�
+− a2 − δ log(16π) − r2
++
+�
+4πr2
++
+�
+a2 + r2
++
+�2
+.
+(18)
+δ=0
+δ=0.4
+δ=0.6
+δ=0.6
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+35
+40
+45
+50
+55
+60
+65
+r+
+F
+Figure 3: Helmholtz free energy versus r+ for a=0.2, Q=0.4..
+The Fig. 3 shows the graph of Helmholtz free energy versus horizon radius. It is observed that the behaviour of
+energy is gradually decreases for the different correction parameter δ values. While the graph of usual entropy shows
+opposite behaviour as the graph is increasing. This behaviour means, the considered system shifts its state towards
+equilibrium, thus, no more work can be extract from it. The expression of internal energy (E = F + T ˜S) for the
+corresponding geometry is given by [37]
+E =
+�
+r+
+�
+a2 + r2
++
+� �
+r+ (Q + r+) − a2�
+�
+δ
+�
+log(16π) − log
+��
+a2 − r+ (Q + r+)
+� 2
+r2
++
+�
+a2 + r2
++
+�
+��
++ π
+�
+a2 + r2
++
+�
+�
+−
+�
+a4 − r2
++
+�
+−4a2 + 2Qr+ + r2
++
+��
+�
+−δ log
+��
+a2 − r+ (Q + r+)
+�2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2 + δ log(16π) + r2
++
+� �
+�
+4πr2
++
+�
+a2 + r2
++
+�2 �−1
+.
+(19)
+
+8
+δ=0.2
+δ=0.4
+δ=0.6
+δ=0.8
+0.0
+0.1
+0.2
+0.3
+0.4
+-2
+0
+2
+4
+6
+r+
+E
+Figure 4: Internal Energy w.r.t r+ for a=0.2, Q=0.4.
+The graphical behaviour of internal energy for the different choices of horizon radius is shown in Fig.
+4.
+It is
+observable that for the small values of radii, the graph is gradually decreases even shifts towards negative side, While
+the corrected internal energy depicts positive behaviour. This mean that the considered BH absorbing more and
+more heat from the surrounding to maintain its state. Since, BHs considered as a thermodynamic system, so there
+is another important thermodynamic quantity that is pressure. In this regard, there is deep connection between
+voulme (V =
+2π(r2
+++a2)(2r2
+++a2)
+3r+
+) and pressure. The Expression of BH pressure (P = − dF
+dV ) under the effect of thermal
+fluctuations takes the form
+P =
+�
+2r2
++
+�
+a2 + r2
++
+��
+− 4a2 + 3Qr+ + 2r2
++
+��
+a2 − r+
+�
+Q + r+
+��2�
+− δ log
+���
+a2 − r+(Q + r+)
+�2�
+(r2
++(a2 + r2
++))−1�
++ a2 + δ log(16π) + r2
++
+�
+− 2
+�
+r+
+�
+Q + r+
+�
+− a2��
+a4 − r2
++
+�
+− 4a2 + 2Qr+ + r2
++
+���
+− a4δ + Qr3
++
+�
+a2 + δ
+�
+− r2
++
+�
+a4 + 3a2δ
+�
++ Qr5
++ + r6
++
+�
++ 4r2
++
+�
+a4 − r2
++
+�
+− 4a2 + 2Qr+ + r2
++
+���
+a2 − r+
+�
+Q + r+
+��2�
+− δ log
+×
+���
+a2 − r+
+�
+Q + r+
+��2��
+r2
++
+�
+a2 + r2
++
+��−1�
++ a2 + δ log(16π) + r2
++
+�
++ 2
+�
+a2 + r2
++
+��
+a4 − r2
++
+�
+− 4a2
++ 2Qr+ + r2
++
+���
+a2 − r+
+�
+Q + r+
+��2�
+− δ log
+���
+a2 − r+
+�
+Q + r+
+��2�
+(r2
++
+�
+a2 + r2
++
+��−1�
++ a2 + δ log(16π)
++ r2
++
+���
+4πr3
++
+�
+a2 + r2
++
+�3�
+a2 − r+
+�
+Q + r+
+��2�−1
+.
+(20)
+δ=0
+δ=0.2
+δ=0.4
+δ=0.6
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+1
+×108
+2
+�108
+3
+�108
+4
+�108
+5
+�108
+r+
+P
+Figure 5: Pressure versus r+ for a=0.2, Q=0.4.
+In the Fig. 5, the graph of pressure is just coincides the state of equilibrium. For the different values of correction
+parameter, the pressure is significantly increases for the considered system.
+Further, there is another important
+thermodynamic quantity enthalpy (H = E + PV ) is given in Appendix B.
+
+9
+δ=0
+δ=0.4
+δ=0.6
+δ=0.8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-10
+-5
+0
+5
+r+
+H
+Figure 6: Enthalpy versus r+ for a=0.2, Q=0.4.
+From Fig. 6, it can observed that the graph of usual enthalpy is coincide with the plots of corrected one and abruptly
+decreases even shifts towards negative side. This means that there exists a exothermic reactions means there will be
+huge amount of energy release into its surroundings. By taking into account the thermal fluctuations, the Gibbs free
+energy (G = H − T ˜S) is expressed in Appendix B.
+δ=0
+δ=0.4
+δ=0.6
+δ=0.8
+0
+5
+10
+15
+20
+5
+10
+15
+20
+r+
+G
+Figure 7: Gibbs free energy versus r+ for a=0.2, Q=0.4.
+The graphical analysis of Gibbs free energy with respect to horizon radius is shows in Fig. 7. The positivity of this
+energy is sign of occurrence of non-spontaneous reactions means this system requires more energy to gain equilibrium
+state. After the detail discussion of thermodynamics quantities, there is another important concept is the stability of
+the system that is checked by specific heat. The specific heat (C ˜S = dE
+dT ) is given as
+C ˜S =
+�
+2
+�
+a2 + 3r2
++
+��
+r+
+�
+r+
+�
+a2�
+− δ(Q − 4) log
+�
+�
+a2 − r+
+�
+Q + r+
+��2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2(πQ − 5) + δ(Q − 4) log(16π)
+�
++ r+
++
+�
+− πa4 − 2δQ log
+�
+�
+a2 − r+
+�
+Q + r+
+��2
+r2
++
+�
+a2 + r2
++
+�
+�
++ r+
+�
+− δ(Q + 1) log
+�
+�
+a2 − r+
+�
+Q + r+
+��2
+r2
++
+�
+a2 + r2
++
+�
+�
++ r+
+�
+− δ log
+×
+�
+�
+a2 − r+
+�
+Q + r+
+��2
+r2
++
+�
+a2 + r2
++
+�
+�
++ πa2 + δ log(16π) + r+
+�
+πQ + πr+ + 1
+�
++ 2Q
+�
++ a2(2πQ − 3) + δ(Q + 1) log(16π)
+�
++ 2a2Q + 2δQ log(16π)
+��
+− a4�
+− δ log
+�
+�
+a2 − r+
+�
+Q + r+
+��2
+r2
++
+�
+a2 + r2
++
+�
+�
++ πa2 + δ log(16π)
+��
+− a4�
+− δ log
+×
+�
+�
+a2 − r+
+�
+Q + r+
+��2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2 + δ log(16π)
+����
+r+
+�
+a2 + r2
++
+��
+− a4 − 4a2r2
++ + 2Qr3
++ + r4
++
+��−1
+.
+(21)
+
+10
+δ=0
+δ=0.4
+δ=0.6
+δ=0.8
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+-0.10
+-0.05
+0.00
+0.05
+r+
+CS
+~
+Figure 8: Specific heat versus r+ for a=0.2, Q=0.4.
+From Fig.
+8, the behaviour of specific heat is observed with respect to horizon radius and different choices of
+correction parameter δ. It can be observed that the uncorrected quantity (black) depicts negative behaviour means
+the system is unstable while the corrected specific heat shows positive behaviour throughout the considered domain.
+The positivity of this plot is indication of stable region. It can be concluded that these correction terms makes the
+system stable under thermal fluctuations.
+IV.
+DISCUSSION AND RESULT
+In this paper, we have utilized Lagrangian gravity equation to observe the tunneling of bosonic particles through
+the horizon of Horndeski like BH. We have used the metric from Ref. [57]. We have considered a new version of
+black hole with Horndeski parameter Q and a rotation parameter a, due to the presence of these parameters, we call
+the metric a new type of spacetime. Our results are also in terms of these parameters, therefore they are different
+from the previous literature related about the thermodynamics of this black hole. Assuming to relativistic quantum
+mechanics and the region of vacuum, where particles are produced continuously in the phenomenon of annihilated.
+The tunneling radiation as a quantum mechanical processes can be observed as a tunneling phenomenon, where
+positive boson particle radiate the horizon and the negative energy boson particle move inward and absorbed by the
+BH. The incoming and outgoing boson particles movement carried out by the action of particle’s is real and complex,
+respectively. The emission rate of these tunneling radiation corresponding to the Horndeski like BH configuration is
+associated to the imaginary part of the action of particles, which is associated to the factor of Boltzmann, this factor
+gives TH for Horndeski like BH. From our investigation, we have observed that, in rotating case of BH, the TH at
+which boson particles tunnel through the BH horizon is not dependent at any types of particles. In special case when
+particles have different (zero or upward or down) spins, the tunneling rate will be alike by assuming the semi-classical
+approach. Thus, their corresponding TH must be similar for all types of particle. Therefore, one can say tunneling
+radiation is independent of all kinds of the particles and this result also holds for different frame of coordinates by
+utilizing the transformations of particular coordinate. For this procedure, the tunneling particles is associated to the
+energy of particles, momentum, quantum gravity, hairy parameter of Horndeski gravity and BH surface gravity, while
+the temperature depends on hairy parameter of Horndeski gravity, rotation and quantum gravity parameters. It is
+very important to mention here that, when β = 0, we obtain the standard temperature for Horndeski like BH. In
+the absence of charge i.e., Q = 0, the above temperature reduces into Kerr BH temperature. For β = 0 and a = 0,
+the temperature reduces into Reissner Nordstr¨om BH. Moreover, when Q = 0 = a, we recover the temperature of
+Schwarzschild BH. For the changing values of β from 10 to 30 in the region 0 ≤ r+ ≤ 5, we have observed that the
+Horndeski like BH is stable and for changing Q from 0.5 to 1.5, the Horndeski like BH is un-stable with negative
+temperature. Moreover, the temperature increases with the increasing values of quantum gravity β.
+Moreover, we have computed TH as well as heat capacity for Horndeski gravity like BHs.
+Firstly, the TH is
+calculated through entropy and the density of state is also calculated with help of inverse Laplace transformation.
+We have observed that the exact entropy of the system depends on Hawking temperature by applying the method of
+steepest descent under different conditions. In graphically, we have studied the monotonically increasing entropy of
+the metric throughout the assumed domain. It is observed that a decrease in entropy for a certain value of the radius,
+the corrected expression of energy is increasing smoothly and also studied that the small BHs are more effective for
+thermal fluctuations.
+It is observed that the behaviour of energy gradually decreases for the different correction parameter δ values and
+the graph of usual entropy shows opposite behaviour as the graph increases. Therefore, the considered system shifts
+its state towards equilibrium. It is observable that for the small values of radii, the graph gradually decreases even
+
+11
+shifts towards negative side, while the corrected internal energy depicts positive behaviour. So, the considered BH
+absorbed more and more heat from the surrounding to maintain its state. The graph of pressure coincides the state
+of equilibrium. For the different values of correction parameter, the pressure significantly increases for the considered
+system.
+It is observed that the graph of usual enthalpy coincides with the plots of corrected one and abruptly decreases even
+shifts towards negative side. It can be concluded that there exists a exothermic reactions means a huge amount of
+energy released into its surroundings. The positivity of this energy is sign of occurrence of non-spontaneous reactions
+means the system required more energy to gain equilibrium state. The behaviour of specific heat with respect to
+horizon radius and different choices of correction parameter δ has been observed. The uncorrected quantity depicts
+negative behaviour means the system is unstable while the corrected specific heat shows positive behaviour throughout
+the considered domain. The positivity of the plots for specific heat is indication of stable region. It can be concluded
+that these correction terms makes the system stable under thermal fluctuations.
+Appendix A
+We have utilized the Lagrangian equation in the approximation of WKB to get following solutions,
+I
+(R2 + fI)g−1
+�
+η1(∂0K0)(∂1K0) + βη1(∂0K0)3(∂1K0) − η0(∂1K0)2 − βη0(∂1K0)4 + η1eA0(∂1K0) +
+η1βeA0(∂0K0)2(∂1K0)
+�
+−
+R
+g−1(fI + R2)
+�
+η3(∂1K0)2 + βη3(∂1K0)4 − η1(∂1K0)(∂3K0) − βη1(∂1K0)(∂3K0)2�
++
+I
+h(fI + R2)
+�
+η2(∂0K0)(∂2K0) + βη2(∂0K0)3(∂2K0) − η0(∂2K0)2 − βη0(∂2K0)4 + η2eA0(∂2K0)
++ η2eA0β(∂0K0)2(∂1K0)
+�
++
+fI
+(fI + R2)2
+�
+η3(∂0K0)(∂3K0) + βη3(∂0K0)3(∂3K0) − η0(∂3K0)2 − βη0(∂3K0)4
++ η3eA0(∂3K0) + η3eA0(∂0K0)2(∂3K0)
+�
+− m2 Iη0 − Rη3
+(fI + R2) = 0,
+(22)
+−I
+g−1(fI + R2)
+�
+η1(∂0K0)2 + βη1(∂0K0)4 − η0(∂0K0)(∂1K0) − βη0(∂0K0)(∂1K0)3 + η1eA0(∂0K0) + βη1eA0(∂0K0)3�
++
+R
+g−1(fI + R2)
+�
+η3(∂0K0)(∂1K0) + βη3(∂0K0)(∂1K0)3 − η1(∂0K0)(∂3K0) − βη1(∂0K0)(∂3K0)3�
++
+1
+g−1h [η2(∂1K0)(∂2K0) + βη2(∂1K0)(∂2K0)3 − η1(∂2K0)2 − βη1(∂2K0)4�
++
+1
+g−1(fI + R2)
+�
+η3(∂1K0)(∂3K0) + βη3(∂1K0)(∂3K0)3 − η1(∂3K0)2 − βη1(∂3K0)4�
+− m2η1
+g−1 +
+eA0I
+g−1(fI + R2)
+�
+η1(∂0K0) + βη1(∂0K0)3 − η0(∂1K0) − βη0(∂1K0)3 + eA0η1 + βη1eA0(∂0K0)2)
+�
++
+eA0R
+g−1(fI + R2)
+�
+η3(∂1K0) + βη3(∂1K0)3 − η1(∂3K0) − βη1(∂1K0)3�
+= 0,
+(23)
+
+12
+I
+h(fI + R2)
+�
+η2(∂0K0)2 + βη2(∂0K0)4 − η0(∂0K0)(∂2K0) − βη0(∂0K0)(∂2K0)3 + η2eA0(∂0K0) + βη2eA0(∂0K0)3�
++
+1
+g−1h
+�
+η2(∂1K0)2 + βη2(∂1K0)4 − η1(∂1K0)(∂2K0) − βη1(∂1K0)(∂2K0)3�
+−
+R
+h(fI + R2)
+�
+η2(∂0K0)(∂3K0)
++ βη2(∂0K0)3(∂3K0) − η0(∂0K0)(∂3K0) − βη0(∂0K0)3(∂3K0) + η2eA0(∂3K0) + βη2eA0(∂3K0)3�
++
+f
+h(fI + R2)
+�
+η3(∂2K0)(∂3K0) + βη3(∂2K0)3(∂3K0) − η2(∂3K0)2 − βη2(∂3K0)4�
++
+eA0I
+h(fI + R2)
+�
+��2(∂0K0)
++ βη2(∂0K0)3 − η0(∂2K0) − βη0(∂2K0)3 + η2eA0 + η2βeA0(∂0K0)2�
+− m2η2
+h
+= 0,
+(24)
+(fI − f 2)
+(fI + R2)2
+�
+η3(∂0K0)2 + βη3(∂0K0)4 − η0(∂0K0)(∂3K0) − βη0(∂0K0)(∂3K0)3 + eA0η3(∂0K0) + βη3eA0(∂0K0)3�
+−
+I
+h(fI + R2)
+�
+η3(∂1K0)2 + βη3(∂1K0)4 − η1(∂1K0)(∂3K0) − βη1(∂1K0)(∂3K0)3�
+−
+R
+h(fI + R2)
+�
+η2(∂0K0)(∂2K0) + βη2(∂0K0)3(∂2K0) − η0(∂2K0)2 + βη0(∂2K0)4 + eA0η2(∂2K0) + βη2eA0∂0K0)2(∂2K0)
+�
+−
+eA0f
+h(fI + R2)
+�
+η3(∂2K0)2 + βη3(∂2K0)4 − η2(∂2K0)(∂3K0) − βη2(∂0K0)(∂3K0)3�
+− m2(Rη0 − fη3
+(fI + R2)
++eA0(fI − f 2)
+(fI + R2)2
+�
+η3(∂0K0) + βη3(∂0K0)3 − η0(∂3K0) − βη0(∂3K0)3 + η3eA0 + η3βeA0(∂0K0)2�
+= 0,
+(25)
+Appendix B
+The thermodynamic quantity enthalpy is given as
+H =
+�
+r+
+�
+a2 + r2
++
+��
+r+
+�
+a2 + r2
++
+��
+r+
+�
+Q + r+
+�
+− a2��
+δ
+�
+log(16π) − log
+�
+�
+a2 − r+
+�
+Q + r+
+��2
+r2
++
+�
+a2 + r2
++
+�
+��
++ π
+�
+a2 + r2
++
+��
+−
+�
+a4 − r2
++
+�
+− 4a2 + 2Qr+ + r2
++
+���
+− δ log
+�
+�
+a2 − r+
+�
+Q + r+
+��2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2 + δ log(16π) + r2
++
+��
++
+�
+2r2
++
+�
+a2 + r2
++
+��
+− 4a2 + 3Qr+ + 2r2
++
+��
+a2 − r+
+�
+Q + r+
+��2�
+− δ log
+�
+�
+a2 − r+
+�
+Q + r+
+��2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2
++ δ log(16π) + r2
++
+�
+− 2
+�
+r+
+�
+Q + r+
+�
+− a2��
+a4 − r2
++
+�
+− 4a2 + 2Qr+ + r2
++
+���
+− a4δ + Qr3
++
+�
+a2 + δ
+�
+− r2
++
+�
+a4 + 3a2δ
+�
++ Qr5
++ + r6
++
+�
++ 4r2
++
+�
+a4 − r2
++
+�
+− 4a2 + 2Qr+ + r2
++
+���
+a2 − r+
+�
+Q + r+
+��2�
+− δ log
+×
+�
+�
+a2 − r+
+�
+Q + r+
+��
+2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2 + δ log(16π) + r2
++
+�
++ 2
+�
+a2 + r2
++
+��
+a4 − r2
++
+�
+− 4a2 + 2Qr+ + r2
++
+��
+×
+�
+a2 − r+
+�
+Q + r+
+��2�
+− δ log
+�
+�
+a2 − r+
+�
+Q + r+
+��2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2 + δ log(16π) + r2
++
+����
+a2 − r+
+�
+Q + r+
+��2�−1
+)
+×
+�
+4πr3
++
+�
+a2 + r2
++
+�3�−1
+.
+(26)
+
+13
+The Gibbs free energy is expressed as
+G =
+�
+− r2
++
+�
+a2 + r2
++
+�
+2�
+r+
+�
+Q + r+
+�
+− a2��
+− δ log
+�
+�
+a2 − r+
+�
+Q + r+
+��
+2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2 + δ log(16π) + r2
++
+�
++ r+
+�
+a2 + r2
++
+�
+×
+�
+r+
+�
+a2 + r2
++
+��
+r+
+�
+Q + r+
+�
+− a2��
+δ
+�
+log(16π) − log
+�
+�
+a2 − r+
+�
+Q + r+
+��
+2
+r2
++
+�
+a2 + r2
++
+�
+��
++ π
+�
+a2 + r2
++
+��
+−
+�
+a4 − r2
++
+×
+�
+− 4a2 + 2Qr+ + r2
++
+���
+− δ log
+�
+�
+a2 − r+
+�
+Q + r+
+��
+2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2 + δ log(16π) + r2
++
+��
++
+�
+2r2
++
+�
+a2 + r2
++
+�
+×
+�
+− 4a2 + 3Qr+ + 2r2
++
+��
+a2 − r+
+�
+Q + r+
+��
+2�
+− δ log
+�
+�
+a2 − r+
+�
+Q + r+
+��
+2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2 + δ log(16π) + r2
++
+�
+− 2
+�
+r+
+�
+Q + r+
+�
+− a2��
+a4 − r2
++
+�
+− 4a2 + 2Qr+ + r2
++
+���
+− a4δ + Qr3
++
+�
+a2 + δ
+�
+− r2
++
+�
+a4 + 3a2δ
+�
++ Qr5
++ + r6
++
+�
++ 4r2
++
+�
+a4 − r2
++
+�
+− 4a2 + 2Qr+ + r2
++
+���
+a2 − r+
+�
+Q + r+
+��2�
+− δ log
+�
+�
+a2 − r+
+�
+Q + r+
+��
+2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2 + δ log(16π)
++ r2
++
+�
++ 2
+�
+a2 + r2
++
+��
+a4 − r2
++
+�
+− 4a2 + 2Qr+ + r2
++
+���
+a2 − r+
+�
+Q + r+
+��2�
+− δ log
+�
+�
+a2 − r+
+�
+Q + r+
+��
+2
+r2
++
+�
+a2 + r2
++
+�
+�
++ a2 + δ log(16π) + r2
++
+����
+a2 − r+
+�
+Q + r+
+��
+2���
+4πr3
++
+�
+a2 + r2
++
+�
+3�−1
+.
+(27)
+[1] M. Sharif, W. Javed, Eur. Phys. J. C 72, 1997(2012).
+[2] M. Sharif, W. Javed, Int. J. Mod. Phys. Conf. Ser. 23(2013)271.
+[3] M. Sharif, W. Javed, Can. J. Phys. 91(2013)236.
+[4] S. W. Hawking, Commun. Math. Phys. 43(1975)199.
+[5] M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85(2000)5042.
+[6] M. K. Parikh, Int. J. Mod. Phys. D 13(2004)2351.
+[7] J. T. Firouzjaee and G. F. R. Ellis, Gen. Rel. Grav. 47(2015)6.
+[8] W. Javed, R. Babar, Adv. High Energy Phys. 2019(2019)2759641.
+[9] W. Javed, R. Babar, Chinese Journal of Phys. 61(2019)138.
+[10] W. Javed, R. Babar, The Fifteenth Marcel Grossmann Meeting, World Scientific 3(2022)905.
+[11] W. Javed, R. Babar, Punjab University Journal of Mathematics 52(2020)6.
+[12] W. Javed, R. Babar, A. ¨Ovg¨un, Mod. Phys. Lett. A 34(2019)1950057.
+[13] R. Babar, W. Javed, A. ¨Ovg¨un, Mod. Phys. Lett. A 35(2020)2050104.
+[14] M. Sharif, W. Javed, J. Korean Phys. Soc. 57(2010)217.
+[15] A. ¨Ovg¨un, K. Jusufi, Eur. Phys. J. Plus. 132(2017)298.
+[16] A. ¨Ovg¨un, Int. J. Theor. Phys. 55(2016)2919.
+[17] K. Jusufi, A. Ovgun, G. Apostolovska, Adv. High Energy Phys. 2017(2017)8798657.
+[18] I. Sakalli, A. ¨Ovg¨un, K. Jusufi, Astrophys Space Sci. 361(2016)330.
+[19] I. Sakalli, A. ¨Ovg¨un, General Relativity and Gravitation 48(2016)1.
+[20] A. ¨Ovg¨un, I. Sakalli, Int. J. Theor. Phys. 57(2018)322.
+[21] A. ¨Ovg¨un, I. Sakalli, J. Saavedra, C. Leiva, Mod. Phys. Lett. A 35(2020)2050163.
+[22] I. Sakalli, A. Ovgun, J. Exp. Theor. Phys. 121(2015)404.
+[23] R. Ali, K. Bamba, M. Asgher, M. F. Malik, S. A. A. Shah, Symmetry. 12(2020)1165.
+[24] R. Ali, K. Bamba, M. Asgher, S. A. A. Shah, Int. J. Mod. Phys. D 30(2021)2150002.
+[25] R. Ali, M. Asgher, M. F. Malik, Mod. Phys. Lett. A 35(2020)2050225.
+[26] W. Javed, R. Ali, R. Babar, A. ¨Ovg¨un, Eur. Phys. J. Plus 134(2019)511.
+[27] W. Javed, R. Ali, R. Babar, A. ¨Ovg¨un, Chinese Phys. C 44(2020)015104.
+[28] K. Jusufi, A. ¨Ovg¨un, Astrophys Space Sci. 361(2016)207.
+
+14
+[29] R. Ali, M. Asgher, New Astronomy 93(2022)101759
+[30] R. Ali, R. Babar and P. K. Sahoo, Physics of the Dark Universe 35(2022)100948.
+[31] R. Ali, R. Babar, M. Asgher and S. A. A. Shah, Int. J. Geom. Methods Mod. Phys. 19(2022)2250017.
+[32] J. M. Bardeen, in Conference Proceedings of GR5 (Tbilisi, URSS, 1968), p. 174.
+[33] M. Faizal and M. M. Khalil, Int. J. Mod. Phys. A 30(2015)1550144.
+[34] B. Pourhassan and M. Faizal, Nucl. Phys. B 913(2016)834.
+[35] A. Jawad and M. U. Shahzad, Eur. Phys. J. C 77(2017)349.
+[36] M. Zhang, Nucl. Phys. B 935(2018)170.
+[37] P. Pradhan, Universe 5(2019)57.
+[38] K. Ghaderi, B. Malakolkalami, Nucl. Phys. B 903(2016)10.
+[39] W. X. Chen, Y. G. Zheng, arXiv preprint arXiv:2204.05470.
+[40] R. Ali, R. Babar, M. Asgher, and S. A. A. Shah, Annals of Physics, 432(2021)168572.
+[41] W. Javed, G. Abbas, R. Ali, Eur. Phys. J. C 77(2017)296.
+[42] A. ¨Ovg¨un, W. Javed, R. Ali, Adv. High Energy Phys. 2018(2018)11.
+[43] W. Javed, R. Ali, G. Abbas, Can. J. Phys. 97(2018)176.
+[44] R. Ali, K. Bamba, S. A. A. Shah, Symmetry. 631(2019)11.
+[45] M. Sharif and Z. Akhtar, Phys. Dark Universe 29(2020)100589.
+[46] M. Sharif and Z. Akhtar, Chin. J. Phys 71(2021)669.
+[47] W. Javed, Z. Yousaf and Z. Akhtar, Mod. Phys. Lett. A 33(2018) 1850089.
+[48] Z. Yousaf, K. Bamba, Z. Akhtar and W. Javed, Int. J. Geom. Methods Mod. 19(2022)2250102.
+[49] K. Bamba, M. Ilyas, M.Z.Bhatti, and Z. Yousaf, General Relativity and Gravitation, 49(8) (2017), pp.1-17.
+[50] M. Ilyas, Eur. Phys. J. C 78, 757 (2018).
+[51] M. Ilyas, International Journal of Modern Physics A 36.24 (2021): 2150165.
+[52] M. Ilyas, International Journal of Geometric Methods in Modern Physics 16.10 (2019): 1950149.
+[53] A. Ditta et al., Eur. Phys. J. C (2022) 82:756.
+[54] L. Hui, A. Nicolis, Phys. Rev. Lett. 110, 241104 (2013)
+[55] E. Babichev, C. Charmousis, A. Leh´ebel, JCAP 04, 027 (2017)
+[56] A. Nicolis, R. Rattazzi, E. Trincherini, Phys. Rev. D 79, 064036 (2009)
+[57] R. K. Walia, S. D. Maharaj and S. G. Ghosh, Eur. Phys. J. C 82(2022)547.
+[58] R. Kumar, S. G. Ghosh, A. Wang, Phys. Rev. D 101(2020)104001.
+[59] R. Kumar, S. G. Ghosh, Eur. Phys. J. C 78(2018)750.
+[60] D. Y. Chen, Q. Q. Jiang, X. T. Zua, Phys. Lett. B 665(2008)106.
+[61] R. P. Kerr, Phys. Rev. Lett. 11(1963)237.
+

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+Springer Nature 2021 LATEX template
+Optimal subsampling algorithm for composite
+quantile regression with distributed data
+Xiaohui Yuan1, Shiting Zhou1† and Yue Wang1*†
+1*School of Mathematics and Statistics, Changchun University of
+Technology, Changchun, 130012, Jilin, China.
+*Corresponding author(s). E-mail(s): [email protected];
+Contributing authors: [email protected];
+[email protected];
+†These authors contributed equally to this work.
+Abstract
+For massive data stored at multiple machines, we propose a dis-
+tributed subsampling procedure for the composite quantile regres-
+sion. By establishing the consistency and asymptotic normality of
+the
+composite
+quantile
+regression
+estimator
+from
+a
+general
+sub-
+sampling algorithm, we derive the optimal subsampling probabili-
+ties and the optimal allocation sizes under the L-optimality cri-
+teria. A two-step algorithm to approximate the optimal subsam-
+pling
+procedure
+is
+developed.
+The
+proposed
+methods
+are
+illus-
+trated through numerical experiments on simulated and real datasets.
+Keywords: Composite quantile regression, Distributed data, Massive data,
+Optimal subsampling
+1 Introduction
+With the rapid development of science and technology, extremely large datasets
+are ubiquitous and lays heavy burden on storage and computation facilities.
+Many efforts have been made to deal with these challenge. There are three
+main directions from the view of statistical applications: divide-and-conquer,
+online updating, and subsampling. Among them, subsampling has been found
+1
+arXiv:2301.02448v1  [stat.CO]  6 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+Optimal subsampling algorithm for CQR with distributed data
+to be useful for reducing computational burden and extracting information
+from massive data.
+The idea of subsampling was first proposed by Jones (1956)[5]. A key
+tactic of subsampling methods is to specify nonuniform sampling probabil-
+ities to include more informative data points with higher probabilities. For
+example, the leverage score-based subsampling in Ma et al. (2015)[6], the
+information based optimal subdata selection in Wang et al. (2019)[12], and
+the optimal subsampling method under the A-optimality criterion in Wang et
+al. (2018)[11]. Recently, Fang et al. (2021)[2] applied subsampling to a weak-
+signal-assisted procedure for variable selection and statistical inference. Ai et
+al. (2021)[1] studied the optimal subsampling method for generalized linear
+models under the A-optimality criterion. Shao et al. (2022)[8] employed the
+optimal subsampling method to ordinary quantile regression.
+Due to the large scale and fast arrival speed of data stream, massive data
+are often partitioned across multiple servers. For example, Walmart stores
+produce a large number of data sets from different locations around the world,
+which need to be processed. However, it is difficult to transmit these datasets to
+a central location. For these datasets, it is common to analyze them on multiple
+machines. Qiu et al. (2020)[7] constructed a data stream classification model
+based on distributed processing. Sun et al. (2021)[10] proposed a data mining
+scheme for edge computing based on distributed integration strategy. Zhang
+and Wang (2021)[17] proposed a distributed subdata selection method for
+big data linear regression model. Zuo et al. (2021)[19] proposed a distributed
+subsampling procedure for the logistic regression. Yu et al. (2022)[16] derived
+a optimal distributed Poisson subsampling procedure for the maximum quasi-
+likelihood estimators with massive data.
+In the paper, we investigate the optimal distributed subsampling for com-
+posite quantile regression (CQR; Zou and Yuan (2008)[18]) in massive data.
+In a linear model, composite quantile regression can uniformly estimate the
+regression coefficients under heavy tail error. Moreover, since the asymptotic
+variance of the composite quantile regression estimate does not depend on
+the moment of the error distribution, the CQR estimator is robust. The CQR
+method is widely used in many fields. For massive data, Jiang et al. (2018)[3]
+proposed a divide-and-conquer CQR method. Jin and Zhao (2021)[4] proposed
+a divide-and-conquer CQR neural network method. Wang et al. (2021)[13]
+proposed a distributed CQR method for the massive data. Shao and Wang
+(2022)[9] and Yuan et al. (2022)[15] developed the subsampling for composite
+quantile regression. To the best of our knowledge, there is almost no work on
+random subsampling for composite quantile regression with distributed data.
+Based on the above motivation, we investigate the optimal subsampling
+for the composite quantile regression in massive data when the datasets are
+stored at different sites. We propose a distributed subsampling method in the
+context of CQR, and then study the optimal subsampling technology for data
+in each machine. The main advantages of our method are as follows: First,
+we establish the convergence rate of the subsample-based estimator, which
+
+Springer Nature 2021 LATEX template
+Optimal subsampling algorithm for CQR with distributed data
+3
+ensures the consistency of our proposed method. Second, it avoids the impact
+of different intercept items in data sets stored at different sites. Third, the
+computational speed of our subsampling method is much faster than the full
+data approach.
+The rest of this article is organized as follows. In Section 2, we propose the
+distributed subsampling algorithm based on composite quantile regression. The
+asymptotic properties of estimators based on subsamples are also established.
+We present a subsampling strategy with optimal subsampling probability and
+optimal allocation size. The simulation studies are given in Section 3. In Section
+4, we study the real data sets. The content of the article is summarized in
+Section 5. All proofs are given in the Appendix.
+2 Methods
+2.1 Model and notation
+Consider the following linear model
+yik = xT
+ikβ0 + εik, i = 1, · · · , nk, k = 1, · · · , K,
+(1)
+where xik denotes a p-dimensional covariate vector, β0 = (β1, · · · , βp)T ∈ Θ
+is a p-dimensional vector of regression coefficients, nk is the sample size of
+the kth dataset, n = �K
+k=1 nk is the total sample size, and K is the number
+of distributed datasets. Assume that the random error εik has cumulative
+distribution function F(·) and probability density function f(·).
+Let M be the composite level of composite quantile regression, which does
+not depend on the sample size n. Given M, let τm, m = 1, · · · , M be the speci-
+fied quantile levels such that τ1 < · · · < τM. Write θ0 = (θ01, · · · , θ0(p+M))T =
+(βT
+0 , bT
+0 )T and b0 = (b01, · · · , b0M)T, where b0m = inf{u : F(u) ≥ τm} for
+m = 1, · · · , M. In this paper, we assume that xik’s are nonrandom and are
+interested in inferences about the unknown θ0 from the observed dataset
+Dn = {Dkn = {(xT
+ik, yik), i = 1 · · · , n}, k = 1, · · · , K}.
+For τ ∈ (0, 1), u ∈ Rp, let ρτ(u) = u{τ − I(u < 0)} be the check loss function
+for the τ-th quantile level. The CQR estimator of θ based on the full dataset
+Dn is given by
+ˆθF = (ˆβ
+T
+F , ˆb
+T
+F )T = arg min
+β,b
+K
+�
+k=1
+nk
+�
+i=1
+M
+�
+m=1
+ρτm(yik − bm − xT
+ikβ),
+(2)
+Our aim is to construct a subsample-based estimator, which can be used to
+effectively approximate the full data estimator ˆθF .
+
+Springer Nature 2021 LATEX template
+4
+Optimal subsampling algorithm for CQR with distributed data
+2.2
+Subsampling algorithm and asymptotic properties
+In this subsection, we propose a distributed subsampling algorithm to approx-
+imate the ˆθF . First we propose a subsampling method in Algorithm 1, which
+can reasonably select a subsample from distributed data.
+Algorithm 1 Distributed Subsampling Algorithm£º
+• Sampling: Assign subsampling probabilities {πik}nk
+i=1 for the kth dataset
+Dk = {(yik, xik), i = 1, · · · , nk} with �nk
+i=1 πik = 1, where k = 1, · · · , K.
+Given total sampling size r, draw a random subsample of size rk with replace-
+ment from Dk according to {πik}nk
+i=1, where {rk}K
+k=1 are allocation sizes
+with �K
+k=1 rk = r. For i = 1, · · · , nk and k = 1, · · · , K, we denote the cor-
+responding responses, covariates, and subsampling probabilities as y∗
+ik, x∗
+ik
+and π∗
+ik, respectively.
+• Estimation: Based on the subsamples {(y∗
+ik, x∗
+ik, π∗
+ik), i = 1, · · · , rk}K
+k=1, and
+calculate the estimate ˜θs = (˜βs, ˜bs) = arg minθ Q∗(θ), where
+Q∗(θ) = 1
+n
+K
+�
+k=1
+r
+rk
+rk
+�
+i=1
+M
+�
+m=1
+ρτm(y∗
+ik − βTx∗
+ik − bm)
+π∗
+ik
+.
+To establish asymptotic properties of the subsample-based estimator ˜θs,
+we need the following assumptions:
+(A.1) Assume that f(t) is continuous with respect to t and 0 < f(b0m) <
++∞ for 1 ≤ m ≤ M. Let ˜xik,m = (xT
+ik, eT
+m)T, where em denotes a M ×1 vector,
+which has a one only in its mth coordinate and is zero elsewhere. Define
+En = 1
+n
+K
+�
+k=1
+nk
+�
+i=1
+M
+�
+m=1
+f(b0m)˜xik,m(˜xik,m)T.
+(3)
+Assume that there exist positive definite matrices E, such that
+En −→ E,
+and
+max
+1≤k≤K,1≤i≤nk ∥xik∥ = o(n1/2).
+(A.2) Assume that, for k = 1, · · · , K.
+max
+1≤k≤K,1≤i≤nk
+∥xik∥ + 1
+rkπik
+= op
+� n
+r1/2
+�
+.
+(4)
+Define
+V π = 1
+n2
+K
+�
+k=1
+r
+rk
+nk
+�
+i=1
+1
+πik
+� M
+�
+m=1
+{I(εik < b0m) − τm}˜xik,m
+�⊗2
+,
+(5)
+
+Springer Nature 2021 LATEX template
+Optimal subsampling algorithm for CQR with distributed data
+5
+where for a vector a, a⊗2 = aaT. Assume that there exist positive definite
+matrices V such that
+V π
+p
+−→ V ,
+where
+p
+−→ means convergence in probability.
+Theorem 1. If Assumptions (A.1) and (A.2) hold, conditional on Dn, as
+n → ∞ and r → ∞, if r/n = o(1), then we have
+Σ−1/2√r(˜θs − θ0)
+d
+−→ N(0, I),
+(6)
+where
+d
+−→ denotes convergence in distribution, Σ = E−1
+n V πE−1
+n .
+2.3 Optimal subsampling strategy
+Given r, we specify the subsampling probablities {πik}nk
+i=1, and the alloca-
+tion sizes {rk}K
+k=1 in Algorithm 1. A naive choice is the uniform subsampling
+strategy with {πik = 1/nk}nk
+i=1 and {rk = [rnk/n]}K
+k=1, where [·] denotes the
+rounding operation. However, this uniform subsampling method is not opti-
+mal. As suggested by Wang et al. (2018)[11], we adopted the nonuniform
+subsampling strategy to determine the optimal allocation sizes and optimal
+subsampling probabilities by minimizing the trace of Σ in Theorem 1.
+Since Σ = E−1
+n V πE−1
+n , the optimal allocation sizes and subsampling prob-
+abilities require the calculation of En, which depend on the unknown density
+function f(·). Following Wang and Ma (2021)[14], we derive optimal subsam-
+pling probabilities under the L-optimality criterion. Note that En and V π are
+nonnegative definite. Simple matrix algebra yields that tr(Σ) = tr(V πE−2
+n ) =
+tr(E−2
+n )tr(V π). Σ depends on rk and πik only through V π, and En is free of
+rk and πik. Hence, we suggest to determine the optimal allocation sizes and
+optimal subsampling probabilites by directly minimizing tr(V π) rather than
+tr(Σ), which can effectively speed up our subsampling algorithm.
+Theorem 2. If rk and πik, i = 1, · · · , nk, k = 1, · · · , K, are chosen as
+πLopt
+ik
+= πLopt
+ik
+(θ0) =
+∥ �M
+m=1{τm − I(εik < b0m)}˜xik,m ∥
+�nk
+i=1 ∥ �M
+m=1{τm − I(εik < b0m)}˜xik,m ∥
+,
+(7)
+and
+rLopt
+k
+= r
+�nk
+i=1 ∥ �M
+m=1{τm − I(εik < b0m)}˜xik,m ∥
+�K
+k=1
+�nk
+i=1 ∥ �M
+m=1{τm − I(εik < b0m)}˜xik,m ∥
+,
+(8)
+then tr(V π)/n attains its minimum.
+
+Springer Nature 2021 LATEX template
+6
+Optimal subsampling algorithm for CQR with distributed data
+2.4 Two-step algorithm
+Note that the optimal subsampling probabilities and allocation sizes depend
+depends on εik = yik − xT
+ikβ0 and b0m, m = 1, · · · , M. The L-optimal weight
+result is not directly implementable. To deal with this problem, we use a pilot
+estimator ˜θ to replace θ0. In the following, we propose a two-step subsampling
+procedure in Algorithm 2.
+Algorithm 2 Two-Step Algorithm£º
+• Step 1: Given r0, we run Algorithm 1 with subsampling size rk = [r0
+nk
+n ] to
+obtain a pilot estimator ˜θ, using πik = 1/nk, where [·] denotes the rounding
+operation. Replace θ0 with ˜θ0 in (7) and (8) to get the allocation sizes rk(˜θ)
+and subsampling probabilities πik(˜θ), for i = 1, · · · , nk and k = 1, · · · , K,
+respectively.
+• Step 2: Based on {rk(˜θ)}K
+k=1 and {πik(˜θ)}nk
+i=1 in Step 1, we can select a sub-
+sample {(y∗
+ik, x∗
+ik, π∗
+ik) : i = 1, · · · , rk}K
+k=1 from the full data Dn. Minimizes
+the following weighted function
+Q∗(θ) =
+K
+�
+k=1
+r
+rk(˜θ)
+rk(˜θ)
+�
+i=1
+M
+�
+m=1
+ρτm(y∗
+ik − βTx∗
+ik − bm)
+π∗
+ik
+,
+to get a two-step subsample estimate ˆθLopt, where ˆθLopt = (ˆβLopt, ˆbLopt) =
+arg min Q∗(θ).
+For the subsample-based estimator ˆθLopt in Algorithm 2, we give its
+asymptotic distribution in the following theorem.
+Theorem 3. If Assumptions (A.1) and (A.2) hold, then as r0 → ∞, r →
+∞, and n → ∞, then we have
+Σ−1/2√r(ˆθLopt − θ0)
+d
+−→ N(0, I),
+(9)
+where
+d
+−→ denotes convergence in distribution, Σ = E−1
+n V πE−1
+n . Here
+V π = 1
+n2
+K
+�
+k=1
+r
+rLopt
+k
+nk
+�
+i=1
+1
+πLopt
+ik
+� M
+�
+m=1
+{I(εik < b0m) − τm}˜xik,m
+�⊗2
+, (10)
+where
+πLopt
+ik
+=
+∥ �M
+m=1{τm − I(εik < b0m)}˜xik,m ∥
+�nk
+i=1 ∥ �M
+m=1{τm − I(εik < b0m)}˜xik,m ∥
+,
+and
+rLopt
+k
+= r
+�nk
+i=1 ∥ �M
+m=1{τm − I(εik < b0m)}˜xik,m ∥
+�K
+k=1
+�nk
+i=1 ∥ �M
+m=1{τm − I(εik < b0m)}˜xik,m ∥
+.
+For the statistical inference about θ0, to avoid estimating f(b0m), we
+propose the following iterative sampling procedure.
+
+Springer Nature 2021 LATEX template
+Optimal subsampling algorithm for CQR with distributed data
+7
+Firstly, using {πLopt
+ik
+(˜θ)}nk
+i=1 proposed in Algorithm 2, we sample with
+replacement to obtain B subsamples, {(y∗,j
+ik , x∗,j
+ik , π∗,j
+ik ), i = 1, · · · , rLopt
+k
+(˜θ), k =
+1, · · · , K} for j = 1, · · · , B. Next, we calculate the jth estimate of θ0 through
+ˆθLopt,j = (ˆβLopt,j, ˆbLopt,j)
+= arg min
+θ
+K
+�
+k=1
+r
+rLopt
+k
+(˜θ)
+rLopt
+k
+(˜θ)
+�
+i=1
+M
+�
+m=1
+ρτm(y∗,j
+ik − βTx∗,j
+ik − bm)
+π∗,j
+ik
+.
+The combined estimate can be obtained by
+ˆθL = (ˆβ
+T
+L, ˆb
+T
+L)T = 1
+B
+B
+�
+j=1
+ˆθLopt,j
+(11)
+and its variance-covariance matrix Ω = cov(ˆθL) can be estimated by
+ˆΩ =
+1
+refB(B − 1)
+B
+�
+j=1
+(ˆθLopt,j − ˆθL)⊗2,
+(12)
+where ref is the effective subsample size ratio (Wang & Ma, 2021[14]) given by
+ref = 1
+K
+K
+�
+k=1
+�
+1 − rkB − 1
+2
+nk
+�
+i=1
+{πLopt
+ik
+(˜θ)}2
+�
+.
+From Theorem 3, for any fixed B, the conditional distribution of
+√
+rB(ˆθL−
+θ0) satisfies
+{E−1
+n V πE−1
+n }−1/2���
+rB(ˆθL − θ0)
+d
+−→ N(0, I).
+The distribution of ˆθLopt can be approximated by the empirical distribution
+of {˜θLopt,j}B
+j=1. For s = 1, · · · , p + K, the 100 × (1 − α)% confidence interval
+of θ0s can be approximated by [ˆθL,s − ˆω1/2
+ss z1−α/2, ˆθL,s + ˆω1/2
+ss z1−α/2], where
+ˆθL,s is the sth element of ˆθL, ˆωss is the (s, s)th element of ˆΩ and z1−α/2 is
+the 1 − α/2 quantile of the standard normal distribution.
+3 Numerical studies
+In this section, we conduct a simulation study to evaluate the performances of
+the proposed optimal subsampling algorithm. Simulations were performed on
+a laptop running Window 10 with an Intel i7 processor and 16 GB memory.
+Full data are generated from the model
+yik = xT
+ikβ0 + εik, i = 1, · · · , nk, k = 1, · · · , K,
+
+Springer Nature 2021 LATEX template
+8
+Optimal subsampling algorithm for CQR with distributed data
+with the true parameter β0 = (1, 1, 1, 1, 1)T . We consider the following four
+cases for the error term ε: (1) the standard normal distribution, N(0, 1); (2)
+the mixture normal distribution, 0.5N(0, 1) + 0.5N(0, 9); (3) the Student¡¯s
+t distribution with three degrees of freedom, t(3); (4) the standard Cauchy
+distribution, Cauchy(0,1).
+We consider the following four cases for the covariate x:
+Case I: xik ∼ N(0, Σ), where Σ = (0.5|s−t|)s,t.
+Case II: xik ∼ N(0, Σ), where Σ = (0.5I(s̸=t))s,t.
+Case III: xik ∼ t3(0, Σ) with three degrees of freedom and Σ = (0.5|s−t|)s,t.
+Case IV: Set K = 5, xi1 ∼ N5(0, I), xi2 ∼ N5(0, Σ1), xi3 ∼ N5(0, Σ2),
+xi4 ∼ t3(0, Σ1) and xi5 ∼ t5(0, Σ1), where Σ1 = (0.5|s−t|)s,t, Σ2 =
+(0.5I(s̸=t))s,t.
+Note that in Cases I-III, the covariate distributions are identical for all
+distributed datasets. In Case IV, the covariates have different distributions for
+distributed datasets.
+All the simulation are based on 1000 replications. We set the sample size
+of each datasets as {nk = [nuk/ �K
+k=1 uk]}K
+k=1, where [·] denotes the rounding
+operation, uk are generated from the uniform distribution over (1, 2) with K =
+5 and 10, respectively. We use the quantile levels τm = m/16, m = 1, · · · , 15
+for the composite quantile regression.
+In Tables 1, we report the simulation results on subsample-based estimator
+for β1 (other βi’s are similar and omitted) with K = 5 and K = 10 respec-
+tively, including the estimated bias (Bias) and the standard deviation (SD) of
+the estimates where r0 = 200, n = 106 in Case I. The bias and SDs of the pro-
+posed subsample estimate for Case IV with n = 106 and n = 107 are presented
+in Tabel 2. The subsample sizes r = 200, 400, 600, 800 and 1000, respectively.
+It can be seen from the results that the subsample-based estimator is unbi-
+ased. The performance of our estimator becomes better as r increases, which
+confirms the theoretical result on consistency of the subsampling methods.
+For comparison, we consider the uniform subsampling method (Uniform)
+with πik =
+1
+nk , and rk = [rnk/n] for i = 1, · · · , nk and k = 1, · · · , K. We cal-
+culate empirical mean square error (MSE) of uniform subsampling estimator
+(Unif) and our optimal subsampling estimator (Lopt) based on 1000 repeti-
+tions of the simulation. Figures 1 and 2 present the MSEs of each method for
+Case I with K = 5 and K = 10, where n = 106. Figures 3 presents the MSEs of
+the subsampling estimator for Case IV with n = 106, n = 107 and ε ∼ N(0, 1).
+From the above results, we can see that the MSEs of our method (Lopt) are
+much smaller than those of Uniform subsampling method (Unif). The results
+indicate that our method also works well with heterogeneous covariates, i.e.,
+the covariates can have different distributions in different data blocks.
+In the following, we evaluate the computational efficiency of our two-step
+subsampling algorithm. The mechanism of data generation is the same as
+the above mentioned situation. For fair comparison, we count the CPU time
+with one core based on the mean calculation time of 1000 repetitions of each
+subsample-based method. In Table 3, we report the results for Case I and
+
+Springer Nature 2021 LATEX template
+Optimal subsampling algorithm for CQR with distributed data
+9
+the normal error with n = 106, K = 5, r0 = 200 and different r, respectively.
+The computing time for the full data method is also given in the last row.
+Note that the uniform subsampling requires the least computing time, because
+its subampling probabilities πik =
+1
+nk , and allocation sizes rk = [rnk/n], do
+not take time to compute. Our subsampling algorithm has great computation
+advantage over the full data method. To further investigate the computational
+gain of the subsampling approach, we increase the dimension p to 30 with the
+true parameter β0 = (0.5, · · · , 0.5)T. Table 4 presents the computing time for
+Case I and normal error with r0 = 200, r = 1000, K = 5, n = 104, 105, 106 and
+107, respectively. It is clear that both subsampling methods take significantly
+less computing times than the full data approach.
+To investigate the performance of ˆΩ in (12), we compare the empirical mean
+square error (EMSE, s−1 �1000
+s=1 ∥ ˆβ
+s
+L−β0 ∥2) and the average estimated mean
+square error(AMSE) of ˆβL in (11) with different B. In Tables 5, we report the
+average length of the confidence intervals and 95% coverage probabilities (CP)
+of our subsample-based estimator for β1 (other βi’s are similar and omitted)
+with n = 106, r = 1000 and K = 5. Figures 4-7 present the EMSEs and AMSEs
+of ˆβL. For all cases, the AMSEs are very close to the EMSEs, and the EMSEs
+and AMSEs become smaller as B increases.
+4 A real data example
+In this section, we apply our method to the USA airline data, which are pub-
+licly available at http://stat-computing.org/datastore/2009/the-data.html.
+The data include detailed information on the arrivals and departures of all
+commercial flights in the USA from 1987 to 2008, and they are stored in 22
+separate files (K = 22). The raw dataset is as large as 10 GB on a hard
+drive. We use the composite regression to model the relationship between the
+arrival delay time, y, and three covariate variables: x1, weekend/weekday sta-
+tus (binary; 1 if departure occurred during the weekend, 0 otherwise), x2, the
+departure delay time and x3, the distance. Since the y, x2 and x3 in the data
+set are on different scales, we normalize them first. In addition, we drop the
+NA values in the dataset and we have n = 115, 257, 291 observations with
+completed information on y and x. Table 6 shows the cleaned data.
+We use the quantile levels τm = m/16, m = 1, · · · , 15 for the composite
+quantile regression. For comparison, the full-data estimate of the regression
+parameters is given by ˆβF = (−0.0451, 0.9179, −0.0248)T. The proposed point
+estimate ˆβL and corresponding confident intervals with different r and B are
+presented in Table 7. It can be seen from Table 7 that the subsample estimator
+ˆβL is close to ˆβF . In Figure 8, we present the MSEs of both subsampling
+methods based on 1000 subsamples with r = 200, 400, 600, 800 and 1000,
+respectively. The MSEs of the the optimal subsampling estimator are smaller
+than those of the uniform subsampling estimator.
+
+Springer Nature 2021 LATEX template
+10
+Optimal subsampling algorithm for CQR with distributed data
+5 Conclusion
+We have studied the statistical properties of a subsampling algorithm for the
+composite quantile regression model with distributed massive data. We derived
+the optimal subsampling probabilities and optimal allocation sizes. The asymp-
+totic properties of the subsample estimator were established. Some simulations
+and a real data example were provided to check the performance of our method.
+Appendix
+Proof of Theorem 1
+Define
+A∗
+r(u) = 1
+n
+K
+�
+k=1
+r
+rk
+rk
+�
+i=1
+M
+�
+m=1
+1
+π∗
+ik
+A∗
+ik,m(u),
+where A∗
+ik,m(u) = ρτm(ε∗
+ik − b0m − uT˜x∗
+ik,m/√r) − ρτm(ε∗
+ik − b0m), ˜x∗
+ik,m =
+(x∗T
+ik , eT
+m)T, and ε∗
+ik = y∗
+ik − βT
+0 x∗
+ik, i = 1, · · · , rk. Since A∗
+r(u) is a convex
+function of u, its minimizer is √r(˜θs − θ0), we can focus on A∗
+r(u) when
+evaluating the properties of √r(˜θs − θ0).
+Let ψτ(u) = τ − I(u < 0). By Knight’s identity (Knight, 1998),
+ρτ(u − v) − ρτ(u) = −vψτ(u) +
+� v
+0
+{I(u ≤ s) − I(u ≤ 0)}ds,
+we can rewrite A∗
+ik,m(u) as
+A∗
+ik,m(u) = ρτm(ε∗
+ik − b0m − uT˜x∗
+ik,m/√r) − ρτm(ε∗
+ik − b0m)
+= − 1
+√ruT˜x∗
+ik,m{τm − I(ε∗
+ik − b0m < 0)}
++
+� uT ˜x∗
+ik,m/√r
+0
+{I(ε∗
+ik − b0m ≤ s) − I(ε∗
+ik − b0m ≤ 0)}ds.
+Thus, we have
+A∗
+r(u)
+= −uT 1
+√r
+1
+n
+K
+�
+k=1
+r
+rk
+M
+�
+m=1
+rk
+�
+i=1
+1
+π∗
+ik
+{τm − I(ε∗
+ik − b0m < 0)}˜x∗
+ik,m
++ 1
+n
+K
+�
+k=1
+r
+rk
+M
+�
+m=1
+rk
+�
+i=1
+1
+π∗
+ik
+� uT ˜xik,m/√r
+0
+{I(ε∗
+ik − b0m ≤ s) − I(ε∗
+ik − b0m ≤ 0)}ds
+= uTZ∗
+r + A∗
+2r(u),
+(1)
+
+Springer Nature 2021 LATEX template
+Optimal subsampling algorithm for CQR with distributed data
+11
+where
+Z∗
+r = − 1
+√r
+1
+n
+K
+�
+k=1
+r
+rk
+M
+�
+m=1
+rk
+�
+i=1
+1
+π∗
+ik
+{τm − I(ε∗
+ik − b0m < 0)}˜x∗
+ik,m,
+A∗
+2r(u) = 1
+n
+K
+�
+k=1
+r
+rk
+rk
+�
+i=1
+1
+π∗
+ik
+A∗
+k,i(u),
+A∗
+k,i(u) =
+M
+�
+m=1
+� uT ˜x∗
+ik,m/√r
+0
+{I(ε∗
+ik − b0m ≤ s) − I(ε∗
+ik − b0m ≤ 0)}ds.
+Firstly, we prove the asymptotic normality of Z∗
+r. Denote
+η∗
+ik = −
+r
+rknπ∗
+ik
+M
+�
+m=1
+{τm − I(ε∗
+ik − b0m < 0)}˜x∗
+ik,m,
+then Z∗
+r can be written as Z∗
+r =
+1
+√r
+�K
+k=1
+�rk
+i=1 η∗
+ik. Direct calculation yields
+E(η∗
+ik | Dn) = − r
+rkn
+nk
+�
+i=1
+M
+�
+m=1
+{τm − I(εik − b0m < 0)}˜xik,m = Op
+�
+rn−1/2
+k
+rkn
+�
+,
+cov(η∗
+ik | Dn) = E{(η∗
+ik)⊗2 | Dn} − {E(η∗
+ik | Dn)}⊗2
+=
+nk
+�
+i=1
+r2
+r2
+kn2πik
+�
+M
+�
+m=1
+[τm − I(εik − b0m < 0)]˜xik,m
+�⊗2
+− {E(η∗
+ik | Dn)}⊗2
+=
+nk
+�
+i=1
+r2
+r2
+kn2πik
+�
+M
+�
+m=1
+[τm − I(εik − b0m < 0)]˜xik,m
+�⊗2
+− op(1).
+It is easy to verify that
+E{E(η∗
+ik | Dn)} = 0,
+cov{E(η∗
+ik | Dn)} =
+r2
+r2
+kn2
+nk
+�
+i=1
+cov
+� M
+�
+m=1
+[τm − I(εik < b0m)] ˜xik,m
+�
+.
+Denote the (s, t) th element of cov{E(η∗
+ik | Dn)} as σst. Using the Cauchy
+inequality, it is easy to obtain
+| σst |≤ √σss
+√σtt ≤
+r2
+r2
+kn2
+nk
+�
+i=1
+M(∥xi∥2 + 1) = Op
+�r2nk
+r2
+kn2
+�
+.
+
+Springer Nature 2021 LATEX template
+12
+Optimal subsampling algorithm for CQR with distributed data
+By Assumption 1 and Chebyshev’s inequality,
+E(η∗
+ik | Dn) = Op
+�
+rn1/2
+k
+rkn
+�
+.
+Under the conditional distribution given Dn, we check Lindeberg’s condi-
+tions (Theorem 2.27 of van der Vaart, 1998). Specifically, for ϵ > 0, we want
+to prove that
+K
+�
+k=1
+rk
+�
+i=1
+E{∥r−1/2η∗
+ik∥2I(∥η∗
+ik∥ > √rϵ) | Dn} = op(1).
+(2)
+Note that
+K
+�
+k=1
+rk
+�
+i=1
+E{∥r−1/2η∗
+ik∥2I(∥η∗
+ik∥ > √rϵ) | Dn}
+=
+K
+�
+k=1
+rk
+�
+i=1
+E
+�����
+r1/2
+rknπ∗
+ik
+M
+�
+m=1
+˜x∗
+ik,m{τm − I(εik − b0m < 0)}
+����
+2
+×I
+�����
+r−1/2
+rknπ∗
+ikϵ
+M
+�
+m=1
+˜x∗
+ik,m{τm − I(εik − b0m < 0)}
+���� > 1
+�����Dn
+�
+=
+K
+�
+k=1
+nk
+�
+i=1
+r
+rkn2πik
+����
+M
+�
+m=1
+{τm − I(εik − b0m < 0)}˜xik,m
+����
+2
+×I
+�
+r1/2
+rknπikϵ
+����
+M
+�
+m=1
+{τm − I(εik − b0m < 0)}˜xik,m
+���� > 1
+�
+.
+(3)
+By Assumption (A.2),
+max
+1≤k≤K max
+1≤i≤nk
+∥xik∥ + 1
+rkπik
+= op
+� n
+r1/2
+�
+,
+M 2
+K
+�
+k=1
+nk
+�
+i=1
+(1 + ∥xik∥)2
+n2πik
+= Op(1),
+the right hand side of (3) satisfies
+K
+�
+k=1
+nk
+�
+i=1
+r
+rkn2πik
+����
+M
+�
+m=1
+{τm − I(εik < b0m)}˜xik,m
+����
+2
+×I
+�
+r1/2
+rknπikϵ
+����
+M
+�
+m=1
+{τm − I(εik < b0m)}˜xik,m
+���� > 1
+�
+
+Springer Nature 2021 LATEX template
+Optimal subsampling algorithm for CQR with distributed data
+13
+≤ M 2
+K
+�
+k=1
+n
+�
+i=1
+r
+rkn2πik
+(1 + ∥xik∥)2I
+�M(1 + ∥xik∥)r1/2
+rknπikϵ
+> 1
+�
+≤ I
+�
+max
+1≤k≤K max
+1≤i≤nk
+∥xik∥ + 1
+rkπik
+>
+nϵ
+r1/2M
+�
+×M 2
+K
+�
+k=1
+nk
+�
+i=1
+r(1 + ∥xik∥)2
+rkn2πik
+= op(1).
+(4)
+Thus, the Lindeberg’s conditions hold with probability approaching one.
+Note that η∗
+ik, i = 1, · · · , rk, are independent and identically distributed
+with mean E(η∗
+ik | Dn) and the covariance cov(η∗
+ik | Dn) when given Dn.
+Based on this result, as r, n → ∞, we get
+V −1/2
+π
+{Z∗
+r − √r
+K
+�
+k=1
+E(η∗
+ik | Dn)}
+d
+−→ N(0, I).
+Since √r �K
+k=1 E(η∗
+ik | Dn) = Op
+�
+r1/2
+n1/2
+�K
+k=1
+rn1/2
+k
+rkn1/2
+�
+= op(1), it is easy
+to verify that
+V −1/2
+π
+Z∗
+r
+d
+−→ N(0, I).
+(5)
+Next, we prove that
+A∗
+2r(u) = 1
+2uTEu + op(1).
+Write the conditional expectation of A∗
+2r(u) as
+E{A∗
+2r(u) | Dn}
+= r
+n
+K
+�
+k=1
+nk
+�
+i=1
+E{Ak,i(u)} + r
+n
+K
+�
+k=1
+nk
+�
+i=1
+[Ak,i(u) − E{A2r,i(u)}].
+(6)
+By Assumption (A.1),
+max
+1≤k≤K max
+1≤i≤nk ∥xik∥ = o(max(n1/2
+1
+, · · · , n1/2
+K )) = o(n1/2),
+we can get
+r
+n
+K
+�
+k=1
+nk
+�
+i=1
+E(Ak,i(u))
+
+Springer Nature 2021 LATEX template
+14
+Optimal subsampling algorithm for CQR with distributed data
+= r
+n
+K
+�
+k=1
+nk
+�
+i=1
+M
+�
+m=1
+� uT ˜xik,m/√r
+0
+{F(b0m + s) − F(b0m)}ds
+=
+√r
+n
+K
+�
+k=1
+nk
+�
+i=1
+M
+�
+m=1
+� uT ˜xik,m
+0
+{F(b0m + t/√r) − F(b0m)}dt
+= 1
+2uT
+�
+1
+n
+K
+�
+k=1
+nk
+�
+i=1
+M
+�
+m=1
+f(b0m)˜xik,m˜xT
+ik,m
+�
+u + o(1)
+= 1
+2uTEu + o(1).
+(7)
+Furthermore, we have
+E
+�
+r
+n
+K
+�
+k=1
+nk
+�
+i=1
+�
+Ak,i(u) − E{Ak,i(u)}
+��
+= 0,
+and
+var
+� r
+n
+K
+�
+k=1
+nk
+�
+i=1
+[Ak,i(u) − E{Ak,i(u)}]
+�
+≤ r2
+n2
+K
+�
+k=1
+nk
+�
+i=1
+E{A2
+k,i(u)}.
+(8)
+Since Ak,i(u) is nonnegative, it is easy to obtain
+Ak,i(u) ≤
+����
+M
+�
+m=1
+� uT ˜xik,m/√r
+0
+{I(εik ≤ b0m + s) − I(εik ≤ b0m)}ds
+����
+≤
+M
+�
+m=1
+� uT ˜xik,m/√r
+0
+����{I(εik ≤ b0m + s) − I(εik ≤ b0m)}
+����ds
+≤
+1
+√r
+M
+�
+m=1
+| uT˜xik,m | .
+(9)
+By Assumption (A.1),
+max
+1≤k≤K max
+1≤i≤nk ∥xik∥ = o(max(n1/2
+1
+, · · · , n1/2
+K )) = o(n1/2),
+together with (8) and (9), we get
+var
+� r
+n
+K
+�
+k=1
+nk
+�
+i=1
+[Ak,i(u) − E{Ak,i(u)}]
+�
+≤
+�
+M ∥u∥
+√n (1 + max
+1≤k≤K max
+1≤i≤nk ∥xik∥)
+� K
+�
+k=1
+r3/2
+n3/2
+nk
+�
+i=1
+E{Ak,i(u)}
+
+Springer Nature 2021 LATEX template
+Optimal subsampling algorithm for CQR with distributed data
+15
+= o(1).
+(10)
+Combining the Chebyshev’s inequality, it follows from (6), (7) and (10) that
+E {A∗
+2r(u) | Dn} = 1
+2uTEu + op(1).
+(11)
+Next, we derive the conditional variance of A∗
+2r(u), i.e., var {A∗
+2r(u) | Dn}.
+Observing that A∗
+k,i(u), i = 1, · · · , rk are independent and identically dis-
+tributed when given Dn,
+var {A∗
+2r(u) | Dn} =
+K
+�
+k=1
+r2
+(rkn)2
+rk
+�
+i=1
+var
+�A∗
+k,i(u)
+π∗
+ik
+����Dn
+�
+≤
+K
+�
+k=1
+r2rk
+r2
+kn2 E
+��A∗
+k,i(u)
+π∗
+ik
+�2����Dn
+�
+.
+(12)
+By (9), the right hand of (12) satisfies
+K
+�
+k=1
+r2rk
+r2
+kn2
+nk
+�
+i=1
+A2
+k,i(u)
+πik
+≤ r2
+n2
+K
+�
+k=1
+nk
+�
+i=1
+Ak,i(u)
+� 1
+√r
+M
+�
+m=1
+| uT˜xik,m |
+rkπik
+�
+≤
+�r1/2
+n M∥u∥ max
+1≤k≤K max
+1≤i≤nk
+∥xik∥ + 1
+rkπik
+� r
+n
+K
+�
+k=1
+nk
+�
+i=1
+Ak,i(u).
+(13)
+Together with (7), (13) and Assumption (A.2), we have
+var
+�
+A∗
+2r(u) | Dn
+�
+= op(1).
+(14)
+Together with (9), (14) and Chebyshev’s inequality, we can obtain
+A∗
+2r(u) = 1
+2uTEu + op|Dn(1),
+(15)
+Here op|Dn(1) means if a = op|Dn(1), then a converges to 0 in conditional
+probability given Dn in probability, in other words, for any δ > 0, P(| a |>
+δ | Dn)
+p
+−→ 0 as n → +∞. Since 0 ≤ P(| a |> δ | Dn) ≤ 1, then it converges
+to 0 in probability if and only P(| a |> δ) = E{P(| a |> δ | Dn)} → 0. Thus,
+a = op|Dn(1) is equivalent to a = op(1).
+
+Springer Nature 2021 LATEX template
+16
+Optimal subsampling algorithm for CQR with distributed data
+It follows from (1) and (15) that
+A∗
+2r(u) = uTZ∗
+r + 1
+2uTEu + op(1).
+Since A∗
+2r(u) is a convex function, we have
+√r(˜θs − θ0) = −E−1
+n Z∗
+r + op(1).
+Based on the above results, we can prove that
+{E−1
+n V πE−1
+n }−1/2√r(˜θs − θ0) = −{E−1
+n V πE−1
+n }−1/2E−1
+n Z∗
+r + op(1).
+By Slutsky’s Theorem, for any a ∈ Rp+M, from (5) we have that
+P[{E−1
+n V πE−1
+n }−1/2√r(˜θs − θ0) ≤ a | Dn]
+p
+−→ Φp+M(a),
+(16)
+where Φp+M(a) denotes the standard p + M dimensional multivariate normal
+distribution function. And the conditional probability in (16) is a bounded
+random variable, then convergence in probability to a constant implies
+convergence in the mean. Therefore, for any a ∈ Rp+M,
+P[{E−1
+n V πE−1
+n }−1/2√r(˜θs − θ0) ≤ a]
+= E(P[{E−1
+n V πE−1
+n }−1/2√r(˜θs − θ0) ≤ a | Dn])
+→ Φp+M(a).
+We complete the proof of Theorem 1.
+Proof the Theorem 2
+We can prove that
+tr(V π) = 1
+n2
+K
+�
+k=1
+r
+rk
+nk
+�
+i=1
+1
+πik
+tr
+�
+�
+� M
+�
+m=1
+{I(εik < b0m) − τm}˜xik,m
+�⊗2�
+�
+= 1
+n2
+K
+�
+k=1
+r
+rk
+� nk
+�
+i=1
+πik
+� � nk
+�
+i=1
+1
+πik
+����
+M
+�
+m=1
+[I(εik < b0m) − τm]˜xik,m
+����
+2�
+≥ 1
+n2
+K
+�
+k=1
+r
+rk
+� nk
+�
+i=1
+����
+M
+�
+m=1
+{I(εik < b0m) − τm}˜xik,m
+����
+2�
+= 1
+n2
+� K
+�
+k=1
+rk
+� K
+�
+k=1
+1
+rk
+� nk
+�
+i=1
+����
+M
+�
+m=1
+[I(εik < b0m) − τm]˜xik,m
+����
+2�
+≥ 1
+n2
+K
+�
+k=1
+nk
+�
+i=1
+����
+M
+�
+m=1
+{I(εik < b0m) − τm}˜xik,m
+����
+2
+,
+
+Springer Nature 2021 LATEX template
+Optimal subsampling algorithm for CQR with distributed data
+17
+with Cauchy-Schwarz inequality and the equality in it holds if and only if
+when πik ∝ ∥ �M
+m=1[I(εik < b0m)−τm]˜xik,m∥ and rk ∝ �nk
+i=1 ∥ �M
+m=1[I(εik <
+b0m) − τm]˜xik,m∥, respectively. We complete the proof of Theorem 2.
+
+Springer Nature 2021 LATEX template
+18
+REFERENCES
+References
+Ai M, Yu J, Zhang H, Wang H (2019) Optimal subsampling algorithms for
+big data regressions. Statistica Sinica 31: 749-772
+Fang F, Zhao J, Ahmed S E, Qu A (2021) A weak-signal-assisted proce-
+dure for variable selection and statistical inference with an informative
+subsample. Biometrics 77(3): 996-1010
+Jiang R, Hu X, Yu K, Qian W (2018) Composite quantile regression for
+massive datasets. Statistics 52(5): 980-1004
+Jin J, Zhao Z (2021) Composite Quantile Regression Neural Network for
+Massive Datasets. Mathematical Problems in Engineering 2021
+Jones H L (1956) Investigating the properties of a sample mean by employ-
+ing random subsample means. Journal of the American Statistical
+Association 51(273): 54-83
+Ma P, Mahoney M W, Yu B (2015) A statistical perspective on algorithmic
+leveraging. Journal of Machine Learning Research 16: 861-919
+Qiu Y, Du G, Chai S (2020) A novel algorithm for distributed data stream
+using big data classification model. International Journal of Information
+Technology and Web Engineering 15(4): 1-17
+Shao L, Song S, Zhou Y (2022) Optimal subsampling for large-sample
+quantile regression with massive data. Canadian Journal of Statistics
+https://doi.org/10.1002/cjs.11697
+Shao Y, Wang L (2022) Optimal subsampling for composite quantile regres-
+sion model in massive data. Statistical Papers 63(4): 1139¨C1161
+Sun X, Xu R, Wu L, Guan Z (2021) A differentially private distributed data
+mining scheme with high efficiency for edge computing. Journal of Cloud
+Computing 10(1): 1-12
+Wang H Y, Zhu R, Ma P (2018) Optimal subsampling for large sample logistic
+regression. Journal of the American Statistical Association 113(522): 829-
+844
+Wang H Y, Yang M, Stufken J (2019) Information-based optimal sub-
+data selection for big data linear regression. Journal of the American
+Statistical Association 114(525): 393-405
+Wang K, Li S, Zhang B (2021) Robust communication-efficient distributed
+composite quantile regression and variable selection for massive data.
+Computational Statistics & Data Analysis 161: 107262
+Wang H, Ma Y (2021) Optimal subsampling for quantile regression in big
+data. Biometrika 108: 99-112
+Yuan X, Li Y, Dong X, Liu T (2022) Optimal subsampling for composite
+quantile regression in big data. Statistical Papers 63(5): 1649-1676
+Yu J, Wang H, Ai M, Zhang H (2022) Optimal Distributed Subsampling for
+Maximum Quasi-Likelihood Estimators With Massive Data. Journal of
+the American Statistical Association 117(537): 265-276
+Zhang H, Wang H (2021) Distributed subdata selection for big data via
+sampling-based approach. Computational Statistics and Data Analysis
+153: 107072
+
+Springer Nature 2021 LATEX template
+REFERENCES
+19
+Zou H, Yuan M (2008) Composite quantile regression and the oracle model
+selection theory. Annals of Statistics 36(3): 1108-1126
+Zuo L, Zhang H, Wang H Y, Sun L (2021) Optimal subsample selection
+for massive logistic regression with distributed data. Computational
+Statistics 36(4): 2535-2562
+
+Springer Nature 2021 LATEX template
+20
+REFERENCES
+Table 1: The proposed subsample estimate of β1 with n = 106 in Case I.
+K = 5
+K = 10
+Error
+r
+Bias
+SD
+Bias
+SD
+200
+0.0006
+0.0769
+0.0010
+0.0737
+400
+-0.0009
+0.0554
+-0.0008
+0.0531
+N(0, 1)
+600
+0.0025
+0.0425
+0.0008
+0.0423
+800
+0.0009
+0.0379
+0.0004
+0.0388
+1000
+0.0004
+0.0348
+-0.0014
+0.0338
+200
+0.0023
+0.1405
+0.0049
+0.1336
+400
+-0.0023
+0.0970
+0.0006
+0.0934
+mixNormal
+600
+-0.0033
+0.0797
+-0.0004
+0.0822
+800
+0.0028
+0.0688
+-0.0019
+0.0707
+1000
+-0.0002
+0.0600
+-0.0033
+0.0621
+200
+-0.0021
+0.0961
+0.0009
+0.0914
+400
+0.0006
+0.0665
+-0.0004
+0.0645
+t(3)
+600
+-0.0015
+0.0552
+-0.0002
+0.0505
+800
+-0.0003
+0.0477
+0.0005
+0.0462
+1000
+0.0024
+0.0415
+0.0013
+0.0423
+200
+-0.0108
+0.1312
+0.0070
+0.1373
+400
+0.0040
+0.0959
+0.0003
+0.0954
+Cauchy
+600
+0.0023
+0.0793
+-0.0008
+0.0778
+800
+0.0011
+0.0700
+-0.0005
+0.0674
+1000
+-0.0014
+0.0612
+-0.0018
+0.0637
+Table 2: The proposed subsample estimate of β1 for Case IV and ε ∼ N(0, 1).
+n = 106
+n = 107
+r
+Bias
+SD
+Bias
+SD
+200
+0.0004
+0.0551
+0.0005
+0.0555
+400
+-0.0003
+0.0394
+0.0003
+0.0392
+600
+0.0002
+0.0313
+-0.0020
+0.0312
+800
+0.0012
+0.0273
+-0.0005
+0.0267
+1000
+0.0012
+0.0242
+-0.0011
+0.0256
+Table 3: The CPU time for Case I and ε ∼ N(0, 1) with K = 5, n = 106 (seconds)
+r
+Methods
+200
+400
+600
+800
+1000
+Uniform
+0.077
+0.098
+0.145
+0.170
+0.217
+Proposed
+0.446
+0.494
+0.552
+0.615
+0.689
+Full data
+421.03
+Table 4: The CPU time for Case I and ε ∼ N(0, 1) with r = 1000, K = 5 and
+p = 30 (seconds)
+n
+Methods
+104
+105
+106
+107
+Uniform
+0.411
+0.417
+0.447
+0.490
+Proposed
+0.586
+0.620
+0.922
+5.393
+Full data
+4.43
+61.60
+676.08
+4667.22
+
+Springer Nature 2021 LATEX template
+REFERENCES
+21
+Table 5: The CPs and the average lengths (in parenthesis) of the confident interval
+of β1 with n = 106, r = 1000 and K = 5.
+Error
+B
+Case I
+Case II
+Case III
+Case IV
+20
+0.930(0.030)
+0.948(0.034)
+0.932(0.014)
+0.920(0.021)
+40
+0.928(0.021)
+0.924(0.024)
+0.936(0.010)
+0.954(0.015)
+N(0, 1)
+60
+0.952(0.018)
+0.942(0.020)
+0.942(0.009)
+0.944(0.013)
+80
+0.918(0.015)
+0.934(0.017)
+0.926(0.008)
+0.914(0.011)
+100
+0.936(0.014)
+0.934(0.016)
+0.930(0.007)
+0.916(0.010)
+20
+0.926(0.054)
+0.920(0.060)
+0.938(0.026)
+0.930(0.038)
+40
+0.932(0.038)
+0.934(0.044)
+0.922(0.019)
+0.954(0.027)
+mixNormal
+60
+0.924(0.031)
+0.936(0.036)
+0.930(0.015)
+0.934(0.023)
+80
+0.928(0.027)
+0.928(0.031)
+0.934(0.014)
+0.946(0.020)
+100
+0.930(0.025)
+0.934(0.028)
+0.932(0.012)
+0.948(0.018)
+20
+0.940(0.037)
+0.940(0.041)
+0.928(0.018)
+0.954(0.026)
+40
+0.944(0.026)
+0.960(0.030)
+0.946(0.013)
+0.916(0.019)
+t(3)
+60
+0.946(0.022)
+0.968(0.025)
+0.936(0.010)
+0.936(0.016)
+80
+0.940(0.019)
+0.944(0.021)
+0.946(0.009)
+0.940(0.013)
+100
+0.948(0.017)
+0.944(0.019)
+0.934(0.008)
+0.914(0.012)
+20
+0.932(0.053)
+0.944(0.060)
+0.918(0.026)
+0.936(0.038)
+40
+0.926(0.037)
+0.932(0.043)
+0.922(0.018)
+0.944(0.027)
+Cauchy
+60
+0.924(0.031)
+0.942(0.036)
+0.930(0.015)
+0.926(0.022)
+80
+0.938(0.027)
+0.946(0.031)
+0.934(0.013)
+0.924(0.020)
+100
+0.942(0.024)
+0.952(0.028)
+0.926(0.012)
+0.928(0.018)
+Table 6: The number of yearly data and allocation sizes (r = 1000)
+Years
+nk
+rk
+Years
+nk
+rk
+1987
+1,287,333
+11
+1998
+5,227,051
+45
+1988
+5,126,498
+47
+1999
+5,360,018
+45
+1989
+4,925,482
+45
+2000
+5,481,303
+45
+1990
+5,110,527
+46
+2001
+4,873,031
+42
+1991
+4,995,005
+46
+2002
+5,093,462
+45
+1992
+5,020,651
+47
+2003
+6,375,689
+56
+1993
+4,993,587
+46
+2004
+6,987,729
+59
+1994
+5,078,411
+46
+2005
+6,992,838
+58
+1995
+5,219,140
+46
+2006
+7,003,802
+57
+1996
+5,209,326
+44
+2007
+7,275,288
+58
+1997
+5,301,999
+47
+2008
+2,319,121
+19
+
+Springer Nature 2021 LATEX template
+22
+REFERENCES
+Table 7: The estimator and the length of confident interval for ˆβL with different r
+and B for the airline data.
+B
+r
+40
+100
+200
+β1
+-0.0524 (-0.0675,-0.0373)
+-0.0458 (-0.0545,-0.0370)
+β2
+0.9232 (0.9164, 0.9299)
+0.9183 (0.9142,0.9225)
+β3
+-0.0242 (-0.0320, -0.0164)
+-0.0221 (-0.0261,-0.0181)
+600
+β1
+-0.0450 (-0.0539,-0.0361)
+-0.0479 (-0.0537,-0.0421)
+β2
+0.9172 (0.9127,0.9217)
+0.9203 (0.9179,0.9227)
+β3
+-0.0268 (-0.0309,-0.0228)
+-0.0264 (-0.0288,-0.0240)
+1000
+β1
+-0.0446 (-0.0509,-0.0383)
+-0.0404 (-0.0445,-0.0363)
+β2
+0.9192 (0.9163,0.9220)
+0.9205 (0.9184,0.9226)
+β3
+-0.0238 (-0.0269,-0.0208)
+-0.0277 (-0.0297,-0.0257)
+Fig. 1: The MSEs for different subsampling methods with K = 5 and n = 106
+(Case 1).
+
+E~N(0,1)
+E-mixNormal
+0.040
+Unif-MSE
+0.14
+Unif-MSE
+Lopt-MSE
+Lopt-MSE
+0.10
+MSE
+0.025
+MSE
+0.06
+0.010
+0.02
+200
+400
+600
+800
+1000
+200
+400
+600
+800
+1000
+E~t(3)
+E-Cauchy
+90'0
+Unif-MSE
+0.14
+Unif-MSE
+Lopt-MSE
+Lopt-MSE
+0.04
+0.10
+MSE
+MSE
+0.06
+0.02
+0
+200
+400
+600
+800
+1000
+200
+400
+600
+800
+1000
+-Springer Nature 2021 LATEX template
+REFERENCES
+23
+Fig. 2: The MSEs for different subsampling methods with K = 10 and n =
+106(Case 1).
+
+E~N(0,1)
+E-mixNormal
+90'0
+Unif-MSE
+0.14
+Unif-MSE
+Lopt-MSE
+Lopt-MSE
+0.10
+MSE
+0.03
+MSE
+0.06
+0.01
+0.02
+200
+400
+600
+800
+1000
+200
+400
+600
+800
+1000
+r
+(
+E~t(3)
+E~Cauchy
+20'0
+Unif-MSE
+0.14
+Unif-MSE
+90'0
+Lopt-MSE
+Lopt-MSE
+0.10
+MSE
+MSE
+0.03
+0.06
+.01
+0.02
+200
+400
+600
+800
+1000
+200
+400
+600
+800
+1000
+-Springer Nature 2021 LATEX template
+24
+REFERENCES
+Fig. 3: The MSEs for different subsampling methods with ε ∼ N(0, 1)(Case
+IV).
+
+n=106
+n=107
+0.025
+Unif-MSE
+0.025
+Unif-MSE
+Lopt-MSE
+Lopt-MSE
+MSE
+0.015
+MSE
+0.015
+.005
+200
+400
+600
+800
+1000
+200
+400
+600
+800
+1000
+rSpringer Nature 2021 LATEX template
+REFERENCES
+25
+Fig. 4: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000
+(Case 1).
+
+E~N(0,1)
+E~mixNormal
+4e-04
+0.0020
+EMSE
+EMSE
+AMSE
+AMSE
+MSE
+2e-04
+MSE
+0.0010
+00+a0
+0000'0
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100
+B
+8
+E~t(3)
+E~Cauchy
+6e-04
+0.0020
+EMSE
+EMSE
+AMSE
+AMSE
+MSE
+3e-04
+MSE
+0.0010
+00+a0
+0000'0
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100
+8
+8Springer Nature 2021 LATEX template
+26
+REFERENCES
+Fig. 5: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000
+(Case II).
+
+E~N(0,1)
+E-mixNormal
+4e-04
+0.0020
+EMSE
+EMSE
+AMSE
+AMSE
+MSE
+2e-04
+MSE
+0.0010
+00+a0
+0000'0
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100
+8
+8
+E~t(3)
+E~Cauchy
+6e-04
+0.0020
+EMSE
+EMSE
+AMSE
+AMSE
+MSE
+3e-04
+MSE
+0.0010
+00+a0
+0000'0
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100
+8
+BSpringer Nature 2021 LATEX template
+REFERENCES
+27
+Fig. 6: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000
+(Case III).
+
+E~N(0,1)
+E-mixNorma
+05000'0
+90-a8
+EMSE
+EMSE
+AMSE
+AMSE
+MSE
+MSE
+0.00015
+4e-05
+00+a0
+00000'0
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100
+B
+B
+E~t(3)
+E~Cauchy
+0.00020
+05000'0
+EMSE
+EMSE
+AMSE
+AMSE
+0.00010
+0.00015
+MSE
+MSE
+00000'0
+00000'0
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100
+8
+8Springer Nature 2021 LATEX template
+28
+REFERENCES
+Fig. 7: The EMSEs and AMSEs of ˆθL with different values of B and r = 1000
+(Case IV).
+
+E~N(0,1)
+E-mixNormal
+0.00020
+6e-04
+EMSE
+EMSE
+AMSE
+AMSE
+0.00010
+SE
+MSE
+3e-04
+0.00000
+00+a0
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100
+8
+8
+E~t(3)
+E-Cauchy
+05000'0
+6e-04
+EMSE
+EMSE
+AMSE
+AMSE
+0.00015
+WSE
+MSE
+3e-04
+00000'0
+00+a0
+20
+40
+60
+80
+100
+20
+40
+60
+80
+100
+8
+8Springer Nature 2021 LATEX template
+REFERENCES
+29
+Fig. 8: The results of MSEs for the airline data.
+
+800'0
+Unif-MSE
+Lopt-MSE
+MSE
+0.004
+000'0
+200
+400
+600
+800
+1000
+r

KNFOT4oBgHgl3EQfyzQ_/content/tmp_files/2301.12929v1.pdf.txt

@@ -0,0 +1,2263 @@
+Can Persistent Homology provide an efficient alternative for
+Evaluation of Knowledge Graph Completion Methods?
+Anson Bastos
+[email protected]
+IIT, Hyderabad
+India
+Kuldeep Singh
+[email protected]
+Zerotha Research and
+Cerence GmbH
+Germany
+Abhishek Nadgeri
+[email protected]
+Zerotha Research and
+RWTH Aachen
+Germany
+Johannes Hoffart
+[email protected]
+SAP
+Germany
+Toyotaro Suzumura
+[email protected]
+The University of Tokyo
+Japan
+Manish Singh
+[email protected]
+IIT Hyderabad
+India
+ABSTRACT
+In this paper we present a novel method, Knowledge Persistence
+(KP), for faster evaluation of Knowledge Graph (KG) completion
+approaches. Current ranking-based evaluation is quadratic in the
+size of the KG, leading to long evaluation times and consequently a
+high carbon footprint. KP addresses this by representing the topol-
+ogy of the KG completion methods through the lens of topological
+data analysis, concretely using persistent homology. The character-
+istics of persistent homology allow KP to evaluate the quality of
+the KG completion looking only at a fraction of the data. Experi-
+mental results on standard datasets show that the proposed metric
+is highly correlated with ranking metrics (Hits@N, MR, MRR). Per-
+formance evaluation shows that KP is computationally efficient:
+In some cases, the evaluation time (validation+test) of a KG com-
+pletion method has been reduced from 18 hours (using Hits@10)
+to 27 seconds (using KP), and on average (across methods & data)
+reduces the evaluation time (validation+test) by ≈ 99.96%.
+ACM Reference Format:
+Anson Bastos, Kuldeep Singh, Abhishek Nadgeri, Johannes Hoffart, Toyotaro
+Suzumura, and Manish Singh. 2023. Can Persistent Homology provide
+an efficient alternative for Evaluation of Knowledge Graph Completion
+Methods?. In Proceedings of the Web Conference 2023 (WWW ’23), APRIL 30 -
+MAY 4, 2023, Texas, USA. WWW, Texas, USA, 13 pages. https://doi.org/10.
+XXXXX/YYYYY.3449917
+1
+INTRODUCTION
+Publicly available Knowledge Graphs (KGs) find broad applicability
+in several downstream tasks such as entity linking, relation extrac-
+tion, fact-checking, and question answering [22, 41]. These KGs are
+large graph databases used to express facts in the form of relations
+between real-world entities and store these facts as triples (subject,
+Permission to make digital or hard copies of part or all of this work for
+personal or classroom use is granted without fee provided that copies are
+not made or distributed for profit or commercial advantage and that copies
+bear this notice and the full citation on the first page. Copyrights for third-
+party components of this work must be honored. For all other uses, contact
+the owner/author(s).
+WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+© 2023 Copyright held by the owner/author(s).
+ACM ISBN 978-Y-4500-YYYY-7/21/04.
+https://doi.org/10.XXXXX/YYYYY.3449917
+relation, object). KGs must be continuously updated because new en-
+tities might emerge or facts about entities are extended or updated.
+Knowledge Graph Completion (KGC) task aims to fill the missing
+piece of information into an incomplete triple of KG [5, 18, 22].
+Several Knowledge Graph Embedding (KGE) approaches have
+been proposed to model entities and relations in vector space for
+missing link prediction in a KG [55]. KGE methods infer the connec-
+tivity patterns (symmetry, asymmetry, etc.) in the KGs by defining a
+scoring function to calculate the plausibility of a knowledge graph
+triple. While calculating plausibility of a KG triple τ = (𝑒ℎ,𝑟,𝑒𝑡),
+the predicted score by scoring function affirms the confidence of a
+model that entities 𝑒𝑡 and 𝑒ℎ are linked by 𝑟.
+For evaluating KGE methods, ranking metrics have been widely
+used [22] which is based on the following criteria: given a KG triple
+with a missing head or tail entity, what is the ability of the KGE
+method to rank candidate entities averaged over triples in a held-
+out test set [28]? These ranking metrics are useful as they intend to
+gauge the behavior of the methods in real world applications of KG
+completion. Since 2019, over 100 KGE articles have been published
+in various leading conferences and journals that use ranking metrics
+as evaluation protocol1.
+Limitations of Ranking-based Evaluation: The key challenge
+while computing ranking metrics for model evaluation is the time
+taken to obtain them. Since the (most of) KGE models aim to rank
+all the negative triples that are not present in the KG [8, 9], comput-
+ing these metrics takes a quadratic time in the number of entities
+in the KG. Moreover, the problem gets alleviated in the case of
+hyper-relations [62] where more than two entities participate, lead-
+ing to exponential computation time. For instance, Ali et al. [2]
+spent 24,804 GPU hours of computation time while performing a
+large-scale benchmarking of KGE methods.
+There are two issues with high model evaluation time. Firstly,
+efficiency at evaluation time is not a widely-adapted criterion for
+assessing KGE models alongside accuracy and related measures.
+There are efforts to make KGE methods efficient at training time
+[52, 54]. However, these methods also use ranking-based protocols
+resulting in high evaluation time. Secondly, the need for signifi-
+cant computational resources for the KG completion task excludes a
+large group of researchers in universities/labs with restricted GPU
+1https://github.com/xinguoxia/KGE#papers
+arXiv:2301.12929v1  [cs.LG]  30 Jan 2023
+
+WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+Bastos, et al.
+availability. Such preliminary exclusion implicitly challenges the ba-
+sic notion of various diversity and inclusion initiatives for making
+the Web and its related research accessible to a wider community.
+In past, researchers have worked extensively towards efficient Web-
+related technologies such as Web Crawling [12], Web Indexing [25],
+RDF processing [17], etc. Hence, for the KG completion task, similar
+to other efficient Web-based research, there is a necessity to develop
+alternative evaluation protocols to reduce the computation com-
+plexity, a crucial research gap in available KGE scientific literature.
+Another critical issue in ranking metrics is that they are biased
+towards popular entities and such popularity bias is not captured
+by current evaluation metrics [28]. Hence, we need a metric which
+is efficient than popular ranking metrics and also omits such biases.
+Motivation and Contribution: In this work, we focus on ad-
+dressing above-mentioned key research gaps and aim for the first
+study to make KGE evaluation more efficient. We introduce Knowl-
+edge Persistence(KP), a method for characterizing the topology
+of the learnt KG representations. It builds upon Topological Data
+Analysis [58] based on the concepts from Persistent Homology(PH)
+[15], which has been proven beneficial for analyzing deep networks
+[29, 36]. PH is able to effectively capture the geometry of the mani-
+fold on which the representations reside whilst requiring fraction of
+data [15]. This property allows to reduce the quadratic complexity
+of considering all the data points (KG triples in our case) for rank-
+ing. Another crucial fact that makes PH useful is its stability with
+respect to perturbations making KP robust to noise [19] mitigating
+the issues due to the open-world problem. Thus we use PH due to
+its effectiveness for limited resources and noise [50]. Concretely,
+the following are our key contributions:
+(1) We propose (KP), a novel approach along with its theoreti-
+cal foundations to estimate the performance of KGE models
+through the lens of topological data analysis. This allows us
+to drastically reduce the computation factor from order of
+O(|E|2) to O(|E|). The code is here.
+(2) We run extensive experiments on families of KGE methods
+(e.g., Translation, Rotation, Bi-Linear, Factorization, Neural
+Network methods) using standard benchmark datasets. The
+experiments show that KP correlates well with the stan-
+dard ranking metrics. Hence, KP could be used for faster
+prototyping of KGE methods and paves the way for efficient
+evaluation methods in this domain.
+In the remainder of the paper, related work is in section 2. Section
+3 briefly explains the concept of persistent homology. Section 4
+describes the proposed method. Later, section 5 shows associated
+empirical results and we conclude in section 7.
+2
+RELATED WORK
+Broadly, KG embeddings are classified into translation and semantic
+matching models [55]. Translation methods such as TransE [8],
+TransH [57], TransR [26] use distance-based scoring functions.
+Whereas semantic matching models (e.g., ComplEx [48], Distmult
+[60], RotatE [44]) use similarity-based scoring functions.
+Kadlec et al. [23] first pointed limitations of KGE evaluation
+and its dependency on hyperparameter tuning. [45] with exhaus-
+tive evaluation (using ranking metrics) showed issues of scoring
+functions of KGE methods whereas [31] studied the effect of loss
+function of KGE performance. Jain et al. [20] studied if KGE meth-
+ods capture KG semantic properties. Work in [35] provides a new
+dataset that allows the study of calibration results for KGE mod-
+els. Speranskaya et al. [43] used precision and recall rather than
+rankings to measure the quality of completion models. Authors pro-
+posed a new dataset containing triples such that their completion
+is both possible and impossible based on queries. However, queries
+were build by creating a tight dependency on such queries for the
+evaluation as pointed by [47]. Rim et al. [37] proposed a capability-
+based evaluation where the focus is to evaluate KGE methods on
+various dimensions such as relation symmetry, entity hierarchy,
+entity disambiguation, etc. Mohamed et al. [28] fixed the popularity
+bias of ranking metrics by introducing modified ranking metrics.
+The geometric perspective of KGE methods was introduced by [40]
+and its correlation with task performance. Berrendorf et al. [6] sug-
+gested the adjusted mean rank to improve reciprocal rank, which
+is an ordinal scale. Authors do not consider the effect of negative
+triples available for a given triple under evaluation. [47] propose
+to balances the number of negatives per triple to improve rank-
+ing metrics. Authors suggested the preparation of training/testing
+splits by maintaining the topology. Work in [24] proposes efficient
+non-sampling techniques for KG embedding training, few other
+initiatives improve efficiency of KGE training time [52–54], and
+hyperparameter search efficiency of embedding models [49, 56, 63].
+Overall, the literature is rich with evaluations of knowledge
+graph completion methods [4, 21, 38, 46]. However, to the best
+of our knowledge, extensive attempts have not been made to im-
+prove KG evaluation protocols’ efficiency, i.e., to reduce run-time
+of widely-used ranking metrics for faster prototyping. We position
+our work orthogonal to existing attempts such as [40], [47], [28],
+and [37]. In contrast with these attempts, our approach provides a
+topological perspective of the learned KG embeddings and focuses
+on improving the efficiency of KGE evaluations.
+3
+PRELIMINARIES
+We now briefly describe concepts used in this paper.
+Ranking metrics have been used for evaluating KG embedding
+methods since the inception of the KG completion task [8]. These
+metrics include the Mean Rank (MR), Mean Reciprocal Rank (MRR)
+and the cut-off hit ratio (Hits@N (N=1,3,10)). MR reports the average
+predicted rank of all the labeled triples. MRR is the average of the
+inverse rank of the labelled triples. Hits@N evaluates the fraction
+of the labeled triples that are present in the top N predicted results.
+Persistent Homology (PH) [15, 19]: studies the topological
+features such as components in 0-dimension (e.g., a node), holes in
+1-dimension (e.g., a void area bounded by triangle edges) and so
+on, spread over a scale. Thus, one need not choose a scale before-
+hand. The number(rank) of these topological features(homology
+group) in every dimension at a particular scale can be used for
+downstream applications. Consider the simplicial complex ( e.g.,
+point is a 0-simplex, an edge is a 1-simplex, a triangle is a 2-simplex
+) 𝐶 with weights 𝑎0 ≤ 𝑎1 ≤ 𝑎2 . . . 𝑎𝑚−1, which could represent the
+edge weights, for example, the triple score from the KG embed-
+ding method in our case. One can then define a Filtration process
+[15], which refers to generating a nested sequence of complexes
+𝜙 ⊆ 𝐶1 ⊆ 𝐶2 ⊆ . . .𝐶𝑚 = 𝐶 in time/scale as the simplices below
+
+Can Persistent Homology provide an efficient alternative
+WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+Figure 1: Calculating Knowledge Persistence(KP) score from the given KG and KG embedding method. The KG is sampled for
+positive(G+) and negative(G−) triples (step one), keeping the order O(|E|). The edge weights represent the score obtained from
+the KG embedding method. In step two, the persistence diagram (PD) is computed using filtration process explained in Figure
+2. In final step, a Sliced Wasserstein distance (SW) is obtained between the PDs of G+ and G− to get the KP score. However,
+ranking metrics run the KGE methods over all the O(|E|2) triples as explained in bottom left part of the figure(red box).
+the threshold weights are added in the complex. The filtration pro-
+cess [15] results in the creation(birth) and destruction(death) of
+components, holes, etc. Thus each structure is associated with a
+birth-death pair (𝑎𝑖,𝑎𝑗) ∈ 𝑅2 with 𝑖 ≤ 𝑗. The persistence or life-
+time of each component can then be given by 𝑎𝑗 − 𝑎𝑖. A persistence
+diagram (PD) summarizes the (birth,death) pair of each object on
+a 2D plot, with birth times on the x axis and death times on the y
+axis. The points near the diagonal are shortlived components and
+generally are considered noise (local topology), whereas the persis-
+tent objects (global topology) are treated as features. We consider
+local and global topology to compare two PDs (i.e., positive and
+negative triple graphs in our case).
+4
+PROBLEM STATEMENT AND METHOD
+4.1
+Problem Setup
+We define a KG as a tuple 𝐾𝐺 = (E, R, T +) where E denotes
+the set of entities (vertices), R is the set of relations (edges), and
+T + ⊆ E × R × E is a set of all triples. A triple τ = (𝑒ℎ,𝑟,𝑒𝑡) ∈ T +
+indicates that, for the relation 𝑟 ∈ R, 𝑒ℎ is the head entity (origin
+of the relation) while 𝑒𝑡 is the tail entity. Since 𝐾𝐺 is a multigraph;
+𝑒ℎ = 𝑒𝑡 may hold and |{𝑟𝑒ℎ,𝑒𝑡 }| ≥ 0 for any two entities. The KG
+completion task predicts the entity pairs ⟨𝑒𝑖,𝑒𝑗⟩ in the KG that have
+a relation 𝑟𝑐 ∈ R between them.
+4.2
+Proposed Method
+In this section we describe our approach for evaluating KG embed-
+ding methods using the theory of persistent homology (PH) . This
+process is divided into three steps ( Figure 1), namely: (i) Graph con-
+struction, (ii) Filtration process and (iii) Sliced Wasserstein distance
+computation. The first step creates two graphs (one for positive
+triples, another for negative triples) using sampling(O(V) triples),
+with scores calculated by a KGE method as edge weights. The
+second step considers these graphs and, using a process called "fil-
+tration," converts to an equivalent lower dimension representation.
+The last step calculates the distance between graphs to provide a
+final metric score. We now detail the approach.
+4.2.1
+Graph Construction. We envisioned KGE from the topolog-
+ical lens while proposing an efficient solution for its evaluation.
+Previous works such as [40] proposed a KGE metric only consider-
+ing embedding space. However, we intend to preserve the topology
+(graph structure and its topological feature) along with the KG em-
+bedding features. We first construct graphs of positive and negative
+triples. We denote a graph as (V, E) where V is the set of 𝑁 nodes
+and E represents the edges between them. Consider a KG embed-
+ding method M that takes as input the triple τ = (ℎ,𝑟,𝑡) ∈ T and
+gives the score 𝑠τ of it being a right triple. We construct a weighted
+directed graph G+ from positive triples τ ∈ T + in the train set,
+with the entities as the nodes and the relations between them as the
+edges having 𝑠τ as the edge weights. Here, 𝑠τ is the score calculated
+by KGE method for a triple and we propose to use it as the edge
+weights. Our idea is to capture topology of graph (G+) with repre-
+sentation learned by a KG embedding method. We sample an order
+of O(|E|) triples, |E| being the number of entities to keep compu-
+tational time linear. Similarly, we construct a negative graph G−
+by sampling the same number of unknown triples as the positive
+samples. One question may arise if KP is robust to sampling, that
+we answer theoretically in Theorem 4.4 and empirically in section
+6. Note, here we do not take all the negative triples in the graphs
+and consider only a fraction of what the ranking metrics need. This
+is a fundamental difference with ranking metrics. Ranking metrics
+use all the unlabeled triples as negatives for ranking, thus incurring
+a computational cost of 𝑂(|E|2).
+4.2.2
+Filtration Process. Having constructed the Graphs G+ and
+G−, we now need some definition of a distance between them
+
+1. Graph Construction
+2. Filtration Process
+Winneror
+gt
+a=0
+Einstein(HAE)
+Hans
+Prize(NP)
+Hans
+KG Embedding
+Einstein
+GrandSonof
+method
+Alfred
+Sonot
+n SupervisedBy
+r(A
+a=2
+3Albert
+Einstein
+Homen
+Alfred
+Birth
+3. Sliced Wasserstein
+Distance Computation
+D+
+O(E*)Graph with scores from the
+KGE method onthe edges
+HAE
+KP(G+, G-) = SW(D+, D-)
+0.3
+Ranking
+Ranking
+metric
+Albert
+Einstein
+HE
+AK
+Hermann
+Einstein
+Birth
+Ranking Metrics
+D-
+process
+Sampled O(E) Graphs
+with scores from the KGE
+method on the edgesWWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+Bastos, et al.
+Figure 2: For a KGE method, the positive triple graph G+ is used as input (leftmost graph with edge weights) and filtration
+process is applied on the edge weights (calculated by KGE method) for the graph. The filtration starts with only nodes as first
+step, and based on the edge weights, edges are added to the nodes. The persistence diagram is given on the right with red dots
+indicating 0-dimensional homology (components) and the blue dots indicating 1-dimensional homology (cycles). Persistent
+Diagram generated from this filtration process is a condensed 2D representation of G+. A similar process is repeated for G−.
+to define a metric. However, since the KGs could be large with
+many entities and relations, directly comparing the graphs could
+be computationally challenging. Therefore, we allude to the theory
+of persistent homology (PH) to summarize the structures in the
+graphs in the form of the persistence diagram (PD). Such summa-
+rizing is obtained by a process known as filtration [64]. One can
+imagine a PD as mapping of higher dimensional data to a 2D plane
+upholding the representation of data points and we can then derive
+computational efficiency for distance comparison between 2D repre-
+sentations. Specifically, we compute the 0-dimensional topological
+features (i.e., connected-nodes/components) for each graph (G−
+and G+) to keep the computation time linear. We also experimented
+using the 1-dimensional features without much empirical benefit.
+Consider the positive triple graph G+ as input (cf., Figure 2).
+We would need a scale (as pointed in section 3) for the filtration
+process. Once the filtration process starts, initially, we have a graph
+structure containing only the nodes (entities) and no edges of G+.
+For capturing topological features at various scales, we define a
+variable 𝑎 which varies from −∞ to +∞ and it is then compared
+with edge weights (𝑠τ). A scale allows to capture topology at various
+timesteps. Thus, we use the edge weights obtained from the scores
+(𝑠τ) of the KGE methods for filtration. As the filtration proceeds,
+the graph structures (components) are generated/removed. At a
+given scale 𝑎, the graph structure ((G+
+𝑠𝑢𝑏)𝑎) contains those edges
+(triples) for which 𝑠τ ≤ 𝑎. Formally, this is expressed as:
+(G+
+𝑠𝑢𝑏)𝑎 = {(V, E+
+𝑎)|E+
+𝑎 ⊆ E,𝑠τ ≤ 𝑎 ∀τ ∈ E+
+𝑎 }
+Alternatively, we add those edges for which score of the triple is
+greater than or equal to the filtration value, i.e., 𝑠τ ≥ 𝑎 defined as
+(G+
+𝑠𝑢𝑝𝑒𝑟)𝑎 = {(V, E𝑎+)|E𝑎+ ⊆ E,𝑠τ ≥ 𝑎 ∀τ ∈ E𝑎+}
+One can imagine that for filtration, graph G+ is subdivided into
+(G+
+𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎 as the filtration adds/deletes edges for cap-
+turing topological features. Hence, specific components in a sub-
+graphs will appear and certain components will disappear at differ-
+ent scale levels (timesteps) 𝑎 = 1, 3, 5 and so on. Please note, Figure 2
+explains creation of PD for (G+
+𝑠𝑢𝑏)𝑎. A similar process is repeated for
+(G+𝑠𝑢𝑝𝑒𝑟)𝑎. This expansion/contraction process enables capturing
+topology at different time-steps without worrying about defining an
+optimal scale (similar to hyperparameter). Next step is the creation
+of persistent diagrams of (G+
+𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎 where the x-axis
+and y-axis denotes the timesteps of appearance/disappearance of
+components. For creating a 2D representation graph, components of
+graphs which appear(disappear) during filtration process at 𝑎𝑥 (𝑎𝑦)
+are plotted on (𝑎𝑥,𝑎𝑦). The persistence or lifetime of each compo-
+nent can then be given by 𝑎𝑦 − 𝑎𝑥. At implementation level, one
+can view PDs(∈ 𝑅𝑁×2) of (G+
+𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎 as tensors which
+are concatenated into one common tensor representing positive
+triple graph G+. Hence, final PD of G+ is a concatenation of PDs
+of (G+
+𝑠𝑢𝑏)𝑎 and (G+𝑠𝑢𝑝𝑒𝑟)𝑎. This final persistent diagram represents
+a summary of the local and global topological features of the graph
+G+. Following are the benefits of a persistent diagram against con-
+sidering the whole graph: 1) a 2D summary of a higher dimensional
+graph structure data is highly beneficial for large graphs in terms
+of the computational efficiency. 2) The summary could contain
+fewer data points than the original graph, preserving the topologi-
+cal information. Similarly, the process is repeated for negative triple
+graph G− for creating its persistence diagram. Now, the two newly
+created PDs are used for calculating the proposed metric score.
+4.2.3
+Sliced Wasserstein distance computation. To compare two
+PDs, generally the Wasserstein distance between them is computed
+[16]. As the Wasserstein distance could be computationally costly,
+we find the sliced Wasserstein distance [13] between the PDs, which
+we empirically observe to be eight times faster on average. The
+Sliced Wasserstein distance(𝑆𝑊 ) between measures 𝜇 and 𝜈 is:
+𝑆𝑊𝑝 (𝜇,𝜈) =
+�∫
+𝑆𝑑−1 𝑊 𝑝
+𝑝 (𝑅𝜇 (.,𝜃), 𝑅𝜈 (.,𝜃))
+� 1
+𝑝
+where 𝑅𝜇 (.,𝜃) is the projection of 𝜇 along 𝜃,𝑊 is initial Wasserstein
+distance. Generally a Monte Carlo average over 𝐿 samples is done
+instead of the integral. The 𝑆𝑊 distance takes O(𝐿𝑁𝑑 +𝐿𝑁𝑙𝑜𝑔(𝑁))
+time which can be improved to linear time O(𝑁𝑑) for 𝑆𝑊2 (i.e.,
+Euclidean distance) as a closed form solution [30]. Thus,
+KP(G+, G−) = 𝑆𝑊 (𝐷+, 𝐷−)
+(1)
+where 𝐷+, 𝐷− are the persistence diagrams for G+, G− respectively.
+Since the metric is obtained by summarizing the Knowledge graph
+using Persistence diagrams we term it as Knowledge Persistence(KP).
+As KP correlates well with ranking metrics (sections 4.2.4 and 5),
+higher KP signifies a better performance of the KGE method.
+4.2.4
+Theoretical justification. This section briefly states the theo-
+retical results justifying the proposed method to approximate the
+
+1Can Persistent Homology provide an efficient alternative
+WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+ranking metrics. We begin the analysis by assuming two distribu-
+tions: One for the positive graph’s edge weights(scores) and the
+other for the negative graph. We define a metric "PERM" (Figure 3),
+that is a proxy to the ranking metrics while being continuous(for
+the definition of integrals and derivatives) for ease of theoretical
+analysis. The proof sketches are given in the appendix.
+Figure 3: Figure gives an intuition of the metric PERM which
+is designed to be a proxy to the ranking metrics for ease of
+theoretical analysis. For a given positive triple 𝜏 with score
+𝑥𝜏 the expected rank(𝐸𝑅(𝜏)) is defined as the area under the
+curve of the negative distribution from 𝑥𝜏 to ∞(shown in the
+shaded area above). PERM is then defined as the expectation
+of the expected rank under the positive distribution.
+Definition 4.1 (Expected Ranking(ER)). Consider the positive triples
+to have the distribution 𝐷+ and the negative triples to have the
+distribution 𝐷−. For a positive triple with score 𝑎 its expected rank-
+ing(ER) is defined as, 𝐸𝑅(𝑎) =
+∫ 𝑥=∞
+𝑥=𝑎
+𝐷−(𝑥)𝑑𝑥
+Definition 4.2 (PERM). Consider the positive triples to have the
+distribution 𝐷+ and the negative triples to have the distribution 𝐷−.
+The PERM metric is then defined as, 𝑃𝐸𝑅𝑀 =
+∫ 𝑥=∞
+𝑥=−∞ 𝐷+(𝑥)𝐸𝑅(𝑥)𝑑𝑥
+It is easy to see that PERM has a monotone increasing corre-
+spondence with the actual ranking metrics. That is, as many of
+the negative triples get a higher score than the positive triples, the
+distribution of the negative triples will shift further right of the pos-
+itive distribution. Hence, the area under the curve would increase
+for a given triple(x=a). We just established a monotone increasing
+correspondence of PERM with the ranking metrics, we now need
+show that there exists a one-one correspondence between PERM
+and KP. For closed-form solutions, we work with normalised dis-
+tributions (can be extended to other distributions using [39]) of
+KGE score under the following mild consideration: As the KGE
+method converges, the mean statistic(𝑚𝜈) of the scores of the posi-
+tive triples consistently lies on one side of the half-plane formed
+by the mean statistic(𝑚𝜇) of the negative triples, irrespective of the
+data distribution.
+Lemma 4.1. KP has a monotone increasing correspondence with
+the Proxy of the Expected Ranking Metrics(PERM) under the above
+stated considerations as 𝑚𝜈 deviates from 𝑚𝜇
+The above lemma shows that there is a one-one correspondence
+between KP and PERM and by definition PERM has a one-one cor-
+respondence with the ranking metrics. Therefore, the next theorem
+follows as a natural consequence:
+Theorem 4.3. KP has a one-one correspondence with the ranking
+metrics under the above stated considerations.
+The above theorem states that, with high probability, there exists
+a correlation between KP and the ranking metrics under certain
+considerations and proof details are in the appendix. In an ideal
+case, we seek a linear relationship between the proposed mea-
+sure and the ranking metric. This would help interpret whether
+an increase/decrease in the measure would cause a corresponding
+increase/decrease in the ranking metric we wish to simulate. Such
+interpretation becomes essential when the proposed metric has
+different behavior from the existing metric. While the correlation
+could be high, for interpretability of the results, we would also like
+the change in KP to be bounded for a change in the scores(ranking
+metrics). The below theorem gives a sense for this bound.
+Theorem 4.4. Under the considerations of theorem 4.3, the relative
+change in KP on addition of random noise to the scores is bounded
+by a function of the original and noise-induced covariance matrix
+as ΔK P
+K P ≤ 𝑚𝑎𝑥((1 − |Σ+1
+𝜇1 Σ−1
+𝜇2 |
+3
+2 ), (1 − |Σ+1
+𝜈1 Σ−1
+𝜈2 |
+3
+2 )), where Σ𝜇1 and
+Σ𝜈1 are the covariance matrices of the positive and negative triples’
+scores respectively and Σ𝜇2 and Σ𝜈2 are that of the corrupted scores.
+Theorem 4.4 gives a bound on the change in KP while inducing
+noise in the KGE predictions. Ideally, the error/change would be 0,
+and as the noise is increased(and the ranking changed), gradually,
+the KP value also changes in a bounded manner as desired.
+5
+EXPERIMENTAL SETUP
+For de-facto KGC task (c.f., section 4.1), we use popular KG embed-
+ding methods from its various categories: (1) Translation: TransE
+[8], TransH [57], TransR [26] (2) Bilinear, Rotation, and Factoriza-
+tion: RotatE [44] TuckER [3], and ComplEx [48], (3) Neural Network
+based: ConvKB [32]. The method selection and evaluation choices
+are similar to [28, 37] that propose new metrics for KG embeddings.
+All methods run on a single P100 GPU machine for a maximum of
+100 epochs each and evaluated every 5 epochs. For training/testing
+the KG embedding methods we make use of the pykg2vec [61]
+library and validation runs are executed 20 times on average. We
+use the standard/best hyperparameters for these datasets that the
+considered KGE methods reported [3, 8, 26, 44, 48, 57, 61].
+5.1
+Datasets
+We use standard English KG completion datasets: WN18, WN18RR,
+FB15k237, FB15k, YAGO3-10 [2, 44]. The WN18 dataset is obtained
+from Wordnet [27] containing lexical relations between English
+words. WN18RR removes the inverse relations in the WN18 dataset.
+FB15k is obtained from the Freebase [7] knowledge graph, and
+FB15k237 was created from FB15k by removing the inverse relations.
+The dataset details are in the Table 1. For scaling experiment, we
+rely on large scale YAGO3-10 dataset [2] and due to brevity, results
+for Yago3-10 are in appendix ( cf., Figure 6 and table 9).
+5.2
+Comparative Methods
+Considering ours is the first work of its kind, we select some com-
+petitive baselines as below and explain "why" we chose them. For
+evaluation, we report correlation [14] between KP and baselines
+with ranking metrics (Hits@N (N= 1,3,10), MRR and MR).
+Conicity [40]: It finds the average cosine of the angle between an
+embedding and the mean embedding vector. In a sense, it gives
+
+PERM =E.(ER(T)
+Distribution
+Distribution
+of positive
+ER(T)
+of neqative
+triples
+triples
+Score of a positive triple TWWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+Bastos, et al.
+spread of a KG embedding method in space. We would like to
+observe instead of topology, if calculating geometric properties of
+a KG embedding method be an alternative for ranking metric.
+Average Vector Length: This metric was also proposed by Sharma
+et al. [40] to study the geometry of the KG embedding methods. It
+computes the average length of the embeddings.
+Graph Kernel (GK): we use graph kernels to compare the two
+graphs(G+, G−) obtained for our approach. The rationale is to check
+if we could get some distance metric that correlates with the ranking
+metrics without persistent homology. Hence, this baseline empha-
+sizes a direct comparison for the validity of persistent homology in
+our proposed method. As an implementation, we employ the widely
+used shortest path kernel [10] to compare how the paths(edge
+weights/scores) change between the two graphs. Since the method
+is computationally expensive, we sample nodes [11] and apply the
+kernel on the sampled graph, averaging multiple runs.
+Table 1: (Open-Source)Benchmark Datasets for Experi-
+ments.
+Dataset
+Triples
+Entities
+Relations
+FB15K
+592,213
+14.951
+1,345
+FB15K-237
+272,115
+14,541
+237
+WN18
+151,442
+40,943
+18
+WN18RR
+93,003
+40,943
+11
+Yago3-10
+1,089,040
+123,182
+37
+6
+RESULTS AND DISCUSSION
+We conduct our experiments in response to the following research
+questions: RQ1: Is there a correlation between the proposed metric
+and ranking metrics for popular KG embedding methods? RQ2:
+Can the proposed metric be used to perform early stopping during
+training? RQ3: What is the computational efficiency of proposed
+metric wrt ranking metrics for KGE evaluation?
+KP for faster prototyping of KGE methods: Our core hypoth-
+esis in the paper is to develop an efficient alternative (proxy) to
+the ranking metrics. Hence, for a fair evaluation, we use the triples
+in the test set for computing KP. Ideally, this should be able to
+simulate the evaluation of the ranking metrics on the same (test)
+set. If true, there exists a high correlation between the two mea-
+sures, namely the KP and the ranking metrics. Table 2 shows the
+linear correlations between the ranking metrics and our method &
+baselines. We report the linear(Pearson’s) correlation because we
+would like a linear relationship between the proposed measure and
+the ranking metric (for brevity, other correlations are in appendix
+Tables 7, 8). This would help interpret whether an increase/decrease
+in the measure would cause a corresponding increase/decrease in
+the ranking metric that we wish to simulate. Specifically we train all
+the KG embedding methods for a predefined number of epochs and
+evaluate the finally obtained models to get the ranking metrics and
+KP. The correlations are then computed between KP and each
+of the ranking metrics. We observe that KP(test) configuration
+(triples are sampled from the test set) achieves the highest correla-
+tion coefficient value among all the existing geometric and kernel
+baseline methods in most cases. For instance, on FB15K, KP(test)
+reports high correlation value of 0.786 with Hits@1, whereas best
+baseline for this dataset (AVL) has corresponding correlation value
+as 0.339. Similarly for WN18RR, KP(test) has correlation value
+of 0.482 compared to AVL with -0.272 correlation with Hits@1.
+Conicity and AVL that provide geometric perspective shows mostly
+low positive correlation with ranking metrics whereas the Graph
+Kernel based method shows highly negative correlations, making
+these methods unsuitable for direct applicability. It indicates that
+the topology of the KG induced by the learnt representations seems
+a good predictor of the performance on similar data distributions
+with high correlation with ranking metric (answering RQ1).
+Furthermore, the results also report a configuration KP(train) in
+which we compute KP on the triples of the train set and find the
+correlation with the ranking metrics obtained from the test set.
+Here our rationale is to study whether the proposed metric would
+be able to capture the generalizability of the unseen test (real world)
+data that is of a similar distribution as the training data. Initial re-
+sults in Table 2 are promising with high correlation of KP(train)
+with ranking metric. Hence, it may enable the use of KP in settings
+without test/validation data while using the available (possibly lim-
+ited) data for training, for example, in few-shot scenarios. We leave
+this promising direction of research for future.
+6.1
+KP as a criterion for early stopping
+Does KP hold correlation while early stopping? To know
+when to stop the training process to prevent overfitting, we must
+be able to estimate the variance of the model. This is generally done
+by observing the validation/test set error. Thus, to use a method as
+a criterion for early stopping, it should be able to predict this gen-
+eralization error. Table 2 explains that KP(Train) can predict the
+generalizability of methods on the last epoch, it remains to empiri-
+cally verify that KP also predicts the performance at every interval
+during the training process. Hence, we study the correlations of
+the proposed method with the ranking metrics for individual KG
+embedding methods in the intra-method setting. Specifically, for
+a given method, we obtain the KP score and the ranking metrics
+on the test set and compute the correlations at every evaluation
+interval. Results in Table 3 suggest that KP has a decent correla-
+tion in the intra-method setting. It indicates that KP could be used
+in place of the ranking metrics for deciding a criterion on early
+stopping if the score keeps persistently falling (answering RQ2).
+What is the relative error of early stopping between KP
+and Ranking Metric? To further cross-validate our response to
+RQ2, we now compute the absolute relative error between the rank-
+ing metrics of the best models selected by KP and the expected
+ranking metrics. Ideally, we would expect the performance of the
+model obtained using this process on unseen test data(preferably
+of the same distribution) to be close to the best achievable result,
+i.e., the relative error should be small. This is important as if we
+were to use any metric for faster prototyping, it should also be
+a good criterion for model selection(selecting a model with less
+generalization error) and being efficient. Table 4 shows that the
+relative error is marginal, of the order of 10−2, in most cases(with
+few exceptions), indicating that KP could be used for early stop-
+ping. The deviation is higher for some methods, such as ConvKB,
+
+Can Persistent Homology provide an efficient alternative
+WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+Metrics
+FB15K
+FB15K237
+WN18
+WN18RR
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Conicity
+-0.156
+-0.170
+-0.202
+0.085
+-0.183
+0.509
+0.379
+0.356
+-0.352
+0.424
+-0.052
+-0.096
+-0.123
+0.389
+-0.096
+-0.267
+-0.471
+-0.510
+0.266
+-0.448
+AVL
+0.339
+0.325
+0.261
+-0.423
+0.308
+-0.527
+-0.149
+-0.158
+0.188
+-0.284
+0.805
+0.825
+0.856
+-0.884
+0.840
+-0.272
+-0.456
+-0.488
+0.303
+-0.438
+GK(train)
+-0.825
+-0.852
+-0.815
+0.952
+-0.843
+-0.903
+-0.955
+-0.972
+0.970
+-0.965
+-0.645
+-0.648
+-0.669
+0.611
+-0.663
+-0.518
+-0.808
+-0.840
+0.591
+-0.779
+GK(test)
+-0.285
+-0.318
+-0.247
+0.629
+-0.300
+-0.031
+-0.130
+-0.123
+0.101
+-0.095
+-0.579
+-0.565
+-0.569
+0.412
+-0.575
+-0.276
+-0.589
+-0.658
+0.470
+-0.549
+KP (Train)
+0.482
+0.418
+0.449
+-0.072
+0.433
+0.773
+0.711
+0.702
+-0.714
+0.745
+0.769
+0.769
+0.782
+-0.682
+0.780
+0.500
+0.809
+0.852
+-0.755
+0.777
+KP (Test)
+0.786
+0.731
+0.661
+-0.669
+0.721
+0.825
+0.870
+0.864
+-0.861
+0.871
+0.875
+0.887
+0.909
+-0.884
+0.899
+0.482
+0.816
+0.863
+-0.683
+0.776
+Table 2: Pearson’s linear correlation (𝑟) scores computed from the metric scores with respect to the ranking metrics on the
+standard KG embedding datasets. The KG methods are evaluated after training. Green values are the best.
+Datasets
+FB15K237
+WN18RR
+KG methods
+r
+𝜌
+𝜏
+r
+𝜌
+𝜏
+TransE
+0.955
+0.861
+0.709
+0.876
+0.833
+0.722
+TransH
+0.688
+0.570
+0.409
+0.864
+0.717
+0.555
+TransR
+0.975
+0.942
+0.811
+0.954
+0.967
+0.889
+Complex
+0.938
+0.788
+0.610
+0.833
+0.933
+0.833
+RotatE
+0.896
+0.735
+0.579
+0.774
+0.983
+0.944
+TuckER
+0.906
+0.676
+0.527
+0.352
+0.25
+0.167
+ConvKB
+0.086
+0.012
+0.007
+0.276
+0.569
+0.422
+Table 3: Correlation scores computed between KP and the
+ranking metric(Hits@10) on the standard KG embedding
+datasets with the methods evaluated at every interval as the
+training progresses. Here, r: Pearson correlation co-efficient,
+𝜌: Spearman’s correlation co-efficient, 𝜏: Kendall’s Tau.
+which had convergence issues. We infer from observed behavior
+that if the KG embedding method has not converged(to good re-
+sults), the correlation and, thus, the early stopping prediction may
+suffer. Despite a few outliers, the promising results shall encourage
+the community to research, develop, and use KGE benchmarking
+methods that are also computationally efficient.
+Datasets
+FB15K237
+WN18RR
+KG methods
+hits@1
+hits@10
+MRR
+hits@1
+hits@10
+MRR
+TransE
+0.006
+0.006
+0.007
+0.000
+0.007
+0.004
+TransH
+0.045
+0.015
+0.019
+0.130
+0.018
+0023
+TransR
+0.074
+0.045
+0.053
+0.242
+0.062
+0.016
+Complex
+0.001
+0.002
+0.003
+0.317
+0.021
+0.028
+RotatE
+0.022
+0.009
+0.007
+0.017
+0.005
+0.009
+TuckER
+0.008
+0.006
+0.002
+0.293
+0.022
+0.101
+ConvKB
+0.000
+0.043
+0.043
+0.659
+0.453
+0.569
+Table 4: Early stopping using KP. The values depict the ab-
+solute relative error between the metrics of the best models
+selected using KP and ranking metrics.
+6.2
+Timing analysis and carbon footprint
+We now study the time taken for running the evaluation (including
+evaluation at intervals) of the same methods as in section 6.1 on
+the standard datasets. Table 5 shows the evaluation times (valida-
+tion+test) and speedup for each method on the respective datasets.
+The training time is constant for ranking metric and KP. In some
+cases (ConvKB), we observe KP achieves a speedup of up to 2576
+times on model evaluation time drastically reducing evaluation time
+from 18 hours to 27 seconds; the latter is even roughly equal to the
+carbon footprint of making a cup of coffee2. Furthermore, Figure
+2https://tinyurl.com/4w2xmwry
+Figure 4: Figure shows a study on the carbon footprint on
+WN18RR when using KP vs Hits@10. The x-axis shows the
+the carbon footprint in g eq 𝐶𝑂2.
+4 illustrates the carbon footprints [33, 59] of the overall process
+(training + evaluation) for the methods when using KP vs ranking
+metrics. Due to evaluation time drastically reduced by KP, it also
+reduces overall carbon footprints. The promising results validate
+our attempt to develop alternative method for faster prototyping
+of KGE methods, thus saving carbon footprint (answering RQ3).
+6.3
+Ablation Studies
+We systematically provide several studies to support our evaluation
+and characterize different properties of KP.
+Robustness to noise induced by sampling: An important
+property that makes persistent homology worthwhile is its stability
+concerning perturbations of the filtration function. This means that
+persistent homology is robust to noise and encodes the intrinsic
+topological properties of the data [19]. However, in our applica-
+tion of predicting the performance of KG embedding methods, one
+source of noise is because of sampling the negative and positive
+triples. It could cause perturbations in the graph topology due to the
+addition and deletion of edges (cf., Figure 2). Therefore, we would
+like the proposed metric to be stable concerning perturbations. To
+understand the behavior of KP against this noise, we conduct a
+study by incrementally adding samples to the graph and observing
+the mean and standard deviation of the correlation at each stage.
+In an ideal case, assuming the KG topology remains similar, the
+mean correlations should be in a narrow range with slight standard
+deviations. We observe a similar effect in Figure 5 where we report
+the mean correlation at various fractions of triples sampled, with
+the standard deviation(error bands). Here, the mean correlation
+coefficients are within the range of 0.06(0.04), and the average stan-
+dard deviations are about 0.02(0.02) for the FB15K237(WN18RR)
+
+Carbon Footprint of the Overall KGE prototyping process
+Carbon Footprint using KP
+Carbon Footprint using ranking metrics
+TransE
+TransH
+TransR
+Complex
+RotatE
+ConvKB
+TuckER
+0
+100
+200
+300
+400WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+Bastos, et al.
+Metrics
+Hits@10
+KP
+Speedup ↑
+Dataset
+FB15K237
+WN18RR
+FB15K237
+WN18RR
+FB15K237
+WN18RR
+split
+val + test
+val + test
+val + test
+val + test
+Avg
+Avg
+TransE
+103.6
+86.1
+0.337
+0.120
+x 322.8
+x 754.1
+TransH
+37.1
+21.2
+0.333
+0.099
+x 117.0
+x 224.4
+TransR
+192.0
+137.1
+0.352
+0.135
+x 572.0
+x 1066.4
+Complex
+136.1
+151.4
+0.340
+0.142
+x 420.1
+x 1121.7
+RotatE
+174.2
+155.2
+0.359
+0.142
+x 509.5
+x 1145.6
+TuckER
+94.8
+22.1
+0.332
+0.098
+x 300.0
+x 241.9
+ConvKB
+1106.0
+138.1
+0.451
+0.139
+x 2576.6
+x 1044.3
+Table 5: Evaluation Metric Comparison wrt Computing
+Time (in minutes, for 100 epochs). Column 1 denotes
+popular KGE methods. Depicted values denote evalua-
+tion(validation+test) time for computing a metric and corre-
+sponding speedup using KP. KP significantly reduces the
+evaluation time (green).
+dataset. This shows that KP inherits the robustness of the topo-
+logical data analysis techniques, enabling linear time by sampling
+from the graph for dense KGs while keeping it robust.
+Figure 5: Effect of sample size on the correlation coefficient
+between KP and the ranking metrics on FB15K237 (left dia-
+gram) and WN18RR datasets. The correlations for the differ-
+ent sampling fractions are comparable. Also, the standard
+deviation is less, indicating the method’s robustness due to
+changes in local topology while doing sampling.
+Generalizability Study- Correlation with Stratified Rank-
+ing Metric: Mohamed et al. [28] proposed a new stratified metric
+(strat-metric) that can be tuned to focus on the unpopular entities,
+unlike the standard ranking metrics, using certain hyperparameters
+(𝛽𝑒 ∈ (−1, 1), 𝛽𝑟 ∈ (−1, 1)). Special cases of these hyperparame-
+ters give the micro and macro ranking metrics. Goal here is to
+study whether our method can predict strat-metric for the spe-
+cial case of 𝛽𝑒 = 1, 𝛽𝑟 = 0, which estimates the performance for
+unpopular(sparse) entities. Also, we aim to observe if KP holds
+a correlation with variants of the ranking metric concerning its
+generalization ability. The results (cf., Table 6) shows that KP has
+a good correlation with each of the stratified ranking metrics which
+indicate KP also takes into account the local geometry/topology
+[1] of the sparse entities and relations.
+6.4
+Summary of Results and Open Directions
+To sum up, following are key observations gleaning from empirical
+studies: 1) KP shows high correlation with ranking metrics (Ta-
+ble 2) and its stratified version (Table 6). It paves the way for the
+use of KP for faster prototyping of KGE methods. 2) KP holds
+Datasets
+FB15K237
+WN18RR
+Metrics
+r
+𝜌
+𝜏
+r
+𝜌
+𝜏
+Strat-Hits@1 (↑)
+0.965
+0.857
+0.714
+0.513
+0.482
+0.411
+Strat-Hits@3 (↑)
+0.898
+0.821
+0.619
+0.691
+0.714
+0.524
+Strat-Hits@10 (↑)
+0.871
+0.821
+0.619
+0.870
+0.750
+0.619
+Strat-MR (↓)
+-0.813
+-0.679
+-0.524
+-0.701
+-0.821
+-0.619
+Strat-MRR (↑)
+0.806
+0.679
+0.524
+0.658
+0.714
+0.524
+Table 6: KP correlation with stratified ranking metrics as
+proposed in [28].
+a high correlation at every interval during the training process
+(Table 3) with marginal relative error; hence, it could be used for
+early stopping of a KGE method. 3) KP inherits key properties of
+persistent homology, i.e., it is robust to noise induced by sampling.
+4) The overall carbon footprints of the evaluation cycle is drastically
+reduced if KP is preferred over ranking metrics.
+What’s Next? We show that topological data analysis based on
+persistent homology can act as a proxy for ranking metrics with
+conclusive empirical evidence and supporting theoretical founda-
+tions. However, it is the first step toward a more extensive research
+agenda. We believe substantial work is needed collectively in the
+research community to develop strong foundations, solving scal-
+ing issues (across embedding methods, datasets, KGs, etc.) until
+persistent homology-based methods are widely adopted.
+For example, there could be methods/datasets where the correla-
+tion turns out to be a small positive value or even negative, in which
+case we may not be able to use KP in the existing form to simu-
+late the ranking metrics for these methods/datasets. In those cases,
+some alteration may exist for the same and seek further exploration
+similar to what stratified ranking metric [28] does by fixing issues
+encountered in the ranking metric. Furthermore, theorem 4.4 would
+be a key to understand error bounds when interpreting limited per-
+formance (e.g., when the correlation is a small positive). However,
+this does not limit the use of KP for KGE methods as it captures and
+contrasts the topology of the positive and negative sampled graphs
+learned from these methods, which could be a useful metric by
+itself. In this paper, the emphasis is on the need for evaluation and
+benchmarking methods that are computationally efficient rather
+than providing an exhaustive one method fits all metric. We believe
+that there is much scope for future research in this direction. Some
+promising directions include 1) better sampling techniques(instead
+of the random sampling used in this paper), 2) rigorous theoretical
+analysis drawing the boundaries on the abilities/limitations across
+settings (zero-shot, few-shot, etc.), 3) using KP (and related metrics)
+in continuous spaces, that could be differentiable and approximate
+the ranking metrics, in the optimization process of KGE methods.
+7
+CONCLUSION
+We propose Knowledge Persistence (KP), first work that uses tech-
+niques from topological data analysis, as a predictor of the ranking
+metrics to efficiently evaluate the performance of KG embedding
+approaches. With theoretical and empirical evidences, our work
+brings efficiency at center stage in the evaluation of KG embedding
+methods along with traditional way of reporting their performance.
+Finally, with efficiency as crucial criteria for evaluation, we hope
+
+Hits@10
+0.90
+MRR
+0.88
+0.86
+0.84
+0.82
+0.80
+0.78
+0.2
+0.4
+0.6
+0.8
+1.00.95
+Hits@10
+MRR
+0.90
+0.85
+0.80
+0.75
+0.2
+0.4
+0.6
+0.8
+1.0Can Persistent Homology provide an efficient alternative
+WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+KGE research becomes more inclusive and accessible to the broader
+research community with limited computing resources.
+Acknowledgment This work was partly supported by JSPS
+KAKENHI Grant Number JP21K21280.
+REFERENCES
+[1] Henry Adams and Michael Moy. 2021. Topology Applied to Machine
+Learning: From Global to Local. Frontiers in Artificial Intelligence 4
+(2021), 54.
+[2] Mehdi Ali, Max Berrendorf, Charles Tapley Hoyt, Laurent Vermue,
+Mikhail Galkin, Sahand Sharifzadeh, Asja Fischer, Volker Tresp, and
+Jens Lehmann. 2021. Bringing light into the dark: A large-scale evalua-
+tion of knowledge graph embedding models under a unified framework.
+IEEE Transactions on Pattern Analysis and Machine Intelligence (2021).
+[3] Ivana Balažević, Carl Allen, and Timothy Hospedales. 2019. TuckER:
+Tensor Factorization for Knowledge Graph Completion. In Proceedings
+of the 2019 Conference on Empirical Methods in Natural Language Pro-
+cessing and the 9th International Joint Conference on Natural Language
+Processing (EMNLP-IJCNLP). 5185–5194.
+[4] Iti Bansal, Sudhanshu Tiwari, and Carlos R Rivero. 2020. The impact
+of negative triple generation strategies and anomalies on knowledge
+graph completion. In Proceedings of the 29th ACM International Con-
+ference on Information & Knowledge Management. 45–54.
+[5] Anson Bastos, Kuldeep Singh, Abhishek Nadgeri, Saeedeh Shekarpour,
+Isaiah Onando Mulang, and Johannes Hoffart. 2021. Hopfe: Knowl-
+edge graph representation learning using inverse hopf fibrations. In
+Proceedings of the 30th ACM International Conference on Information &
+Knowledge Management. 89–99.
+[6] Max Berrendorf, Evgeniy Faerman, Laurent Vermue, and Volker Tresp.
+2020. Interpretable and Fair Comparison of Link Prediction or Entity
+Alignment Methods. In 2020 IEEE/WIC/ACM International Joint Con-
+ference on Web Intelligence and Intelligent Agent Technology (WI-IAT).
+IEEE, 371–374.
+[7] Kurt D. Bollacker, Robert P. Cook, and Patrick Tufts. 2007. Freebase: A
+Shared Database of Structured General Human Knowledge. In AAAI.
+[8] Antoine Bordes, Nicolas Usunier, Alberto Garcia-Duran, Jason Weston,
+and Oksana Yakhnenko. 2013. Translating embeddings for modeling
+multi-relational data. In NeurlPS. 1–9.
+[9] Antoine Bordes, Jason Weston, Ronan Collobert, and Yoshua Bengio.
+2011. Learning structured embeddings of knowledge bases. In Twenty-
+fifth AAAI conference on artificial intelligence.
+[10] Karsten M Borgwardt and Hans-Peter Kriegel. 2005. Shortest-path
+kernels on graphs. In Fifth IEEE international conference on data mining
+(ICDM’05). IEEE, 8–pp.
+[11] Karsten M. Borgwardt, Tobias Petri, S. V. N. Vishwanathan, and Hans-
+Peter Kriegel. 2007. An Efficient Sampling Scheme For Comparison of
+Large Graphs. In Mining and Learning with Graphs, MLG.
+[12] Andrei Z Broder, Marc Najork, and Janet L Wiener. 2003. Efficient
+URL caching for world wide web crawling. In Proceedings of the 12th
+international conference on World Wide Web. 679–689.
+[13] Mathieu Carrière, Marco Cuturi, and Steve Oudot. 2017. Sliced Wasser-
+stein Kernel for Persistence Diagrams. In Proceedings of the 34th In-
+ternational Conference on Machine Learning (Proceedings of Machine
+Learning Research), Vol. 70. PMLR, 664–673.
+[14] Nian Shong Chok. 2010. Pearson’s versus Spearman’s and Kendall’s cor-
+relation coefficients for continuous data. Ph.D. Dissertation. University
+of Pittsburgh.
+[15] Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. 2000.
+Topological persistence and simplification. In Proceedings 41st annual
+symposium on foundations of computer science. IEEE, 454–463.
+[16] Brittany Fasy, Yu Qin, Brian Summa, and Carola Wenk. 2020. Compar-
+ing Distance Metrics on Vectorized Persistence Summaries. In NeurIPS
+2020 Workshop on Topological Data Analysis and Beyond.
+[17] Luis Galárraga, Katja Hose, and Ralf Schenkel. 2014. Partout: a dis-
+tributed engine for efficient RDF processing. In Proceedings of the 23rd
+International Conference on World Wide Web. 267–268.
+[18] Genet Asefa Gesese, Russa Biswas, Mehwish Alam, and Harald Sack.
+2019. A survey on knowledge graph embeddings with literals: Which
+model links better literal-ly? Semantic Web Preprint (2019), 1–31.
+[19] Felix Hensel, Michael Moor, and Bastian Rieck. 2021. A survey of topo-
+logical machine learning methods. Frontiers in Artificial Intelligence 4
+(2021), 52.
+[20] Nitisha Jain, Jan-Christoph Kalo, Wolf-Tilo Balke, and Ralf Krestel.
+2021. Do Embeddings Actually Capture Knowledge Graph Semantics?.
+In European Semantic Web Conference. Springer, 143–159.
+[21] Prachi Jain, Sushant Rathi, Soumen Chakrabarti, et al. 2020. Knowl-
+edge base completion: Baseline strikes back (again). arXiv preprint
+arXiv:2005.00804 (2020).
+[22] Shaoxiong Ji, Shirui Pan, Erik Cambria, Pekka Marttinen, and Philip S
+Yu. 2021. A survey on knowledge graphs: Representation, acquisition
+and applications. EEE Transactions on Neural Networks and Learning
+Systems (2021).
+[23] Rudolf Kadlec, Ondřej Bajgar, and Jan Kleindienst. 2017. Knowledge
+Base Completion: Baselines Strike Back. In Proceedings of the 2nd
+Workshop on Representation Learning for NLP. 69–74.
+[24] Zelong Li, Jianchao Ji, Zuohui Fu, Yingqiang Ge, Shuyuan Xu, Chong
+Chen, and Yongfeng Zhang. 2021. Efficient non-sampling knowledge
+graph embedding. In Proceedings of the Web Conference 2021. 1727–
+1736.
+[25] Lipyeow Lim, Min Wang, Sriram Padmanabhan, Jeffrey Scott Vitter,
+and Ramesh Agarwal. 2003. Dynamic maintenance of web indexes
+using landmarks. In Proceedings of the 12th international conference on
+World Wide Web. 102–111.
+[26] Yankai Lin, Zhiyuan Liu, Maosong Sun, Yang Liu, and Xuan Zhu. 2015.
+Learning entity and relation embeddings for knowledge graph com-
+pletion. In Proceedings of the AAAI Conference on Artificial Intelligence,
+Vol. 29.
+[27] George A Miller. 1995. WordNet: a lexical database for English. Com-
+mun. ACM 38, 11 (1995), 39–41.
+[28] Aisha Mohamed, Shameem Parambath, Zoi Kaoudi, and Ashraf Aboul-
+naga. 2020. Popularity agnostic evaluation of knowledge graph em-
+beddings. In Conference on Uncertainty in Artificial Intelligence. PMLR,
+1059–1068.
+[29] Michael Moor, Max Horn, Bastian Rieck, and Karsten Borgwardt. 2020.
+Topological autoencoders. In International conference on machine learn-
+ing. PMLR, 7045–7054.
+[30] Kimia Nadjahi, Alain Durmus, Pierre E Jacob, Roland Badeau, and
+Umut Simsekli. 2021. Fast Approximation of the Sliced-Wasserstein
+Distance Using Concentration of Random Projections. Advances in
+Neural Information Processing Systems 34 (2021).
+[31] Mojtaba
+Nayyeri,
+Chengjin
+Xu,
+Yadollah
+Yaghoobzadeh,
+Hamed Shariat Yazdi, and Jens Lehmann. 2019.
+Toward Un-
+derstanding The Effect Of Loss function On Then Performance Of
+Knowledge Graph Embedding. arXiv preprint arXiv:1909.00519 (2019).
+[32] Tu Dinh Nguyen, Dat Quoc Nguyen, Dinh Phung, et al. 2018. A
+Novel Embedding Model for Knowledge Base Completion Based on
+Convolutional Neural Network. In NAACL. 327–333.
+[33] David Patterson, Joseph Gonzalez, Urs Hölzle, Quoc Le, Chen Liang,
+Lluis-Miquel Munguia, Daniel Rothchild, David So, Maud Texier, and
+Jeff Dean. 2022. The Carbon Footprint of Machine Learning Training
+Will Plateau, Then Shrink. arXiv preprint arXiv:2204.05149 (2022).
+[34] Xutan Peng, Guanyi Chen, Chenghua Lin, and Mark Stevenson. 2021.
+Highly Efficient Knowledge Graph Embedding Learning with Orthog-
+onal Procrustes Analysis. In NAACL. 2364–2375.
+[35] Pouya Pezeshkpour, Yifan Tian, and Sameer Singh. 2020. Revisit-
+ing evaluation of knowledge base completion models. In Automated
+Knowledge Base Construction.
+[36] Bastian Rieck, Matteo Togninalli, Christian Bock, Michael Moor, Max
+Horn, Thomas Gumbsch, and Karsten Borgwardt. 2018. Neural Per-
+sistence: A Complexity Measure for Deep Neural Networks Using
+Algebraic Topology. In International Conference on Learning Represen-
+tations.
+[37] Wiem Ben Rim, Carolin Lawrence, Kiril Gashteovski, Mathias Niepert,
+and Naoaki Okazaki. 2021. Behavioral Testing of Knowledge Graph
+Embedding Models for Link Prediction. In 3rd Conference on Automated
+Knowledge Base Construction.
+[38] Tara Safavi and Danai Koutra. 2020. CoDEx: A Comprehensive Knowl-
+edge Graph Completion Benchmark. In Proceedings of the 2020 Confer-
+ence on Empirical Methods in Natural Language Processing (EMNLP).
+8328–8350.
+[39] Remi M Sakia. 1992. The Box-Cox transformation technique: a review.
+Journal of the Royal Statistical Society: Series D (The Statistician) 41, 2
+(1992), 169–178.
+[40] Aditya Sharma, Partha Talukdar, et al. 2018. Towards understanding
+the geometry of knowledge graph embeddings. In Proceedings of the
+56th Annual Meeting of the Association for Computational Linguistics
+(Volume 1: Long Papers). 122–131.
+
+WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+Bastos, et al.
+[41] Kuldeep Singh, Arun Sethupat Radhakrishna, Andreas Both, Saeedeh
+Shekarpour, Ioanna Lytra, Ricardo Usbeck, Akhilesh Vyas, Akmal
+Khikmatullaev, Dharmen Punjani, Christoph Lange, et al. 2018. Why
+reinvent the wheel: Let’s build question answering systems together.
+In Proceedings of the 2018 world wide web conference. 1247–1256.
+[42] C. Spearman. 1907. Demonstration of Formulæ for True Measurement
+of Correlation. The American Journal of Psychology 18, 2 (1907), 161–
+169. http://www.jstor.org/stable/1412408
+[43] Marina Speranskaya, Martin Schmitt, and Benjamin Roth. 2020. Rank-
+ing vs. Classifying: Measuring Knowledge Base Completion Quality.
+In Automated Knowledge Base Construction.
+[44] Zhiqing Sun, Zhi-Hong Deng, Jian-Yun Nie, and Jian Tang. 2018. Ro-
+tatE: Knowledge Graph Embedding by Relational Rotation in Complex
+Space. In International Conference on Learning Representations.
+[45] Zhiqing Sun, Shikhar Vashishth, Soumya Sanyal, Partha Talukdar, and
+Yiming Yang. 2020. A Re-evaluation of Knowledge Graph Completion
+Methods. In Proceedings of the 58th Annual Meeting of the Association
+for Computational Linguistics. 5516–5522.
+[46] Pedro Tabacof and Luca Costabello. 2019. Probability Calibration for
+Knowledge Graph Embedding Models. In International Conference on
+Learning Representations.
+[47] Sudhanshu Tiwari, Iti Bansal, and Carlos R Rivero. 2021. Revisiting
+the evaluation protocol of knowledge graph completion methods for
+link prediction. In Proceedings of the Web Conference 2021. 809–820.
+[48] Théo Trouillon, Johannes Welbl, Sebastian Riedel, Éric Gaussier, and
+Guillaume Bouchard. 2016.
+Complex embeddings for simple link
+prediction. In International Conference on Machine Learning. PMLR,
+2071–2080.
+[49] Ke Tu, Jianxin Ma, Peng Cui, Jian Pei, and Wenwu Zhu. 2019. Au-
+tone: Hyperparameter optimization for massive network embedding.
+In Proceedings of the 25th ACM SIGKDD International Conference on
+Knowledge Discovery & Data Mining. 216–225.
+[50] Renata Turkeš, Guido Montúfar, and Nina Otter. 2022. On the effec-
+tiveness of persistent homology. https://doi.org/10.48550/ARXIV.2206.
+10551
+[51] C. Villani. 2009. Optimal transport. Grundlehren der Mathematischen
+Wissenschaften [Fundamental Principles of Mathematical Sciences] 338
+(2009).
+[52] Haoyu Wang, Yaqing Wang, Defu Lian, and Jing Gao. 2021. A light-
+weight knowledge graph embedding framework for efficient inference
+and storage. In Proceedings of the 30th ACM International Conference
+on Information & Knowledge Management. 1909–1918.
+[53] Kai Wang, Yu Liu, Qian Ma, and Quan Z Sheng. 2021. Mulde: Multi-
+teacher knowledge distillation for low-dimensional knowledge graph
+embeddings. In Proceedings of the Web Conference 2021. 1716–1726.
+[54] Kai Wang, Yu Liu, and Quan Z Sheng. 2022. Swift and Sure: Hardness-
+aware Contrastive Learning for Low-dimensional Knowledge Graph
+Embeddings. In Proceedings of the ACM Web Conference 2022. 838–849.
+[55] Quan Wang, Zhendong Mao, Bin Wang, and Li Guo. 2017. Knowledge
+graph embedding: A survey of approaches and applications. IEEE
+Transactions on Knowledge and Data Engineering 29, 12 (2017), 2724–
+2743.
+[56] Xin Wang, Shuyi Fan, Kun Kuang, and Wenwu Zhu. 2021. Explain-
+able automated graph representation learning with hyperparameter
+importance. In International Conference on Machine Learning. PMLR,
+10727–10737.
+[57] Zhen Wang, Jianwen Zhang, Jianlin Feng, and Zheng Chen. 2014.
+Knowledge graph embedding by translating on hyperplanes. In Pro-
+ceedings of the AAAI Conference on Artificial Intelligence, Vol. 28.
+[58] Larry Wasserman. 2018. Topological data analysis. Annual Review of
+Statistics and Its Application 5 (2018), 501–532.
+[59] Carole-Jean Wu, Ramya Raghavendra, Udit Gupta, Bilge Acun, New-
+sha Ardalani, Kiwan Maeng, Gloria Chang, Fiona Aga, Jinshi Huang,
+Charles Bai, et al. 2022. Sustainable ai: Environmental implications,
+challenges and opportunities. Proceedings of Machine Learning and
+Systems 4 (2022), 795–813.
+[60] Bishan Yang, Wen-tau Yih, Xiaodong He, Jianfeng Gao, and Li Deng.
+2015. Embedding Entities and Relations for Learning and Inference in
+Knowledge Bases. In 3rd International Conference on Learning Repre-
+sentations, ICLR.
+[61] Shih-Yuan Yu, Sujit Rokka Chhetri, Arquimedes Canedo, Palash Goyal,
+and Mohammad Abdullah Al Faruque. 2021. Pykg2vec: A Python
+Library for Knowledge Graph Embedding. J. Mach. Learn. Res. 22
+(2021), 16–1.
+[62] Yufeng Zhang, Weiqing Wang, Wei Chen, Jiajie Xu, An Liu, and Lei
+Zhao. 2021. Meta-Learning Based Hyper-Relation Feature Modeling for
+Out-of-Knowledge-Base Embedding. In Proceedings of the 30th ACM
+International Conference on Information & Knowledge Management.
+2637–2646.
+[63] Yongqi Zhang, Zhanke Zhou, Quanming Yao, and Yong Li. 2022. KG-
+Tuner: Efficient Hyper-parameter Search for Knowledge Graph Learn-
+ing. CoRR abs/2205.02460 (2022).
+[64] Afra Zomorodian and Gunnar Carlsson. 2005. Computing persistent
+homology. Discrete & Computational Geometry 33, 2 (2005), 249–274.
+8
+APPENDIX
+Figure 6: Study on the carbon footprint of the evaluation
+phase of the KGE methods on YAGO3-10 when using KP vs
+Hits@10. The x-axis shows the the carbon footprint in g eq
+𝐶𝑂2 in log scale.
+Figure 7: Study on the carbon footprint of the evaluation
+phase of the KGE methods on Wikidata when using KP vs
+Hits@10. The x-axis shows the the carbon footprint in g eq
+𝐶𝑂2 in log scale.
+8.1
+Extended Evaluation
+Effect of KP on Efficient KGE Methods Evaluation: The re-
+search community has recently proposed several KGE methods
+to improve training efficiency [34, 52, 54]. Our idea in this experi-
+ment is to perceive if efficient KGE methods improve their overall
+carbon footprint using KP. For the same, we selected state-of-the-
+art efficient KGE methods: Procrustes [34] and HalE [54]. Figure
+8 illustrates that using KP for evaluation drastically reduces the
+carbon footprints of already efficient KGE methods. For instance,
+the carbon footprint of HalE is reduced from 110g (using hits@10)
+to 20g of CO2 (using KP).
+
+Carbon Footprint of the Evaluation phase of the KGE
+prototyping process (YAGO3_10)
+Carbon Footprint using Kp
+Carbon Footprint using ranking metrics
+TransE
+TransH
+TransR
+Method
+Complex
+RotatE
+ConvKB
+TuckER
+0.5
+L0Carbon Footprint of the Evaluation phase of the KGE
+prototyping process (Wikidata)
+Carbon Footprint using KP
+Carbon Footprint using ranking metrics
+TransE
+TransH
+TransR
+Method
+Complex
+RotatE
+ConvkB
+TuckER
+LOCan Persistent Homology provide an efficient alternative
+WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+Metrics
+FB15K
+FB15K237
+WN18
+WN18RR
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Conicity
+0.071
+-0.071
+0.000
+-0.036
+-0.214
+0.393
+0.250
+-0.071
+0.036
+0.250
+0.600
+0.600
+0.600
+-0.143
+0.600
+-0.393
+-0.607
+-0.607
+0.357
+-0.607
+AVL
+0.607
+0.214
+0.250
+-0.679
+0.179
+-0.107
+-0.321
+-0.429
+0.393
+-0.250
+0.886
+0.886
+0.886
+-0.771
+0.886
+0.321
+-0.143
+-0.464
+0.607
+-0.143
+Graph Kernel (Train)
+-0.536
+-0.321
+-0.357
+0.964
+-0.393
+-0.929
+-0.714
+-0.607
+0.643
+-0.821
+-0.943
+-0.943
+-0.943
+0.714
+-0.943
+-0.357
+-0.786
+-0.607
+0.571
+-0.786
+Graph Kernel (Test)
+-0.107
+0.107
+0.000
+0.893
+0.036
+-0.429
+-0.607
+-0.679
+0.464
+-0.607
+-0.657
+-0.657
+-0.657
+0.086
+-0.657
+-0.393
+-0.714
+-0.821
+0.786
+-0.714
+KP (Train)
+0.214
+0.536
+0.750
+0.000
+0.607
+0.893
+0.750
+0.643
+-0.679
+0.786
+0.829
+0.829
+0.829
+-0.600
+0.829
+0.286
+0.714
+0.643
+-0.750
+0.714
+KP (Test)
+0.964
+0.750
+0.750
+-0.536
+0.714
+0.714
+0.821
+0.857
+-0.750
+0.857
+0.943
+0.943
+0.943
+-0.829
+0.943
+0.286
+0.714
+0.643
+-0.643
+0.714
+Table 7: Spearman’s ranked correlation (𝜌) scores computed from the metric scores with respect to the ranking metrics on the
+standard KG embedding datasets. The KG methods are evaluated after training.
+Metrics
+FB15K
+FB15K237
+WN18
+WN18RR
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Hits1(↑)
+Hits3(↑)
+Hits10(↑)
+MR(↓)
+MRR(↑)
+Conicity
+0.048
+-0.048
+-0.048
+0.048
+-0.143
+0.238
+0.143
+-0.048
+-0.048
+0.143
+0.467
+0.467
+0.467
+-0.333
+0.467
+-0.238
+-0.333
+-0.429
+0.238
+-0.333
+AVL
+0.429
+0.143
+0.143
+-0.524
+0.048
+-0.143
+-0.238
+-0.429
+0.333
+-0.238
+0.733
+0.733
+0.733
+-0.600
+0.733
+0.238
+-0.048
+-0.333
+0.524
+-0.048
+Graph Kernel (Train)
+-0.429
+-0.143
+-0.143
+0.905
+-0.238
+-0.810
+-0.524
+-0.333
+0.429
+-0.714
+-0.867
+-0.867
+-0.867
+0.467
+-0.867
+-0.333
+-0.619
+-0.333
+0.333
+-0.619
+Graph Kernel (Test)
+-0.143
+0.143
+-0.048
+0.810
+0.048
+-0.429
+-0.524
+-0.524
+0.429
+-0.524
+-0.600
+-0.600
+-0.600
+0.200
+-0.600
+-0.238
+-0.524
+-0.619
+0.619
+-0.524
+KP (Train)
+0.143
+0.429
+0.619
+-0.048
+0.524
+0.714
+0.619
+0.429
+-0.524
+0.619
+0.600
+0.600
+0.600
+-0.467
+0.600
+0.238
+0.524
+0.429
+-0.619
+0.524
+KP (Test)
+0.905
+0.619
+0.619
+-0.429
+0.524
+0.619
+0.714
+0.714
+-0.619
+0.714
+0.867
+0.867
+0.867
+-0.733
+0.867
+0.238
+0.524
+0.429
+-0.429
+0.524
+Table 8: Kendall’s tau (𝜏) scores computed from the metric scores with respect to the ranking metrics on the standard KG
+embedding datasets. The KG methods are evaluated after training.
+Figure 8: Study on efficient KGE methods and their the car-
+bon footprint on WN18RR when using KP vs Hits@10. The
+x-axis shows the the carbon footprint in g eq 𝐶𝑂2.
+Robustness and Efficiency on large KGs: This ablation study
+aims to gauge the correlation behavior of KP and ranking metric
+on a large-scale KG. For the experiment, we use Yago3-10 dataset.
+A key reason to select the Yago-based dataset is that besides being
+large-scale, it has rich semantics. Results in Table 9 illustrate KP
+shows a stable and high correlation with the ranking metric, con-
+firming the robustness of KP. We show carbon footprint results in
+Figure 6 for the yago dataset. Further we also study the efficiency
+of KP on the wikidata dataset in Figure 7 which reaffirms that KP
+maintains its efficiency on large scale datasets.
+Efficiency comparison of Sliced Wasserstein vs Wasserstein
+as distance metric in KP: In this study we empirically provide
+a rationale for using sliced wasserstein as a distance metric over
+the wasserstein distance in KP. The results are in table 10. We see
+that KP using sliced wasserstein distance provides a significant
+computational advantage over wasserstein distance, while having
+a good performance as seen in the previous experiments. Thus
+we need an efficient approximation such as the sliced wasserstein
+distance as the distance metric in place of wasserstein distance in
+KP.
+Metrics
+Hits@1(↑)
+Hits@3(↑)
+Hits@10(↑)
+MR(↓)
+MRR(↑)
+r
+0.657
+0.594
+0.414
+-0.920
+0.572
+𝜌
+0.679
+0.679
+0.5
+-0.714
+0.643
+𝜏
+0.524
+0.524
+0.333
+-0.524
+0.429
+Table 9: KP correlations on the YAGO dataset.
+Metrics
+KP(W)
+KP(SW)
+Speedup ↑
+Dataset
+FB15K237
+WN18RR
+FB15K237
+WN18RR
+FB15K237
+WN18RR
+split
+val + test
+val + test
+val + test
+val + test
+Avg
+Avg
+TransE
+1136.766
+9.655
+0.321
+0.114
+x 3540.3
+x 84.6
+TransH
+2943.869
+7.549
+0.317
+0.095
+x 9278.5
+x 79.8
+TransR
+1734.576
+4.423
+0.336
+0.129
+x 5168.3
+x 34.4
+Complex
+1054.721
+13.089
+0.324
+0.135
+x 3255.3
+x 97.0
+RotatE
+865.417
+12.783
+0.342
+0.136
+x 2531.1
+x 94.3
+TuckER
+1021.649
+3.840
+0.316
+0.098
+x 3230.0
+x 39.1
+ConvKB
+719.310
+5.154
+0.429
+0.132
+x 1675.7
+x 39.0
+Table 10: Evaluation Metric Comparison wrt Computing
+Time (in minutes, for 100 epochs). Column 1 denotes
+popular KGE methods. Depicted values denote evalua-
+tion(validation+test) time for computing a metric and cor-
+responding speedup using KP(𝑆𝑊 ). KP(𝑆𝑊 ) with sliced
+wasserstein as the distance metric significantly reduces the
+evaluation time (green) in comparison with KP(𝑊 ) which
+uses the wasserstein distance.
+8.2
+Theoretical Proof Sketches
+We work under the following considerations: As the KGE method
+converges the mean statistic(𝑚𝜈) of the scores of the positive triples
+consistently lies on one side of the half plane formed by the mean
+statistic(𝑚𝜇) of the negative triples, irrespective of the data distri-
+bution. The detail proofs are here.
+Lemma 8.1. KP has a monotone increasing correspondence with
+the Proxy of the Expected Ranking Metrics(PERM) under the above
+stated considerations as 𝑚𝜈 deviates from 𝑚𝜇
+
+Carbon Footprint of the Overall KGE prototyping process using
+methods that save on training time(WN18RR)
+Carbon Footprint using KP
+ Carbon Footprint using ranking metrics
+HaLE
+Method
+ProcrustEs
+0
+25
+50
+75
+100
+125WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+Bastos, et al.
+Proof Sketch. Considering the 0-dimensional PD as used by
+KP and a normal distribution for the edge weights (can be extended
+to other distributions using techniques like [39]) of the graph(scores
+of the triples), we have univariate gaussian measures [40] 𝜇 and 𝜈
+for the positive and negative distributions respectively. Denote by
+𝑚𝜇 and 𝑚𝜈 the means of the distributions 𝜇 and 𝜈 respectively and
+by Σ𝜇, Σ𝜈 the respective covariance matrices.
+𝑊 2
+2 (𝜇,𝜈) = ∥𝜇 − 𝜈∥2 + 𝐵(Σ𝜇, Σ𝜈)2
+(2)
+where 𝐵(Σ𝜇, Σ𝜈)2 = 𝑡𝑟 (Σ𝜇 + Σ𝜈 − 2(Σ
+1
+2𝜇 Σ𝜈Σ
+1
+2𝜇 )
+1
+2 ).
+Next we see how that changing the means of the distribution(and
+also variance) changes PERM and KP. We can show that,
+𝑃 =
+∫ 𝑥=∞
+𝑥=−∞
+𝐷+(𝑥)
+�∫ 𝑦=∞
+𝑦=𝑥
+𝐷−(𝑥)𝑑𝑦
+�
+𝑑𝑥
+𝜕𝑃
+𝜕𝑚𝜈
+=
+∫ 𝑥=∞
+𝑥=−∞
+𝐷+(𝑥)
+�∫ 𝑦=∞
+𝑦=𝑥
+𝜕𝐷−(𝑥)
+𝜕𝑚𝜈
+𝑑𝑦
+�
+𝑑𝑥
+≥ 0
+𝜕𝑃
+𝜕Σ𝜈
+=
+∫ 𝑥=∞
+𝑥=−∞
+𝐷+(𝑥)
+�∫ 𝑦=∞
+𝑦=𝑥
+𝜕𝐷−(𝑦)
+𝜕Σ𝜈
+𝑑𝑦
+�
+𝑑𝑥
+≤ 0
+𝜕𝑃
+𝜕Σ𝜇
+=
+∫ 𝑥=∞
+𝑥=−∞
+𝜕𝐷+(𝑥)
+𝜕Σ𝜈
+�∫ 𝑦=∞
+𝑦=𝑥
+𝐷−(𝑦)𝑑𝑦
+�
+𝑑𝑥
+Since KP is the (sliced) wasserstein distance between PDs we can
+show the respective gradients are as below,
+𝜕𝑊 2
+2 (𝜇,𝜈)
+𝜕𝑚𝜈
+= 2|𝑚𝜇 − 𝑚𝜈 |
+≥ 0
+𝜕𝑊 2
+2 (𝜇,𝜈)
+𝜕Σ𝜈
+= 𝐼 − Σ
+1
+2𝜇 (Σ
+1
+2𝜇 Σ𝜈Σ
+1
+2𝜇 )
+−1
+2 Σ
+1
+2𝜇
+As the generating process of the scores changes the gradient of
+PERM along the direction (𝑑𝑚𝜈,𝑑𝜎𝜇,𝑑𝜎𝜈) can be shown to be the
+following
+�
+(𝑑𝑚𝜈,𝑑𝜎,𝑑𝜎) ,
+� 𝜕𝑃𝐸𝑅𝑀
+𝜕𝑚𝜈
+, 𝜕𝑃𝐸𝑅𝑀
+𝜕Σ𝜇
+, 𝜕𝑃𝐸𝑅𝑀
+𝜕Σ𝜈
+��
+≥ 0
+Similarly the gradient of KP along the direction (𝑑𝑚𝜇,𝑑𝜎𝜇,𝑑𝜎𝜈)
+is
+�
+(𝑑𝑚𝜈,𝑑𝜎,𝑑𝜎), (
+𝜕𝑊 2
+2 (𝜇,𝜈)
+𝜕𝑚𝜈
+,
+𝜕𝑊 2
+2 (𝜇,𝜈)
+𝜕Σ𝜇
+,
+𝜕𝑊 2
+2 (𝜇,𝜈)
+𝜕Σ𝜈
+)
+�
+≥ 0
+Since both PERM and and KP vary in the same manner as the
+distribution changes, the two have a one-one correspondence [42].
+□
+The above lemma shows that there is a one-one correspondence
+between KP and PERM and by definition PERM has a one-one cor-
+respondence with the ranking metrics. Therefore, the next theorem
+follows as a natural consequence
+Theorem 8.1. KP has a one-one correspondence with the Ranking
+Metrics under the above stated considerations
+Theorem 8.2. Under the considerations of theorem 8.1, the relative
+change in KP on addition of random noise to the scores is bounded
+by a function of the original and noise-induced covariance matrix
+as ΔK P
+K P ≤ 𝑚𝑎𝑥((1 − |Σ+1
+𝜇1 Σ−1
+𝜇2 |
+3
+2 ), (1 − |Σ+1
+𝜈1 Σ−1
+𝜈2 |
+3
+2 )), where Σ𝜇1 and
+Σ𝜈1 are the covariance matrices of the positive and negative triples’
+scores respectively and Σ𝜇2 and Σ𝜈2 are that of the corrupted scores.
+Proof Sketch. Consider a zero mean random noise to simulate
+the process of varying the distribution of the scores of the KGE
+method. Let 𝑚𝜇1 and 𝑚𝜈1 be the means of the positive and negative
+triples’ scores of the original method and Σ𝜇1, Σ𝜈1 be the respective
+covariance matrices. Let 𝑚𝜇2 and 𝑚𝜈2 be the means of the positive
+and negative triples’ scores of the corrupted method and Σ𝜇2, Σ𝜈2
+be the respective covariance matrices. Considering the kantorovich
+duality [51] and taking the difference between the two measures
+we have
+KP1 − KP2
+=
+𝑖𝑛𝑓
+𝛾1∈Π(𝑥,𝑦)
+∫
+𝛾1
+𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑥,𝑦)𝑑𝛾1(𝑥,𝑦)
+−
+𝑖𝑛𝑓
+𝛾2∈Π(𝑥,𝑦)
+∫
+𝛾2
+𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑥,𝑦)𝑑𝛾2(𝑥,𝑦)
+≤ 𝑠𝑢𝑝
+Φ,Ψ
+∫
+𝑥
+Φ(𝑥)𝑑𝜇1(𝑥) +
+∫
+𝑦
+Ψ(𝑦)𝑑𝜈1(𝑦)
+−
+∫
+𝑥
+Φ(𝑥)𝑑𝜇2(𝑥) −
+∫
+𝑦
+Ψ(𝑦)𝑑𝜈2(𝑦)
+≤ 𝑠𝑢𝑝
+Φ,Ψ
+∫
+𝑥
+Φ(𝑥)(𝑑𝜇1(𝑥) − 𝑑𝜇2(𝑥)) +
+∫
+𝑦
+Ψ(𝑦)(𝑑𝜈1(𝑦) − 𝑑𝜈2(𝑦))
+Now by definition of the measure 𝜇1 we have
+𝜕𝜇1
+𝜕𝑥 = −𝜇1Σ−1
+𝜇1 (𝑥 − ���𝜇1)
+𝑑𝜇1(𝑥𝑖) = −(𝜇1Σ−1
+𝜇1 (𝑥 − 𝑚𝜇1))[𝑖]𝑑𝑥𝑖
+∴ 𝑑𝜇1(𝑥) = 𝑑𝑒𝑡(𝑑𝑖𝑎𝑔(−𝜇1Σ−1
+𝜇1 (𝑥 − 𝑚𝜇1)))𝑑𝑥
+From the above results we can show the following
+KP1 − KP2
+≤ 𝑚𝑎𝑥((1 − 𝑑𝑒𝑡(Σ𝜇1Σ−1
+𝜇2 )
+𝑛
+2 +1), (1 − 𝑑𝑒𝑡(Σ𝜈1Σ−1
+𝜈2 )
+𝑛
+2 +1))KP1
+∴ ΔKP
+KP
+≤ 𝑚𝑎𝑥
+��
+1 − 𝑑𝑒𝑡(Σ𝜇1Σ−1
+𝜇2 )
+𝑛
+2 +1�
+,
+�
+1 − 𝑑𝑒𝑡(Σ𝜈1Σ−1
+𝜈2 )
+𝑛
+2 +1��
+In our case as we work in the univariate setting 𝑛 = 1 and thus we
+have ΔK P
+K P ≤ 𝑚𝑎𝑥
+��
+1 − 𝑑𝑒𝑡(Σ𝜇1Σ−1
+𝜇2 )
+3
+2
+�
+,
+�
+1 − 𝑑𝑒𝑡(Σ𝜈1Σ−1
+𝜈2 )
+3
+2
+��
+, as
+required.
+□
+
+Can Persistent Homology provide an efficient alternative
+WWW ’23, APRIL 30 - MAY 4, 2023, Texas, USA
+Theorem 8.2 shows that as noise is induced gradually, the KP
+value changes in a bounded manner as desired.
+