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+arXiv:2301.04436v1  [math.CA]  11 Jan 2023
+OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
+WITH TWO VARIABLES
+ISROIL A. IKROMOV, MICHAEL RUZHANSKY, AKBAR R. SAFAROV∗
+Abstract. In this paper we consider the problem of estimation of oscillatory in-
+tegrals with Mittag-Leffler functions in two variables. The generalisation is that
+we replace the exponential function with the Mittag-Leffler-type function, to study
+oscillatory type integrals.
+Contents
+1.
+Introduction
+1
+2.
+Preliminaries
+2
+3.
+Auxiliary statements
+4
+4.
+Proof of the main result
+7
+Acknowledgements
+9
+Data availability
+9
+References
+9
+1. Introduction
+The function Eα(z) is named after the Swedish mathematican G¨osta Magnus
+Mittag-Leffler (1846-1927) who defined it by a power series
+Eα(z) =
+∞
+�
+k=0
+zk
+Γ(αk + 1),
+α ∈ C, Re(α) > 0,
+(1.1)
+and studied its properties in 1902-1905 in several subsequent notes [18, 19, 20, 21] in
+connection with his summation method for divergent series.
+A classical generalization of the Mittag-Leffler function, namely the two-parametric
+Mittag-Leffler function is
+Eα,β(z) =
+∞
+�
+k=0
+zk
+Γ(αk + β),
+α, β ∈ C, Re(α) > 0,
+(1.2)
+which was deeply investigated independently by Humbert and Agarval in 1953 ([1,
+10, 11]) and by Dzherbashyan in 1954 ([4, 5, 6]) as well as in [9].
+∗Corresponding author
+2010 Mathematics Subject Classification. 35D10, 42B20, 26D10.
+Key words and phrases. Mittag-Leffler functions, phase function, amplitude.
+All authors contributed equally to the writing of this paper. All authors read and approved the
+final manuscript.
+1
+
+2
+I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV
+It has the property that
+E1,1(x) = ex, and we can refer to [23] for other properties.
+(1.3)
+In harmonic analysis one of the most important estimates for oscillatory integral is
+van der Corput lemma [24, 25, 26, 34]. Estimates for oscillatory integrals with poly-
+nomial phases can be found, for instance, in papers [2, 15, 29, 30, 31]. In the current
+paper we replace the exponential function with the Mittag-Leffler-type function and
+study oscillatory type integrals (2.3). In the papers [26] and [27] analogues of the van
+der Corput lemmas involving Mittag-Leffler functions for one dimensional integrals
+have been considered. We extend results of [26] and [27] for two-dimensional inte-
+grals with phase having some simple singularities. Analogous problem on estimates
+for Mittag-Leffler functions with the smooth phase functions of two variables having
+simple singularities was considered in [28] and [32].
+2. Preliminaries
+Definition 2.1. An oscillatory integral with phase f and amplitude a is an integral
+of the form
+J(λ, f, a) =
+�
+Rn a(x)eiλf(x)dx,
+(2.1)
+where a ∈ C∞
+0 (Rn) and λ ∈ R.
+If the support of a lies in a sufficiently small neighborhood of the origin and f is
+an analytic function at x = 0, then for λ → ∞ the following asymptotic expansion
+holds ([17]):
+J(λ, f, a) ≈ eiλf(0) �
+s
+n−1
+�
+k=0
+bs,k(a)λs(ln λ)k,
+(2.2)
+where s belongs to a finite number of arithmetic progressions, independent of a,
+composed of negative rational numbers, bs,k is a distribution with support in the
+critical set {x : ∇f(x) = 0}.
+Inspired by the terminology from [3], we refer to the maximal value of s, denoting
+it by α in this case, as the growth index of f, or the oscillation index at the origin,
+and the corresponding value of k is referred to as the multiplicity.
+More precisely, the multiplicity of the oscillation index of an analytic phase at a
+critical point is the maximal number k possessing the property: for any neighbour-
+hood of the critical point there is an amplitude with support in this neighbourhood
+for which in the asymptotic series (2.2) the coefficient bs,k(a) is not equal to zero.
+The multiplicity of the oscillation index will be denoted by m (see [3]).
+Let f be a smooth real-valued function defined on a neighborhood of the origin in
+R2 with f(0, 0) = 0, ∇f(0, 0) = 0, and consider the associated Taylor series
+f(x1, x2) ∼
+∞
+�
+j,k=0
+cjkxj
+1xk
+2
+of f centered at the origin. The set
+ℑ(f) := {(j, k) ∈ N2 : cjk =
+1
+j!k!∂j
+x1∂k
+x2f(0, 0) ̸= 0}
+
+OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
+3
+is called the Taylor support of f at (0, 0). We shall always assume that
+ℑ(f) ̸= ∅,
+i.e., that the function f is of finite type at the origin. If f is real analytic, so that the
+Taylor series converges to f near the origin, this just means that f ̸= 0. The Newton
+polyhedron ℵ(f) of f at the origin is defined to be the convex hull of the union of
+all the quadrants (j, k) + R2
++, with (j, k) ∈ ℑ(f). The associated Newton diagram
+ℵd(f) in the sense of Varchenko [33] is the union of all compact faces of the Newton
+polyhedron; here, by a face, we mean an edge or a vertex.
+We shall use coordinates (t1, t2) for points in the plane containing the Newton
+polyhedron, in order to distinguish this plane from the (x1, x2) - plane.
+The distance d = d(f) between the Newton polyhedron and the origin in the sense
+of Varchenko is given by the coordinate d of the point (d, d) at which the bisectrix
+t1 = t2 intersects the boundary of the Newton polyhedron.
+The principal face π(f) of the Newton polyhedron of f is the face of minimal
+dimension containing the point (d, d). Deviating from the notation in [33], we shall
+call the series
+fp(x1, x2) :=
+�
+j,k∈π(f)
+cjkxj
+1xk
+2
+the principal part of f. In the case that π(f) is compact, fπ is a mixed homogeneous
+polynomial; otherwise, we shall consider fπ as a formal power series.
+Note that the distance between the Newton polyhedron and the origin depends
+on the chosen local coordinate system in which f is expressed. By a local analytic
+(respectively smooth) coordinate system at the origin we shall mean an analytic (re-
+spectively smooth) coordinate system defined near the origin which preserves 0. If
+we work in the category of smooth functions f, we shall always consider smooth co-
+ordinate systems, and if f is analytic, then one usually restricts oneself to analytic
+coordinate systems (even though this will not really be necessary for the questions we
+are going to study, as we will see). The height of the analytic (respectively smooth)
+function f is defined by
+h := h(f) := sup{dx},
+where the supremum is taken over all local analytic (respectively smooth) coordinate
+systems x at the origin, and where dx is the distance between the Newton polyhedron
+and the origin in the coordinates x.
+A given coordinate system x is said to be adapted to f if h(f) = dx.
+Let π be the principal face. We assume that π is a point or a compact edge, then
+fπ is a weighted homogeneous polynomial. Denote by ν the maximal order of roots
+of fπ on the unit circle at the origin, so
+ν := max
+S1 ord(fπ).
+If there exists a coordinate system x such that ν = dx then we set m = 1. It can
+be shown that in this case x is adapted to f (see [12]). Otherwise we take m = 0.
+Following A. N. Varchenko we call m the multiplicity of the Newton polyhedron.
+In the classical paper by A. N. Varchenko [33], he obtained the sharp estimates
+for oscillatory integrals in terms of the height. Also in the paper [13] the height was
+used to get the sharp bound for maximal operators associated to smooth surfaces in
+
+4
+I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV
+R3. It turns out that analogous quantities can be used for oscillatory integrals with
+the Mittag-Leffler function.
+We consider the following integral with phase f and amplitude ψ, of the form
+Iα,β =
+�
+U
+Eα,β(iλf(x))ψ(x)dx,
+(2.3)
+where 0 < α < 1, β > 0, U is a sufficiently small neighborhood of the origin. We
+are interested in particular in the behavior of Iα,β when λ is large, as for small λ the
+integral is just bounded. In particular if α = 1 and β = 1 we have oscillatory integral
+(2.1).
+The main result of the work is the following.
+Theorem 2.2. Let f be a smooth finite type function of two variables defined in a
+sufficiently small neighborhood of the origin and let ψ ∈ C∞
+0 (U).
+Let h be the height of the function f, and let m = 0, 1 be the multiplicity of its
+Newton polyhedron. If 0 < α < 1, β > 0, h > 1, and λ ≫ 1 then we have the
+estimate
+����
+�
+U
+Eα,β(iλf(x1, x2))ψ(x)dx
+���� ≤
+C| ln λ|m∥ψ∥L∞(U)
+λ
+1
+h
+.
+(2.4)
+If 0 < α < 1, β > 0, h = 1 and λ ≫ 1, then we have following estimate
+����
+�
+U
+Eα,β(iλf(x1, x2))ψ(x)dx
+���� ≤
+C| ln λ|2∥ψ∥L∞(U)
+λ
+,
+(2.5)
+where the constants C are independent of the phase, amplitude and λ.
+3. Auxiliary statements
+We first recall some useful properties.
+Proposition 3.1. If 0 < α < 2, β is an arbitrary real number, µ is such that πα/2 <
+µ < min{π, πα}, then there is C > 0, such that we have
+|Eα,β(z)| ≤
+C
+1 + |z|, z ∈ C, µ ≤ | arg(z)| ≤ π.
+(3.1)
+See [4], [9], [23].
+Proposition 3.2. Let Ω be an open, bounded subset of
+R2, and let f : Ω → R be a
+measurable function such that for all λ ≫ 1 and for some positive δ ̸= 1, we have
+����
+�
+Ω
+eiλf(x)dx
+���� ≤ C|λ|−δ| ln λ|m,
+(3.2)
+with m ≥ 0. Then, we have
+��x ∈ Ω : |f(x)| ≤ ε| ≤ Cδεδ| ln ε|m, for δ < 1,
+for 0 < ε ≪ 1, and for δ > 1, |x ∈ Ω : |f(x)| ≤ ε| ≤ Cδε ,
+for δ = 1,
+��x ∈ Ω : |f(x)| ≤ ε| ≤ Cδε| ln ε|m+1,
+where Cδ depends only on δ, |A| means the Lebesgue measure of a set A. See [7].
+
+OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
+5
+Proof. For the convenience of the reader we give an independent proof of Proposition
+3.2. We consider an even non-negative smooth function
+ω(x) =
+�
+1,
+when |x| ≤ 1,
+0,
+when |x| ≥ 2.
+For the characteristic function of Ω with Ω ⊂ U, the following inequality holds true
+|x ∈ Ω : |f(x)| ≤ ε| =
+�
+Ω
+χ[0,1]
+�|f(x)|
+ε
+�
+dx ≤
+�
+Ω
+ω
+�f(x)
+ε
+�
+dx.
+Now we will use the Fourier inversion formula, and rewrite the last integral as
+�
+Ω
+ω
+�f(x)
+ε
+�
+dx = 1
+2π
+�
+Ω
+� ∞
+−∞
+ˇω(ξ)eiξ f(x)
+ε dξdx.
+As ˇω(ξ) is a Schwartz function, we can use Fubini theorem and change the order of
+integration. So we have
+�
+Ω
+� ∞
+−∞
+ˇω(ξ)eiξ f(x)
+ε dξdx =
+� ∞
+−∞
+ˇω(ξ)
+�
+Ω
+eiξ f(x)
+ε dxdξ.
+We use inequality (3.2) for the inner integral and get
+����
+�
+Ω
+eiξ f(x)
+ε dx
+���� ≤ C| ln(2 + ξ
+ε)|m
+(1 + | ξ
+ε|)δ
+.
+As ˇω(ξ) is a Schwartz function, we also have
+|ˇω(ξ)| ≤
+C
+1 + |ξ|.
+So
+�����
+� ∞
+−∞
+C ˇω(ξ)| ln(2 + ξ
+ε)|m
+(2 + | ξ
+ε|)δ
+dξ
+����� ≲
+� ∞
+0
+2C| ln( ξ
+ε)|m
+(1 + |ξ|)(2 + | ξ
+ε|)δ dξ.
+Now we change the variable as ξ = ηε, and we get
+� ∞
+0
+| ln( ξ
+ε)|m
+(1 + |ξ|)(2 + | ξ
+ε|)δ dξ =
+� ∞
+0
+ε| ln η|m
+(1 + |εη|)(2 + |η|)δ dη.
+Now we estimate the last integral for different values of δ.
+If δ < 1 then we have
+� ∞
+0
+ε| ln η|m
+(1 + |εη|)(2 + |η|)δ dη ≤ Cε
+�
+1
+ε
+0
+| ln η|mdη
+(2 + η)δ + Cε
+� ∞
+1
+ε
+| ln η|mdη
+εηδ+1
+.
+We represent
+1
+(2+η)δ =
+1
+ηδ(1+ 2
+η )δ =
+1
+ηδ + O(
+1
+ηδ+1). So
+Cε
+�
+1
+ε
+0
+| ln η|mdη
+(2 + η)δ = ε
+� 2
+0
+| ln η|mdη
+(2 + η)δ + ε
+�
+1
+ε
+2
+| ln η|mdη
+(2 + η)δ .
+Integrating by parts we obtain
+ε
+�
+1
+ε
+2
+| ln η|mdη
+(2 + η)δ ≤ ε
+�
+1
+ε
+2
+| ln η|mdη
+ηδ
+≤ Cεδ| ln ε|m.
+
+6
+I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV
+As δ < 1, the integrals
+� 2
+0
+| ln η|mdη
+(2+η)δ
+and
+� ∞
+1
+ε
+| ln η|mdη
+εηδ+1
+convergence.
+If δ > 1 then we trivially obtain
+����
+� ∞
+0
+Cε| ln η|m
+(1 + |εη|)(2 + |η|)δ dη
+���� ≤ Cε.
+If δ = 1 then assuming 0 < ε < 1
+2 we get |εη| < 1 (for |η| < 2), then write the integral
+as the sum of three integrals and obtain
+����
+� ∞
+0
+Cε| ln η|m
+(1 + |εη|)(1 + |η|)dη
+���� ≤
+����
+� 2
+0
+Cε| ln η|mdη
+���� +
+�����
+�
+1
+ε
+2
+Cε| ln η|m
+η
+dη
+����� +
+�����
+� ∞
+1
+ε
+Cε| lnη|m
+η
+dη
+����� .
+Then we have
+����
+� 2
+0
+Cε| ln η|mdη
+���� ≤ Cε,
+and we get with simple calculating that
+�����
+�
+1
+ε
+2
+Cε| lnη|m
+η
+dη
+����� ≤ Cε| ln ε|m+1.
+We use the formula of integrating by parts several times, to get
+�����
+� ∞
+1
+ε
+Cε| ln η|m
+η
+dη
+����� ≤ Cε| ln ε|m,
+completing the proof.
+□
+From Proposition 3.2 we get the following corollaries.
+Corollary 3.3. Let f(x1, x2) be a smooth function with f(0, 0) = 0, ∇f(0, 0) = 0,
+and h be the height of the function f(x1, x2), and let m = 0, 1 be the multiplicity of
+its Newton polyhedron. Let also a(x) =
+�
+1,
+when |x| ≤ σ,
+0,
+when |x| ≥ 2σ,
+σ > 0, and a(x) ≥ 0
+with a ∈ C∞
+0 (R2). If for all real λ ≫ 1 and for any positive δ ̸= 1, the following
+inequality holds
+����
+�
+R2 eiλf(x)a(x)dx
+���� ≤ C|λ|−δ| ln λ|m,
+(3.3)
+then we have
+||x| ≤ σ : |f(x)| ≤ ε| ≤ Cεδ| ln ε|m,
+where m ≥ 0. See [8, 12, 14, 22].
+Corollary 3.4. Let f(x1, x2) be a smooth function with f(0, 0) = 0, ∇f(0, 0) = 0,
+and let Ω be a sufficiently small compact set around the origin. Let also h be the
+height of the function f(x1, x2), and let m = 0, 1 be the multiplicity of its Newton
+polyhedron. Then for all 0 < ε ≪ 1 we have
+|x ∈ Ω : |f(x)| ≤ ε| ≤ Cε
+1
+h| ln ε|m,
+where h is the height of f and m is its multiplicity [8].
+
+OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
+7
+4. Proof of the main result
+Proof of Theorem 2.2. As for λ < 2 the integral (2.3) is just bounded, we
+consider the case λ ≥ 2. Without loss of generality, we can consider the integral over
+U. Using inequality (3.1), we have
+|Eα,β(iλf(x))| ≤
+C
+1 + λ|f(x)|.
+(4.1)
+We then use (4.1) for the integral (2.3), and get that
+|Iα,β| ≤
+����
+�
+U
+Eα,β(iλf(x))ψ(x)dx
+���� ≤ C
+�
+U
+|ψ(x)|dx
+1 + λ|f(x)|.
+(4.2)
+Now we represent the integral Iα,β over the union of sets Ω1 := Ω ∩ {λ|f(x1, x2)| <
+M} and Ω2 := Ω ∩ {λ|f(x1, x2)| ≥ M} respectively, where M is a positive real
+number.
+We estimate the integral Iα,β over the sets Ω1 and Ω2, respectively,
+|Iα,β| ≤ C
+�
+U
+|ψ(x)|dx
+1 + λ|f(x)| = J1 + J2 := C
+�
+Ω1
+|ψ(x)|dx
+1 + λ|f(x)| + C
+�
+Ω2
+|ψ(x)|dx
+1 + λ|f(x)|.
+First we estimate the integral over the set Ω1. Using the results of the paper ([17]
+page 31) (see also Corollary 3.4) we obtain
+|J1| = C
+�
+Ω1
+|ψ(x)|dx
+1 + λ|f(x)| ≤
+C| ln λ|m∥ψ∥L∞(Ω1)
+λ
+1
+h
+.
+Lemma 4.1. Let f ∈ C∞ and h be the height of the function f, and let m = 0, 1 be
+the multiplicity of its Newton polyhedron. For any smooth function a = a(x, y) with
+sufficiently small support and for h > 1 the following inequality holds
+I :=
+�
+{|f(x,y)|≥ M
+λ }
+a(x, y)
+1 + λ|f(x, y)|dxdy ≤
+C| ln ��|m∥a∥L∞(U)
+λ
+1
+h
+,
+(4.3)
+where supp{a(x, y)} = U.
+Proof. Let h > 1. Consider the sets
+Ak =
+�
+x ∈ U : 2k
+λ ≤ |f(x)| ≤ 2k+1
+λ
+�
+.
+For the measure of a set of smaller values we use Lemma 1
+′ in the paper [16] (see also
+Corollary 3.4), and we have
+µ
+�
+|f(x)| ≤ 2k+1
+λ , x ∈ U
+�
+≤ C
+�2k+1
+λ
+� 1
+h �
+ln
+����
+λ
+2k+1
+����
+�m
+.
+Let
+Ik :=
+�
+Ak
+a(x, y)
+1 + λ|f(x, y)|dxdy.
+For the integral
+�
+2k≤λ|f(x)|≤2k+1
+Ik =
+�
+Ω2
+a(x, y)
+1 + λ|f(x, y)|dxdy,
+
+8
+I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV
+we find the following estimate:
+|Ik| =
+����
+�
+Ak
+a(x, y)
+1 + λ|f(x, y)|dxdy
+���� ≤ C∥a∥L∞(U)
+�2k+1
+λ
+� 1
+h ����ln 2k+1
+λ
+����
+m
+2−k.
+From here we find the sum of Ik and, by estimating the integral I, we get
+I ≤ ∥a∥L∞(U)
+∞
+�
+k=1
+Ik ≤ ∥a∥L∞(U)
+∞
+�
+k=1
+�2k+1
+λ
+� 1
+h ����ln 2k+1
+λ
+����
+m
+2−k
+≤ ∥a∥L∞(U)
+| ln λ|m
+λ
+1
+h
+∞
+�
+k=1
+2
+k+1
+h −kkm.
+As h > 1, the last series is convergent, proving the lemma.
+□
+Remark 4.2. Consider the case h = 1.
+The smooth function has non-degenerate
+critical point at the origin if and only if h = 1. As f(x, y) is a smooth function with
+∇f(0, 0) = 0, using Morse lemma we have f ∼ x2 ± y2. So in this case we estimate
+two sets ∆ = ∆1 ∪ ∆2, where ∆1 := {(x, y) : λ|x2 ± y2| ≤ M, |x| ≤ 1, |y| ≤ 1} and
+∆2 := {(x, y) : λ|x2 ± y2| > M, |x| ≤ 1, |y| ≤ 1}. First we consider the integral over
+the set ∆1. Then we have
+����
+�
+∆1
+a(x, y)
+1 + λ|x2 ± y2|dxdy
+���� ≤ C∥a∥L∞(∆1)
+����
+�
+∆1
+dxdy
+����.
+Now we estimate the last integral as
+����
+�
+λ|x2+y2|≤M
+dxdy
+���� ≤ C
+λ .
+Then we estimate the measure of the set {|x2 − y2| ≤ εM}, where ε = 1
+λ. We have,
+for simplicity putting M = 1,
+����
+�
+|x2−y2|≤εM
+dxdy
+���� ≤ C
+�����
+� √1−ε
+√ε
+dy
+� √
+y2+ε
+√
+y2−ε
+dx
+����� =
+�����
+� √1−ε
+√ε
+��
+y2 + ε −
+�
+y2 − ε
+�
+dy
+����� =
+=
+�y
+2
+�
+y2 + ε + ε
+2 ln |y +
+�
+y2 + ε|
+� ���
+√1−ε
+√ε
+−
+�y
+2
+�
+y2 − ε − ε
+2 ln |y +
+�
+y2 − ε|
+� ���
+√1−ε
+√ε
+=
+=
+�����
+√1 − ε
+2
++ ε
+2 ln
+√1 − ε + 1
+√ε
+−
+√
+2
+2 ε − ε
+2 ln |√ε(1 +
+√
+2)|−
+−
+��
+(1 − ε)(1 − 2ε)
+2
+− ε
+2 ln |
+√
+1 − ε +
+√
+1 − 2ε| + ε
+2 ln √ε|
+������ ≤ Cε ln ε.
+Now we consider the integral over the set ∆2. In this case we change the variables
+to polar coordinate system and with easy calculating we get
+����
+�
+{λ|x2+y2|≥M}
+a(x, y)
+1 + λ|x2 + y2|dxdy
+���� ≤ C| ln λ|∥a∥L∞(∆2)
+λ
+(4.4)
+and
+����
+�
+{λ|x2−y2|≥M}
+a(x, y)
+1 + λ|x2 − y2|dxdy
+���� ≤ C| ln λ|2∥a∥L∞(∆2)
+λ
+.
+(4.5)
+
+OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
+9
+Now we continue the proof of Theorem 2.2. Let h > 1. We use Proposition 3.2 for
+the integral J1, to get
+|J1| ≤
+C| ln λ|m∥a∥L∞(U)
+λ
+1
+h
+.
+Let consider the integral J2. If h > 1, then using Lemma 4.1 we get
+|J2| ≤
+C| ln λ|∥a∥L∞(U)
+λ
+1
+h
+.
+If h = 1, using the Remark 4.2 we get the inequality (2.5). The proof is complete.
+The proof of Theorem 2.2 shows that if h = 1, we can get a more precise result.
+Proposition 4.3. If h = 1 and f has an extremal point at the point (0,0) (then f is
+diffeomorhic equivalent to x2
+1 + x2
+2 or −x2
+1 − x2
+2), then we have
+|Iα,β| ≤
+C| ln λ|∥ψ∥L∞(U)
+λ
+,
+for all λ ≥ 2.
+Declaration of competing interest
+This work does not have any conflicts of interest.
+Acknowledgements
+The second author was supported in parts by the FWO Odysseus 1 grant G.0H94.18N:
+Analysis and Partial Differential Equations and by the Methusalem programme of the
+Ghent University Special Research Fund (BOF) (Grant number 01M01021) and also
+supported by EPSRC grant EP/R003025/2.
+Data availability. The manuscript has no associated data.
+References
+[1] R. P. Agarwal, A propos d’une note de M.Pierre Humbert, C. R. Acad. Sci. Paris, 236, 2031-2032
+(1953).
+[2] G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov, Theory of multiple trigonometric sums, -
+Moscow. Nauka, 1987, p. 357.
+[3] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of Differentiable Maps,
+Birkhauser, Boston Basel · Stuttgart, 1985.
+[4] M. M. Dzherbashyan, On the asymtotic expansion of a function of Mittag-Leffler type, Akad.
+Nauk Armjan. SSR Doklady. 19, 65-72 (1954, in Russian).
+[5] M. M. Dzherbashyan, On integral representation of functions continuous on given rays (gener-
+alization of the Fourier integrals), Izvestija Akad. Nauk SSSR Ser. Mat. 18, 427-448 (1954, in
+Russian).
+[6] M. M. Dzherbashyan, On Abelian summation of the eneralized integral transform, Akad. Nauk
+Armjan. SSR Izvestija, fiz-mat. estest. techn.nauki. 7(6), 1-26 (1954, in Russian).
+[7] J. Green, Uniform oscillatory integral estimates for convex phases via sublevel set estimates,
+arxiv: 2111.05395v1.
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+Math. Annalen. - 2010. - V.346, № 4. - p.857-890.
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+applications, Springer Monographs in Mathematics, Springer-Verlag Berlin Heidelberg (2014).
+[10] P. Humbert, Quelques r´esultats relatifs `a la fonction de Mittag-Leffler, C. R. Acad. Sci. Paris,
+236, 1467-1468 (1953).
+
+10
+I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV
+[11] P. Humbert,
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+[13] I. A. Ikromov, M. Kempe, D. M¨uller, Estimates for maximal functions associated with hyper-
+surfaces in R3 and related problems of harmonic analysis, Acta mathematica, 2010, 204 (2),
+151–271.
+[14] I. A. Ikromov and D. M¨uller, Fourier Restriction for Hypersurfaces in Three Dimensions and
+Newton Polyhedra, Annals of Mathematics Studies 194, Princeton Univ. Press, Princeton and
+Oxford, 2016.
+[15] I. A. Ikromov, Invariant estimates of two-dimensional trigonometric integrals, Math. USSR.
+Sb. 76 (1990), 473–488.
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+// Proceedings of the I. G. Petrovsky Seminar. Vol.9. 1983. P. 3-39.(Russian)
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+939 (1902).
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+Sci. Paris 136, 537-539 (1903).
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+558 (1903).
+[21] M. G. Mittag-Leffler, Sopra la funzione Eα(x), Rend.R.Acc.Lincei, (Ser.5)13, 3-5 (1904).
+[22] D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operator, Acta
+Math. 179(1), 1997, 105-152.
+[23] I. Podlubny, Fractional Differensial Equations, Academic Press, New York, 1999.
+[24] M. Ruzhansky, Pointwise van der Corput Lemma for Functions of Several Variables, Functional
+Analysis and Its Applications, 43 (2009), no.1, 75–77.
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+Calculus and Applied Analysis, 23 (6), (2021), 1663–1677.
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+Van der Corput lemmas for Mittag-Leffler functions. II.
+α−directions , Bull. Sci. Math., 171 (2021), 103016, 23 pp.
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+homogeneous polynomial phases of degree 4, Analysis and Mathematical Physics, 12(130),
+(2022).
+[29] A. Safarov, Invariant estimates of two-dimensional oscillatory integrals // Math. Notes. 104,
+2018. P.293–302.
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+Fed. Univ. Math. Phys. 9 (2016), P.102–107.
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+Mathematics, 63 (4), P.57-63.
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+Variables, J. Sib. Fed. Univ. Math. Phys., 15(4) (2022), P.459-–466.
+[33] A. N. Varchenko, Newton polyhedra and estimation of oscillating integrals //Functional Analysis
+and Its Applications, vol. 10, pages 175-–196 (1976).
+[34] Van der Korput, Zur Methode der stationaren phase// Compositio Math. V.1. 1934. P. 15-38.
+
+OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
+11
+Isroil A. Ikromov
+V.I. Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan
+Olmazor district, University 46, Tashkent, Uzbekistan
+Samarkand State University
+Department of Mathematics, 15 University Boulevard
+Samarkand, 140104, Uzbekistan
+Email address: [email protected]
+Michael Ruzhansky
+Department of Mathematics: Analysis, Logic and Discrete Mathematics
+Ghent University,
+Krijgslaan 281, Ghent, Belgium,
+School of Mathematical Sciences, Queen Mary University of London,
+United Kingdom
+Email address: [email protected]
+Akbar R.Safarov
+Uzbek-Finnish Pedagogical Institute
+Spitamenshox 166, Samarkand, Uzbekistan
+Samarkand State University
+Department of Mathematics, 15 University Boulevard
+Samarkand, 140104, Uzbekistan
+Email address: [email protected]
+

-9E3T4oBgHgl3EQfSwmi/content/tmp_files/load_file.txt

@@ -0,0 +1,458 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf,len=457
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='04436v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='CA]  11 Jan 2023  OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS  WITH TWO VARIABLES  ISROIL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' IKROMOV, MICHAEL RUZHANSKY, AKBAR R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' SAFAROV∗  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' In this paper we consider the problem of estimation of oscillatory in-  tegrals with Mittag-Leffler functions in two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' The generalisation is that  we replace the exponential function with the Mittag-Leffler-type function, to study  oscillatory type integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Introduction  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Preliminaries  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Auxiliary statements  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Proof of the main result  7  Acknowledgements  9  Data availability  9  References  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Introduction  The function Eα(z) is named after the Swedish mathematican G¨osta Magnus  Mittag-Leffler (1846-1927) who defined it by a power series  Eα(z) =  ∞  �  k=0  zk  Γ(αk + 1),  α ∈ C, Re(α) > 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1)  and studied its properties in 1902-1905 in several subsequent notes [18, 19, 20, 21] in  connection with his summation method for divergent series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  A classical generalization of the Mittag-Leffler function, namely the two-parametric  Mittag-Leffler function is  Eα,β(z) =  ∞  �  k=0  zk  Γ(αk + β),  α, β ∈ C, Re(α) > 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2)  which was deeply investigated independently by Humbert and Agarval in 1953 ([1,  10, 11]) and by Dzherbashyan in 1954 ([4, 5, 6]) as well as in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  ∗Corresponding author  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' 35D10, 42B20, 26D10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Mittag-Leffler functions, phase function, amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  All authors contributed equally to the writing of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' All authors read and approved the  final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  1  2  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='IKROMOV, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='RUZHANSKY, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='SAFAROV  It has the property that  E1,1(x) = ex, and we can refer to [23] for other properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='3)  In harmonic analysis one of the most important estimates for oscillatory integral is  van der Corput lemma [24, 25, 26, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Estimates for oscillatory integrals with poly-  nomial phases can be found, for instance, in papers [2, 15, 29, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' In the current  paper we replace the exponential function with the Mittag-Leffler-type function and  study oscillatory type integrals (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' In the papers [26] and [27] analogues of the van  der Corput lemmas involving Mittag-Leffler functions for one dimensional integrals  have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' We extend results of [26] and [27] for two-dimensional inte-  grals with phase having some simple singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Analogous problem on estimates  for Mittag-Leffler functions with the smooth phase functions of two variables having  simple singularities was considered in [28] and [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Preliminaries  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' An oscillatory integral with phase f and amplitude a is an integral  of the form  J(λ, f, a) =  �  Rn a(x)eiλf(x)dx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1)  where a ∈ C∞  0 (Rn) and λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  If the support of a lies in a sufficiently small neighborhood of the origin and f is  an analytic function at x = 0, then for λ → ∞ the following asymptotic expansion  holds ([17]):  J(λ, f, a) ≈ eiλf(0) �  s  n−1  �  k=0  bs,k(a)λs(ln λ)k,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2)  where s belongs to a finite number of arithmetic progressions, independent of a,  composed of negative rational numbers, bs,k is a distribution with support in the  critical set {x : ∇f(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Inspired by the terminology from [3], we refer to the maximal value of s, denoting  it by α in this case, as the growth index of f, or the oscillation index at the origin,  and the corresponding value of k is referred to as the multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  More precisely, the multiplicity of the oscillation index of an analytic phase at a  critical point is the maximal number k possessing the property: for any neighbour-  hood of the critical point there is an amplitude with support in this neighbourhood  for which in the asymptotic series (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2) the coefficient bs,k(a) is not equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  The multiplicity of the oscillation index will be denoted by m (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Let f be a smooth real-valued function defined on a neighborhood of the origin in  R2 with f(0, 0) = 0, ∇f(0, 0) = 0, and consider the associated Taylor series  f(x1, x2) ∼  ∞  �  j,k=0  cjkxj  1xk  2  of f centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' The set  ℑ(f) := {(j, k) ∈ N2 : cjk =  1  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='∂j  x1∂k  x2f(0, 0) ̸= 0}  OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS  3  is called the Taylor support of f at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' We shall always assume that  ℑ(f) ̸= ∅,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=', that the function f is of finite type at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' If f is real analytic, so that the  Taylor series converges to f near the origin, this just means that f ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' The Newton  polyhedron ℵ(f) of f at the origin is defined to be the convex hull of the union of  all the quadrants (j, k) + R2  +, with (j, k) ∈ ℑ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' The associated Newton diagram  ℵd(f) in the sense of Varchenko [33] is the union of all compact faces of the Newton  polyhedron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' here, by a face, we mean an edge or a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  We shall use coordinates (t1, t2) for points in the plane containing the Newton  polyhedron, in order to distinguish this plane from the (x1, x2) - plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  The distance d = d(f) between the Newton polyhedron and the origin in the sense  of Varchenko is given by the coordinate d of the point (d, d) at which the bisectrix  t1 = t2 intersects the boundary of the Newton polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  The principal face π(f) of the Newton polyhedron of f is the face of minimal  dimension containing the point (d, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Deviating from the notation in [33], we shall  call the series  fp(x1, x2) :=  �  j,k∈π(f)  cjkxj  1xk  2  the principal part of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' In the case that π(f) is compact, fπ is a mixed homogeneous  polynomial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' otherwise, we shall consider fπ as a formal power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Note that the distance between the Newton polyhedron and the origin depends  on the chosen local coordinate system in which f is expressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' By a local analytic  (respectively smooth) coordinate system at the origin we shall mean an analytic (re-  spectively smooth) coordinate system defined near the origin which preserves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' If  we work in the category of smooth functions f, we shall always consider smooth co-  ordinate systems, and if f is analytic, then one usually restricts oneself to analytic  coordinate systems (even though this will not really be necessary for the questions we  are going to study, as we will see).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' The height of the analytic (respectively smooth)  function f is defined by  h := h(f) := sup{dx},  where the supremum is taken over all local analytic (respectively smooth) coordinate  systems x at the origin, and where dx is the distance between the Newton polyhedron  and the origin in the coordinates x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  A given coordinate system x is said to be adapted to f if h(f) = dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Let π be the principal face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' We assume that π is a point or a compact edge, then  fπ is a weighted homogeneous polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Denote by ν the maximal order of roots  of fπ on the unit circle at the origin, so  ν := max  S1 ord(fπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  If there exists a coordinate system x such that ν = dx then we set m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' It can  be shown that in this case x is adapted to f (see [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Otherwise we take m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Following A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Varchenko we call m the multiplicity of the Newton polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  In the classical paper by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Varchenko [33], he obtained the sharp estimates  for oscillatory integrals in terms of the height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Also in the paper [13] the height was  used to get the sharp bound for maximal operators associated to smooth surfaces in  4  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='IKROMOV, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='RUZHANSKY, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='SAFAROV  R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' It turns out that analogous quantities can be used for oscillatory integrals with  the Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  We consider the following integral with phase f and amplitude ψ, of the form  Iα,β =  �  U  Eα,β(iλf(x))ψ(x)dx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='3)  where 0 < α < 1, β > 0, U is a sufficiently small neighborhood of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' We  are interested in particular in the behavior of Iα,β when λ is large, as for small λ the  integral is just bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' In particular if α = 1 and β = 1 we have oscillatory integral  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  The main result of the work is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Let f be a smooth finite type function of two variables defined in a  sufficiently small neighborhood of the origin and let ψ ∈ C∞  0 (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Let h be the height of the function f, and let m = 0, 1 be the multiplicity of its  Newton polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' If 0 < α < 1, β > 0, h > 1, and λ ≫ 1 then we have the  estimate  ����  �  U  Eα,β(iλf(x1, x2))ψ(x)dx  ���� ≤  C| ln λ|m∥ψ∥L∞(U)  λ  1  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='4)  If 0 < α < 1, β > 0, h = 1 and λ ≫ 1, then we have following estimate  ����  �  U  Eα,β(iλf(x1, x2))ψ(x)dx  ���� ≤  C| ln λ|2∥ψ∥L∞(U)  λ  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='5)  where the constants C are independent of the phase, amplitude and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Auxiliary statements  We first recall some useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' If 0 < α < 2, β is an arbitrary real number, µ is such that πα/2 <  µ < min{π, πα}, then there is C > 0, such that we have  |Eα,β(z)| ≤  C  1 + |z|, z ∈ C, µ ≤ | arg(z)| ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1)  See [4], [9], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Let Ω be an open, bounded subset of  R2, and let f : Ω → R be a  measurable function such that for all λ ≫ 1 and for some positive δ ̸= 1, we have  ����  �  Ω  eiλf(x)dx  ���� ≤ C|λ|−δ| ln λ|m,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2)  with m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Then, we have  ��x ∈ Ω : |f(x)| ≤ ε| ≤ Cδεδ| ln ε|m, for δ < 1,  for 0 < ε ≪ 1, and for δ > 1, |x ∈ Ω : |f(x)| ≤ ε| ≤ Cδε ,  for δ = 1,  ��x ∈ Ω : |f(x)| ≤ ε| ≤ Cδε| ln ε|m+1,  where Cδ depends only on δ, |A| means the Lebesgue measure of a set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' See [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS  5  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' For the convenience of the reader we give an independent proof of Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' We consider an even non-negative smooth function  ω(x) =  �  1,  when |x| ≤ 1,  0,  when |x| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  For the characteristic function of Ω with Ω ⊂ U, the following inequality holds true  |x ∈ Ω : |f(x)| ≤ ε| =  �  Ω  χ[0,1]  �|f(x)|  ε  �  dx ≤  �  Ω  ω  �f(x)  ε  �  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Now we will use the Fourier inversion formula, and rewrite the last integral as  �  Ω  ω  �f(x)  ε  �  dx = 1  2π  �  Ω  � ∞  −∞  ˇω(ξ)eiξ f(x)  ε dξdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  As ˇω(ξ) is a Schwartz function, we can use Fubini theorem and change the order of  integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' So we have  �  Ω  � ∞  −∞  ˇω(ξ)eiξ f(x)  ε dξdx =  � ∞  −∞  ˇω(ξ)  �  Ω  eiξ f(x)  ε dxdξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  We use inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2) for the inner integral and get  ����  �  Ω  eiξ f(x)  ε dx  ���� ≤ C| ln(2 + ξ  ε)|m  (1 + | ξ  ε|)δ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  As ˇω(ξ) is a Schwartz function, we also have  |ˇω(ξ)| ≤  C  1 + |ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  So  �����  � ∞  −∞  C ˇω(ξ)| ln(2 + ξ  ε)|m  (2 + | ξ  ε|)δ  dξ  ����� ≲  � ∞  0  2C| ln( ξ  ε)|m  (1 + |ξ|)(2 + | ξ  ε|)δ dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Now we change the variable as ξ = ηε, and we get  � ∞  0  | ln( ξ  ε)|m  (1 + |ξ|)(2 + | ξ  ε|)δ dξ =  � ∞  0  ε| ln η|m  (1 + |εη|)(2 + |η|)δ dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Now we estimate the last integral for different values of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  If δ < 1 then we have  � ∞  0  ε| ln η|m  (1 + |εη|)(2 + |η|)δ dη ≤ Cε  �  1  ε  0  | ln η|mdη  (2 + η)δ + Cε  � ∞  1  ε  | ln η|mdη  εηδ+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  We represent  1  (2+η)δ =  1  ηδ(1+ 2  η )δ =  1  ηδ + O(  1  ηδ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' So  Cε  �  1  ε  0  | ln η|mdη  (2 + η)δ = ε  � 2  0  | ln η|mdη  (2 + η)δ + ε  �  1  ε  2  | ln η|mdη  (2 + η)δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Integrating by parts we obtain  ε  �  1  ε  2  | ln η|mdη  (2 + η)δ ≤ ε  �  1  ε  2  | ln η|mdη  ηδ  ≤ Cεδ| ln ε|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  6  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='IKROMOV, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='RUZHANSKY, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='SAFAROV  As δ < 1, the integrals  � 2  0  | ln η|mdη  (2+η)δ  and  � ∞  1  ε  | ln η|mdη  εηδ+1  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  If δ > 1 then we trivially obtain  ����  � ∞  0  Cε| ln η|m  (1 + |εη|)(2 + |η|)δ dη  ���� ≤ Cε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  If δ = 1 then assuming 0 < ε < 1  2 we get |εη| < 1 (for |η| < 2), then write the integral  as the sum of three integrals and obtain  ����  � ∞  0  Cε| ln η|m  (1 + |εη|)(1 + |η|)dη  ���� ≤  ����  � 2  0  Cε| ln η|mdη  ���� +  �����  �  1  ε  2  Cε| ln η|m  η  dη  ����� +  �����  � ∞  1  ε  Cε| lnη|m  η  dη  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Then we have  ����  � 2  0  Cε| ln η|mdη  ���� ≤ Cε,  and we get with simple calculating that  �����  �  1  ε  2  Cε| lnη|m  η  dη  ����� ≤ Cε| ln ε|m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  We use the formula of integrating by parts several times, to get  �����  � ∞  1  ε  Cε| ln η|m  η  dη  ����� ≤ Cε| ln ε|m,  completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  □  From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2 we get the following corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Let f(x1, x2) be a smooth function with f(0, 0) = 0, ∇f(0, 0) = 0,  and h be the height of the function f(x1, x2), and let m = 0, 1 be the multiplicity of  its Newton polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Let also a(x) =  �  1,  when |x| ≤ σ,  0,  when |x| ≥ 2σ,  σ > 0, and a(x) ≥ 0  with a ∈ C∞  0 (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' If for all real λ ≫ 1 and for any positive δ ̸= 1, the following  inequality holds  ����  �  R2 eiλf(x)a(x)dx  ���� ≤ C|λ|−δ| ln λ|m,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='3)  then we have  ||x| ≤ σ : |f(x)| ≤ ε| ≤ Cεδ| ln ε|m,  where m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' See [8, 12, 14, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Let f(x1, x2) be a smooth function with f(0, 0) = 0, ∇f(0, 0) = 0,  and let Ω be a sufficiently small compact set around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Let also h be the  height of the function f(x1, x2), and let m = 0, 1 be the multiplicity of its Newton  polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Then for all 0 < ε ≪ 1 we have  |x ∈ Ω : |f(x)| ≤ ε| ≤ Cε  1  h| ln ε|m,  where h is the height of f and m is its multiplicity [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS  7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Proof of the main result  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' As for λ < 2 the integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='3) is just bounded, we  consider the case λ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Without loss of generality, we can consider the integral over  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Using inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1), we have  |Eα,β(iλf(x))| ≤  C  1 + λ|f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1)  We then use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1) for the integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='3), and get that  |Iα,β| ≤  ����  �  U  Eα,β(iλf(x))ψ(x)dx  ���� ≤ C  �  U  |ψ(x)|dx  1 + λ|f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2)  Now we represent the integral Iα,β over the union of sets Ω1 := Ω ∩ {λ|f(x1, x2)| <  M} and Ω2 := Ω ∩ {λ|f(x1, x2)| ≥ M} respectively, where M is a positive real  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  We estimate the integral Iα,β over the sets Ω1 and Ω2, respectively,  |Iα,β| ≤ C  �  U  |ψ(x)|dx  1 + λ|f(x)| = J1 + J2 := C  �  Ω1  |ψ(x)|dx  1 + λ|f(x)| + C  �  Ω2  |ψ(x)|dx  1 + λ|f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  First we estimate the integral over the set Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Using the results of the paper ([17]  page 31) (see also Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='4) we obtain  |J1| = C  �  Ω1  |ψ(x)|dx  1 + λ|f(x)| ≤  C| ln λ|m∥ψ∥L∞(Ω1)  λ  1  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Let f ∈ C∞ and h be the height of the function f, and let m = 0, 1 be  the multiplicity of its Newton polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' For any smooth function a = a(x, y) with  sufficiently small support and for h > 1 the following inequality holds  I :=  �  {|f(x,y)|≥ M  λ }  a(x, y)  1 + λ|f(x, y)|dxdy ≤  C| ln λ|m∥a∥L∞(U)  λ  1  h  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='3)  where supp{a(x, y)} = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Let h > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Consider the sets  Ak =  �  x ∈ U : 2k  λ ≤ |f(x)| ≤ 2k+1  λ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  For the measure of a set of smaller values we use Lemma 1  ′ in the paper [16] (see also  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='4), and we have  µ  �  |f(x)| ≤ 2k+1  λ , x ∈ U  �  ≤ C  �2k+1  λ  � 1  h �  ln  ����  λ  2k+1  ����  �m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Let  Ik :=  �  Ak  a(x, y)  1 + λ|f(x, y)|dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  For the integral  �  2k≤λ|f(x)|≤2k+1  Ik =  �  Ω2  a(x, y)  1 + λ|f(x, y)|dxdy,  8  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='IKROMOV, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='RUZHANSKY, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='SAFAROV  we find the following estimate:  |Ik| =  ����  �  Ak  a(x, y)  1 + λ|f(x, y)|dxdy  ���� ≤ C∥a∥L∞(U)  �2k+1  λ  � 1  h ����ln 2k+1  λ  ����  m  2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  From here we find the sum of Ik and, by estimating the integral I, we get  I ≤ ∥a∥L∞(U)  ∞  �  k=1  Ik ≤ ∥a∥L∞(U)  ∞  �  k=1  �2k+1  λ  � 1  h ����ln 2k+1  λ  ����  m  2−k  ≤ ∥a∥L∞(U)  | ln λ|m  λ  1  h  ∞  �  k=1  2  k+1  h −kkm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  As h > 1, the last series is convergent, proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Consider the case h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  The smooth function has non-degenerate  critical point at the origin if and only if h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' As f(x, y) is a smooth function with  ∇f(0, 0) = 0, using Morse lemma we have f ∼ x2 ± y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' So in this case we estimate  two sets ∆ = ∆1 ∪ ∆2, where ∆1 := {(x, y) : λ|x2 ± y2| ≤ M, |x| ≤ 1, |y| ≤ 1} and  ∆2 := {(x, y) : λ|x2 ± y2| > M, |x| ≤ 1, |y| ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' First we consider the integral over  the set ∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Then we have  ����  �  ∆1  a(x, y)  1 + λ|x2 ± y2|dxdy  ���� ≤ C∥a∥L∞(∆1)  ����  �  ∆1  dxdy  ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Now we estimate the last integral as  ����  �  λ|x2+y2|≤M  dxdy  ���� ≤ C  λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Then we estimate the measure of the set {|x2 − y2| ≤ εM}, where ε = 1  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' We have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  for simplicity putting M = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  ����  �  |x2−y2|≤εM  dxdy  ���� ≤ C  �����  � √1−ε  √ε  dy  � √  y2+ε  √  y2−ε  dx  ����� =  �����  � √1−ε  √ε  ��  y2 + ε −  �  y2 − ε  �  dy  ����� =  =  �y  2  �  y2 + ε + ε  2 ln |y +  �  y2 + ε|  � ���  √1−ε  √ε  −  �y  2  �  y2 − ε − ε  2 ln |y +  �  y2 − ε|  � ���  √1−ε  √ε  =  =  �����  √1 − ε  2  + ε  2 ln  √1 − ε + 1  √ε  −  √  2  2 ε − ε  2 ln |√ε(1 +  √  2)|−  −  ��  (1 − ε)(1 − 2ε)  2  − ε  2 ln |  √  1 − ε +  √  1 − 2ε| + ε  2 ln √ε|  ������ ≤ Cε ln ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Now we consider the integral over the set ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' In this case we change the variables  to polar coordinate system and with easy calculating we get  ����  �  {λ|x2+y2|≥M}  a(x, y)  1 + λ|x2 + y2|dxdy  ���� ≤ C| ln λ|∥a∥L∞(∆2)  λ  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='4)  and  ����  �  {λ|x2−y2|≥M}  a(x, y)  1 + λ|x2 − y2|dxdy  ���� ≤ C| ln λ|2∥a∥L∞(∆2)  λ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='5)  OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS  9  Now we continue the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Let h > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' We use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2 for  the integral J1, to get  |J1| ≤  C| ln λ|m∥a∥L∞(U)  λ  1  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Let consider the integral J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' If h > 1, then using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1 we get  |J2| ≤  C| ln λ|∥a∥L∞(U)  λ  1  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  If h = 1, using the Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2 we get the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='2 shows that if h = 1, we can get a more precise result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' If h = 1 and f has an extremal point at the point (0,0) (then f is  diffeomorhic equivalent to x2  1 + x2  2 or −x2  1 − x2  2), then we have  |Iα,β| ≤  C| ln λ|∥ψ∥L∞(U)  λ  ,  for all λ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Declaration of competing interest  This work does not have any conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Acknowledgements  The second author was supported in parts by the FWO Odysseus 1 grant G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='0H94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='18N:  Analysis and Partial Differential Equations and by the Methusalem programme of the  Ghent University Special Research Fund (BOF) (Grant number 01M01021) and also  supported by EPSRC grant EP/R003025/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' The manuscript has no associated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  References  [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Agarwal, A propos d’une note de M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='Pierre Humbert, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
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+page_content=' Mittag-Leffler, Une g´en´eralization de l’int´egrale de Laplace-Abel, Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Paris 136, 537-539 (1903).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Mittag-Leffler, Sur la nouvelle fonction Eα(x), Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Paris 137, 554-  558 (1903).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Mittag-Leffler, Sopra la funzione Eα(x), Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='Lincei, (Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='5)13, 3-5 (1904).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  [22] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Phong and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Stein, The Newton polyhedron and oscillatory integral operator, Acta  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
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+page_content=' Podlubny, Fractional Differensial Equations, Academic Press, New York, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
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+page_content=' Ruzhansky, Pointwise van der Corput Lemma for Functions of Several Variables, Functional  Analysis and Its Applications, 43 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1, 75–77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Ruzhansky, Multidimensional decay in the van der Corput Lemma, Studia Mathematica,  208 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1, 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Ruzhansky, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Torebek, Van der Corput lemmas for Mittag-Leffler functions, Fractional  Calculus and Applied Analysis, 23 (6), (2021), 1663–1677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Ruzhansky,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Torebek,  Van der Corput lemmas for Mittag-Leffler functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  α−directions , Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
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+page_content='  [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Ruzhansky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Safarov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Khasanov, Uniform estimates for oscillatory integrals with  homogeneous polynomial phases of degree 4, Analysis and Mathematical Physics, 12(130),  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Safarov, Invariant estimates of two-dimensional oscillatory integrals // Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' 104,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='293–302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Safarov, On invariant estimates for oscillatory integrals with polynomial phase, // J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Sib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' 9 (2016), P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='102–107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Safarov, On a problem of restriction of Fourier transform on a hypersurface // Russian  Mathematics, 63 (4), P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='57-63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Safarov, Estimates for Mittag-–Leffler Functions with Smooth Phase Depending on Two  Variables, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Sib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=', 15(4) (2022), P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
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+page_content='  [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Varchenko, Newton polyhedra and estimation of oscillating integrals //Functional Analysis  and Its Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' 10, pages 175-–196 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  [34] Van der Korput, Zur Methode der stationaren phase// Compositio Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' 1934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' 15-38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='  OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS  11  Isroil A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Ikromov  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content=' Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan  Olmazor district, University 46, Tashkent, Uzbekistan  Samarkand State University  Department of Mathematics, 15 University Boulevard  Samarkand, 140104, Uzbekistan  Email address: ikromov1@rambler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='ru  Michael Ruzhansky  Department of Mathematics: Analysis, Logic and Discrete Mathematics  Ghent University,  Krijgslaan 281, Ghent, Belgium,  School of Mathematical Sciences, Queen Mary University of London,  United Kingdom  Email address: michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='ruzhansky@ugent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='be  Akbar R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
+page_content='Safarov  Uzbek-Finnish Pedagogical Institute  Spitamenshox 166, Samarkand, Uzbekistan  Samarkand State University  Department of Mathematics, 15 University Boulevard  Samarkand, 140104, Uzbekistan  Email address: safarov-akbar@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfSwmi/content/2301.04436v1.pdf'}
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+IDENTITY MASKING WITH EYE ENHANCEMENT
+1
+Identity masking effectiveness and gesture
+recognition: Effects of eye enhancement in seeing
+through the mask
+Madeline Rachow∗, Thomas Karnowski† and Alice J. O’Toole‡
+∗University of Arkansas
+† Oak Ridge National Laboratory
+∗ The University of Texas at Dallas
+Email: ∗[email protected], †[email protected], ‡[email protected]
+Abstract—Face identity masking algorithms developed in re-
+cent years aim to protect the privacy of people in video
+recordings. These algorithms are designed to interfere with
+identification, while preserving information about facial actions.
+An important challenge is to preserve subtle actions in the eye
+region, while obscuring the salient identity cues from the eyes. We
+evaluated the effectiveness of identity-masking algorithms based
+on Canny filters, applied with and without eye enhancement, for
+interfering with identification and preserving facial actions. In
+Experiments 1 and 2, we tested human participants’ ability to
+match the facial identity of a driver in a low resolution video to
+a high resolution facial image. Results showed that both masking
+methods impaired identification, and that eye enhancement did
+not alter the effectiveness of the Canny filter mask. In Experiment
+3, we tested action preservation and found that neither method in-
+terfered significantly with driver action perception. We conclude
+that relatively simple, filter-based masking algorithms, which are
+suitable for application to low quality video, can be used in
+privacy protection without compromising action perception.
+Index Terms—identity-masking, face recognition, privacy, hu-
+man visual perception, driver behavior, de-identification, action
+preservation.
+I. INTRODUCTION
+Video recordings for security and surveillance are now
+ubiquitous in public and private spaces. This has lead to a
+pressing need to develop face identity masking algorithms
+aimed at protecting the privacy of people in the recordings.
+Facial identity masking technology also needs to preserve
+the facial actions (gestures and expressions) of those being
+photographed for applications that require action classification
+without identification. Understanding and measuring the extent
+to which identity-masking algorithms effectively accomplish
+both goals is a challenging problem. Because identification
+and action classification are tasks that can be done accurately
+by humans, the success of masking algorithms cannot be eval-
+uated comprehensively without examining human perception.
+Human identification and gesture categorization of identity-
+masked faces have been examined previously [1]. The effec-
+tiveness of eight different identity masking algorithms was
+evaluated using human perception and a deep convolutional
+neural network (DCNN) trained for face identification. Human
+participants and the DCNN were tested with videos taken of
+drivers actively operating a motor vehicle. For the human ex-
+periment, people studied high-resolution images of the drivers
+to learn their identities and were tested on their recognition
+of those drivers in low-resolution videos. Test videos were
+low resolution and showed drivers actively operating a motor
+vehicle. Videos were either unmasked or masked by one
+of eight algorithms, including methods that rely on Facial
+Action Transfer (FAT) (cf., [2], [3]), a DMask [4], Canny
+filtering [5], and Scharr filtering [6]. The results showed that
+all of the algorithms reduced human face recognition accuracy.
+Moreover, people made their recognition decisions with a
+conservative response bias (i.e., a tendency to indicate that
+they did not recognize drivers, when they were uncertain).
+This bias indicates that the participants had low confidence in
+their identification decisions—supporting the effectiveness of
+the masking methods.
+In the machine evaluation of that test [1], the DCNN
+matched identities between the high-resolution images and
+masked videos, and between the unmasked and masked
+videos. DCNN performance matching high-resolution images
+to masked and unmasked videos showed a pattern of poor
+performance approximately comparable to human behavior—
+echoing the effectiveness of the masking algorithms for both
+humans and the CNN. The results showed that even simple
+methods, such as edge-detection, can impair identification
+performance.
+It is worth noting that more sophisticated methods than
+filtering have been developed for identity masking, including
+generative adversarial networks, GANS (e.g., [7]). However,
+these techniques can only be applied to high quality (frontal)
+images and are computationally intense, which limits their util-
+ity for high volume throughput (e.g., videos). Many important
+applications of face identity masking must deal with large
+quantities of low resolution, poor quality video. Therefore,
+there is a need to consider the effectiveness of simpler methods
+that can be applied in these less controlled circumstances.
+The present work builds on previous work [1], with the
+goal of looking more carefully at the role the eyes play in
+facilitating face recognition in the context of identity mask-
+ing. Simple filtering operations can preserve eye information,
+which is both valuable for gesture recognition, but may also
+inadvertently boost face recognition by people. Specifically,
+in human perception experiments, the eye region of the face
+is known to support particularly good face recognition (e.g.,
+arXiv:2301.08408v1  [cs.CV]  20 Jan 2023
+
+IDENTITY MASKING WITH EYE ENHANCEMENT
+2
+Fig. 1: Example stimuli from the mask conditions. a. Canny+Eyezoom; b. (left) Unmasked, (right) Canny
+[8]). In this study, we tested whether eye enhancement of an
+identity masked face would increase human face identification
+performance. To that end, we created a set of stimuli in which
+the eye region was localized, expanded in size, and enhanced
+with a Scharr filter [6]. We compared face identification in
+three masking conditions: 1.) unmasked driver videos, 2.)
+driver videos masked with the Canny method [5], and 3.)
+a combination method that showed the Canny-masked video
+with an inset of the Scharr-enhanced eye region. See Figure 1
+for an example of the stimulus conditions. Note that we chose
+the Canny method filter for our masking algorithm, because
+it is relatively simple, easy to implement, and is effective for
+both identity-masking and action preservation [1].
+In the first and second experiments, we focused on the
+effectiveness of identity masking. Videos were either shown
+unmasked (unaltered), masked solely with Canny, or masked
+with Canny and Canny+EyeZoom (see details, section II-B).
+The third experiment examined action preservation in these
+conditions.
+A. Study contributions
+• Masking the face of a driver in a video using a Canny
+filter effectively impairs face identification by comparison
+to an unmasked video.
+• Enhancing and enlarging the eye region (Eyezoom of the
+face) and masking it with a Schaar filter does not alter
+the effectiveness of the Canny filter mask.
+• Facial actions are preserved, in large part, when drivers’
+faces are masked with both the Canny and Canny +
+Eyezoom manipulations.
+II. METHODS
+A. Dataset
+Stimuli for the present experiment came from a set of
+driver videos in the Head Pose Validation (HPV) database.
+The HPV dataset was created to emulate data from the SHRP2-
+Naturalistic Driving Study (SHRP2-NDS) database [9], which
+is not easily available for research applications. The SHRP2-
+NDS database is nearly unique in the range of imaging con-
+ditions encompassed in the data. It includes approximately 2
+petabytes of video from approximately 3, 400 drivers obtained
+over 1 to 2 years of observation. However, the dynamic video
+nature of the dataset provides for highly salient, personally
+identifiable, information about the drivers. The dataset is
+characterized by extreme illumination conditions (e.g., night-
+time shadowing, day-time bright spots, or illumination via
+transient headlights as a car turns). There is also the problem
+of quick driver movements (e.g., head turns and other actions
+which are very common in real-world driving).
+The HPV dataset used in the present study includes low-
+resolution videos of people actively driving a car or performing
+staged actions typical while driving, such as using a cellphone
+and putting on headwear or glasses. The video resolution is
+356 × 240 pixels, with a frame rate of 14.98 frames per second.
+Each video segment was edited to 4s and masks were applied
+to the segments for direct comparison of mask effectiveness
+across conditions. Video length ranged from 1-4s depending on
+the type of action (looking left, looking right, looking down).
+The video segment lengths were identical for each identity
+across conditions.
+B. Conditions
+The three masking conditions tested were implemented, as
+follows:
+
+a
+b.IDENTITY MASKING WITH EYE ENHANCEMENT
+3
+• unmasked - drivers’ faces were unaltered.
+• Canny mask - drivers’ faces were altered by applying
+a series of processes aimed at producing optimal edge
+detection, including the use of a Gaussian smoothing
+filter, a set of gradient-based edge detectors to enhance
+edges in the image, and then non-maximum suppression,
+threshold, and tracking to produce thin, refined edges.
+• Eyezoom condition– drivers’ faces were first masked
+with the Canny process. Then the eyes were detected in
+the original image using the retinaface algorithm [10].
+The original image was then expanded and masked with a
+Schaar filter, and the region around the eye detection was
+cropped. Finally the Canny-masked face was presented
+in an inset showing the Schaar-filtered, zoomed eyes (see
+Fig. II-B).
+III. EXPERIMENT I: EFFECT OF EYEZOOM MASKING
+METHOD
+In Experiment 1, we investigated the effectiveness of the
+Canny and Canny+Eyezoom filters at masking the identities
+of drivers in low-resolution videos.
+A. Participants
+A total of 30 (11 male, 18 female, 1 other) undergraduate
+student volunteers (ages 18-34) from the University of Texas
+at Dallas (UTD) participated in the study in exchange for
+research credit. All human experimental procedures were
+approved by UTD’s Institutional Review Board.
+B. Procedure
+The experiment was composed of 72 trials in which a video
+stimulus was displayed in the top center of the screen. The
+response options were presented below the video and showed
+two faces and silhouette (see Figure 2). Participants were asked
+to select the face image that matched the driver in the video or
+to select the silhouette if neither of the two images matched the
+driver. In target-present trials (n = 36), one of the two faces
+matched the driver. In target-absent trials (n = 36), neither of
+the two faces matched the driver. In all cases, the two face
+images presented as options showed similar-looking identities
+from the dataset. Each of the dataset’s 36 identities was shown
+twice, once with the correct response being one of the target-
+present choices and once with the correct response being the
+target-absent choice.
+The video segments were shown in random order and looped
+until the subjects responded. Subjects were assigned randomly
+to one of the three masking conditions, with the unmasked
+condition serving as a control for general recognition success.
+Subjects were asked to determine whether the identity in the
+video matched one of the two identity images shown or if the
+identity was absent from the identity images shown.
+C. Outcome Measures
+1) Accuracy: Accuracy was assessed in two ways using a
+signal detection-type calculation based on d’. This measure
+depends on the proportion of hits p(hit) and false alarms
+p(false alarms), as follows:
+d′ = z(p(hit)) − z(p(false alarms),
+where the z refers to the z-score.
+In this experiment, hits were defined as target-present trials
+in which participants correctly recognized a driver as the
+matched-identity response choice. The design of the response
+options in the experiment offered two ways to compute false
+alarms. Specifically, false alarms can be defined as: a.) target-
+present trials in which the participant choose the incorrect
+identity; and/or b.) incorrect target-absent trials in which
+neither image showed the identity (i.e., participants chose one
+of the face images, when neither was an identity match to the
+video). Because both options are consistent with the concept
+of a false alarm, in what follows, we included both types of
+false alarms (a and b) in the accuracy computation.
+D. Results
+1) Accuracy: Figure 3 shows the average d’ for each mask
+condition. These values indicate that faces in the unmasked
+condition were identified moderately well, but face recognition
+in both masked conditions was significantly impaired. The
+negative d’ values for the masked conditions are unusual
+and suggest that participants used a systematically incorrect
+decision strategy, which we will investigate further in Section
+III-D2.
+A one-factor Analysis of Variance (ANOVA) was performed
+on accuracy (d’), with mask condition as the independent
+variable. The resulting model yielded a main effect of mask
+condition on d’, F(2, 27) = 11.03, p < .001. When comparing
+the conditions, d’ accuracy was significantly higher in the
+unmasked condition than in the masked conditions, with no
+significant difference between the two masked conditions. This
+suggests that Canny and ORNL masking are not significantly
+less effective when used together than Canny masking alone.
+As is clear from the Fig. 3, participant performance was more
+variable in the Eyezoom condition.
+2) Response Distribution: To further investigate the finding
+of negative d’s, we examined the proportion of responses
+allocated to each response type (face images chosen versus
+no identity chosen). The pattern of responses is shown for
+each mask type in Figure 4, with separate graphs for target-
+present (correct identity was available as a choice) and target-
+absent (correct identity was not available as a choice) trials.
+For the unmasked condition, the graphs show a standard
+(relatively accurate) pattern of responses as a function of
+whether the target was present or absent. The graphs for
+the masked conditions show inaccurate performance, but also
+suggest that participants did not systematically choose the no-
+identity match when a match was present, but instead often
+chose the wrong face as the identity match.
+We conclude tentatively that performance in the masked
+conditions was very poor indicating the effectiveness of the
+masks for preventing identification. However, given the un-
+usual performance in the masked condition (i.e., negative d’s),
+
+IDENTITY MASKING WITH EYE ENHANCEMENT
+4
+Fig. 2: Example trial in Experiment 1.
+Fig. 3: Experiment 1 accuracy, measured as d’ across
+conditions. Results show that both masking algorithms were
+equally effective.
+we retested the conditions with a design that eliminates the
+possibility of response bias.
+IV. EXPERIMENT II: EFFECT OF EYEZOOM MASKING
+METHOD WITH A FORCED-CHOICE TASK
+In this experiment, we used a two-alternative forced choice
+(2AFC) task to test masking effectiveness. In the 2AFC, two
+faces are presented as response options. In all cases, one of
+the two images will be the same identity as the person in the
+video.
+A. Participants
+A total of 30 (7 male, 22 female, 1 other) undergraduate
+student volunteers (ages 18-26) from UTD participated in the
+study in exchange for research credit.
+B. Procedure
+The experiment consisted of 72 trials. The video stimulus
+was displayed in the top center of the screen with the two
+face images beneath it. Participants were asked to determine
+which of the two face images matched the identity shown
+in the video. To make the task challenging, the two faces
+presented had a similar appearance and were of the same race
+and gender. An example trial is shown in Fig. 5.
+Each of the dataset’s 36 identities was shown twice, once
+with the correct response as the left-located option and once
+with the correct response as the right-located option. The
+video segments were shown in random order and looped until
+the subjects responded. Participants were assigned randomly
+to one of the three masking conditions, with the unmasked
+condition serving as a baseline condition for identification
+accuracy.
+C. Results
+Accuracy was assessed as the proportion of correct re-
+sponses. Fig. 6 shows the proportion of correct responses
+for each mask condition. These values indicate that face
+recognition in the unmasked condition was more accurate
+than face recognition in the masked conditions. A one-factor
+ANOVA was performed on the accuracy data (proportion of
+correct responses), with condition as the independent variable.
+The model yielded a main effect of mask condition on
+
+?
+Press "" if the person in the
+Press "2" if the person in the
+Press "3" if the person in the video
+video is the person on the left.
+video is the person in the middle.
+is NOT either of the two people picturedcondition
+unmasked
+canny
+eyezoom
+-1
+-2
+conditionIDENTITY MASKING WITH EYE ENHANCEMENT
+5
+Fig. 4: Proportion of responses by trial type in Experiment I.
+proportion of correct responses, F(2, 27) = 9.68, p < .001.
+As in the first experiment, participants were more accurate in
+the unmasked condition than in the masked conditions, and
+performed comparably for the two masked conditions.
+The results replicate the pattern of performance across
+conditions found for Experiment 1. As expected with a 2AFC
+task, performance was more accurate in all three conditions
+than it was in Experiment 1. Notably, average identification
+was above chance in both masked conditions. Performance in
+the Eyezoom condition was more variable than performance
+in the Canny mask condition—replicating a similar finding in
+Experiment I.
+We conclude that the masks strongly inhibit identification,
+but that when forced to guess between two images (with the
+assurance that one was an identity match), participants fared
+better than chance. Notwithstanding, applications of identity
+masking would rarely if ever be able to assure a human or ma-
+chine system that one of two candidates was an identity match.
+Our goal in applying this method here was to test examine the
+role of response bias in the unusual pattern of results found in
+Experiment 1. The present results suggest that these masking
+algorithms leave behind some residual identity information in
+the face that humans can exploit when the response decision
+is highly constrained. As noted, it is unlikely that that would
+
+Unmasked Target Present Trials
+Unmasked Target Absent Trials
+1.00
+response
+1.00
+response
+responses
+chose either identity
+chose either identity
+0.75
+chose no identity
+chose no identity
+09'0 9.
+prop
+prop
+0.00
+0.00
+response
+response
+Canny Target Present Trials
+Canny Target Absent Trials
+1.00
+response
+1.00
+response
+responses
+chose either identity
+chose either identity
+0.75
+ chose no identity
+chose no identity
+090 9
+pro
+pro
+0.00
+0.00
+response
+response
+Eyezoom Target Present Trials
+Eyezoom Target Absent Trials
+1.00
+response
+1.00
+response
+chose eitheridentity
+chose either identity
+chose noidentit
+chose no identity
+res
+pro
+pro
+0.00
+0.00
+response
+responseIDENTITY MASKING WITH EYE ENHANCEMENT
+6
+Fig. 5: Example trial from Experiment II.
+Fig. 6: Experiment 2 - identification accuracy across
+conditions.
+be the case in any applied scenario, and so we conclude that
+these simple simple filtering procedures provide a reasonably
+high degree of identity protection. Additionally, we conclude,
+albeit more tentatively, that the eyezoom procedure does not
+improve identification significantly over the Canny procedure.
+V. EXPERIMENT III: EFFECT OF EYEZOOM MASKING
+METHOD ON ACTION PRESERVATION
+The effectiveness of the identity protection provided by
+these masks opens the question of whether this protection
+comes at the cost of preserving information about facial
+actions. In this experiment, we examined whether the Canny
+and Canny+Eyezoom mask conditions impaired driver facial
+action perception.
+A. Participants
+A total of 30 (6 male, 23 female, 1 nonbinary) undergradu-
+ate student volunteers (ages 18-30) from UTD participated in
+the study in exchange for research credit.
+B. Procedure
+The experiment consisted of 100 trials, each with three
+response options: a.) driver looking to the left, 2.) driver
+looking to the right, and 3.) driver looking down. Each of the
+36 identities in the dataset appeared between two and three
+times, each with a different action (looking right, left, down).
+Prior to the start of the main experiment, a pilot test with
+only the unmasked condition was conducted to ensure that the
+actions were identifiable in all videos. This test resulted in the
+elimination of eight (of 108) videos segments in which actions
+were not recognizable at sufficiently high levels of accuracy
+for inclusion in the action preservation study.
+The participants were assigned randomly to one of three
+masking conditions with the unmasked condition providing a
+baseline action recognition accuracy and were asked to identify
+whether the driver was looking to the left, right, or down. The
+video stimuli were shown in the upper center of the screen
+with three written options below. See Fig. 7 for an example
+trial. The clips were played in a random order and looped until
+the participant responded.
+C. Results
+The proportion of correct responses was used to assess accu-
+racy. Fig. 8 shows the proportion of correct responses for each
+mask condition. These values indicate that action preservation
+
+Press"1" if the person in the
+Press "2" if the person in the
+videoisthepersonontheleft
+video is the person in the rightcondition
+0.9 
+unmasked
+canny
+eyezoom
+I of correct response
+0.7
+ortion
+propor
+0.6
+0.5 
+conditionIDENTITY MASKING WITH EYE ENHANCEMENT
+7
+Fig. 7: Example trial from Experiment III.
+was generally high, but also suggest a small advantage for
+action perception in the unmasked condition. A one-factor
+ANOVA, performed on the accuracy (proportion of correct
+responses) data, with the independent variable of condition,
+did not show a significant effect, but was generally consistent
+with this conclusion. The model yielded a marginal main
+effect of mask condition on proportion of correct responses,
+F(2, 27) = 2.69, p = 0.086. This suggests a very slight
+advantage for action perception without stimulus masking.
+In conclusion, although the results did not reach statistical
+significance, there is some indication that masking impaired
+action perception.
+VI. DISCUSSION
+Our goal was to examine the effectiveness of simple
+Canny-filtering based masking methods, with and without eye
+enhancement, for interfering with face identification while
+preserving facial actions. In Experiment I, face recognition
+accuracy was diminished for both mask conditions, relative
+to the unmasked condition. There was no difference between
+the Canny mask alone and the mask with eye enhancement. In
+Experiment II, we replicated this result with a 2AFC procedure
+that controlled for response option bias, which may have been
+a factor in the findings of negative ’. values for both masking
+conditions. In combination, both studies point to the relative
+effectiveness of the masks for interfering with identification.
+They also point to the conclusion that eye enhancement did
+not alter this effectiveness. Experiment III showed that facial
+actions were preserved to a similar degree with both masks,
+Fig. 8: ANOVA of proportion of correct responses.
+though there was a marginal advantage for action perception
+in the unmasked condition.
+In summary, these results indicate that Eyezoom masking
+does not significantly increase identification or alter facial
+action preservation.
+ACKNOWLEDGMENT
+This work was supported through collaboration with Oak
+Ridge National Laboratory and the Federal Highway Admin-
+
+Press "1" if the person looks toward the driver's side.
+Press "2" if the person looks toward the passenger's side.
+Press "3" if the person looks down.condition
+unmasked
+canny
+eyezoom
+f correct responses
+0.96
+0.92
+of
+proportion
+0.88
+un
+ma
+ez
+conditionIDENTITY MASKING WITH EYE ENHANCEMENT
+8
+istration under the Exploratory Advanced Research Program.
+The human experiment and analysis was subcontracted to
+the University of Texas at Dallas from Oak Ridge National
+Laboratory.
+This manuscript has been authored in part by UT-Battelle,
+LLC, under contract DE-AC05-00OR22725 with the US De-
+partment of Energy (DOE). The US government retains and
+the publisher, by accepting the article for publication, acknowl-
+edges that the US government retains a nonexclusive, paid-up,
+irrevocable, worldwide license to publish or reproduce the pub-
+lished form of this manuscript, or allow others to do so, for US
+government purposes. DOE will provide public access to these
+results of federally sponsored research in accordance with the
+DOE Public Access Plan (http://energy.gov/downloads/doe-
+public-access-plan).
+REFERENCES
+[1] K. D. O. Hooge, A. Baragchizadeh, T. P. Karnowski, D. S. Bolme,
+R. Ferrell, P. R. Jesudasen, C. D. Castillo, and A. J. O’toole, “Evaluating
+automated face identity-masking methods with human perception and
+a deep convolutional neural network,” ACM Transactions on Applied
+Perception (TAP), vol. 18, no. 1, pp. 1–20, 2020.
+[2] D. Huang and F. De La Torre, “Facial action transfer with personalized
+bilinear regression,” in Computer Vision–ECCV 2012.
+Springer, 2012,
+pp. 144–158.
+[3] X. Xiong and F. De la Torre, “Supervised descent method and its
+applications to face alignment,” in Proceedings of the IEEE conference
+on computer vision and pattern recognition, 2013, pp. 532–539.
+[4] Federal
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+Project:
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+Program,
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+FHWA-PROJ-
+14-0054, 2014-2016. [Online]. Available: https://highways.dot.gov/
+dmask-reliable-identity-masking-system-driver-safety-video-data
+[5] J. Canny, “A computational approach to edge detection,” IEEE Transac-
+tions on pattern analysis and machine intelligence, no. 6, pp. 679–698,
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+Handbook of computer vision and applications, vol. 2, pp. 125–151,
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+J. Hankey, and T. Dingus, “Transportation safety meets big data: the
+shrp 2 naturalistic driving database,” Journal of the Society of Instrument
+and Control Engineers, no. 55.5, pp. 415–421, 2016.
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+“Retinaface: Single-shot multi-level face localization in the wild,” Pro-
+ceedings of the IEEE/CVF conference on computer vision and pattern
+recognition, 2020.
+

0dFAT4oBgHgl3EQfCRwI/content/tmp_files/load_file.txt

@@ -0,0 +1,348 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf,len=347
+page_content='IDENTITY MASKING WITH EYE ENHANCEMENT  1  Identity masking effectiveness and gesture  recognition: Effects of eye enhancement in seeing  through the mask  Madeline Rachow∗, Thomas Karnowski† and Alice J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' O’Toole‡  ∗University of Arkansas  † Oak Ridge National Laboratory  ∗ The University of Texas at Dallas  Email: ∗mrachow@uark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='edu, †karnowskitp@ornl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='gov, ‡otoole@utdallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='edu  Abstract—Face identity masking algorithms developed in re-  cent years aim to protect the privacy of people in video  recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' These algorithms are designed to interfere with  identification, while preserving information about facial actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  An important challenge is to preserve subtle actions in the eye  region, while obscuring the salient identity cues from the eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' We  evaluated the effectiveness of identity-masking algorithms based  on Canny filters, applied with and without eye enhancement, for  interfering with identification and preserving facial actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In  Experiments 1 and 2, we tested human participants’ ability to  match the facial identity of a driver in a low resolution video to  a high resolution facial image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Results showed that both masking  methods impaired identification, and that eye enhancement did  not alter the effectiveness of the Canny filter mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In Experiment  3, we tested action preservation and found that neither method in-  terfered significantly with driver action perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' We conclude  that relatively simple, filter-based masking algorithms, which are  suitable for application to low quality video, can be used in  privacy protection without compromising action perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Index Terms—identity-masking, face recognition, privacy, hu-  man visual perception, driver behavior, de-identification, action  preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' INTRODUCTION  Video recordings for security and surveillance are now  ubiquitous in public and private spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' This has lead to a  pressing need to develop face identity masking algorithms  aimed at protecting the privacy of people in the recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Facial identity masking technology also needs to preserve  the facial actions (gestures and expressions) of those being  photographed for applications that require action classification  without identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Understanding and measuring the extent  to which identity-masking algorithms effectively accomplish  both goals is a challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Because identification  and action classification are tasks that can be done accurately  by humans, the success of masking algorithms cannot be eval-  uated comprehensively without examining human perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Human identification and gesture categorization of identity-  masked faces have been examined previously [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The effec-  tiveness of eight different identity masking algorithms was  evaluated using human perception and a deep convolutional  neural network (DCNN) trained for face identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Human  participants and the DCNN were tested with videos taken of  drivers actively operating a motor vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' For the human ex-  periment, people studied high-resolution images of the drivers  to learn their identities and were tested on their recognition  of those drivers in low-resolution videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Test videos were  low resolution and showed drivers actively operating a motor  vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Videos were either unmasked or masked by one  of eight algorithms, including methods that rely on Facial  Action Transfer (FAT) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=', [2], [3]), a DMask [4], Canny  filtering [5], and Scharr filtering [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The results showed that  all of the algorithms reduced human face recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Moreover, people made their recognition decisions with a  conservative response bias (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=', a tendency to indicate that  they did not recognize drivers, when they were uncertain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  This bias indicates that the participants had low confidence in  their identification decisions—supporting the effectiveness of  the masking methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  In the machine evaluation of that test [1], the DCNN  matched identities between the high-resolution images and  masked videos, and between the unmasked and masked  videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' DCNN performance matching high-resolution images  to masked and unmasked videos showed a pattern of poor  performance approximately comparable to human behavior—  echoing the effectiveness of the masking algorithms for both  humans and the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The results showed that even simple  methods, such as edge-detection, can impair identification  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  It is worth noting that more sophisticated methods than  filtering have been developed for identity masking, including  generative adversarial networks, GANS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=', [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' However,  these techniques can only be applied to high quality (frontal)  images and are computationally intense, which limits their util-  ity for high volume throughput (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=', videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Many important  applications of face identity masking must deal with large  quantities of low resolution, poor quality video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Therefore,  there is a need to consider the effectiveness of simpler methods  that can be applied in these less controlled circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  The present work builds on previous work [1], with the  goal of looking more carefully at the role the eyes play in  facilitating face recognition in the context of identity mask-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Simple filtering operations can preserve eye information,  which is both valuable for gesture recognition, but may also  inadvertently boost face recognition by people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Specifically,  in human perception experiments, the eye region of the face  is known to support particularly good face recognition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=',  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='08408v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='CV]  20 Jan 2023  IDENTITY MASKING WITH EYE ENHANCEMENT  2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 1: Example stimuli from the mask conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Canny+Eyezoom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' (left) Unmasked, (right) Canny  [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In this study, we tested whether eye enhancement of an  identity masked face would increase human face identification  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' To that end, we created a set of stimuli in which  the eye region was localized, expanded in size, and enhanced  with a Scharr filter [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' We compared face identification in  three masking conditions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=') unmasked driver videos, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=')  driver videos masked with the Canny method [5], and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=')  a combination method that showed the Canny-masked video  with an inset of the Scharr-enhanced eye region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' See Figure 1  for an example of the stimulus conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Note that we chose  the Canny method filter for our masking algorithm, because  it is relatively simple, easy to implement, and is effective for  both identity-masking and action preservation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  In the first and second experiments, we focused on the  effectiveness of identity masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Videos were either shown  unmasked (unaltered), masked solely with Canny, or masked  with Canny and Canny+EyeZoom (see details, section II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  The third experiment examined action preservation in these  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Study contributions  Masking the face of a driver in a video using a Canny  filter effectively impairs face identification by comparison  to an unmasked video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Enhancing and enlarging the eye region (Eyezoom of the  face) and masking it with a Schaar filter does not alter  the effectiveness of the Canny filter mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Facial actions are preserved, in large part, when drivers’  faces are masked with both the Canny and Canny +  Eyezoom manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' METHODS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Dataset  Stimuli for the present experiment came from a set of  driver videos in the Head Pose Validation (HPV) database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  The HPV dataset was created to emulate data from the SHRP2-  Naturalistic Driving Study (SHRP2-NDS) database [9], which  is not easily available for research applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The SHRP2-  NDS database is nearly unique in the range of imaging con-  ditions encompassed in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' It includes approximately 2  petabytes of video from approximately 3, 400 drivers obtained  over 1 to 2 years of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' However, the dynamic video  nature of the dataset provides for highly salient, personally  identifiable, information about the drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The dataset is  characterized by extreme illumination conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=', night-  time shadowing, day-time bright spots, or illumination via  transient headlights as a car turns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' There is also the problem  of quick driver movements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=', head turns and other actions  which are very common in real-world driving).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  The HPV dataset used in the present study includes low-  resolution videos of people actively driving a car or performing  staged actions typical while driving, such as using a cellphone  and putting on headwear or glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The video resolution is  356 × 240 pixels, with a frame rate of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='98 frames per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Each video segment was edited to 4s and masks were applied  to the segments for direct comparison of mask effectiveness  across conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Video length ranged from 1-4s depending on  the type of action (looking left, looking right, looking down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  The video segment lengths were identical for each identity  across conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Conditions  The three masking conditions tested were implemented, as  follows:  a  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='IDENTITY MASKING WITH EYE ENHANCEMENT  3  unmasked - drivers’ faces were unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Canny mask - drivers’ faces were altered by applying  a series of processes aimed at producing optimal edge  detection, including the use of a Gaussian smoothing  filter, a set of gradient-based edge detectors to enhance  edges in the image, and then non-maximum suppression,  threshold, and tracking to produce thin, refined edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Eyezoom condition– drivers’ faces were first masked  with the Canny process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Then the eyes were detected in  the original image using the retinaface algorithm [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  The original image was then expanded and masked with a  Schaar filter, and the region around the eye detection was  cropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Finally the Canny-masked face was presented  in an inset showing the Schaar-filtered, zoomed eyes (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' EXPERIMENT I: EFFECT OF EYEZOOM MASKING  METHOD  In Experiment 1, we investigated the effectiveness of the  Canny and Canny+Eyezoom filters at masking the identities  of drivers in low-resolution videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Participants  A total of 30 (11 male, 18 female, 1 other) undergraduate  student volunteers (ages 18-34) from the University of Texas  at Dallas (UTD) participated in the study in exchange for  research credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' All human experimental procedures were  approved by UTD’s Institutional Review Board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Procedure  The experiment was composed of 72 trials in which a video  stimulus was displayed in the top center of the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The  response options were presented below the video and showed  two faces and silhouette (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Participants were asked  to select the face image that matched the driver in the video or  to select the silhouette if neither of the two images matched the  driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In target-present trials (n = 36), one of the two faces  matched the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In target-absent trials (n = 36), neither of  the two faces matched the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In all cases, the two face  images presented as options showed similar-looking identities  from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Each of the dataset’s 36 identities was shown  twice, once with the correct response being one of the target-  present choices and once with the correct response being the  target-absent choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  The video segments were shown in random order and looped  until the subjects responded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Subjects were assigned randomly  to one of the three masking conditions, with the unmasked  condition serving as a control for general recognition success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Subjects were asked to determine whether the identity in the  video matched one of the two identity images shown or if the  identity was absent from the identity images shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Outcome Measures  1) Accuracy: Accuracy was assessed in two ways using a  signal detection-type calculation based on d’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' This measure  depends on the proportion of hits p(hit) and false alarms  p(false alarms), as follows:  d′ = z(p(hit)) − z(p(false alarms),  where the z refers to the z-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  In this experiment, hits were defined as target-present trials  in which participants correctly recognized a driver as the  matched-identity response choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The design of the response  options in the experiment offered two ways to compute false  alarms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Specifically, false alarms can be defined as: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=') target-  present trials in which the participant choose the incorrect  identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' and/or b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=') incorrect target-absent trials in which  neither image showed the identity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=', participants chose one  of the face images, when neither was an identity match to the  video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Because both options are consistent with the concept  of a false alarm, in what follows, we included both types of  false alarms (a and b) in the accuracy computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Results  1) Accuracy: Figure 3 shows the average d’ for each mask  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' These values indicate that faces in the unmasked  condition were identified moderately well, but face recognition  in both masked conditions was significantly impaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The  negative d’ values for the masked conditions are unusual  and suggest that participants used a systematically incorrect  decision strategy, which we will investigate further in Section  III-D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  A one-factor Analysis of Variance (ANOVA) was performed  on accuracy (d’), with mask condition as the independent  variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The resulting model yielded a main effect of mask  condition on d’, F(2, 27) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='03, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' When comparing  the conditions, d’ accuracy was significantly higher in the  unmasked condition than in the masked conditions, with no  significant difference between the two masked conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' This  suggests that Canny and ORNL masking are not significantly  less effective when used together than Canny masking alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  As is clear from the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 3, participant performance was more  variable in the Eyezoom condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  2) Response Distribution: To further investigate the finding  of negative d’s, we examined the proportion of responses  allocated to each response type (face images chosen versus  no identity chosen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The pattern of responses is shown for  each mask type in Figure 4, with separate graphs for target-  present (correct identity was available as a choice) and target-  absent (correct identity was not available as a choice) trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  For the unmasked condition, the graphs show a standard  (relatively accurate) pattern of responses as a function of  whether the target was present or absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The graphs for  the masked conditions show inaccurate performance, but also  suggest that participants did not systematically choose the no-  identity match when a match was present, but instead often  chose the wrong face as the identity match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  We conclude tentatively that performance in the masked  conditions was very poor indicating the effectiveness of the  masks for preventing identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' However, given the un-  usual performance in the masked condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=', negative d’s),  IDENTITY MASKING WITH EYE ENHANCEMENT  4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 2: Example trial in Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 3: Experiment 1 accuracy, measured as d’ across  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Results show that both masking algorithms were  equally effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  we retested the conditions with a design that eliminates the  possibility of response bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' EXPERIMENT II: EFFECT OF EYEZOOM MASKING  METHOD WITH A FORCED-CHOICE TASK  In this experiment, we used a two-alternative forced choice  (2AFC) task to test masking effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In the 2AFC, two  faces are presented as response options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In all cases, one of  the two images will be the same identity as the person in the  video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Participants  A total of 30 (7 male, 22 female, 1 other) undergraduate  student volunteers (ages 18-26) from UTD participated in the  study in exchange for research credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Procedure  The experiment consisted of 72 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The video stimulus  was displayed in the top center of the screen with the two  face images beneath it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Participants were asked to determine  which of the two face images matched the identity shown  in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' To make the task challenging, the two faces  presented had a similar appearance and were of the same race  and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' An example trial is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Each of the dataset’s 36 identities was shown twice, once  with the correct response as the left-located option and once  with the correct response as the right-located option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The  video segments were shown in random order and looped until  the subjects responded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Participants were assigned randomly  to one of the three masking conditions, with the unmasked  condition serving as a baseline condition for identification  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Results  Accuracy was assessed as the proportion of correct re-  sponses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 6 shows the proportion of correct responses  for each mask condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' These values indicate that face  recognition in the unmasked condition was more accurate  than face recognition in the masked conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' A one-factor  ANOVA was performed on the accuracy data (proportion of  correct responses), with condition as the independent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  The model yielded a main effect of mask condition on  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Press "" if the person in the  Press "2" if the person in the  Press "3" if the person in the video  video is the person on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  video is the person in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  is NOT either of the two people picturedcondition  unmasked  canny  eyezoom  1  2  conditionIDENTITY MASKING WITH EYE ENHANCEMENT  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 4: Proportion of responses by trial type in Experiment I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  proportion of correct responses, F(2, 27) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='68, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  As in the first experiment, participants were more accurate in  the unmasked condition than in the masked conditions, and  performed comparably for the two masked conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  The results replicate the pattern of performance across  conditions found for Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' As expected with a 2AFC  task, performance was more accurate in all three conditions  than it was in Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Notably, average identification  was above chance in both masked conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Performance in  the Eyezoom condition was more variable than performance  in the Canny mask condition—replicating a similar finding in  Experiment I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  We conclude that the masks strongly inhibit identification,  but that when forced to guess between two images (with the  assurance that one was an identity match), participants fared  better than chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Notwithstanding, applications of identity  masking would rarely if ever be able to assure a human or ma-  chine system that one of two candidates was an identity match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Our goal in applying this method here was to test examine the  role of response bias in the unusual pattern of results found in  Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The present results suggest that these masking  algorithms leave behind some residual identity information in  the face that humans can exploit when the response decision  is highly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' As noted, it is unlikely that that would  Unmasked Target Present Trials  Unmasked Target Absent Trials  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  response  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  response  responses  chose either identity  chose either identity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content="75  chose no identity  chose no identity  09'0 9." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  prop  prop  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  response  response  Canny Target Present Trials  Canny Target Absent Trials  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  response  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  response  responses  chose either identity  chose either identity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='75  chose no identity  chose no identity  090 9  pro  pro  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  response  response  Eyezoom Target Present Trials  Eyezoom Target Absent Trials  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  response  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  response  chose eitheridentity  chose either identity  chose noidentit  chose no identity  res  pro  pro  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='00  response  responseIDENTITY MASKING WITH EYE ENHANCEMENT  6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 5: Example trial from Experiment II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 6: Experiment 2 - identification accuracy across  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  be the case in any applied scenario, and so we conclude that  these simple simple filtering procedures provide a reasonably  high degree of identity protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Additionally, we conclude,  albeit more tentatively, that the eyezoom procedure does not  improve identification significantly over the Canny procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' EXPERIMENT III: EFFECT OF EYEZOOM MASKING  METHOD ON ACTION PRESERVATION  The effectiveness of the identity protection provided by  these masks opens the question of whether this protection  comes at the cost of preserving information about facial  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In this experiment, we examined whether the Canny  and Canny+Eyezoom mask conditions impaired driver facial  action perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Participants  A total of 30 (6 male, 23 female, 1 nonbinary) undergradu-  ate student volunteers (ages 18-30) from UTD participated in  the study in exchange for research credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Procedure  The experiment consisted of 100 trials, each with three  response options: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=') driver looking to the left, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=') driver  looking to the right, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=') driver looking down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Each of the  36 identities in the dataset appeared between two and three  times, each with a different action (looking right, left, down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Prior to the start of the main experiment, a pilot test with  only the unmasked condition was conducted to ensure that the  actions were identifiable in all videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' This test resulted in the  elimination of eight (of 108) videos segments in which actions  were not recognizable at sufficiently high levels of accuracy  for inclusion in the action preservation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  The participants were assigned randomly to one of three  masking conditions with the unmasked condition providing a  baseline action recognition accuracy and were asked to identify  whether the driver was looking to the left, right, or down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The  video stimuli were shown in the upper center of the screen  with three written options below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 7 for an example  trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The clips were played in a random order and looped until  the participant responded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Results  The proportion of correct responses was used to assess accu-  racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 8 shows the proportion of correct responses for each  mask condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' These values indicate that action preservation  Press"1" if the person in the  Press "2" if the person in the  videoisthepersonontheleft  video is the person in the rightcondition  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='9  unmasked  canny  eyezoom  I of correct response  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='7  ortion  propor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='5  conditionIDENTITY MASKING WITH EYE ENHANCEMENT  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 7: Example trial from Experiment III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  was generally high, but also suggest a small advantage for  action perception in the unmasked condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' A one-factor  ANOVA, performed on the accuracy (proportion of correct  responses) data, with the independent variable of condition,  did not show a significant effect, but was generally consistent  with this conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The model yielded a marginal main  effect of mask condition on proportion of correct responses,  F(2, 27) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='69, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='086.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' This suggests a very slight  advantage for action perception without stimulus masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  In conclusion, although the results did not reach statistical  significance, there is some indication that masking impaired  action perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' DISCUSSION  Our goal was to examine the effectiveness of simple  Canny-filtering based masking methods, with and without eye  enhancement, for interfering with face identification while  preserving facial actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In Experiment I, face recognition  accuracy was diminished for both mask conditions, relative  to the unmasked condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' There was no difference between  the Canny mask alone and the mask with eye enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In  Experiment II, we replicated this result with a 2AFC procedure  that controlled for response option bias, which may have been  a factor in the findings of negative ’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' values for both masking  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' In combination, both studies point to the relative  effectiveness of the masks for interfering with identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  They also point to the conclusion that eye enhancement did  not alter this effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Experiment III showed that facial  actions were preserved to a similar degree with both masks,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' 8: ANOVA of proportion of correct responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  though there was a marginal advantage for action perception  in the unmasked condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  In summary, these results indicate that Eyezoom masking  does not significantly increase identification or alter facial  action preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  ACKNOWLEDGMENT  This work was supported through collaboration with Oak  Ridge National Laboratory and the Federal Highway Admin-  Press "1" if the person looks toward the driver\'s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Press "2" if the person looks toward the passenger\'s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  Press "3" if the person looks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='condition  unmasked  canny  eyezoom  f correct responses  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='92  of  proportion  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='88  un  ma  ez  conditionIDENTITY MASKING WITH EYE ENHANCEMENT  8  istration under the Exploratory Advanced Research Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  The human experiment and analysis was subcontracted to  the University of Texas at Dallas from Oak Ridge National  Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  This manuscript has been authored in part by UT-Battelle,  LLC, under contract DE-AC05-00OR22725 with the US De-  partment of Energy (DOE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' The US government retains and  the publisher, by accepting the article for publication, acknowl-  edges that the US government retains a nonexclusive, paid-up,  irrevocable, worldwide license to publish or reproduce the pub-  lished form of this manuscript, or allow others to do so, for US  government purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' DOE will provide public access to these  results of federally sponsored research in accordance with the  DOE Public Access Plan (http://energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='gov/downloads/doe-  public-access-plan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content='  REFERENCES  [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Bolme,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Ferrell, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
+page_content=' Jesudasen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFAT4oBgHgl3EQfCRwI/content/2301.08408v1.pdf'}
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+MOAT: Towards Safe BPF Kernel Extension
+Hongyi Lu1,2, Shuai Wang2,∗, Yechang Wu1, Wanning He1, Fengwei Zhang1,∗
+1Southern University of Science and Technology
+2Hong Kong University of Science and Technology
+Abstract
+The Linux kernel makes considerable use of Berkeley Packet
+Filter (BPF) to allow user-written BPF applications to execute
+in the kernel space. BPF employs a verifier to statically check
+the security of user-supplied BPF code. Recent attacks show
+that BPF programs can evade security checks and gain unau-
+thorized access to kernel memory, indicating that the verifica-
+tion process is not flawless. In this paper, we present MOAT,
+a system that isolates potentially malicious BPF programs
+using Intel Memory Protection Keys (MPK). Enforcing BPF
+program isolation with MPK is not straightforward; MOAT
+is carefully designed to alleviate technical obstacles, such
+as limited hardware keys and supporting a wide variety of
+kernel BPF helper functions. We have implemented MOAT
+in a prototype kernel module, and our evaluation shows that
+MOAT delivers low-cost isolation of BPF programs under
+various real-world usage scenarios, such as the isolation of a
+packet-forwarding BPF program for the memcached database
+with an average throughput loss of 6%.
+1
+Introduction
+It is common to extend kernel functionality by allowing user
+applications to download code into the kernel space. In 1993,
+the well-known Berkeley Packet Filter (BPF) was introduced
+for this purpose [4]. The classic BPF is an infrastructure
+that inspects network packets and decides whether or not
+to forward or discard them. With the introduction of its ex-
+tended version (referred to as eBPF) in the Linux kernel, BPF
+soon became more powerful and is now utilized in numerous
+real-life scenarios, such as load balancing, scheduling, and
+auditing [18, 22, 28, 52, 62, 63].
+To ensure security, BPF is equipped with a verifier [6].
+The verifier performs a variety of static analyses to ensure
+the user-supplied code is secure. For instance, the verifier
+tracks the bounds of all pointers to prevent an out-of-bound
+access. Given that BPF code runs directly within the kernel,
+∗Shuai Wang and Fengwei Zhang are the corresponding authors.
+the verifier becomes crucial for the BPF security. Neverthe-
+less, as pointed out by recent studies [25, 31, 32, 50, 60], the
+currently available verifier has various limitations, and is in-
+sufficient for the overall security of BPF. First, the current
+BPF ecosystem supports a variety of kernel functionalities
+with over 200 dedicated APIs [2], resulting in a complicated
+verification process. Even though the verifier’s correctness has
+been formally proved [59], the gap between abstraction and
+implementation may still result in vulnerabilities [35–41, 43].
+Second, BPF Just-In-Time (JIT) is currently supported on
+multiple platforms, including x86, ARM, and RISC-V, whose
+differences frequently result in subtle vulnerabilities [44, 45];
+note that the verifier cannot detect vulnerabilities in the JIT
+stage. Third, due to the rapid expansion of BPF capabilities,
+the verifier has to be frequently updated. Nonetheless, it is
+inherently difficult to frequently update a complex static veri-
+fication tool without introducing new vulnerabilities [42]. To
+date, the BPF subsystem has been constantly exploited. For
+instance, two privileged-escalation vulnerabilities have been
+discovered in the implementation of bpf_ringbuf, a rather
+new BPF feature introduced in 2020 [4]. Further, the veri-
+fier’s register-value tracking is quite complex and has been
+bypassed by several severe vulnerabilities [35–38].
+Given the increasing security threats in BPF and the chal-
+lenge of enforcing safe BPF programs with merely static
+verification, we seek to employ hardware extensions to sand-
+box untrusted BPF programs. In particular, we leverage Intel
+Memory Protection Keys (MPK) [9], an emerging hardware
+extension which partitions memory into distinct permission
+groups by assigning up to 16 keys to their Page Table En-
+trys (PTEs). With the aid of MPK and the BPF verifier’s
+analysis results, we present MOAT, which isolates untrusted
+BPF programs in a low-cost and principled manner. For in-
+stance, two MPK protection keys K and E may be assigned to
+the kernel and a BPF program, respectively. When the kernel
+transfers control to the BPF program, it can set K as access-
+disabled to prevent the potentially malicious BPF program
+from tampering with kernel memory regions.
+Despite its promising potential, we observe that using MPK
+1
+arXiv:2301.13421v1  [cs.CR]  31 Jan 2023
+
+tracepoint
+packet filter
+schduler
+tracepoint
+packet filter
+schduler
+User  
+Application
+Kernel
+packet filter
+schduler
+tracepoint
+BPF Programs
+BPF Bytecode
+Verifier
+Maps
+Helpers
+call bpf_pid
+...
+log next_sched
+ret next_sched
+Kernel
+BPF (Runtime) Utilities
+BPF Bytecode
+BPF Compiler
+Figure 1: BPF overview. We illustrate the BPF compilation procedure, and the execution context of a sample BPF program attached to the
+kernel scheduler. Note that BPF verification is conducted at the BPF bytecode loading time.
+to enforce BPF isolation is not straightforward. MOAT is de-
+liberately designed to overcome two major technical hurdles.
+First, Intel MPK provides a maximum of 16 keys. Thus, it
+becomes challenging to support many BPF programs running
+concurrently with this limited number of hardware keys. Exist-
+ing workarounds like key virtualization [51] are incompatible
+with the BPF scenario and challenging to be implemented in
+kernel. This is because the key virtualization heavily relies on
+scheduling and notification facilities that are only available
+to userspace; directly reusing them in the kernel space may
+largely block kernel threads. To address this hurdle, we pro-
+pose a novel dynamic/fixed key allocation scheme that can
+support multiple BPF programs with a small overhead.
+Second, while MPK-based hardware isolation mitigates ma-
+licious BPF programs, helper functions provided by the BPF
+subsystem may be exploited by attackers. Overall, the growth
+of the BPF ecosystem is accompanied by the expansion of its
+dedicated helper functions; helper functions facilitate various
+tasks commonly conducted by a BPF program. On one hand,
+MOAT should allow benign BPF programs to freely use these
+helpers. On the other hand, MOAT must be cautious enough
+with these APIs to ensure they are not exploited by attackers.
+Given that there are over 200 helpers [2] provided in the latest
+Linux kernel, designing individual security policy for each
+of them is impractical and less extensible. To this end, we
+analyze all existing helpers with static dependency-analysis,
+and propose several general defense schemes, each of which
+is applicable to a group of helpers. We envision that when
+a new helper is added, MOAT can be applied easily without
+introducing new schemes.
+To evaluate the security impact of MOAT, we systemati-
+cally examined how MOAT mitigates attack surfaces due to
+untrusted BPF programs. We also empirically analyze all
+recent CVEs within MOAT’s application scope. The result
+shows that MOAT successfully mitigates each CVE. More-
+over, we evaluate the performance overhead of MOAT under
+representative and edge-case scenarios. First, we examine the
+performance of our dynamic/fixed key allocation policy by
+assessing a use case where multiple programs are executed
+concurrently to use all MPK keys. Then, we build a real-
+life port-forwarding BPF program for the memcached [24]
+database, and secure it with MOAT to measure how MOAT
+influences memcached’s throughput. Furthermore, we apply
+MOAT to several real-world BPF applications [52] to illus-
+trate that MOAT can be directly applied to the current BPF
+ecosystem with minimal engineering effort. MOAT’s worst
+case performance overhead in all these experiments is less
+than 30%, which is acceptable given the security benefits
+MOAT provides. Moreover, MOAT imposes only 6% over-
+head on average to the memcache’s throughput. In sum, we
+make the following contributions.
+• Instead of merely relying on BPF verifiers to statically
+validate untrusted BPF programs, this paper for the first
+time advocates to isolate user-supplied BPF programs with
+an emerging hardware extension, Intel MPK.
+• Technically, MOAT is specially designed to address domain-
+specific challenges including limited hardware keys and
+protecting over 200 helper functions in the BPF ecosystem.
+• We implement a prototype of MOAT as a Loadable Kernel
+Module (LKM) of the latest Linux (v5.19) and conduct a
+thorough evaluation of its security and performance. The
+evaluation shows that MOAT delivers a principled security
+guarantee with moderate performance penalty.
+2
+Background
+2.1
+Berkeley Packet Filter (BPF)
+BPF Overview. BPF [4] was originally introduced to facili-
+tate flexible network package filtering. Instead of inspecting
+packages in the userspace, users can provide BPF instructions
+specifying package filter rules, which are directly executed in
+the kernel. This allows configurable package filtering without
+costly context switching and data copying. Modern Linux
+kernel features extended BPF (eBPF), a Linux subsystem,
+which supports a wide range of use cases such as kernel pro-
+filing, load balancing, and firewalls.1 Popular applications like
+Docker [34], web browsers [30, 48], and kernel debugging
+utilities like Kprobes [8] are built on top of BPF.
+Fig. 1 depicts an overview of how BPF programs are com-
+piled and then deployed. The eBPF subsystem offers ten
+general-purpose 64-bit registers, memory stack, eBPF cus-
+tomized data structures (often referred to as eBPF maps), and
+a set of eBPF helper functions. To use eBPF (e.g., for ker-
+nel profiling), users can first write their own BPF programs
+(in C code) to specify the functionality. The BPF programs
+will then be compiled into BPF bytecode and downloaded
+into the kernel. Given that eBPF code is written by untrusted
+1While there are indeed two variants of BPF: classic BPF (cBPF) and
+eBPF, cBPF is internally converted into the latter variant by the kernel.
+2
+
+users, the kernel employs a verifier to conduct several checks
+during the bytecode loading time (see below). By default,
+the bytecode is executed by the BPF interpreter (omitted in
+Fig. 1). Additionally, depending on the kernel configuration
+and architectural support, an optional JIT compilation may
+be applied to the bytecode for better performance. The BPF
+bytecode is then attached to certain kernel components, based
+on its specific end goal. For instance, as shown in Fig. 1, a
+BPF program attached to kernel scheduler collects relevant
+statistics and decides which thread should be running next to
+improve overall performance.
+- len < INSN_MAX 
+- no loop 
+- no dead code 
+- no OOB jmp
+Unverified
+CFG  
+Check Phase
+Data-Flow
+Check Phase
+- register tracking 
+- access check 
+- helper check 
+- misc fixups
+Verified
+Figure 2: BPF verification process.
+BPF Verifier. BPF programs are written in C, and compiled
+into a RISC-like instruction set. As aforementioned, the kernel
+strictly verifies the BPF programs upon loading to ensure
+that they are safe to execute. Fig. 2 illustrates the verifying
+process in a holistic manner. First, a BPF program is parsed
+into a control flow graph (CFG) by the verifier, which first
+performs a CFG check phase to ensure four key properties:
+1) the program size is within a limit; 2) there exists no back
+edges (loops) on its CFG; 3) there exists no unreachable
+codes; and 4) all jumps are direct jump and they refer to a
+valid destination. Overall, given that BPF programs must be
+terminated, the CFG check phase ensures that all jumps are
+direct jumps and there are no back edges. Given that said,
+loops are still feasible via unrolling at the cost of binary size.
+The verifier further performs finer-grained data flow anal-
+ysis. It first tracks the value flow of every register to deduce
+its value ranges conservatively. Based on these ranges, the
+verifier can decide if a pointer accesses safe memory regions,
+and if a parameter is valid. Since this analysis is performed
+statically prior to execution, there exists possibility that a ma-
+licious BPF program uses certain operations to bypass this
+analysis [35–41, 43]. Last, the verifier also performs some
+miscellaneous fixups, like rewriting certain instructions to
+simplify the follow-up JIT compilation.
+BPF Maps. Out of security concern, the kernel also sets a
+rather strict space limit on BPF programs. Each program
+by default can only use up to 512 bytes stack space and 10
+registers, which is far from enough for certain BPF programs.
+To address this problem, BPF maps can be allocated and
+provide additional space for BPF programs. Up to now, there
+are over 30 types of maps supported by kernel, such as array
+map and hash map [5]. Moreover, as demonstrated in Fig. 1,
+maps may act as a communication channel between BPF
+programs and user applications, since some of these maps can
+be accessed by both the BPF program and the user application.
+BPF Helpers. Kernel also limits the kernel functions a BPF
+program may call. Those functions are dubbed BPF helper
+functions, as shown in Fig. 1. Up to now, there are over 200
+helpers scattered across subsystems of the kernel [2]. Note
+that depending on the specific task, a BPF program can usually
+call a group of relevant helpers. For example, a BPF program
+attached to the scheduler is not allowed to call any helper
+related to kernel probing, but it can call bpf_pid2 to obtain
+the PID of the current process and chooses which process to
+be scheduled next.
+00
+01
+...
+10
+00
+32
+0
+PKR
+PTE[62:59] = 0xF
+PTE[62:59] = 0xE
+PTE[62:59] = 0x1
+PKR Entry Options
+00
+Access Enable     (EN)
+01
+Access Disabled  (AD)
+10
+Write Disabled    (WD)
+11
+Access Disabled  (AD)
+Page Table Entry
+Figure 3: Intel MPK overview.
+Intel MPK. Intel introduces MPK [9] to provide efficient
+page table permissions control. By assigning a MPK protec-
+tion key to each page table entry (PTE) of one process, users
+can enable intra-process isolation and confidential data access
+control [17, 27, 33, 51, 58]. As illustrated in Fig. 3, MPK uses
+four reserved bits [62:59]in each PTE to indicate which pro-
+tection key is attached with this page. Those three PTEs in
+Fig. 3 are assigned with keys 0x1, 0xE and 0xF, respectively.
+Since there are only 4 bits involved, the maximum number
+of keys is 16. Then, a new 32-bit register named Protection
+Key Register (PKR) is introduced to specify the access per-
+mission of these protection keys. Each key occupies two bits
+in PKR, whose values denote either access-disabled (AD) or
+write-disabled (WD), respectively. By writing to certain bits
+in PKR, the access permission of corresponding pages can be
+configured accordingly. It is worth noting that one key may
+be assigned to arbitrary number of pages by modifying their
+PTEs. This facilitates changing the access permission of a
+large number of pages without severe performance penalty.
+Clarification and Notations. As a side note, there are
+actually two versions of Intel MPK. One applies to the
+user-mode while the other applies to the supervisor-mode.
+For brevity, we refer these two versions in their conven-
+tional abbreviations as Protection Key Supervisor (PKS)
+and Protection Key User (PKU), respectively. Most existing
+works [17, 26, 27, 33, 51, 58] are based on PKU. In MOAT,
+we use PKS instead since our goal is to isolate BPF programs,
+which execute in the supervisor-mode. The logistics behind
+these two versions are mostly identical with slight variations.
+For instance, the permission configuration register in PKS is
+a Model Specific Register (MSR) named IA32_PKRS, which
+is inaccessible from userspace, whereas in PKU, this role is
+assigned to a dedicated register PKRU. In addition to PKR,
+there also exists a bit in the control register CR4 that can dis-
+able/enable MPK entirely; for PKU, this bit is CR4.PKE; for
+2Here, bpf_pid refers to bpf_get_current_pid_tgid.
+3
+
+PKS, this bit is CR4.PKS. To avoid potential misleading, the
+rest of the paper directly refers MPK leveraged by MOAT as
+PKS.
+3
+Motivation and Assumptions
+3.1
+Typical Threats to BPF Verifier
+Fast Feature Evolving. As a fast developing technology,
+threats may come from the inconsistency between the con-
+stantly expanding BPF capabilities and the rigorous static
+verification process imposed on them [39, 42]. It is a common
+practice to add corresponding verification procedures simul-
+taneously when introducing new features to BPF programs.
+However, it is very hard to make changes correctly to the BPF
+verifier, a critical security kernel component, which has over
+10K LoC and a variety of functionalities [6].
+Challenging Pointer Tracking. Second type of threats origi-
+nates from the complexity of pointer tracking mechanism. Al-
+though the correctness of the verifier is formally proved [59],
+there still exist gaps between the implementation and the
+abstraction, especially in some corner cases, such as sign ex-
+tension, truncation, and bit operators [35–41, 43].
+The fact that the contemporary BPF verifier only performs
+static analysis is a severe deficiency, as evidenced by the
+threats noted above. Performing sound and complete static
+analysis toward BPF programs to uncover potential threats is
+fundamentally challenging, and from the disclosed BPF vul-
+nerabilities, we find that there frequently exists a gap between
+verifier’s static analysis results and BPF programs’ runtime
+behavior. For instance, the verifier, based on its static analysis
+results, may conclude that a program is benign because it
+only accesses a memory region ranging from [0x0,0x1000].
+However, by leveraging vulnerabilities like noted above, the
+software may behave differently during execution. Therefore,
+a hardware feature, Intel MPK, is utilized to enforce further
+isolation, such that a BPF program is constrained in its own
+memory regions, and any runtime accesses that violate this
+constraint are effectively flagged and terminated by MOAT.
+3.2
+Threat Model
+Assumption. Our threat model considers a practical setting
+which is aligned with existing BPF vulnerabilities [35–41, 43].
+In particular, we assume attackers are non-privileged users
+with BPF access, since a root user already has the control over
+almost the entire kernel. Attackers can download their pre-
+pared BPF code into the kernel space to launch exploitation.
+Attacker Capability and Application Scope. MOAT iso-
+lates user-submitted BPF programs and prevent them from
+accessing privileged kernel memory regions. As will be intro-
+duced in Sec. 4, a BPF program is given the minimum neces-
+sary resources and privileges to complete its task. To clarify,
+there are also other more subtle vulnerabilities (not relevant
+to memory exploitations) such as speculation, race condition,
+and DoS occurred to exploit the BPF subsystem [46, 47]; a
+well-isolated BPF program can still launch these attacks to
+jeopardize the BPF subsystem and the kernel. This research
+views them as common security defects shared by many other
+applications such as SGX enclaves [15, 21]. To date, coun-
+termeasures have been deployed by the BPF verifier [11],
+and we assume the standard BPF verifier can handle them
+properly. In contrast, correctly detecting memory-related BPF
+exploitation requires systematic and rigorous static analysis
+over BPF programs and is fundamentally hard for BPF ver-
+ifiers; MOAT enhances mitigating memory-related exploita-
+tions with hardware-assisted isolation. Next, we present anal-
+ysis of three major components in our research context as
+follows.
+BPF Programs. This includes the BPF bytecodes or the JIT-
+emitted native instructions. Our threat model takes the as-
+sumption that malicious BPF programs are able to bypass
+checks statically performed by the verifier; they may thus
+behave maliciously during runtime. Our threat model deems
+BPF programs as untrusted, and MOAT is designed to isolate
+them from the rest of the kernel. More specifically, every BPF
+program, during its runtime, is only allowed to access its own
+stack, allocated maps, and certain helper functions.
+BPF Helper Functions. These helpers act as the interme-
+diate layer between the BPF subsystem and kernel. Certain
+malicious BPF programs can abuse these helpers to perform
+attacks, and therefore, we assume they are also untrusted.
+MOAT mitigates risks raised by adversarial-manipulated
+helper functions with practical defenses.
+Kernel. Kernel is the target to protect. We assume the kernel
+is functioning normally, and attackers aim to leverage mali-
+cious BPF programs to gain unauthorized access to kernel
+data or executing arbitrary privileged kernel code.
+4
+Design
+MOAT Overview. As motivated in Sec. 3, current security
+design against malicious BPF programs solely relies on the
+static analysis performed by the BPF verifier, which is seen as
+a weak point and exploitable by non-privileged users. MOAT
+instead delivers a principled isolation of BPF programs using
+MPK from the rest part of the kernel and prevent bypasses.
+bpf_lookup_elem
+call bpf_run
+...
+bpf_delete_elem
+mov %rax, $0x1
+...
+call bpf_helper
+st %(rax), $0x10
+Helper 
+Auditor
+BPF Memory
+...
+bpf_get_time
+mov %rax, %rbx
+MOAT
+BPF Payload
+Access
+Rules
+Stack
+...
+Maps
+MPK
+Verifier
+Kernel Memory
+1
+2
+3
+4
+Figure 4: MOAT overview.
+4
+
+Fig. 4 illustrates the lifecycle of a BPF program with the
+presence of MOAT. 1 Given a user-submitted BPF program
+P, MOAT statically derives the minimum necessary memory
+regions the program needs, such as stack, used maps and
+context by reading metadata from P. 2 These regions (“BPF
+Memory” in Fig. 4) are assigned to P using PKS, forming its
+runtime environment. 3 When the kernel invokes P, MOAT
+configures PKS to constrain P to its own regions and forbids
+its access to other memory regions. 4 On the occasions that P
+requires helper calls to interact with the kernel, depending on
+the helper types, MOAT may adjust involved kernel memory
+region permissions and also validate the helper parameter
+values to prevent helpers from being abused.
+Security Guarantees. Overall, MOAT provides the following
+two key secure design guarantees.
+(i) A BPF program is given the minimum necessary ker-
+nel resources and privileges for completing its task,
+preventing any malicious behavior.
+(ii) The interactions (e.g. helper calls) between the BPF
+program and the kernel are audited thus not abused.
+Extensibility. MOAT leverages MPK, a de facto hardware ex-
+tension available on mainstream Intel architectures to isolate
+BPF programs. We view this design choice is consistent with
+recent hardware-assisted security enforcement works [17, 58].
+Nevertheless, we clarify that the design of MOAT is not lim-
+ited to leveraging MPK. There exist similar hardware security
+mechanisms on other platforms and architectures such as the
+Memory Domains [3] on ARM and the Domain Keys [53]
+on RISC-V. These mechanisms can be used to replace MPK
+on these platforms with a small amount of engineering effort;
+see Sec. 8 for our discussion on migration and extension.
+4.1
+General BPF Isolation
+In accordance with the BPF program lifecycle depicted in
+Fig. 4, this section elaborates on the general isolation ap-
+proach offered by MOAT. We further discuss two key techni-
+cal challenges in Sec. 4.2.
+0x0
+...
+59
+62
+Kernel
+BPF 
+BPF 
+0x3
+...
+...
+Shared by 
+ & 
+Kernel Data
+Kernel Code
+Stack
+Maps
+Context
+Code
+Stack
+Context
+Code
+Shared Maps
+Page Table Entries
+Data Regions
+Runtime PKR Value
+Enable
+01
+00
+01
+00
+AD
+EN
+EN
+N/A
+AD
+01
+01
+00
+00
+AD
+EN EN AD
+8
+..
+..
+..
+..
+32
+...
+0x2
+...
+...
+0x1
+...
+...
+AD Access-Disabled
+EN Access-Enabled
+Figure 5: BPF memory regions.
+4.1.1
+BPF Memory Regions
+Fig. 5 depicts the memory regions of BPF programs and
+the kernel. By default, all pages should belong to the kernel
+memory region, and each page is initialized with a default
+MPK key value 0. Then, when a BPF program P is newly
+loaded into the kernel, MOAT decides the minimum pages
+it needs, and assigns these amount of pages to the memory
+region of P. Note that the necessary memory sections of a
+BPF program includes its code, stack, and the context; many
+non-trivial BPF programs also require BPF maps (e.g., array
+and hash maps) to use. After assigning these sections to the
+memory region of P, MOAT restricts P to its own memory
+regions by configuring the PKR register. Take the BPF P1 in
+Fig. 5 as an example, most of its sections (including a number
+of BPF maps) solely belong to itself. Furthermore, P1 and
+P2 share several extra BPF maps. Thus, at its runtime, MOAT
+configures the PKR register of P1 to enable its access (EN;
+denoted as 00 in the runtime PKR value column of Fig. 5)
+to its own region 0x1 and the shared region 0x3. Moreover,
+MOAT disables any accesses from P1 to the kernel region 0x0
+and the P2 memory region 0x2 by setting corresponding bits
+in P1’s PKR register as 01 (denoting AD).
+To clarify, the code and map sections of a BPF program
+requires are trivially known (by reading the metadata in the
+BPF program) once it is loaded. Thus, MOAT can assign these
+pages to its designated region by modifying their PTEs during
+the program loading phase without any runtime overhead. The
+assignment for stack, context and some special types of maps
+will be discussed in the next section.
+4.1.2
+BPF Runtime Environment
+Apart from the program itself and the maps it uses, a BPF
+program requires additional kernel structures to function prop-
+erly. These structures include descriptor tables, stacks, and the
+program’s runtime context. Furthermore, certain maps (such
+as the hash map) are not stored continuously in the kernel
+and cannot be assigned trivially during initialization. MOAT
+assigns entries of this kind of maps on the fly.
+Descriptor Tables. On x86 platforms, descriptor tables such
+as Global Descriptor Table (GDT) and Interrupt Descriptor
+Table (IDT) are essential for basic operations like interrupt.
+These kernel data structures are assigned to a shared region
+that all BPF programs can access. To prevent tampering those
+critical structures, they are made read-only when shared.
+Dedicated Stack. BPF programs require a 512-byte stack
+space to store local variables and function frames. The ver-
+ifier is in charge of determining if a program makes Out of
+Bound (OOB) accesses toward this stack. Thus, when the
+BPF program passes the static checks, its required stack is di-
+rectly allocated from the kernel stack. However, as discussed
+in Sec. 3, certain vulnerabilities may allow BPF programs
+to bypass this check at runtime. Given that this stack is uti-
+lized so frequently, we note that executing dynamic auditing
+5
+
+Table 1: BPF context of common program types.
+Program Type
+Context Type
+Note
+Socket Filter
+__sk_buff *
+Metadata of sk_buff
+Socket Ops
+bpf_sock_ops *
+Socket events (timeout, retransmission, ...)
+XDP
+xdp_md *
+Metadata of xdp_buff
+Kprobe
+pt_regs *
+Register status
+Tracepoints
+Depending on Tracepoint Types
+Relevant Tracepoint information
+Perf Event
+bpf_perf_event_data *
+Perf. event (register status, sample period)
+Cgroup Device
+bpf_cgroup_dev_ctx *
+Device ID, access type (read, write, ...)
+on it, as MOAT does for helper calls (see Sec. 4.2.2), would
+incur an unreasonable level of overhead. Thus, to prevent
+malicious BPF programs from tampering the kernel stack,
+MOAT allocates per-CPU stacks for BPF programs to use. To
+do so, similar to the descriptor tables, these per-CPU stacks
+are shared by all BPF programs running on the same CPU
+core. Consequently, they are also assigned to the shared re-
+gion. To prevent a malicious BPF program from tampering
+stacks of other CPU cores, the stack beginning addresses are
+randomized for each CPU core.
+Runtime Context. The context refers to BPF program param-
+eters, which vary depending on the BPF program types. For
+instance, if the BPF program serves as the filter attached to a
+particular socket, its runtime context is a pointer to the socket
+buffer, which stores packets for the attached socket. Since
+these contexts are not available until runtime, MOAT assigns
+these contexts upon the entry point of each BPF program.
+Table 1 lists common BPF contexts: These contexts are rather
+simple and only a few of them are nested data structures (i.e.,
+containing pointers to other structures). Thus, this assignment
+can be performed efficiently upon each entry point.
+Incontiguous Maps. Despite the fact that there are over 30
+distinct types of maps, their implementations can be roughly
+divided into only two types: Array maps and hash maps.
+The array maps are easy for MOAT to isolate since they
+are stored in a continuous form and of a fixed size. For
+these maps, MOAT determines its isolation when loading
+the BPF programs. The hash maps, however, are stored non-
+contiguously in the memory and can be dynamically expanded
+upon map insertion. This prevents MOAT from determining
+the addresses and sizes of the maps before executing the
+BPF programs. To overcome this issue, MOAT attaches to
+the bpf_map_lookup_elem, which is used to lookup a map
+entry and return its pointer. If the pointer is retrieved from
+an non-contiguous map, the memory to which it points is
+dynamically assigned to the BPF program. These entries are
+returned to the kernel once the program exits.
+4.1.3
+Lifecycle of a BPF Program
+This section has described how MOAT uses PKS to grant a
+BPF program accesses to its minimum necessary memory
+regions required to complete its task. This protects the ker-
+nel from being attacked by malicious BPF programs while
+allowing benign BPF programs to operate smoothly. We sum-
+marize all these details and depict the lifecycle of an isolated
+BPF program in Fig. 6.
+BPF Program
+Used Map
+BPF Program
+Used Map
+Ctx
+Stack
+Entry
+BPF Program
+Used Map
+Ctx
+Stack
+Dynamic 
+Map Entry
+Run
+Exit
+BPF Program
+Load
+1
+4
+2
+3
+1 Load: The program itself and its maps are assigned to its region.
+2 Entry: Context is assigned and stack is swapped.
+3 Runtime: Entries of incontiguous maps are assigned on the fly.
+4 Exit: Memory assigned during runtime is returned.
+Figure 6: BPF program lifecycle under isolation of MOAT.
+4.2
+Challenges for MOAT
+The preceding section illustrates the overall procedure of
+MOAT. However, to effectively isolate a BPF program using
+PKS, MOAT needs to overcome the following obstacles.
+C1: Limited Hardware Regions. In PKS, only 16 hardware
+keys are available. This means there can be no more than
+16 memory regions concurrently, but there may be signifi-
+cantly more than 16 BPF programs running in the kernel. To
+overcome this limitation, we propose a novel dynamical key
+allocation policy in Sec. 4.2.1.
+C2: Helpers. BPF is a complex ecosystem containing over
+200 helper functions [2]. Unlike BPF programs, these helper
+functions must have access to certain kernel memory to func-
+tion properly. Thus, MOAT must ensure that these helper func-
+tions are secure and not being abused. However, designing
+specific isolation policy for every one of these helpers requires
+massive human effort. Even worse, designing individualized
+isolation strategy for each helper may impede the applica-
+bility to helpers added in the future. To this end, we analyze
+these BPF helper functions with static analysis techniques and
+propose three general security isolation schemes in Sec. 4.2.2.
+4.2.1
+Dynamic Key Allocation
+Currently, PKS supports up to 16 memory regions, whose
+permissions are decided by a 32-bit PKR. Although works
+like libmpk [51] propose key virtualization to enable key
+sharing, these works typically focus on isolating userspace
+applications. Therefore, they rely on scheduling and notifica-
+tion mechanisms that are exclusive to userspace. However,
+6
+
+after examining their methods, we conclude that porting these
+userspace mechanisms to kernel is difficult, if at all possible.
+Intuitively, we may explore making key a shared resource;
+each BPF program will dynamically fetch and return a key
+upon its entry point and exit. Our tentative study shows that
+this approach works well with small BPF programs consum-
+ing few pages. Nevertheless, this approach may incur signifi-
+cant runtime overhead, as assigning these pages to a specific
+region upon each entry and exit can be time-consuming, partic-
+ularly if the program is attached with large maps. For instance,
+a 512KB map consists of over 100 pages. If a BPF tracepoint
+program employs this map to log kernel events, there will
+be over 200 page assignments every time this BPF program
+is invoked. These frequent assignments bring unacceptable
+overhead. Overall, given that frequent key retrieval and return
+is too expensive due to the presence of large BPF programs
+with many pages, we propose an adaptive dynamic key allo-
+cation scheme that shares keys across relatively small BPF
+programs and assigns fixed keys to large BPF programs.
+Dynamic keys
+K1
+K2
+Run
+Fixed 
+keys
+Exit
+K1
+Wait
+Wait
+...
+Entry
+Large BPF
+Programs
+1
+2
+3
+4
+Figure 7: Adaptive key allocation.
+As illustrated in Fig. 7, we divide PKS keys into two cate-
+gories — dynamic keys and fixed keys. We allocate dynamic
+keys to small BPF programs, whose allocation procedure are
+specified as follows. 1 Upon a BPF program P’s entry point,
+MOAT fetches a dynamic key and assigns this key to all pages
+of P. 2 During the runtime, MOAT can detect if P accesses
+pages not assigned to it via PKS. 3 When P exits, all of its
+pages are returned to the kernel, and the key is deallocated.
+4 If currently no key available when the kernel launches a
+BPF program, then this program is placed in a queue to wait.
+In contrast, fixed key allocation is straightforward. Once
+a large BPF program is loaded by the kernel, MOAT assigns
+a fixed key to it. In extreme cases where multiple large BPF
+programs are loaded into the kernel, and fixed keys are insuffi-
+cient, the smallest and least frequently invoked BPF program
+running will be evicted to use dynamic keys.
+We need to decide a threshold to determine whether a BPF
+program is “small” or “large.” Note that the current BPF sub-
+system only accepts programs that with fewer than 4,096
+instructions, which occupy about eight pages. Considering
+that the majority of BPF programs use a small map to com-
+municate with userspace, we select ten pages as the threshold
+for dynamic key allocation. That is, a BPF program using up
+to ten pages is configured to use dynamic keys, whereas BPF
+programs with more than ten pages uses fixed keys.
+4.2.2
+Helper Security Mechanism
+As interfaces between kernel and BPF programs, a set of
+BPF helper functions has been provided for kernel interac-
+tion. Since these helper functions serve as interfaces, most of
+them have to access certain kernel memory to function prop-
+erly. Therefore, these helpers may be leveraged by malicious
+BPF programs to launch attacks. Thus, MOAT has to prevent
+these helpers from being abused. However, there are over 200
+helpers provided by the BPF subsystem; it is impractical to
+design individual protection policy for each one. To overcome
+this obstacle, we analyze these helper functions and propose
+three defenses based on their interaction with the kernel. Each
+of these defenses applies to a large number of helpers and can
+be combined to enhance the offered protection guarantee.
+Analyzing all these helpers manually requires a significant
+amount of human effort. We leverage a de facto static pointer
+analysis library, SVF[55, 56], to perform dependency analysis.
+SVF performs sparse value flow analysis to establish value
+flow and pointer analysis results. SVF has been widely used
+to analyze large-size production software [54]. We use the
+default flow-sensitive pointer analysis [55] provided by SVF.
+Specifically, we use it to track the value flow of the parameters
+of these helper functions. Based on the value flow, we can
+scope the usage (read or write) of parameters and decide
+which category (see below) a helper function belongs to. With
+the help of SVF, this categorization process can be conducted
+in a principled way and scalable to analyze all helpers.
+Attackers might manipulate the parameters of these helpers
+to launch attacks. Therefore, based on the above analysis
+results, we divide 260 BPF helper functions into five types.
+As shown in Table 2, the first type (No Arg.) has no argu-
+ments, which does not need any extra protection. The second
+type (Pure Arg.) operates solely on its own arguments and
+does not access kernel memory, which is also safe. The third
+type (Read Only) accesses kernel in a read-only manner, and
+the forth type (Write) may use its argument to modify the
+kernel memory. The fifth type (Other) includes helpers that
+are hard to categorize. For example, bpf_loopis the auxiliary
+function that simplifies the verification process of loops. Note
+that the last three types may interact with the kernel space
+and potentially cause unauthorized access or even kernel ex-
+ploitations by being abused by malicious BPF programs.
+Table 2: BPF helper analysis result. CRP denotes critical region
+protection, ROK denotes read-only kernel space, and DPA denotes
+dynamic parameter auditing.
+Type
+#
+Example
+Applicable Defense
+No Arg.
+30
+bpf_get_retval()
+No Need
+Pure Arg.
+16
+bpf_strncmp()
+No Need
+Read Only
+75
+bpf_get_stackid_tp()
+ROK/CRP/DPA
+Write
+129
+bpf_skb_set_tstamp()
+CRP/DPA
+Other
+10
+bpf_loop()
+CRP/DPA
+With this categorization, we now present three mechanisms
+7
+
+in MOAT that ensure helper security as follows.
+Read-Only Kernel Space (ROK). Our analysis reveals that
+the majority of helpers only access the kernel in a read-only
+manner. These read-only helpers account for near one third
+of all helpers. Even though in most cases, read-only helpers
+do not alter the kernel state and are considered safe, MOAT
+still sets the kernel space as read-only when executing these
+helpers.3 This nullifies possibility of potentially tempering
+kernel spaces, and it does not impose extra runtime overhead.
+Normal Regions
+AD
+Critical Regions
+AD
+PKR
+...
+AD
+Critical Regions
+BPF Region
+EN
+PKR
+...
+Helper Call
+Kernel Address Space
+Kernel Address Space
+AD Access-Disabled
+EN Access-Enabled
+Normal Regions
+BPF Region
+EN
+EN
+Figure 8: Critical region protection (CRP).
+Critical Region Protection (CRP) in Kernel. Further to the
+discussion in ROK, though many helpers only access kernel
+in a read-only manner, they may still be abused to probe sen-
+sitive data of the kernel, such as task_struct. Moreover, a
+considerable number of helpers, as illustrated in Table 2, may
+modify kernel memory. To prevent such abuse, we protect
+these critical kernel regions with PKS. As shown in Fig. 8,
+instead of treating the entire kernel memory as a whole, we
+divide it into normal regions and critical regions. When enter-
+ing helper functions, instead of setting the entire kernel space
+as access-enabled (EN), those critical memory regions remain
+access-disabled (AD), preventing any access to these regions.
+Once the helper finishes, these normal region will be set back
+to access-disabled (AD) to avoid potential security risk. It is
+worth noting these critical memory regions do not vary with
+helpers. That is, only helpers manipulated by attackers (e.g.,
+via deliberately crafted helper parameters) may attempt to
+access these critical regions. These critical regions can be
+specified in the configurations of MOAT.
+r0 = 0x10
+r1 = r0 + 0x1
+call BPF_HELPER
+BPF Instructions
+Static Register Value  
+Inferred by Verifier
+0x10
+0x11
+Runtime Register Values  
+for Each Instruction
+...
+0x10
+0xbe
+0x10
+0x11
+r0
+r1
+r0 = 0x10
+r0 = 0x10   r1 = 0x11
+r0 = 0x10   r1 = 0x11
+...
+...
+Figure 9: Register value tracking of the verifier. While the veri-
+fier can indeed deduce a possible value range of each register, for
+simplicity, we use a value point (e.g., r1 = 0x11) here.
+Dynamic Parameter Auditing (DPA). To further regulate
+helpers, we propose dynamic parameters auditing (DPA),
+which leverages the information obtained from the BPF ver-
+3There exist few functions in this category that rely on synchronization
+facilities like Read-Copy Update (RCU), which cannot be applied with this
+protection scheme.
+ifier to dynamically check if the parameters are within their
+legitimate ranges. As illustrated in Fig. 9, the verifier can
+deduce the value range of each register via static analysis
+(as a practical assumption, we allow the statically deduced
+value ranges to be invalid; see below for clarification). MOAT
+logs such value range information, and during runtime, MOAT
+serves as a “gateway” when the BPF program enters a helper
+function to check if the provided parameter values are within
+the verifier-deduced value ranges. In our example, we can
+check if r0==0x10;r1==0x11 when BPF_HELPER is called.
+If the parameter runtime values do not match with the static
+analysis results, the BPF program is terminated immediately.
+Clarification. In the aforementioned DPA strategy, one may
+question if the “legitimate value ranges” inferred by the veri-
+fier are correct. Recall as discussed in our research motivation
+in Sec. 3, there exist several vulnerabilities that can be lever-
+aged to bypass verifier static checks. Overall, we clarify that
+we do not need the verifier’s static analysis results as always
+correct. Nevertheless, as long as the runtime input values are
+inconsistent with the static analysis results, we terminate the
+BPF program. For such cases, either the verifier is wrong or
+the BPF program is behaving maliciously, both are highly
+severe and we require manual inspection of the triage. We
+assume the chance of both verifier and BPF program being
+unsafe (but still appear to be consistent) is extremely low,
+if at all possible. In fact, for today’s known BPF exploita-
+tions, the verifier’s static analysis results (e.g., deciding the
+value ranges of certain pointers) are safe, though incomplete
+(omitting some data facts on subtle variables) and thus being
+leveraged by malicious BPF programs. Also, even though it
+may be technically feasible to perform dynamic auditing to
+validate the data facts after executing every BPF instruction, it
+is apparently too costly. MOAT thus leverages PKS to deliver
+a low-cost and principled isolation.
+Hybrid Usage. We summarize the applicability of these three
+defense mechanisms in Table 2. On the one hand, DPA pro-
+tects helpers from being abused by ensuring the validity of
+their parameters. On the other hand, even if the helpers are
+already compromised, ROK and CRP can still protect the
+kernel from these compromised helpers. Thus, combining
+these mechanisms together improves the overall security for
+both BPF helpers and the kernel itself. Moreover, we want to
+emphasize that these defenses are not dependent on a partic-
+ular helper. Instead, they are applicable to helper groups, as
+listed in Table 2. Although it can be argued all three defenses
+may be evaded in extreme circumstances, we believe the at-
+tack feasibility is very low (if it exists at all), given that the
+BPF program has been isolated by MOAT and these restricted
+helpers constitute a relatively minor attack surface. Our inves-
+tigation on existing vulnerabilities supports this assumption.
+8
+
+5
+Implementation
+MOAT is written in 2,075 lines of C code, as a loadable kernel
+module.4 It includes three components: a BPF loader, a BPF
+executor, and a key allocator. We explain key points below.
+Portable Implementation. The major components of MOAT
+are implemented as hooks to replace their corresponding ker-
+nel functions. This is accomplished using an existing ker-
+nel hook utility named ftrace [7]. This introduces a small
+amount of overhead, but it allows these major components to
+be kernel-agnostic and can be easily ported across different
+kernel versions. Though the overhead of the current MOAT
+prototype is reasonable (see details in Sec. 6.2), we anticipate
+to further reduce the performance overhead of MOAT, if it is
+implemented via directly modifying kernel.
+Kernel Interrupt Handling. Though the major components
+of MOAT are implemented as loadable modules, certain low-
+level codes still require direct kernel modification. For in-
+stance, during the execution of BPF programs, an interrupt
+may occur and take over the control flow to its handler. Note
+that most interrupt handlers require access to kernel memory
+and as a result, the PKS would presumably raise spurious
+alerts. Thus, we need to temporarily disable PKS inside these
+handlers and re-enable it once the handlers are finished. The
+modified code is shown in Fig. 10. Additionally, the exception
+handler of the kernel is also modified to support terminating
+and detaching malicious BPF programs upon violation.
+1
+mov
+%cr4,%rbx
+2
+push %rbx
+; save CR4
+3
+and
+$0xfffffffffeffffff, %rbx ; clear CR4.PKS
+4
+mov
+%rbx,%cr4
+5
+call \cfunc
+; invoke handler
+6
+pop
+%rbx
+7
+mov
+%rbx,%cr4
+; restore CR4
+Figure 10: The modified kernel interrupt handler in entry_64.S.
+6
+Evaluation
+To evaluate MOAT, we first analyze how MOAT mitigates
+various attack interfaces, and then benchmark its CVEs de-
+tectability in Sec. 6.1. We then assess the performance of
+MOAT under different BPF program setups in Sec. 6.2. Lastly,
+the functionality of MOAT is tested using various types of BPF
+programs and under different scenarios in Sec. 6.3.
+6.1
+Security Evaluation
+6.1.1
+Analysis of Attack Surface Mitigation
+We first systematically analyze how MOAT mitigates five rep-
+resentative attack interfaces presented in the BPF ecosystem.
+These potential attack interfaces are illustrated in Fig. 11.
+4We will release the codebase of MOAT once this paper is published. We
+will maintain MOAT to benefit the community and follow-up research.
+PTEs
+IDT/GDT
+Memory
+BPF
+Program
+Helper 
+Auditor
+BPF 
+Helper
+IA32_PKRS
+CR4.PKS
+3
+4
+1
+2
+5
+PKS Region
+Write Disabled
+Access Disabled
+Figure 11: Analysis of mitigating potential attack surfaces.
+1 Arbitrary Kernel Accesses. Currently, the most prevalent
+threat to the BPF ecosystem is the ability of malicious BPF
+programs to arbitrarily modify kernel memory. In order to
+accomplish this, these BPF programs typically employ corner-
+case operations to deceive the verifier during the loading
+phase and to behave maliciously during runtime. This type
+of attack is effectively mitigated due to the fact that MOAT
+derives the minimum necessary memory regions of each BPF
+program and uses PKS to prevent any runtime access beyond
+this region (Sec. 4.1), mitigating such illegal accesses.
+2 Helper Function Abuse. Apart from launching attack di-
+rectly from BPF programs, a malicious BPF program may
+carefully prepare parameter values by exploiting similar
+corner-cases operations in 1 and pass them to abuse certain
+helpers. To prevent such abuse, MOAT features three security
+enforcement schemes (Sec. 4.2.2) to dynamically audit helper
+parameters and also protect critical kernel memory regions
+during the execution of these helpers. Thus, the attacker can
+no longer take advantage of these helpers.
+3 PTE Corruption. A page’s PKS region is configured via
+its PTE. Consequently, a malicious BPF program may attempt
+to tamper these PTEs to disable MOAT. However, this is im-
+possible since MOAT sets these PTEs as access-disabled; they
+are thus protected by PKS like other kernel resources.
+4 Descriptor Table Tampering. Descriptor tables like GDT
+and IDT are essential for segmentation and interrupt handling.
+Since they are needed for these critical functions, blindly set-
+ting them as access-disabled would cause system crashes.
+However, since these descriptor tables are only accessed in
+a read-only manner, MOAT sets them as write-disabled to
+thwart any tampering made by malicious BPF programs. This
+effectively prevents malicious BPF programs from compro-
+mising the kernel using these tables.
+5 Hardware Configuration Tampering. Besides memory-
+based attacks discussed above, attackers may also directly
+disable PKS through hardware configurations. As described
+in Sec. 2, CR4.PKS and IA32_PKRS are two critical registers
+for configuring PKS. One may disable PKS via modifying
+these two registers. However, both registers can only be mod-
+ified via special instructions, and BPF instruction sets do not
+include any of these. Therefore, BPF bytecodes containing
+these instructions are rejected immediately. Since the BPF
+programs are set to W ⊕ X (meaning write and executable
+permissions cannot be simultaneously enabled), adding these
+instructions via self-modification is also impossible.
+9
+
+6.1.2
+Real-world CVE Evaluation
+We analyzed all 37 CVEs relating to BPF since 2020 and
+found that nine of them are related to runtime memory corrup-
+tion caused by malicious BPF programs, which falls within
+the application scope of MOAT. Even though these memory
+corruption vulnerabilities only account for about one-forth
+of all CVEs, they all result in privilege escalation and pose a
+severe security threat to the kernel. As listed in Table 3, five
+of these vulnerabilities have PoC exploits available and are
+evaluated at this step.
+We report that MOAT can successfully mitigate all of them.
+We clarify that these five are not cherry-picked; the untested
+four only have high-level text descriptions without further de-
+tails or any PoC, making it extremely hard for us to build
+a workable exploit based on these descriptions alone. In-
+stead, we thoroughly analyze these four vulnerabilities. Due
+to their conceptual similarity to the other five tested cases,
+it should be accurate to conclude that these four can also be
+mitigated by MOAT. For instance, although there is no exploit
+for CVE-2021-3444, it shares the same logistics with CVE-
+2021-31440, albeit with different BPF instructions. Note that
+both originate from incorrect truncation. From the fact that
+CVE-2021-31440 is mitigated by MOAT, we would believe
+the same for CVE-2021-3444.
+Table 3: BPF CVE detectability evaluation.
+denotes experimented
+and mitigated by MOAT.
+denotes the CVEs share conceptually
+identical patterns, though they lack available PoC exploit.
+CVE ID
+Description
+Status
+2022-2785 [43]
+Incorrect Instruction Rewrite
+2022-23222 [42]
+Mischeck *_OR_NULL Pointer
+2021-45402 [41]
+Incorrect MOV32 Bound
+2021-3490 [40]
+Incorrect ALU32 Bound
+2021-31440 [37]
+Incorrect 32-bit Truncation
+2021-3444 [39]
+Incorrect MOD32 Truncation
+2021-33200 [38]
+Incorrect Pointer Arithmetic
+2020-8835 [36]
+Incorrect 32-bit Bound
+2020-27194 [35]
+Incorrect OR32 Bound
+CVE Case Study. To better explain how MOAT mitigates
+these CVEs, we elaborate on the exploit paths for two of
+them, CVE-20222-23222 and CVE-2020-27194.
+CVE-2022-23222 is a pointer mischeck vulnerability intro-
+duced via a rather new feature of BPF named bpf_ringbuf.
+This new feature was brought to BPF in 2020 along with
+a new pointer type named PTR_TO_MEM_OR_NULL. However,
+the verifier had not been updated to track the bounds of this
+new type, resulting in this vulnerability. As illustrated in
+Fig. 12, the malicious payload first retrieves a nullptr via
+bpf_ringbuf_reserve (line 1), which returns this newly-
+added pointer type named PTR_TO_MEM_OR_NULL. Since this
+new type is not tracked by the verifier, the payload can bypass
+pointer checks by convincing the verifier that r1 is 0x0 when
+it is actually 0x1 (line 3). This pointer can then be multiplied
+with any offset to perform arbitrary kernel accesses (line 9).
+However, such arbitrary access violates PKS immediately and
+is terminated by MOAT (line 10).
+1
+r0 = bpf_ringbuf_reserve(fd, INT_MAX, 0)
+2
+r1 = r0
+// R:r0=0;r1=0 V:r0=r1=?
+3
+r1 = r0 + 1
+// R:r0=0;r1=1 V:r0=r1=?
+4
+if (r0 != nullptr) {
+// R:r0=0;r1=1 V:r0=r1=?
+5
+ringbuf_discard(r0, 1)
+6
+exit(2)
+7
+}
+8
+off = <OOB addr>
+// R:r0=0;r1=1 V:r0=r1=0
+9
+off = off * r1
+// R:off=<OOB addr> V:off=0
+10
+*(ptr+off) = 0xbad
+// PKS violation!
+Figure 12: Code snippet of CVE-2022-23222. R denotes variable
+runtime statuses. V denotes verifier-deduced values of variables.
+CVE-2020-27194 is a vulnerability due to incorrect trunca-
+tion. As in Fig. 13, the user first inputs an arbitrary value
+in the range of [0,0x600000001] (line 1). Then, two con-
+ditional clauses help the verifier to determine its lower and
+upper bounds (line 3 and line 5). However, when tracking
+the BPF_OR operator (line 7), the verifier performs a wrong
+truncation on its upper bound. After the truncation, the user-
+controlled r5is viewed by the verifier as a legitimate constant
+scalar 0x1(line 7), which can later be used as the offset to per-
+form arbitrary accesses to the kernel (line 8). Similarly, such
+accesses can be detected by MOAT and terminated instantly.
+1
+r5 = <OOB addr>
+2
+r6 = 0x600000002
+3
+if (r5 >= r6)
+// R&V:r5<=0x600000001
+4
+exit(2)
+5
+if (r5 <= 0)
+// R&V:0x1<=r5<=0x600000001
+6
+exit(2)
+7
+r5 = r5 | 0
+// R:r5=<OOB addr> V: r5=0x1
+8
+*(ptr+r5)=0xbad
+// PKS violation!
+Figure 13: Code snippet of CVE-2020-27194. R denotes variable
+runtime statuses. V denotes verifier-deduced values of variables.
+6.2
+Performance Evaluation
+We assess MOAT performance overhead on Linux v5.195 and
+a 16-core Intel 12700H, whose efficiency cores are disabled
+and performance cores are locked to 4 GHz to avoid random-
+ness. As a common setup, the cycle and time statistics are
+measured via the rdtscp instruction and the kernel utility
+get_ktime_raw(), respectively.
+6.2.1
+Micro Benchmark
+For micro benchmark, we measure the CPU cycles of four
+key operations in MOAT. We list the the four operations in
+Table 4. switch_pks() enables/disables PKS by setting/-
+clearing the corresponding control bit in CR4. set_pkrs()
+changes region permissions by changing IA32_PKRS via
+WRMSR. get_pkrs() returns current permission configuration
+by reading IA32_PKRS via RDMSR. assign_page() changes
+5The kernel is slightly modified as described in Sec. 5.
+10
+
+the permission region of one page by modifying its PTE. Each
+operation is measured by averaging ten runs of one million
+invocations to eliminate randomness.
+Table 4: Micro benchmark results. As a reference [51], userspace
+RDPKRU, WRPKRU, and pkey_assign() take 0.5, 23.3, and 1104.9
+cycles, respectively.
+Operation
+# Cycle
+Note
+switch_pks()
+4.2
+Set/Clear CR4.PKS
+set_pkrs()/WRMSR
+71.7
+Set region permissions
+get_pkrs()/RDMSR
+25.8
+Get region permissions
+assign_page()
+1120.4
+Assign a page to region
+As Table 4 shows, the most expensive operation is
+assign_page() which modifies the region one page be-
+longs to, including locating its PTE and changing specific
+bits within. Notably, setting and getting the region permis-
+sions (set_pkrs()/get_pkrs()) in PKS is much more ex-
+pensive than its userspace variant in libmpk [51] (see the
+caption of Table 4). We presume that this is because in PKU,
+the region permission is controlled via a dedicated register
+named PKRU with two special instructions RDPKRU/WRPKRU,
+whereas in PKS employed by MOAT, its region permission
+is stored in an MSR named IA32_PKRS without any special
+instruction. To configure the permission in IA32_PKRS, one
+has to use the general RDMSR/WRMSR instructions with the
+MSR ID 0x6E1, which requires additional cycles to complete.
+Similarly, directly enabling/disabling PKS via switch_pks()
+also takes fewer cycle than set_pkrs().
+Since configuring permission via set_pkrs() is more
+expensive than switch_pks(), on situations where MOAT
+needs to temporarily switch back to kernel regions (e.g. inter-
+rupt handling), it uses switch_pks() to disable PKS instead
+of using set_pkrs(). Then, before returning to BPF pro-
+grams, we reactive PKS to maintain isolation.
+6.2.2
+Macro Benchmark
+To prepare the macro benchmark suite, we consider the fol-
+lowing properties.
+(a) To test the performance of MOAT conducting fixed and
+dynamic key allocation, it is necessary to include BPF
+programs of varying sizes.
+(b) The number of BPF programs should exceed the num-
+ber of available keys to test MOAT in situations where
+hardware keys are insufficient.
+(c) The BPF programs should be highly parallel to evaluate
+the waiting time when dynamic keys are insufficient.
+(d) The execution order should reflect actual system behav-
+ior with high enough frequency to stress MOAT.
+To simultaneously fulfill these requirements, we prepare
+macro benchmark as follows. We choose seven different
+events frequently triggered in the kernel, which are sys_open,
+sys_close,
+sys_read,
+sys_write,
+sched_switch,
+page_fault_user, and page_fault_kernel. These events
+are of high frequency (e.g., sched_switch occurs on every
+context switch) and can reflect actual BPF running behavior.
+For each of these events, we attach three BPF tracepoints of
+varying sizes to log this event. This ensures that these BPF
+programs are highly parallel.
+MOAT Configuration. In both regular and extreme cases (see
+below), we choose the configuration as follows: the threshold
+for dynamic key allocation is ten pages. The number of fixed
+keys is ten, while the number of dynamic keys is four. Two
+keys are reserved for the kernel memory region and the shared
+region (i.e., for per-CPU stack, IDT, GDT), respectively.
+Regular Case. In the regular case, we attach each one of
+these events with three types of BPF tracepoints, i.e., small (1
+page), medium (10 pages) and large (200 pages). We run
+each setup ten times, and each run consists of 1,000 invoca-
+tions of each tracepoint. The average results are reported in
+Fig. 14. We find that even in the worst case, MOAT imposes a
+moderate overhead of less than 30%. This overhead occurs
+when launching the medium-size BPF program attached to
+the event page_fault_kernel. Since its size (10 pages) does
+not exceed the threshold of dynamic key allocation, it has to
+repetitively assign and return the dynamic key to its pages
+upon every entry point and exit. As reflected on the micro
+benchmark in Sec. 6.2.1, such key assignment is quite costly.
+Overall, we interpret the performance penalty is aligned with
+our expectation, and the overall overhead is reasonable.
+All large-size BPF programs exceed the page number
+threshold of dynamic key allocation. Therefore, MOAT as-
+signs fixed keys to them during their loading phase without
+incurring runtime overhead. The incurred overheads are gen-
+erally moderate: for all cases, the overheads are less than 10%.
+Moreover, the overheads for those small-size BPF programs
+are all less than 22%, which lie between the large-size and
+the medium-size ones. Apart from the total overhead reported
+above, we also investigate the waiting overhead, which is the
+amount of time a BPF program must wait if there is no dy-
+namic key available. Note that in the regular cases above, 14
+programs are smaller than the page number threshold; they
+are configured to use the dynamic key allocation scheme,
+although there are only four dynamic keys available. Their
+waiting statistics are shown in Table 5. It is seen that although
+the average waiting time is near 1µs, less than 1% BPF exe-
+cutions really experience this delay. Considering there are 14
+running processes and only four dynamic keys available, we
+can conclude that the dynamic key allocation policy handles
+parallelism reasonably well. Moreover, this also shows that
+four dynamic keys are sufficient for most scenarios; adding
+more dynamic keys brings marginal benefit.
+Table 5: Waiting time statistics.
+Avg. (ns)
+Waited Avg. (ns)
+Max. (ns)
+# Waited
+7.1
+915.2
+2559
+0.8%
+Extreme Cases. The above regular cases only evaluate MOAT
+under situations where dynamic keys are limited but fixed
+keys are sufficient. Here, we further explore MOAT’s overhead
+via extreme cases. Instead of attaching three BPF programs
+11
+
+1.00 
+1.00 
+1.00 
+1.00 
+1.00 
+1.00 
+1.00 
+1.19 
+1.22 
+1.07 
+1.04 
+1.14 
+1.06 
+1.06 
+1.25 
+1.29 
+1.25 
+1.12 
+1.18 
+1.22 
+1.24 
+1.08 
+1.09 
+1.03 
+1.04 
+1.05 
+1.03 
+1.03 
+0.00
+0.50
+1.00
+1.50
+pf_u
+pf_k
+sched
+open
+close
+read
+write
+Relative Time
+Base
+Small
+Medium
+Large
+Figure 14: Regular macro benchmark.
+of varying sizes, as we did in the regular cases above, in the
+extreme case evaluation we attach three large (200 pages)
+BPF programs to each tracepoint. Under this setting, there are
+only ten fixed keys available, although there are 21 large-size
+BPF programs, requiring dynamic key allocation for over half
+of these programs. Since each of these programs contains
+over 200 pages, there are a large number of page assignments
+occurring upon their program entry points and exits.
+Table 6: Extreme overhead.
+Static Keys (ns)
+Dynamic Keys (ns)
+Avg.
+Max.
+Avg.
+Max.
+Waited
+# Waited
+140.7
+202.8
+3630
+4401
+1968.1
+4%
+We report the evaluation results of extreme cases in Ta-
+ble 6. We find that MOAT imposes a negligible overhead to
+BPF programs that use fixed keys even under such extreme
+cases. And for those large BPF programs that use dynamic
+keys, the average overhead is still reasonably low (around
+3.6µs). Overall, we point out that real-life scenarios seldomly
+require this many BPF programs with large maps running
+concurrently. Moreover, the currently observed overhead can
+be further reduced by sharing these large maps between BPF
+programs, thereby reducing the need for fixed keys. We also
+report that the waiting time due to the shortage of dynamic
+keys shows a similar pattern to the regular cases. Although
+the average waiting time is near 2µs, less than 5% of the
+executions would experience this delay.
+6.2.3
+Real-world Case Study
+To evaluate the performance of MOAT under real-world sce-
+narios, we setup a BPF port forwarding program which redi-
+rects incoming requests to the memcached [24] memory
+database. To prepare the benchmark, we choose YCSB [19] to
+generate six distinct workloads and test the overall throughput
+of the memcached service. The results are shown in Fig. 15.
+From the figure, we can see that MOAT imposes on average
+6% (up to 14%) slowdown to the overall performance of the
+BPF-based port forwarding, which is acceptable considering
+the security benefits MOAT provides. Note that this overhead
+is far less than the worst overhead we observed from the
+regular/extreme cases above, which further justifies our as-
+sumption that BPF programs are invoked less frequently in
+real-world applications than in extreme cases.
+5586
+5649
+7407
+5649
+14084
+4975
+5464
+5681
+6493
+5050
+13889
+4366
+0
+5000
+10000
+15000
+YCSB_A
+YCSB_B
+YCSB_C
+YCSB_D
+YCSB_E
+YCSB_F
+Throughput 
+(ops/sec)
+Base
+MOAT
+Figure 15: Overall throughput of the memcached case study.
+6.3
+Functionality Evaluation
+To show that MOAT is able to support various BPF features,
+we select seven BPF applications with varying functionalities
+from the famous bcc toolbox [52]. Among them, execsnoop
+and opensnoopare used for kernel profiling, recording differ-
+ent system events; tcptrace and net_monitor are used for
+network monitoring, collecting packet statistics; xdp_drop,
+xdp_cpu and xdp_interface can be used in firewalls and
+various load balancing scenarios, redirecting or dropping
+packages. These applications cover the majority of contem-
+porary BPF ecosystem usage scenarios. After securing these
+applications with MOAT, we examine the runtime status of
+these applications and confirm that they are operating cor-
+rectly and are not affected by MOAT. Furthermore, Fig. 16
+reports the performance evaluation results of these applica-
+tions with MOAT enabled. The extra overhead incurred by
+MOAT under different scenarios is reasonably low. Overall,
+the evaluation shows that MOAT can be smoothly applied to
+secure de facto BPF applications under various scenarios with
+minimal engineering effort and moderate cost.
+1.00 
+1.00 
+1.00 
+1.00 
+1.00 
+1.00 
+1.00 
+1.07 
+1.01 
+1.25 
+1.10 
+1.21 
+1.11 
+1.07 
+0.00
+0.50
+1.00
+1.50
+execsnoop
+opensnoop
+tcptrace
+net_monitor
+xdp_drop
+xdp_cpu
+xdp_interface
+Relative Time
+Base
+MOAT
+Figure 16: Application benchmark.
+7
+Related Work
+In-Kernel Isolation. Most existing works [10, 12–14, 16, 23,
+26, 29, 49, 61, 64] on kernel isolation focuses kernel com-
+ponents like device drivers and file systems, which are dis-
+tinct from BPF programs and hence cannot be reused directly
+in our scenario. Existing works can be roughly divided into
+three categories: virtualization, Software Fault Isolation (SFI),
+and formal methods. Narayanan et al. [49] propose LVD,
+which isolates kernel components in a virtualized environ-
+ment. Based on LVD, Huang et al. [29] split kernel modules
+into individual components for finer-grained isolation. SFI
+12
+
+is employed to instrument programs at the source or binary
+level [13, 14, 23]. These works ensure kernel security by in-
+serting pointer checks prior to memory accesses. Furthermore,
+formal methods enable principled isolation of kernel compo-
+nents, e.g., separating kernel code from untrusted drivers [61],
+or verifying file system correctness [10, 16].
+We believe none of these methods are readily re-usable
+in our BPF scenario. Virtualization method [12, 29, 49, 64]
+require placing the program in a separated address space,
+making it hard for BPF programs to interact with kernel.
+SFI [13, 14, 23] is based on program (compile-time) instru-
+mentation, whose inserted software checks often lead to high
+runtime overhead. Lastly, the BPF verifier itself performs
+formal verification, which shares conceptually similar advan-
+tage and drawbacks with existing formal method-based kernel
+isolation methods [10, 16, 61]; MOAT employs hardware ex-
+tensions to offer more principled BPF isolation.
+MPK-Based Isolation. Prior to PKS, Intel first announced
+its userspace variant PKU. Consequently, most existing
+works [27, 51, 58] using MPK focus on userspace isolation.
+To better utilize PKU as an isolation primitive, Park et al. [51]
+proposed libmpk that resolves the semantic discrepancies
+between PKU and conventional mprotect. There are also
+works [27, 58] that leverage this hardware feature to protect
+confidential data. Apart of using PKU to isolate normal user
+applications, efforts are made to isolate trusted applications
+in SGX via PKU [17, 33]. SGXLock [17] establishes mu-
+tual distrust between kernel and the trusted SGX applications,
+while EnclaveDom [33] enables intra-isolation within one
+enclave. PKU has been used for kernel security [26, 57] as
+well. IskiOS [26] applies PKU to kernel pages by marking
+them as user-owned, while Sung et al. [57] employ PKU to
+protect userspace unikernels. As a new feature introduced in
+2021, research works using PKS are rather rare comparing
+to PKU. Linux community attempted to use PKS to prevent
+stray writes [1], which refers to kernel accidentally writing to
+wrong addresses.
+BPF Security. There also exist many works [25, 31, 32, 50,
+60] on securing the the BPF ecosystem. However, most of
+these works use formal methods to enhance the following
+BPF components: the verifier, the JIT compiler and the BPF
+program itself. To enhance the standard BPF verifier, Ger-
+shuni et al. [25] built PREVAIL based on abstract interpre-
+tation [20], which supports more program structures (e.g.
+loops) and is more efficient comparing to the standard verifier.
+PRSafe [32], on the other hand, designs a new domain-specific
+language based on primitive recursive functions, whose prop-
+erties ensure that all computations must terminate. The ul-
+timate goal of PRSafe is to build a mathematically verifi-
+able compiler for BPF programs. As for BPF JIT compiler,
+Jitk [60] is a classic BPF JIT compiler whose correctness is
+proven manually. Further, Nelson et al. [50] propose Jitterbug
+to generate automated proof for real-world BPF JIT compilers.
+Lastly, Luke Nelson [31] build proof-carrying BPF programs,
+requiring developers to provide a correctness proof alongside
+with the program before loading it into the kernel.
+8
+Discussion
+Platform Migration. The current prototype implementation
+of MOAT is based on MPK, a hardware extension available on
+Intel platforms. Below, we discuss migrating MOAT to other
+platforms with similar hardware extensions.
+ARM Memory Domains. “Domain” is a MPK-like feature
+supported since ARMv7 [3]. It employs 4-bit domain keys in
+PTEs and a Domain Access Control Register (DACR) in su-
+pervisor mode. Following a similar rationale to MPK, DACR
+allows accesses to be configured as denied, fully-allowed, or
+the same as PTEs. Since this feature is only supported on first-
+level and section-level PTEs, the domain boundaries must
+be aligned to 1 megabyte. Due to the similarity between this
+feature and MPK, we expect MOAT to be implemented on
+ARM with a moderate effort using this feature.
+RISC-V Domain Keys. As an open-source architecture, there
+exists a hardware extension on the RISC-V platform that
+supports similar features as MPK named Donky [53]. Donky
+leverages ten unused bits in the PTEs as a protection key,
+hence supporting 1,024 permission regions. Since Donky
+supports 1,024 keys, it is no longer possible to control permis-
+sions for all these regions using a single register, like MPK
+does. Donky thus introduces a 64-bit DKRU register with four
+key slots. Each slot can be loaded with a 10-bit protection key.
+Only when a key is loaded in DKRU can its associated region
+be written to or read from. From the description above, we
+interpret that Donky is quite flexible, and therefore, MOAT
+may be smoothly implemented on RISC-V using Donky.
+BPF JIT Support. As described in Sec. 2, there are two ways
+of executing a BPF program: directly interpreting the BPF
+bytecode, or using a JIT compiler for improved performance.
+Our prototype implementation of MOAT is based on the BPF
+interpreter. However, we note that the design of MOAT is
+compatible with the JIT compiler. First, the PKS is config-
+ured at the entry and exit points of running a BPF program,
+which is independent of the BPF program execution method.
+Second, the operations that MOAT performs during the BPF
+execution, such as helper auditing, are implemented as part
+of BPF helpers and also decoupled from how BPF programs
+are executed. Therefore, MOAT is essentially agnostic about
+the BPF program execution method, and it is adaptive to the
+native code produced by the BPF JIT compiler. Moreover,
+unlike the JIT compiler in Java virtual machine (JVM), which
+compiles only hotspot code chunks of Java bytecode each
+time, the BPF JIT compiler compiles the entire BPF program
+bytecode into native code once. This further reduces the effort
+of adapting MOAT to BPF programs compiled by JIT.
+13
+
+9
+Conclusion
+Despite the increasing popularity of using BPF to extend
+kernel functionality, the security of BPF programs is still a
+concern. Recent attacks reveal that BPF applications can by-
+pass static security checks and conduct unauthorized kernel
+memory accesses. This paper has presented MOAT, which iso-
+lates potentially malicious BPF applications from the kernel
+using Intel MPK. MOAT addresses technical challenges and
+delivers a practical and extensible protection mechanism, in
+compensation to the contemporary BPF verifiers. Our evalua-
+tion reveals that MOAT can isolate (malicious) BPF programs
+in various real-world circumstances at a low cost.
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+Multilingual Sentence Transformer as A Multilingual Word Aligner
+Weikang Wang1∗ Guanhua Chen2∗
+Hanqing Wang1
+Yue Han1
+Yun Chen1†
+1Shanghai University of Finance and Economics
+2Southern University of Science and Technology
+[email protected]
+[email protected]
+{whq,hanyue}@163.sufe.edu.cn
+[email protected]
+Abstract
+Multilingual
+pretrained
+language
+models
+(mPLMs) have shown their effectiveness in
+multilingual word alignment induction. How-
+ever, these methods usually start from mBERT
+or XLM-R. In this paper, we investigate
+whether multilingual sentence Transformer
+LaBSE is a strong multilingual word aligner.
+This idea is non-trivial as LaBSE is trained
+to
+learn
+language-agnostic
+sentence-level
+embeddings, while the alignment extraction
+task requires the more fine-grained word-
+level embeddings to be language-agnostic.
+We
+demonstrate
+that
+the
+vanilla
+LaBSE
+outperforms other mPLMs currently used
+in the alignment task, and then propose to
+finetune LaBSE on parallel corpus for further
+improvement.
+Experiment results on seven
+language pairs show that our best aligner
+outperforms previous state-of-the-art models
+of all varieties.
+In addition, our aligner
+supports different language pairs in a single
+model, and even achieves new state-of-the-art
+on zero-shot language pairs that does not
+appear in the finetuning process.
+1
+Introduction
+Word alignment aims to find the correspondence
+between words in parallel texts (Brown et al., 1993).
+It is useful in a variety of natural language process-
+ing (NLP) applications such as noisy parallel cor-
+pus filtering (Kurfalı and Östling, 2019), bilingual
+lexicon induction (Shi et al., 2021), code-switching
+corpus building (Lee et al., 2019; Lin et al., 2020)
+and incorporating lexical constraints into neural
+machine translation (NMT) models (Hasler et al.,
+2018; Chen et al., 2021b).
+Recently, neural word alignment approaches
+have developed rapidly and outperformed statistical
+word aligners like GIZA++ (Och and Ney, 2003)
+and fast-align (Dyer et al., 2013). Some works
+∗The first two authors contribute equally.
+†Corresponding author.
+Figure 1: Cosine similarities between subword repre-
+sentations in a parallel sentence pair from 8th layer of
+mBERT (left) and 6th layer of LaBSE (right).
+Red
+boxes denote the gold alignments.
+(Garg et al., 2019; Li et al., 2019; Zenkel et al.,
+2019, 2020; Chen et al., 2020b; Zhang and van Gen-
+abith, 2021; Chen et al., 2021a) induce alignments
+from NMT model or its variants. However, these
+bilingual models only support the language pair
+involved in the training process. They also treat the
+source and target side differently, thus two models
+are required for bidirectional alignment extraction.
+Another line of works (Jalili Sabet et al., 2020; Dou
+and Neubig, 2021) build multilingual word aligners
+with contextualized embeddings from the multilin-
+gual pretrained language model (Wu and Dredze,
+2019; Conneau et al., 2020, mPLM). Thanks to
+the language-agnostic representations learned with
+multilingual masked language modeling task, these
+methods are capable of inducing word alignments
+even for language pairs without any parallel corpus.
+Different from previous methods, in this pa-
+per we present AccAlign, a more accurate mul-
+tilingual word aligner with the multilingual sen-
+tence Transformer LaBSE (Feng et al., 2022, see
+Figure 1). The LaBSE is trained on large scale
+parallel corpus of various language pairs to learn
+language-agnostic sentence embeddings with con-
+trastive learning. However, it is unclear whether
+LaBSE has learned language-agnostic word-level
+arXiv:2301.12140v1  [cs.CL]  28 Jan 2023
+
+0.81
+0.68
+0.55
+0.51
+0.55
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+0.62
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+0.28
+0.33
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+0.61
+0.66
+0.74
+0.53
+0.56
+0.51
+0.55
+0.58
+0.47
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+0.87
+0.36
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+0.25
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+0.57
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+0.34
+0.25
+0.36
+0.32
+0.29
+、
+0.56
+0.56
+0.77
+0.79
+0.64
+0.54
+0.54
+0.54
+0.42
+0.50
+0.67
+0.40
+0.33
+0.42
+0.46
+0.30
+0.48
+0.52
+0.53
+0.73
+0.70
+0.58
+0.50
+0.52
+0.24
+0.28
+0.33
+0.90
+0.52 
+0.36
+0.34
+0.28
+e
+0.54
+0.52
+0.56
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+0.63
+0.61
+0.59
+0.25
+0.27
+0.31
+0.49
+0.88
+0.38
+0.41
+0.29
+verstehen
+0.50
+0.54
+0.52
+0.55
+0.62
+0.69
+0.77
+0.54
+0.28
+0.28
+0.33
+0.34
+0.39
+0.52
+0.89
+0.29
+0.63
+0.62
+0.56
+0.57
+0.60
+0.58
+0.57
+0.94
+0.21
+0.26
+0.19
+0.18
+0.20
+0.23
+0.20
+0.61
+Jno
+That
+can
+That
+our
+can
+nderstand
+understand
+mBERT
+LaBSEembeddings, which is the key for the success of
+word alignment extraction. Specifically, we first
+direct induce word alignments from LaBSE and
+demonstrate that LaBSE outperforms other mPLMs
+currently used in the alignment task. This indi-
+cates that LaBSE has implicitly learned language-
+agnostic word-level embeddings at some intermedi-
+ate layer. Then we propose a simple and effective
+finetuning method to further improve performance.
+Empirical results on seven language pairs show that
+our best aligner outperforms previous SOTA mod-
+els of all varieties. In addition, our aligner supports
+different language pairs in a single model, and even
+achieves new SOTA on zero-shot language pairs
+that does not appear in finetuning process.1
+2
+AccAlign
+2.1
+Background: LaBSE
+LaBSE (Feng et al., 2022) is the state-of-the-art
+model for the cross-lingual sentence retrieval task.
+Given an input sentence, the model can retrieve the
+most similar sentence from candidates in a different
+language. LaBSE is first pretrained on a combina-
+tion of masked language modeling (Devlin et al.,
+2019) and translation language modeling (Conneau
+and Lample, 2019) tasks. After that, it is effec-
+tively finetuned with contrastive loss on 6B parallel
+sentences across 109 languages. We leave the train-
+ing detail of LaBSE in the appendix. However, as
+LaBSE does not include any word-level training
+loss when finetuning with contrastive loss, it is un-
+clear whether the model has learned high-quality
+language-agnostic word-level embeddings, which
+is the key for a multilingual word aligner.
+2.2
+Alignment Induction from LaBSE
+To investigate whether LaBSE is a strong multilin-
+gual word aligner, we first induce word alignments
+from vanilla LaBSE without any modification or
+finetuning. This is done by utilizing the contextual
+embeddings from LaBSE. Specifically, consider
+a bilingual sentence pair x = ⟨x1, x2, ..., xn⟩ and
+y = ⟨y1, x2, ..., ym⟩, we denote the contextual em-
+beddings from LaBSE as hx = ⟨hx1, ..., hxn⟩ and
+hy = ⟨hy1, ..., hym⟩, respectively. Following pre-
+vious work (Dou and Neubig, 2021; Jalili Sabet
+et al., 2020), we get the similarity matrix from the
+contextual embeddings:
+S = hxhT
+y.
+(1)
+1Code is available at https://github.com/sufenlp/
+AccAlign.
+Figure 2: The framework of adapter-based finetuning.
+The blue blocks are kept frozen, while the red adapter
+blocks are updated during finetuning.
+The similarity matrix is normalized for each row to
+get Sxy. Sxy is treated as the probability matrix as
+its i-th row represents the probabilities of aligning
+xi to all tokens in y. The reverse probability ma-
+trix Syx is computed similarly by normalizing each
+column of S. Taking intersection of the two prob-
+ability matrices yields the final alignment matrix:
+A = (Sxy > c) ∗ (ST
+yx > c),
+(2)
+where c is a threshold and Aij = 1 indicates that
+xi and yj are aligned. The above method induces
+alignments on the subword level, which are con-
+verted into word-level alignments by aligning two
+words if any of their subwords are aligned follow-
+ing (Zenkel et al., 2020; Jalili Sabet et al., 2020).
+2.3
+Finetuning LaBSE for Better Alignments
+Inspired by (Dou and Neubig, 2021), we propose a
+finetuning method to further improve performance
+given parallel corpus with alignment labels.
+Adapter-based Finetuning
+Adapter-based fine-
+tuning (Houlsby et al., 2019; Bapna and Firat, 2019;
+He et al., 2021) is not only parameter-efficient,
+but also benefits model performance, especially
+for low-resource and cross-lingual tasks (He et al.,
+2021). Figure 2 illustrates our overall framework,
+where the adapters are adopted from (Houlsby et al.,
+2019). For each layer of LaBSE, we introduce
+an adapter for each sublayer, which maps the in-
+put vector of dimension d to dimension m where
+m < d, and then re-maps it back to dimension d.
+Let h and h′ denote the input and output vector,
+
+Add & Norm
+Adapter
+Feed-forward
+00000
+Add & Norm
+000
+Adapter
+00000
+Feed-forward
+Self-attention
+Adapter
+XL
+AccAlignerModel
+Setting
+de-en
+sv-en
+fr-en
+ro-en
+ja-en
+zh-en
+fa-en
+avg
+Bilingual Statistical Methods
+fast-align (Dyer et al., 2013)
+scratch
+27.0
+-
+10.5
+32.1
+51.1
+38.1
+-
+-
+eflomal (Östling and Tiedemann, 2016)
+22.6
+-
+8.2
+25.1
+47.5
+28.7
+-
+-
+GIZA++ (Och and Ney, 2003)
+20.6
+-
+5.9
+26.4
+48.0
+35.1
+-
+-
+Bilingual Neural Methods
+MTL-FULLC-GZ (Garg et al., 2019)
+scratch
+16.0
+-
+4.6
+23.1
+-
+-
+-
+-
+BAO-GUIDE (Zenkel et al., 2020)
+16.3
+-
+5.0
+23.4
+-
+-
+-
+-
+SHIFT-AET (Chen et al., 2020b)
+15.4
+-
+4.7
+21.2
+-
+17.2
+-
+-
+MASK-ALIGN (Chen et al., 2021a)
+14.4
+-
+4.4
+19.5
+-
+13.8
+-
+-
+BTBA-FCBO-SST (Zhang and van Genabith, 2021)
+14.3
+-
+6.7
+18.5
+-
+-
+-
+-
+Multilingual Neural Methods
+SimAlign (Jalili Sabet et al., 2020)
+no ft
+18.8
+11.2
+7.6
+27.2
+46.6
+21.6
+32.7
+23.7
+AwesomeAlign (Dou and Neubig, 2021)
+no ft
+17.4
+9.7
+5.6
+27.9
+45.6
+18.1
+33.0
+22.5
+self-sup ft
+15.9
+7.9
+4.4
+26.2
+42.4
+14.9
+27.1
+19.8
+sup ft
+15.2
+7.2
+4.0
+25.5
+40.6
+13.4
+25.8
+18.8
+AccAlign
+no ft
+16.0
+7.3
+4.5
+20.8
+43.3
+16.2
+23.4
+18.8
+self-sup ft
+14.3
+5.8
+3.9
+21.6
+39.2
+13.0
+22.6
+17.2
+sup ft
+13.6
+5.2
+2.8
+20.8
+36.9
+11.5
+22.2
+16.1
+Table 1: AER comparison between AccAlign and the baselines on test set of 7 language pairs. self-sup and
+sup mean finetuning the model with parallel corpus of self-supervised and human-annotated alignment labels,
+respectively. All multilingual methods are tested on zero-shot language pairs.
+respectively. The output vector h′ is calculated as:
+h
+′ = Wup · tanh(Wdown · h) + h.
+(3)
+Note that a skip-connection is employed to approx-
+imate an identity function if parameters of the pro-
+jection matrices are near zero. During finetuning,
+only parameters of the adapters are updated.
+Training Objective
+Let ˆA denote the alignment
+labels for the given sentence pair x and y. We
+define the learning objective as:
+L =
+�
+ij
+ˆAij
+1
+2
+�
+(Sxy)ij
+n
++ (ST
+yx)ij
+m
+�
+,
+(4)
+where Sxy and Syx are the alignment probabil-
+ity matrices, n and m are the length of sentence
+x and y, respectively. Intuitively, this objective
+encourages the gold aligned words to have closer
+contextualized representations. In addition, as both
+Sxy and ST
+yx are encouraged to be close to ˆA, it im-
+plicitly encourages the two alignment probability
+matrices to be symmetrical to each other as well.
+Our framework can be easily extended to cases
+where alignment labels are unavailable, by replac-
+ing ˆA with pseudo labels A (Equation 2) and train-
+ing in a self-supervised manner.
+3
+Experiments
+3.1
+Setup
+As we aim at building an accurate multilingual
+word aligner, we evaluate AccAlign on a di-
+verse alignment test set of seven language pairs:
+de/sv/ro/fr/ja/zh/fa-en. For finetuning LaBSE, we
+use nl/cs/hi/tr/es/pt-en as the training set and cs-en
+as the validation set. To reduce the alignment anno-
+tation efforts and the finetuning cost, our training
+set only contains 3, 362 annotated sentence pairs.
+To simulate the most difficult use cases where the
+test language pair may not included in training, we
+set the test language pairs different from training
+and validation. Namely, LaBSE is tested in a zero-
+shot manner. We denote this dataset as ALIGN6.
+We induce alignments from 6-th layer of LaBSE,
+which is selected on the validation set. We use
+Alignment Error Rate (AER) as the evaluation met-
+ric. Our model is not directly comparable to the
+bilingual baselines, as they build model for each
+test language pair using large scale parallel corpus
+of that language pair. In contrast, our method is
+more efficient as it supports all language pairs in
+a single model and our finetuning only requires
+3, 362 sentence pairs.
+Appendix B show more
+dataset, model, baselines and other setup details.
+3.2
+Main Results
+Table 1 shows the comparison of our methods
+against baselines. AccAlign-supft achieves new
+SOTA on word alignment induction, outperforming
+all baselines in 6 out of 7 language pairs. AccAlign
+is also simpler than AwesomeAlign, which is the
+best existing multilingual word aligner, as Awe-
+someAlign finetunes with a combination of five
+objectives, while AccAlign only has one objective.
+The vanilla LaBSE is a strong multilingual word
+
+Model
+fi-el
+fi-he
+SimAglin
+noft
+69.3
+85.8
+AwesomeAlign
+noft
+69.8
+84.4
+self-sup ft
+68.8
+87.7
+sup ft
+67.4
+86.1
+AccAlign
+noft
+47.0
+81.2
+self-sup ft
+40.8
+76.1
+sup ft
+36.7
+71.7
+Table 2: AER comparison between AccAlign and mul-
+tilingual baselines on non-English zero-shot language
+pairs. The best AER for each column is bold and un-
+derlined.
+aligner (see AccAlign-noft). It performs better than
+SimAlign-noft and AwesomeAlign-noft, and com-
+parable with AwesomeAlign-supft, indicating that
+LaBSE has learned high-quality language-agnostic
+word embeddings. Our finetuning method is ef-
+fective as well, improving AccAlign-noft by 1.6
+and 2.7 AER with self-supervised and supervised
+alignment labels, respectively. Our model improves
+multilingual baselines even more significantly on
+non-English language pairs. See Table 2 of ap-
+pendix for detailed results.
+3.3
+Analysis
+Performance on non-English Language Pair
+We conduct experiments to evaluate AccAlign
+against multilingual baselines on non-English test
+language pairs. The fi-el (Finnish-Greek) and fi-he
+(Finnish-Hebrew) test set contains 791 and 2,230
+annotated sentence pairs, respectively. Both test
+sets are from ImaniGooghari et al. (2021)2. The
+results are shown in Table 2. As can be seen, Ac-
+cAlign in all three settings significantly improves
+all multilingual baselines. The improvements is
+much larger compared with zero-shot English lan-
+guage pairs, demonstrating the effectiveness of Ac-
+cAlign on non-English language pairs. We also
+observe that finetuning better improves AccAlign
+than AwesomeAlign. This verifies the strong cross-
+lingual transfer ability of LaBSE , even between
+English-centric and non-English language pairs.
+Adapter-based vs. Full Finetuning
+We com-
+pare full and adapter-based fine-tuning in Table 3.
+Compared with full finetuning, adapter-based fine-
+tuning updates much less parameters and obtains
+better performance under both supervised and self-
+supervised settings, demonstrating its efficiency
+and effectiveness for zero-shot word alignments.
+2https://github.com/cisnlp/graph-align
+Ft type
+full
+adapter
+Ft mode
+self-supervised (avg.)
+17.4
+17.2
+supervised (avg.)
+16.2
+16.1
+Number of ft param.
+428M
+2.4M
+Table 3:
+AER comparison of full finetuning and
+adapter-based finetuning.
+Bilingual Finetuning
+To better understand our
+method, we compare with AwesomeAlign under
+bilingual finetuning setup where the model is fine-
+tuned and tested in the same single language pair.
+We follow the setup in (Dou and Neubig, 2021) and
+use finetuning corpus without human-annotated la-
+bels. As shown in Table 4, LaBSE outperforms
+AwesomeAlign in the finetuning language pair
+(18.8 vs. 18.2). The performance gap becomes
+larger for zero-shot language pairs (21.3 vs. 18.8).
+The results demonstrate that AccAlign is an effec-
+tive zero-shot aligner, as LaBSE has learned more
+language-agnostic representations which benefit
+cross-lingual transfer.
+Different Multilingual Pretrained Models
+We
+investigate the performance of AccAlign-noft when
+replacing LaBSE with other mPLMs, including
+XLM-R, mBERT and four other multilingual sen-
+tence Transformer from HuggingFace. LaBSE out-
+performs other mPLMs by 3.5 to 9.6 averaged AER.
+Table 9 in appendix shows more details.
+Performance across Layer
+We investigate the
+performance of AccAlign-noft when extracts align-
+ments from different layers. Layer 6, which is the
+layer we use for all experiments, outperforms other
+layers by 0.1 to 26.0 averaged AER. Please refer to
+Table 10 in appendix for more details.
+Representation Analysis
+To succeed in multi-
+lingual word alignment, the contextual embed-
+dings should prefer two following properties: (1)
+language-agnostic: two aligned bilingual words
+should be mapped to nearby features in the
+same language-agnostic feature space. (2) word-
+identifiable: the embeddings of two random tokens
+from the same sentence should be distinguishable.
+Therefore, we analyze the embeddings from dif-
+ferent layers of AccAlign under different settings
+by computing cosine similarity for aligned word
+pairs and word pairs randomly sampled from the
+same sentence, denoted as sbi and smono (see ap-
+pendix for more experiment details). Intuitively,
+bigger sbi and smaller smono are preferred as we
+
+Model
+Test lang.
+Ft lang.
+de-en
+fr-en
+ro-en
+ja-en
+zh-en
+avg.
+AwesomeAlign
+ft lang.
+14.9
+4.0
+22.9
+38.1
+14.1
+18.8
+zero-shot langs (avg.)
+16.3
+4.7
+26.6
+43.7
+15.0
+21.3
+AccAlign
+ft lang.
+14.2
+3.8
+21.0
+38.0
+13.8
+18.2
+zero-shot langs (avg.)
+14.8
+3.9
+20.7
+40.5
+13.8
+18.8
+Table 4: AER results with bilingual finetuning.
+Figure 3: sbi (↑) and smono (↓) of AccAlign without
+finetuning (noft), with self-supervised finetuning (self-
+sup ft) and supervised finetuning (sup ft).
+expect the features of aligned words to be similar
+while that of two different words to be different.
+The results on de-en test set are presented in Fig-
+ure 3. For vanilla LaBSE (green curves), we find
+that features from 6-th layer, namely the best layer
+to induce alignment, successfully trades off these
+two properties as it obtains the biggest sbi − smono
+among all layers. In addition, adapter-based fine-
+tuning improves performance mainly by making
+features more word-identifiable, as it significantly
+decreases smono while almost maintaining sbi .
+4
+Conclusion
+In this paper, we introduce AccAlign, a novel multi-
+lingual word aligner based on multilingual sentence
+Transformer LaBSE. The best proposed approach
+finetunes LaBSE on a few thousands of annotated
+parallel sentences and achieves state-of-the-art per-
+formance even for zero-shot language pairs. Ac-
+cAlign is believed to be a valuable alignment tool
+that can be used out-of-the-box for other NLP tasks.
+Limitations
+AccAlign has shown to extract high quality word
+alignments when the input texts are two well-paired
+bilingual sentences.
+However, the condition is
+not always met. In lexically constrained decod-
+ing of NMT (Hasler et al., 2018; Song et al., 2020;
+Chen et al., 2021b), the aligner takes a full source-
+language sentence and a partial target-language
+translation as the input at each step to determine
+the right position to incorporate constraints. In cre-
+ating translated training corpus in zero-resource
+language for sequence tagging or parsing (Ni et al.,
+2017; Jain et al., 2019; Fei et al., 2020), the aligner
+extracts alignments from the labelled sentence and
+its translation to conduct label projection. Both
+cases deviate from our current settings as the input
+sentence may contain translation error or even be
+incomplete. We leave exploring the robustness of
+AccAlign as the future work.
+At the same time, our proposed method only
+supports languages included in LaBSE. This hin-
+ders applying AccAlign to more low-resource lan-
+guages. Future explorations are needed to rapidly
+adapt AccAlign to new languages (Neubig and Hu,
+2018; Garcia et al., 2021).
+Acknowledgements
+This project was supported by National Natural
+Science Foundation of China (No. 62106138) and
+Shanghai Sailing Program (No. 21YF1412100).
+We thank the anonymous reviewers for their in-
+sightful feedbacks on this work.
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+sociation for Computational Linguistics.
+
+A
+LaBSE
+LaBSE (Feng et al., 2022) is the state-of-the-art
+model for the cross-lingual sentence retrieval task.
+Given an input sentence, the model can retrieve the
+most similar sentence from candidates in a differ-
+ent language. It has 471M parameters and supports
+109 languages. The model is first pretrained on a
+combination of masked language modeling (De-
+vlin et al., 2019) and translation language model-
+ing (Conneau and Lample, 2019) tasks on the 17B
+monolingual data and 6B bilingual translation pairs,
+respectively. After that, it is effectively finetuned
+with contrastive loss on 6B bilingual translation
+pairs across 109 languages.
+Specifically, given a bilingual sentence pair
+⟨xi, yi⟩, we use exi and eyi to denote their sen-
+tence embeddings from LaBSE. Then the model is
+finetuned using contrative loss with in-batch nega-
+tives (Chen et al., 2020a):
+ℓ = − 1
+N
+N
+�
+i=1
+�
+log
+exp
+�
+φ(exi, eyi)
+�
+�N
+j=1 exp
+�
+φ(exi, eyj)
+�+
+log
+exp
+�
+φ(exi, eyi)
+�
+�N
+j=1 exp
+�
+φ(exj, eyi)
+�
+�
+,
+(5)
+where φ(exi, eyj) measures the similarity of sen-
+tence xi and yj in the embedding space:
+φ
+�
+exi, eyj
+�
+=
+�
+e⊤
+xieyj − b
+if i = j
+e⊤
+xieyj
+if i ̸= j .
+(6)
+Note that a margin b is introduced to improve the
+separation between positive and negative pairs.
+B
+Experiments Setup
+B.1
+Language Code
+We refer to the language information in Table 1 of
+(Fan et al., 2021). The information of the languages
+used in this paper is listed in Table 5.
+B.2
+Dataset
+Table 6 shows the detailed data statistics of
+ALIGN6. The ja and zh sentences are preprocessed
+by Dou and Neubig (2021) and Liu and Sun (2015),
+respectively. For finetuning AccAlign and multilin-
+gual baselines, we use the training and validation
+set from ALIGN6. As bilingual baselines are not
+capable of zero-shot alignment induction, they are
+trained from scratch with parallel corpus of the
+test language pair using the same dataset as Dou
+ISO
+Name
+Family
+en
+English
+Germanic
+nl
+Dutch
+Germanic
+cs
+Czech
+Slavic
+hi
+Hindi
+Indo-Aryan
+tr
+Turkish
+Turkic
+es
+Spanish
+Romance
+pt
+Portuguese
+Romance
+de
+German
+Germanic
+sv
+Swedish
+Germanic
+fr
+French
+Romance
+ro
+Romanian
+Romance
+ja
+Japanese
+Japonic
+zh
+Chinese
+Chinese
+fa
+Persian
+Iranian
+Table 5: The information of the languages used in this
+paper.
+and Neubig (2021). The bilingual training data
+set of de/fr/ro/ja/zh-en contain 1.9M, 1.1M, 450K,
+444K and 40K parallel sentence pairs, respectively,
+which are much larger than the training dataset of
+ALIGN6.
+B.3
+Model Setup
+We use the contextual word embeddings from the
+6-th layer of the official LaBSE3, which have 768
+dimensions. We set the threshold in Equation 2 to
+0.1, which is selected on validation set by manual
+tuning among [0, 0.2]. For adapter-based finetun-
+ing, we set the hidden dimension of the adapters to
+be 128. The adapters have 2.4M parameters, which
+account 0.5% of the parameters of LaBSE. We use
+the AdamW optimizer with learning rate of 1e-4,
+and do not use warmup or dropout. The batch size
+is set to 40 and maximum updates number is 1500
+steps. We use a single NVIDIA V100 GPU for all
+experiments.
+B.4
+Baselines
+Besides three statistical baselines fast-align (Dyer
+et al., 2013), eflomal (Östling and Tiedemann,
+2016) and GIZA++ (Och and Ney, 2003), we com-
+pare AccAlign with the following neural baselines:
+MTL-FULLC-GZ (Garg et al., 2019). This model
+supervises an attention head in Transformer-based
+NMT model with GIZA++ word alignments in a
+multitask learning framework.
+BAO-GUIDE (Zenkel et al., 2020). This model
+3https://huggingface.co/sentence-transformers/LaBSE
+
+Type
+Lang.
+Source
+Link
+# Sents
+Training set
+cs-en
+Mareˇcek (2011)
+http://ufal.mff.cuni.cz/
+czech-english-manual-word-alignment
+2400
+nl-en
+Macken (2010)
+http://www.tst.inl.nl
+372
+hi-en
+Aswani and Gaizauskas (2005)
+http://web.eecs.umich.edu/~mihalcea/wpt05/
+90
+tr-en
+Cakmak et al. (2012)
+http://web.itu.edu.tr/gulsenc/resources.htm
+300
+es-en
+Graca et al. (2008)
+https://www.hlt.inesc-id.pt/w/Word_Alignments
+100
+pt-en
+Graca et al. (2008)
+https://www.hlt.inesc-id.pt/w/Word_Alignments
+100
+Validation set
+cs-en
+Mareˇcek (2011)
+http://ufal.mff.cuni.cz/
+czech-english-manual-word-alignment
+101
+Test set
+de-en
+Vilar et al. (2006)
+http://www-i6.informatik.rwth-aachen.de/
+goldAlignment/
+508
+sv-en
+Holmqvist and Ahrenberg (2011)
+https://www.ida.liu.se/divisions/hcs/nlplab/
+resources/ges/
+192
+fr-en
+Mihalcea and Pedersen (2003)
+http://web.eecs.umich.edu/~mihalcea/wpt/
+447
+ro-en
+Mihalcea and Pedersen (2003)
+http://web.eecs.umich.edu/~mihalcea/wpt05/
+248
+ja-en
+Neubig (2011)
+http://www.phontron.com/kftt
+582
+zh-en
+Liu and Sun (2015)
+https://nlp.csai.tsinghua.edu.cn/~ly/systems/
+TsinghuaAligner/TsinghuaAligner.html
+450
+fa-en
+Tavakoli and Faili (2014)
+http://eceold.ut.ac.ir/en/node/940
+400
+Table 6: Training, validation and test dataset of ALIGN6. Note that this is a zero-shot setting as the test language
+pairs do not appear in training and validation.
+adds an extra alignment layer to repredict the to-be-
+aligned target token and further improves perfor-
+mance with Bidirectional Attention Optimization.
+SHIFT-AET (Chen et al., 2020b). This model
+trains a separate alignment module in a self-
+supervised manner, and induce alignments when
+the to-be-aligned target token is the decoder input.
+MASK-ALIGN (Chen et al., 2021a). This model
+is a self-supervised word aligner which makes use
+of the full context on the target side.
+BTBA-FCBO-SST (Zhang and van Genabith,
+2021). This model has similar idea with Chen
+et al. (2021a), but with different model architecture
+and training objectives.
+SimAlign (Jalili Sabet et al., 2020). This model is a
+multilingual word aligner which induces alignment
+with contextual word embeddings from mBERT
+and XLM-R.
+AwesomeAlign (Dou and Neubig, 2021). This
+model improves over SimAlign by designing new
+alignment induction method and proposing to fur-
+ther finetune the mPLM on parallel corpus.
+Among them, SimAlign and AwesomeAlign are
+multilingual aligners which support multiple lan-
+guage pairs in a single model, while others are
+bilingual word aligners which require training from
+scratch with bilingual corpus for each test lan-
+guage pair. We re-implement SimAlign and Awe-
+someAlign, while quote the results from (Dou and
+Neubig, 2021) for the three statistical baselines and
+the corresponding paper for other baselines.
+B.5
+Sentence Transformer
+We compare LaBSE with four other multilingual
+sentence Transformer in HuggingFace. The de-
+tailed information of these models are:
+distiluse-base-multilingual-cased-v2.4
+This
+model is a multilingual knowledge distilled version
+of m-USE (Yang et al., 2020), which has 135M
+parameters and supports more than 50+ languages.
+paraphrase-xlm-r-multilingual-v1.5 This model
+is a multilingual version of paraphrase-distilroberta-
+base-v1 (Reimers and Gurevych, 2019), which has
+278M parameters and supports 50+ languages. It
+initializes the student model with an mPLM and
+trains it to imitate monolingual sentence Trans-
+former on parallel data with knowledge distillation.
+paraphrase-multilingual-MiniLM-L12-v2.6
+This model is a multilingual version of paraphrase-
+MiniLM-L12-v2 (Reimers and Gurevych, 2019),
+which has 118M parameters and supports 50+
+languages. It trains similarly as paraphrase-xlm-
+r-multilingual-v1, but with different teacher and
+student model initialization.
+paraphrase-multilingual-mpnet-base-v2.7 This
+model is a multilingual version of paraphrase-
+mpnet-base-v2 (Reimers and Gurevych, 2019),
+4https://huggingface.co/sentence-transformers/distiluse-
+base-multilingual-cased-v2
+5https://huggingface.co/sentence-
+transformers/paraphrase-xlm-r-multilingual-v1
+6https://huggingface.co/sentence-
+transformers/paraphrase-multilingual-MiniLM-L12-v2
+7https://huggingface.co/sentence-
+transformers/paraphrase-multilingual-mpnet-base-v2
+
+which has 278M parameters and supports 50+ lan-
+guages. It trains similarly as paraphrase-xlm-r-
+multilingual-v1, but with different teacher model
+initialization.
+B.6
+Bilingual Finetuning
+We use the same dataset as bilingual baselines for
+bilingual finetuning following (Dou and Neubig,
+2021). At each time, we finetune LaBSE with one
+language pair among de/fr/ro/ja/zh-en and test on
+all seven language pairs. For Awesome-align, we
+follow the setup in their paper, while for AccAlign,
+we use the same hyperparameters as the main ex-
+periments.
+B.7
+Representation Analysis
+We conduct representation analysis on de-en test
+set. To compute sbi, we calculate the averaged co-
+sine similarity of all gold aligned bilingual word
+pairs. To compute smono, we randomly permute a
+given sentence x = ⟨x1, x2, ..., xn⟩ to get x′ =
+⟨x′
+1, x′
+2, ..., x′
+n⟩ and then create n word pairs as
+{⟨xi-x′
+i⟩}n
+i=1. We go through all de and en test
+sentences and report the averaged cosine similarity
+of all created word pairs as smono.
+C
+Experiment Results
+Detailed results for each test language in Sec-
+tion 3.3 are shown in Table 7 to Table 10.
+
+Ft mode
+Ft type
+de-en
+sv-en
+fr-en
+ro-en
+ja-en
+zh-en
+fa-en
+avg
+Self-supervised
+full
+14.7
+5.8
+3.7
+21.6
+39.9
+13.3
+22.7
+17.4
+adapter
+14.3
+5.8
+3.9
+21.6
+39.2
+13.0
+22.6
+17.2
+Supervised
+full
+13.6
+5.3
+2.8
+21.0
+37.1
+11.0
+22.5
+16.2
+adapter
+13.6
+5.2
+2.7
+20.8
+36.8
+11.5
+22.2
+16.1
+Table 7: AER comparison of full finetuning and adapter-based finetuning. The best AER for each column is bold
+and underlined.
+Model
+Ft lang.
+Test lang.
+de-en
+fr-en
+ro-en
+ja-en
+zh-en
+sv-en
+fa-en
+AwesomeAlign
+de-en
+14.9
+4.7
+26.2
+43.6
+14.6
+7.1
+28.2
+fr-en
+16.4
+4.0
+26.9
+44.6
+15.7
+7.6
+28.0
+ro-en
+15.8
+4.7
+22.9
+44.2
+15.1
+7.8
+27.0
+ja-en
+16.8
+4.9
+27.0
+38.1
+15.2
+8.5
+30.0
+zh-en
+16.2
+4.6
+26.2
+42.4
+14.1
+8.1
+28.0
+AccAlign
+de-en
+14.2
+3.8
+20.9
+39.3
+13.1
+5.7
+22.5
+fr-en
+14.6
+3.8
+20.8
+41.0
+14.1
+6.0
+22.5
+ro-en
+15.2
+4.0
+21.0
+42.1
+14.4
+6.5
+23.2
+ja-en
+14.8
+3.9
+20.3
+38.0
+13.5
+6.3
+22.5
+zh-en
+14.6
+3.9
+20.7
+38.9
+13.4
+5.9
+22.4
+Table 8: AER results with bilingual finetuning. The results where the model is trained and tested on the same
+language pair are bold and underlined.
+layer
+de-en
+sv-en
+fr-en
+ro-en
+ja-en
+zh-en
+fas-en
+avg
+mBERT
+8
+17.4
+8.7
+5.6
+27.9
+45.6
+18.1
+33.0
+22.3
+XLM-R
+8
+23.1
+13.3
+9.2
+28.6
+62.0
+30.3
+28.6
+27.9
+distiluse-base-multilingual-cased-v2
+3
+23.7
+17.2
+9.8
+29.2
+56.3
+29.2
+33.5
+28.4
+paraphrase-xlm-r-multilingual-v1
+6
+17.4
+8.7
+4.9
+24.7
+53.8
+26.1
+26.5
+23.2
+paraphrase-multilingual-MiniLM-L12-v2
+6
+19.4
+9.4
+6.2
+26.0
+57.7
+29.7
+27.4
+25.1
+paraphrase-multilingual-mpnet-base-v2
+5
+18.0
+8.9
+5.4
+24.1
+54.9
+25.7
+25.5
+23.2
+LaBSE
+6
+16.0
+7.3
+4.5
+20.8
+43.3
+16.2
+23.4
+18.8
+Table 9: AER comparison of LaBSE and other multilingual pretrained model. All are without finetuning. We
+determine the best layer of alignment induction for each model using the validation set. The best AER for each
+column is bold and underlined.
+Layer
+de-en
+sv-en
+fr-en
+ro-en
+ja-en
+zh-en
+fa-en
+avg
+0
+32.4
+27.7
+20.5
+44.2
+65.5
+40.1
+38.7
+38.4
+1
+27.3
+19.7
+12.8
+35.6
+64.0
+33.9
+35.4
+32.7
+2
+22.3
+14.0
+8.6
+28.8
+58.0
+25.0
+31.3
+26.9
+3
+18.5
+9.9
+6.0
+24.0
+50.3
+17.9
+26.8
+21.9
+4
+17.7
+8.7
+5.9
+23.3
+48.4
+16.3
+25.7
+20.9
+5
+15.8
+7.4
+4.5
+21.5
+43.7
+15.4
+23.8
+18.9
+6
+16.0
+7.3
+4.5
+20.8
+43.3
+16.2
+23.4
+18.8
+7
+16.5
+7.6
+4.8
+22.4
+43.4
+15.0
+23.7
+19.1
+8
+16.2
+7.3
+5.0
+21.6
+42.7
+16.7
+23.4
+19.0
+9
+16.8
+7.6
+5.3
+21.5
+42.7
+17.9
+23.2
+19.3
+10
+17.7
+9.0
+5.6
+23.0
+44.4
+20.4
+24.4
+20.6
+11
+36.7
+27.0
+24.2
+43.6
+61.3
+35.0
+46.2
+39.1
+12
+43.1
+33.2
+30.5
+46.0
+65.7
+42.6
+52.4
+44.8
+Table 10: AER comparison of vanilla LaBSE across layers. Layer 0 is the embedding layer. The best AER for
+each column is bold and underlined.
+

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+arXiv:2301.01465v1  [hep-ph]  4 Jan 2023
+Spin Polarization and Anomalous Magnetic Moment in a (2 +
+1)-flavor Nambu-Jona-Lasinio model in the thermomagnetic
+background
+Yi-Wei Qiu1 and Sheng-Qin Feng1, 2, 3, ∗
+1College of Science, China Three Gorges University, Yichang 443002, China
+2Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics,
+Central China Normal University, Wuhan 430079, China
+3Center for Astronomy and Space Sciences,
+China Three Gorges University, Yichang 443002, China
+(Dated: January 5, 2023)
+Abstract
+Abstract: We investigate the magnetized QCD matter and chiral phase transition in a (2 + 1)-
+flavor Nambu–Jona-Lasinio (NJL) model at finite temperature and chemical potential by comparing
+the contributions from the tensor spin polarization (TSP) and anomalous magnetic moment (AMM)
+of quarks. For light u and d quarks, when TSP and AMM are not considered, the magnetized system
+is characterized by magnetic catalysis. The introduction of TSP will further enhance the magnetic
+catalytic characteristics.
+On the other hands, when AMM is introduced, the phase transition
+temperature decreases with the magnetic field, which is the feature of inverse magnetic catalysis.
+The phase diagram of u and d quarks will change from the crossover phase transition to the first
+order phase transition with the increase of magnetic field and chemical potential when AMM is
+induced. The phase diagram will not change from the crossover phase transition to the first order
+phase transition when TSP is induced. For the phase diagram of strange s quark, whether TSP
+or AMM is induced, the phase diagram will keep a crossover phase transition with the increase of
+magnetic field and chemical potential.
+∗ Corresponding author: [email protected]
+1
+
+I.
+INTRODUCTION
+Comprehending properties of QCD matter under a strong magnetic field is of essential
+importance to further investigate the evolution of the early universe [1], non-central heavy-
+ion collisions [2–5], neutron-star merges [6, 7], and the interior of magnestar [8, 9]. The
+exploration of the QCD vacuum and strongly interacting matter under external strong mag-
+netic fields has fascinated much attention, see reviews, e.g., Refs. [10–14]. Here we stress
+the study of the magnetic field of non-central heavy-ion collisions, which comes from the
+laboratory of mankind. The magnetic field reaches up to
+√
+eB ∼ 0.1GeV for RHIC and
+√
+eB ∼ 0.5 GeV for LHC in non-central heavy-ion collisions. This magnetic field is external
+since it is generated by the spectators, and though it has a very short lifetime(of the order of
+1 fm/c). However, as taken in Refs. [15–18], the presence of the quark-gluon plasma (QGP)
+medium response effect, substantially delays the decay of these time-dependent magnetic
+fields. This is why in the most cases, the effect of constant and uniform magnetic fields on
+quark matter is discussed in the literature. The magnetic field coincides with the produc-
+tion of the QGP and thus may have a fairly important effect on the properties of the phase
+transition. For example, the chiral magnetic effect (CME) [16, 19–22], magnetic cataly-
+sis (MC) in the vacuum [23–25], inverse magnetic catalysis (IMC) around the chiral phase
+transition [26–29].
+The magnetic field can lead to spin polarization, that is, the condensation of quark
+anti-quark (¯qq) pairs with spin parallel. Ref.[30] shows that a tensor-type interaction ∼
+� ¯ψΣ3ψ
+�2 +
+� ¯ψiγ5Σ3ψ
+�2 produces a spin polarization (SP)
+� ¯ψiγ1γ2ψ
+�
+, which is very similar
+to the anomalous magnetic moment (AMM) produced by quarks in a magnetic field. The
+tensor polarization operator ¯ψσµνψ can also be named as the spin polarization operator, or
+the spin density since ¯ψσ12ψ = ψγ0Σ3ψ. If the quark spinor ψ is projected into the sub-spin
+space ψ = ψ↑ + ψ↓ , corresponding to ¯ψσ12ψ ∼
+� ¯ψ↑ψ↑
+�
+−
+� ¯ψ↓ψ↓
+�
+, which can be used to
+measure the difference between the spin-up quark pair and the spin-down quark pair.
+We investigate the magnetized QCD matter in a (2 + 1)-flavor Nambu–Jona-Lasinio
+(NJL) model at finite temperature and chemical potential by comparing the contributions
+from the tensor spin polarization (TSP) and AMM of quarks. For a particle with charge
+e, mass m and spin ⃗s, its corresponding magnetic moment (MM) is µ. Corresponding to
+¯qq pair with antiparallel spin pairs, it has a net magnetic moment (MM), so the chiral
+2
+
+condensation triggers a dynamic AMM. Under the action of the magnetic field, the net MM
+tends to be parallel to the magnetic field. For SP with ¯qq pair parallel spin pairing, the
+MM of spin-aligned quarks and anti-quarks cancel each other, and the spin polarization
+pairing does not present a net MM. Therefore, compared with the chiral condensation with
+a nonzero net MM, the total MM of the system considering SP condensation will reduce.
+Therefore, systems with spin polarization are expected to exhibit relative diamagnetism. At
+high temperatures, the pair of ¯qq dissociates, and all charged quarks become a single small
+magnet, which is arranged in turn along the magnetic field; Therefore, QCD matter at high
+temperature manifests paramagnetism.
+The catalysis of chiral symmetry breaking induced by a magnetic field, namely the MC
+effect, can be easily understood from dimension reduction. On the other hand, IMC effect,
+the critical temperature of the chiral phase transition decreases with the increasing mag-
+netic field, which is intuitively contradictory to the MC effect and is still a puzzle. Although
+there are many publications trying to explain IMC by considering running coupling constant
+generated by the magnetic field [31] and chiral imbalance caused by sphaleron transition or
+instanton anti-instanton pairing [32]. Some interesting and novel properties of magnetized
+QCD materials have recently been presented by lattice calculations, for example, magne-
+tized materials exhibit paramagnetism (positive susceptibility) at high temperatures and
+diamagnetism (negative susceptibility) at low temperatures [33, 34].
+The effect of an AMM of quark has drawn quite a lot of interest recently [35–41] in order
+to investigate the IMC effect. The dynamical chiral symmetry broken is known as one of
+the most important characteristics of QCD, which makes quarks achieve a dynamical mass
+of QCD. Refs. [42, 43] pointed out that quarks’ AMM can also be dynamically produced
+like the dynamic quark mass. Therefore, once quarks achieve dynamic mass, they should
+also achieve dynamical AMM [42, 44–46]. The coefficient κ of quarks’ AMM in the magnetic
+field by the effective interaction 1
+2qκFµν ¯ψσµνψ = 1
+2 [γµ, γν] is introduced and the IMC effect
+at finite temperature is proposed by Ref. [47]. For QCD, both explicit and spontaneous
+chiral symmetry breaking is dedicated to the AMM of quarks, which is also called dynamical
+AMM [43].
+In this paper, we investigate the magnetism of QCD matter and chiral phase transition
+under a magnetic field with the contribution from the TSP and the AMM of quarks re-
+spectively. This paper is organized as follows: in Sec. II, we introduce the (2 + 1)-flavor
+3
+
+NJL models by including the AMM and the TSP in the external magnetic field respectively.
+in Sec. III, we investigate MC and IMC by the AMM and TSP, respectively. Then the
+dependencies of dynamical mass, entropy, sound-velocity, and critical point on the magnetic
+field by comparing the contributions from the TSP and the AMM of quarks are studied in
+Sec. III. Finally, we make the summaries and conclusions in Sec. IV.
+II.
+THE 2 + 1 FLAVORS NJL MODEL UNDER A MAGNETIC FIELD
+The Lagrangian density of the (2 + 1)-flavor NJL model [48, 49] in the presence of an
+external magnetic field is given as:
+L = ¯ψ
+�
+iγµDµ + γ0µ − m
+�
+ψ + Gs
+8
+�
+a=0
+�� ¯ψλaψ
+�2 +
+� ¯ψiγ5λaψ
+�2�
+− K
+�
+det ¯ψ (1 + γ5) ψ + det ¯ψ (1 − γ5) ψ
+�
+,
+(1)
+where the quark field ψ carries three flavors (f = u, d, s) and three colors (c = r, g, b ), and
+λa(a = 1, · · ·N2
+f − 1) represents the SU(3) Gell-Mann matrices in the three flavor space.
+Current quark mass m is considered as mu = md for isospin symmetry of light quarks,
+strange quark mass ms is different from the other light quark (mu and md) masses. The
+difference between the strange and non-strange quark masses obviously breaks the SU(3)
+flavor symmetry. µ is the quark chemical potential, and we assume that the quark chemical
+potentials of the strange and non-strange quarks are the same. A covariant derivative with
+magnetic field is introduced as Du = ∂µ + i QAext
+µ , and the charge matrix in flavor space is
+Q = diag (qu, qd, qs) = diag
+�2
+3, −1
+3, −1
+3
+�
+.
+(2)
+In general, if one chooses the gauge field Aext
+µ
+= (0, 0, Bx1, 0), a constant magnetic field
+should point at the x3-direction. The K term of Eq. (1) is the term of Kobayashi-Maskawa-
+t’Hooft interaction [49–51].
+A.
+The introduction of a (2 + 1)- flavors NJL model with TSP
+It is shown that [30, 35] the breaking of the rotational symmetry by a uniform magnetic
+field induces a separation between longitudinal and transverse fermion modes along the
+direction of the magnetic field. This separation gives rise to the effective splitting of the
+4
+
+couplings in the one-gluon exchange interactions on which the NJL models are usually based.
+This splitting is therefore reported in the four-fermion couplings of a QCD-inspired NJL
+model in a magnetic field, and we can use the Fierz identities in a magnetic field [30, 31, 52]
+to propose the interactions of scalar and tensor of the (2 + 1)-flavor NJL Lagrangian:
+LTSP = ¯ψ
+�
+iγµDµ + γ0µ − m
+�
+ψ + Gs
+8
+�
+a=0
+�� ¯ψλaψ
+�2 +
+� ¯ψiγ5λaψ
+�2�
++ Gt
+8
+�
+a=0
+�� ¯ψΣ3λaψ
+�2 +
+� ¯ψΣ3iγ5λaψ
+�2�
+− K
+�
+det
+� ¯ψ (1 + γ5) ψ
+�
++ det
+� ¯ψ (1 − γ5) ψ
+��
+.
+(3)
+The coupling constant Gs in the scalar/pseudo-scalar channel is closely related to the
+spontaneously chiral symmetry breaking, which produces a dynamical quark mass, and
+the tensor/ pseudo-tensor channels term Gt
+8�
+a=0
+�� ¯ψc
+fΣ3λaψc
+f
+�2 +
+� ¯ψc
+fiΣ3γ5λaψc
+f
+�2�
+is closely
+related to the spin-spin interaction, which causes spin polarization condensation.
+For the (2 + 1)-flavor NJL model, tensor-type interaction at the mean field level leads to
+two types of spin polarization as
+F3 = −2Gt
+� ¯ψΣ3λ3ψ
+�
+,
+F8 = −2Gt
+� ¯ψΣ3λ8ψ
+�
+.
+(4)
+In general, F3 contains only u and d quark spin polarization condensates, on the other
+hand, F8 is associated with the strange quark spin polarization condensate. The running
+coupling constants are divided into longitudinal (g∥) and transverse (g⊥) components due
+to the existence of the magnetic field. In our current study, the couplings of the above NJL
+interactions relevant to quark gluon vertex coupling are expressed as Gs =
+�
+g2
+|| + g2
+⊥
+�
+/Λ2
+and Gt =
+�
+g2
+|| − g2
+⊥
+�
+/Λ2. The distinguishing transverse and parallel Fierz identities auto-
+matically create a new channel of four-fermion interaction term with second order tensor
+structure in Lagrangian density during the transformation from splitting quark-gluon cou-
+pling to the scalar and pseudoscalar bilinear quantity [30]. Gs and Gt can be considered as
+the scalar and tensor channel interaction couplings, respectively.
+5
+
+The effective potential by using standardized process is given
+ΩTSP =Gs
+�
+f=u,d,s
+�
+ψψ
+�2
+f + Gt
+�
+ψλ3Σ3ψ
+�2 + Gt
+�
+ψλ8Σ3ψ
+�2 − Nc
+2π
+�
+f=u,d,s
+|qfB|
+∞
+�
+l=0
+αl
+∞
+�
+−∞
+dpz
+2π
+×
+�
+εf,l,η + T ln
+�
+1 + exp
+�−εf,l,η − µ
+T
+��
++ T ln
+�
+1 + exp
+�−εf,l,η + µ
+T
+���
++ 4K
+�
+ψψ
+�
+u
+�
+ψψ
+�
+d
+�
+ψψ
+�
+s
+(5)
+where l= 0, 1, 2 ...
+represents the quantum number of Landau level,and η = ±1 cor-
+responds to the two kinds of spin direction of quark-antiquark(¯qq) pair. Contribution of
+non-degenerate particles due to spin difference at non-lowest Landau energy levels can be
+taken into account with the definition of this new operator αl = δ0,l +∆ (l) �
+η=±1
+, where ∆ (l)
+is denoted by
+∆ (l) =
+
+
+
+0
+1
+l = 0
+l > 0
+(6)
+and the energy spectrum of the lowest Landau Level ( LLL) (l = 0) and non-LLL (l ̸= 0)
+are given as
+ε2
+u,l=0 = p2
+z +
+�
+Mf +
+�
+F3 + F8
+√
+3
+��2
+,
+ε2
+u,l̸=0,η=±1 = p2
+z +
+��
+Mf
+2 + 2|qfB|l + η
+�
+F3 + F8
+√
+3
+��2
+,
+ε2
+d,l=0 = p2
+z +
+�
+Mf +
+�
+F3 − F8
+√
+3
+��2
+,
+ε2
+d,l̸=0,η=±1 = p2
+z +
+��
+Mf
+2 + 2|qfB|l + η
+�
+F3 − F8
+√
+3
+��2
+,
+ε2
+s,l=0 = p2
+z +
+�
+Mf +
+�2F8
+√
+3
+��2
+,
+ε2
+s,l̸=0,η=±1 = p2
+z +
+��
+Mf
+2 + 2|qfB|l + η
+�2F8
+√
+3
+��2
+.
+(7)
+Note that the breaking of energy spectrum degeneracy caused by spin known as Zeeman
+effect. Therefore, the contributions of spin come not only from the ground state of Landau
+level, but also from the whole excited states of Landau level. The tensor condensate param-
+eter F3 and F8 are self-consistently satisfied the minimum of the thermodynamic potential,
+6
+
+which are similar to dynamical quark mass Mf. At first, one can obtain three gap equations
+for Mf (f = u, d, s)
+∂ΩTSP (Mf, F3, F8)
+∂Mf
+= 0,
+(8)
+and the other two gap equations for F3 and F8 is given as
+∂ΩTSP (Mf, F3, F8)
+∂F3
+= 0,
+∂ΩTSP (Mf, F3, F8)
+∂F8
+= 0.
+(9)
+To ensure that the thermodynamic potential in vacuum returns to zero, we define the
+normalized thermodynamic potential as effective potential
+Ωeff (T, µ, eB) = Ω (T, µ, eB) − Ω (0, 0, eB) .
+(10)
+Some of the relevant thermodynamical quantities can be evaluated by the effective po-
+tential. The quark number density is
+ρf =
+�
+l,η
+Nc |qfeB|
+4π2
+∞
+�
+-∞
+dpz
+�
+n+ − n−�
+,
+(11)
+where n± = 1/(exp [(εf,l,η ∓ µ) /T] + 1) is quark (antiquark) number distribution.
+The
+entropy density Sf = −∂Ωeff
+∂T
+is given as
+Sf = −
+�
+l,η
+Nc |qfeB|
+4π2
+∞
+�
+-∞
+dpz
+�
+ln
+�
+1 − n+�
++ ln
+�
+1 − n−�
+− εf,l,η
+T
+�
+n+ + n−�
++ µ
+T (n+ − n−)
+�
+.
+(12)
+The energy density is given as
+ε = T ∂P
+∂T +µ∂P
+∂µ − P,
+(13)
+where P is pressure. The square of sound-speed are defined as
+c2
+s = ∂P
+∂ε =
+� µ
+Sf
+∂ρf
+∂T + T
+Sf
+∂Sf
+∂T
+�-1
+.
+(14)
+7
+
+B.
+the introduction of the (2 + 1)- flavor NJL model with AMM
+The effective Lagrangian density of the (2 + 1)- flavor with AMM [48, 49] is given as
+LAMM = ¯ψ
+�
+iγµDµ + γ0µ − m+1
+2qfκσµνFµν
+�
+ψ
++ Gs
+8
+�
+a=0
+�� ¯ψλaψ
+�2 +
+� ¯ψiγ5λaψ
+�2�
+− K
+�
+det ¯ψ (1 + γ5) ψ + det ¯ψ (1 − γ5) ψ
+�
+.
+(15)
+The e���ective potential with AMM can be taken as
+ΩAMM =Gs
+�
+f=u,d,s
+�
+ψψ
+�2
+f + 4K
+�
+ψψ
+�
+u
+�
+ψψ
+�
+d
+�
+ψψ
+�
+s − Nc
+2π
+�
+f=u,d,s
+|qfB|
+∞
+�
+l=0
+�
+t=±1
+∞
+�
+−∞
+dpz
+2π
+×
+�
+Ef,l,t + T ln
+�
+1 + exp
+�−Ef,l,t − µ
+T
+��
++ T ln
+�
+1 + exp
+�−Ef,l,t + µ
+T
+���
+,
+(16)
+where
+Ef,l,t =
+�
+p2
+z +
+��
+Mf
+2 + 2|qfB|l
+�1/2 − tκfqfeB
+�2
+(17)
+is the energy spectrum under different Landau energy levels, and t = ±1 corresponds to the
+two kinds of spin direction of ¯qq pair. One can obtain three coupling gap equations for each
+order parameter as
+∂ΩAMM
+∂Mf
+= 0,
+(18)
+where f = u, d, s for the three different flavors. Thus we can obtain three dynamical quark
+masses of u, d, and s as
+Mu = mu − 4Gs
+� ¯ψψ
+�
+u + 2K
+�
+ψψ
+�
+d
+�
+ψψ
+�
+s,
+Md = md − 4Gs
+� ¯ψψ
+�
+d + 2K
+�
+ψψ
+�
+u
+�
+ψψ
+�
+s,
+Ms = ms − 4Gs
+� ¯ψψ
+�
+s + 2K
+�
+ψψ
+�
+u
+�
+ψψ
+�
+d,
+(19)
+where
+� ¯ψψ
+�
+f = NcGs
+2π
+∞
+�
+l=0
+αl|qfB|
++∞
+�
+−∞
+dpz
+2π
+Mf
+εf,l,t
+�
+1 − sκfqfB
+ˆ
+Mf,l,t
+� �
+1 −
+1
+e
+εf,l,t+µ
+T
++ 1
+−
+1
+e
+εf,l,t−µ
+T
++ 1
+�
+(20)
+corresponds to chiral condensation of different quark flavors.
+8
+
+III.
+RESULTS AND DISCUSSIONS
+To calibrate sets of parameters to applicable observable, parameters are referred [49, 53]
+to be chosen as: Λ = 631.4 MeV, mu = md = 5.6 MeV, ms = 135.7 MeV, Λ2Gs = 1.835
+and KΛ5 = 9.29.
+The empirical values are given as fπ = 93 MeV, mπ = 138 MeV,
+mK = 495.7 MeV, and mη′ = 957.5 MeV.
+The tensor channel coupling constant Gt restricted by the magnetic fields ought to be
+zero in the case of the vanished magnetic field, and equals the value of Gs when eB → ∞.
+At the following study, the value of Gt is taken as Gt = Gs/2.
+FIG. 1.
+The dependence of dynamical quark mass (M) on temperature (T) for four different
+magnetic fields ( eB = 0.05, 0.10, 0.15 and 0.20 GeV2 ) , which does not consider TSP and AMM.
+Fig 1.(a) is for µ = 0.0 GeV; and Fig 1.(b) is for µ = 0.25 GeV.
+In order to investigate the effect of AMM on the phase transition, we make comparisons
+between the two AMM sets. The compatible results obtained in [54] we define it as AMM1
+set as κu = κd = 0.38, κs = 0.25, while the defined AMM2 set chosen as κu = 0.123, κd =
+9
+
+0.6
+Ms
+eB = 0.05GeV2
+-eB = 0.10GeV2
+.. eB = 0.15GeV2
+0.5
+eB = 0.20GeV2
+(GeV)
+0.4
+M
+0.3
+0.2
+eB = 0.05GeV2
+ eB = 0.10GeV2
+Mu
+eB = 0.15GeV2
+0.1
+-eB = 0.20GeV2
+0.6
+(b)
+Ms
+eB = 0.05GeV2
+eB = 0.10GeV2
+0.5
+..eB = 0.15GeV2
+-eB = 0.20GeV2
+(GeV)
+0.4
+M
+0.3
+0.2
+eB = 0.05GeV2
+ -eB = 0.10GeV2
+Mu
+0.1
+...eB = 0.15GeV2
+---eB = 0.20GeV2
+0
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+T (GeV)0.555, κs = 0.329 fixed by [55].
+Due to the NJL model is non-renormalizable, the divergent vacuum terms merged in gap
+equation regularized by using the magnetic-field-independent regularization (MIFR) scheme
+[56, 57], which gets rid of the nonphysical part by separating the vacuum term form the
+integrals. The scheme dealing with the sums of all Landau level within the integrals by
+means of Hurwitz zeta function are presented.
+FIG. 2.
+The dependence of dynamical quark mass (M) on temperature (T) for four different
+magnetic fields ( eB = 0.05, 0.10, 0.15 and 0.20 GeV2 ) by considering TSP. Fig 2.(a) is for
+µ = 0.0 GeV; and Fig 2.(b) is for µ = 0.25 GeV.
+The dynamical mass or the quark condensate plays as an order parameter for the chiral
+phase transition. Chiral restoration happens at high temperatures and/or high chemical
+potentials. In Fig. 1(a, b), the dynamical quark masses M of u, d and s quarks without
+considering AMM and TSP are manifested as decreasing smooth functions of temperatures
+at µ = 0 and µ = 0.25 GeV, which indicates a chiral crossover. The dynamical mass M
+is apparently enhanced by increasing the magnetic field. The magnetic field is shown at
+eB = 0.05, 0.1, 0.15, and 0.2 GeV2 with µ = 0 and µ = 0.25 GeV respectively. Since we
+10
+
+(a)
+eB = 0.05GeV2
+0.6
+Ms
+eB =0.10GeV2
+•eB = 0.15GeV2
+0.5
+ieB = 0.20GeV2
+(GeV)
+0.4
+M
+0.3
+0.2
+eB = 0.05GeV2
+Mu
+ eB= 0.10GeV2
+... eB = 0.15GeV2
+0.1
+-eB = 0.20GeV2
+0
+(q)
+0.6
+Ms
+eB = 0.05GeV2
+- eB = 0.10GeV2
+. eB = 0.15GeV2
+0.5
+eB = 0.20GeV2
+(GeV)
+0.4
+M
+0.3
+0.2
+eB = 0.05GeV2
+Mu
+-eB=0.10GeV2
+. eB = 0.15GeV2
+0.1
+---eB = 0.20GeV2
+0
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+T (GeV)have considered non-vanishing current quark mass, the chiral symmetry is never restored
+fully. Since the dynamical mass is proportional to chiral condensate, it can be seen from
+Fig.1 that the larger the magnetic field is, the larger the corresponding chiral condensation
+is. This phenomenon is manifested as magnetic catalysis [19, 23, 24, 58], which accounts for
+the magnetic field has a strong tendency to enhance (or catalyze) spin-zero ¯qq condensates.
+By considering TSP of quarks, we investigate the temperature dependence of constituent
+quark mass for eB = 0.05, 0.10, 0.15 and 0.20
+GeV2 respectively shown in Fig.2(a, b).
+The dynamical mass M by considering TSP of quarks is manifested as a decreasing smooth
+function of temperatures for different magnetic fields and chemical potentials, which cor-
+responds to a chiral crossover. The dynamical mass M is apparently enhanced with the
+increase of magnetic field, It is suggested that the introduction of TSP will enhance the
+magnetic catalysis effect.
+FIG. 3. Fig.3(a, b) shows the contour plots of the F3 and F8 distributions with zero chemical
+potential in the T − eB plane, and Fig.3(c, d) shows similar plots of the F3 and F8 distributions
+but with non-zero chemical potential µ = 0.25 GeV.
+11
+
+(a) F, with u = O Gev
+(b) F。with u = 0 GeV
+0.5
+0.5
+0.4
+0.12
+0.35
+0.4
+0.4
+0.1
+0.3
+(GeV3)
+0.3
+0.08
+(GeV
+0.3
+0.25
+0.2
+eB
+eB
+0.06
+0.2
+0.2
+0.15
+0.04
+0.1
+0.1
+0.1
+0.02
+0.05
+0
+0
+0
+0
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+T (GeV)
+T (GeV)
+(c) F, with μ = 0.25 GeV
+(d) F。with u = 0.25 GeV
+0.5
+0.5
+0.14
+0.06
+0.12
+0.4
+0.4
+0.05
+0.1
+ (Gev2)
+0.04
+0.3
+0.3
+0.08
+0.03
+8
+8
+0.06
+0.2
+0.2
+0.02
+0.04
+0.1
+0.01
+0.1
+0.02
+0
+0
+0
+0
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+T (GeV)
+T (GeV)In the T − eB plane of the Fig.3, the corresponding temperature range is
+0 ≤ T ≤
+0.3 GeV, and the magnetic field range is 0 ≤ eB ≤ 0.5 GeV2. Fig.3 (a, b) displays the
+contour plots of the F3 and F8 distributions with zero chemical potential in the T − eB
+plane, and Fig.3 (c, d) shows similar plots of the F3 and F8 distributions but with non-zero
+chemical potential µ = 0.25 GeV. The (2 + 1)-flavor spin polarization is different from
+that of two flavor spin polarization because of an additional term F8 = −2Gt
+� ¯ψΣ3λ8ψ
+�
+associated with the λ8 flavor generator.
+The spin condensates affect dynamical quark masses and quark dispersion relation. It is
+found that the nonzero values of the two spin condensates F3 and F8 exist in the restored
+chiral symmetry phase with high temperature and large magnetic field, but F3 and F8 are
+almost zero in the chiral symmetry broken phase. We also noticed that F8 decreases sharply
+with the increase of chemical potential, but F3 changes slightly with the chemical potential.
+FIG. 4. The dynamical quark mass (M) as a function of temperature (T) for four different magnetic
+fields (eB = 0.05, 0.10, 0.15 and 0.20 GeV2) by considering the different sets of AMM. Fig.4(a,
+b) are for µ = 0 and µ = 0.25 GeV respectively with AMM1 set as κu = κd = 0.38, κs = 0.25.
+Fig.4(c, d) is same as Fig.4 (a, b) but for AMM2 set as κu = 0.123, κd = 0.555, κs = 0.329.
+Figure 4. displays the dependence of dynamical quark mass (M) on temperature (T)
+12
+
+0.6F (b)
+(a)
+eB = 0.05GeV2
+0.6
+Ms
+M.
+eB = 0.05 GeV2
+- eB = 0.10GeV2
+-eB = 0.10GeV2
+...eB = ..5 GeV2.
+0.5
+0.5
+-eB = 0.20GeV2
+eB = 0.15GeV2
+M 0.4
+(GeV)
+0.4
+(Gev
+M
+0.3
+0.3
+M
+Mu
+eB = 0.05GeV2
+0.2
+0.2
+eB = 0.05GeV2
+- eB = 0.10GeV2
+eB = 0.15GeV2
+- -eB = 0.10GeV2
+Mu
+0.1
+0.1
+- eB = 0.20GeV2
+:eB = 0.15GeV2
+0
+0
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.05
+0.1
+0.15
+0.2
+0
+T (GeV)
+T (GeV)
+0.6
+0.6
+(d)
+(c)
+M:
+M.
+eB = 0.05 GeV2
+eB = 0.05 GeV2
+eB = 0.10GeV2
+0.5
+-eB = 0.10GeV2
+0.5
+eB = 0.15GeV2
+eB = 0.20GeV2
+eB = 0.15GeV2
+0.4
+(GeV
+0.3
+M
+M
+0.3
+Mu
+一
+0.2
+0.2
+Mu
+eB = 0.05GeV2
+eB = 0.05GeV2
+- eB = 0.10GeV2
+-eB = 0.10GeV2
+0.1
+0.1
+-eB = 0.15GeV2
+-
+-eB = 0.15GeV2
+.eB = 0.20GeV2
+0
+0
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0
+0.05
+0.1
+0.15
+0.2
+T (GeV)
+T (GeV)for four different magnetic fields (eB = 0.05, 0.10, 0.15 and 0.20 GeV2) by considering the
+two AMM’s sets. Fig.4(a, b) are for µ = 0 GeV and µ = 0.25 GeV with AMM1 set as
+κu = κd = 0.38 and κs = 0.25. Fig.4(c, d) is same as Fig.4(a, b) but with AMM2 set as
+κu = 0.123, κd = 0.555 and κs = 0.329. Contrary to the behavior of the zero AMM in Fig.1,
+the mass-decreasing behavior of u and d quarks in the chiral restoration is not a smooth
+slope but a sudden drop, which indicates the existence of a first-order transition. However,
+the smooth slope of the dynamical mass for the crossover can be still present in the weak
+field eB = 0.05 GeV2 for the non-zero AMM. The mass-decreasing behavior of s quark in the
+chiral restoration is still a smooth slope, which suggests a chiral crossover for s quark. From
+Fig.4, it is found that the dynamical quark mass of u and d quarks have the characteristics
+of inverse magnetic catalysis in the chiral restoration phase (T ≥ TC) by using the AMM
+sets.
+FIG. 5. The critical temperature of u and d quarks as a function of the magnetic field at µ = 0
+(a) and = 0.25 GeV (b).
+In Fig. 5, the critical temperature is shown as a function of the magnetic field with the
+chemical potentials µ = 0 and 0.25 GeV respectively. It is found that the critical temperature
+decreases with the magnetic field for the AMM1 and AMM2 sets, which indicates an inverse
+magnetic catalysis which qualitatively agrees with lattice result in [33].
+On the contrary, with the TSP, TC enhances as a function of the magnetic field, which
+is the extension of the magnetic catalysis effect from vacuum to finite temperature. The
+different effects of AMM and TSP on chiral condensate can be easily understood from the
+dispersion relations in Eq. (7) and Eq. (17), the AMM reduces the LLL energy and the
+TSP lifts up the LLL energy, which causes the different effects.
+13
+
+(a)
+(b)
+0.13
+0.19
+- no AMM&TSP
+0.11
+(GeV)
+ no AMM&TSP
+.TSP
+.TSP
+AMM1
+-AMM1
+-AMM2
+- AMM2
+0.15
+0.07
+0.13
+0.05
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
+0.15
+0.2
+eB (GeV2)
+eB (GeV2)FIG. 6. The same as Fig. 5, but for the s-quark.
+The critical temperature of chiral phase transition of s quark as a function of eB is man-
+ifested in Fig.6. Compared with light quarks of u and d, the phase transition temperature
+TC of s quark with TSP increases significantly with the increase of magnetic field, which
+corresponds to the characteristics of magnetic catalysis.
+The introduction of AMM sets
+corresponds to inverse magnetic catalytic characteristics.
+Figure 7 displays the dependencies of the entropy density of u , d and s quarks on
+temperature at zero chemical potential. It can be noted that the introduction of the AMM
+makes the crossover phase transition sharp.
+It is worth noting that the AMM in Fig.7
+corresponds to three different settings, which are AMM0, AMM1 and AMM2, respectively.
+AMM0 means that the AMM is not considered, that is, all κ values in Eq. (17) are set
+to zero. AMM1 and AMM2 sets have been mentioned above. When eB = 0.05 GeV2, the
+magnetic field is not big enough to excite the effect on entropy. When eB = 0.2 GeV2, some
+of the effects of the magnetic field on entropy for different AMM sets and TSP can be excited.
+It is found that the entropy shows a sharp change near the phase transition temperature
+after adding AMM sets, and this sharp change is more obvious with the magnetic field
+increases and chemical potential, showing a first-order phase characteristic. The change of
+entropy with temperature near the phase transition temperature is relatively smooth after
+adding TSP, and it behaves like the crossover transition.
+14
+
+0.34
+0.34 F
+(b)
+0.3
+- no AMM&TSP
+ no AMM&TSP
+0.3
+(GeV)
+(GeV)
+.TSP
+. TSP
+AMM1
+0.26
+-AMM1
+c
+C
+AMM2
+-AMM2
+0.26
+0.22
+0.22
+0.18
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
+0.15
+0.2
+eB (GeV2)
+eB (GeV2)FIG. 7. The dependence of S/T 3 on temperature T at µ = 0GeV with different magnetic field.
+Fig.7 (a) is for eB = 0.05 GeV2 and Fig.7 (b) is for eB = 0.2 GeV2.
+The dependence of square of sound-velocity c2
+s on temperature T is manifested in Fig.8.
+Fig.8(a) and Fig.8(b) are for zero chemical potential µ = 0 and µ = 0.25 GeV respectively.
+The square of sound-velocity shows a sudden rapid rise inflection near the phase transition
+after adding AMM sets, and this rapid rise is more obvious with the magnetic field increases,
+showing a obviously first-order phase characteristic.
+On the other hands, the change of
+square of sound-velocity with temperature near the phase transition is relatively smooth
+inflection after adding TSP, showing a obviously cross-over transition characteristic. The
+result obtained by using the square of sound velocity is completely consistent with the result
+of entropy analysis.
+Compared with u and d quarks, the square of sound-velocity of s quark with temperature
+is relatively smooth inflection after adding TSP and AMM sets. It is proposed that s quarks
+have always maintained obvious cross-over characteristics. In the high-temperature region,
+the square of sound-velocity c2
+s increases with temperature and obtains the saturation value
+15
+
+16
+(a)
+12
+S
+8
+no AMM&TSP, eB = 0.05GeV2
+" AMM1,eB = 0.05GeV2
+4
+AMM2,eB = 0.05GeV2
+--TSP,eB = 0.05GeV2
+Stefan-Boltzmann limit
+0
+(b)
+16
+12
+2
+S
+8
+ no AMM&TSP,eB = 0.20GeV2
+. AMM1, eB = 0.20GeV2
+4
+-AMM2,eB = 0.20GeV2
+--TSP,eB = 0.20GeV2
+Stefan-Boltzmann limit
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+T (GeV)c2
+s = 1/3 to satisfy the relativistic requirement. This suggests that the equation of state
+in the chiral restoration phase at high temperatures is close to the Stefan-Boltzmann limit
+ε = 3P.
+FIG. 8. The sound-velocity square C2
+s of u and d with s quarks as a function of the temperature
+T with different chemical potential. Fig.8 (a, b) is for u and d quarks with zero chemical potential
+µ = 0, and µ = 0.25 GeV, and Fig.8 (c, d) is for s quarks
+IV.
+SUMMARY AND CONCLUSIONS
+In this work, we thoroughly study the effect from TSP and AMM on the vacuum, phase
+transition and thermal magnetized QCD in the (2 + 1)-flavor Nambu-Jona-Lasinio (NJL)
+model with nonzero current quark masses at finite temperature and chemical potential. An
+unified physical mechanism to illustrate the novel consequences from recent lattice QCD as
+magnetic catalysis and inverse magnetic catalysis effect proposed in the paper.
+We focus on two topics: the AMM and TSP. For these two topics, we should pay special
+attention to the dispersion relation, especially the lowest Landau level, which determines
+16
+
+(a) u and d quarks with u = O GeV
+(b) u and d quarks with u = 0.25 GeV
+0.5
+0.5
+0.4
+0.4
+0.3
+0.3
+2s
+TSP, eB = 0.05GeV2
+0.2
+0.2
+TSP,eB = 0.05 GeV2
+- AMM1, eB = 0.05GeV2
+AMM1, eB = 0.05GeV2
+AMM2, eB = 0.05GeV2
+- AMM2, eB = 0.05GeV2
+0.1
+—TSP,eB = 0.20GeV2
+0.1
++ -AMM1,eB = 0.20GeV2
++- AMM1,eB = 0.20GeV2
+- AMM2, eB = 0.20GeV2
++-- AMM2, eB = 0.20GeV2
+0:
+01
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0
+0.05
+0.1
+0.15
+0.2
+T (GeV)
+T (GeV)
+(c) s quarks with u = O GeV
+(d) s quarks with u = 0.25 GeV
+0.4
+0.4
+TSP,eB = 0.05GeV2
+TSP,eB = 0.05GeV2
+- AMM1,eB = 0.05GeV2
+- AMM1,eB = 0.05GeV2
+ AMM2, eB = 0.05GeV2
+- AMM2,eB = 0.05GeV2
+0.3
+0.3
+TSP, eB = 0.20GeV2
+-TSP,eB = 0.15GeV2
++ AMM1,eB = 0.15GeV2
+- AMM1,eB = 0.20GeV2
+2s
+2s
+- AMM2,eB = 0.20GeV2
++- AMM2, eB = 0.15GeV2
+0.2
+0.2
+0.1
+0.1
++0
+0+
+0.05
+0.1
+0.15
+0.2
+0
+0.25
+0
+0.05
+0.1
+0.15
+0.2
+T (GeV)
+T (GeV)the properties of the magnetized quark matter system.
+The TSP lifts up the LLL en-
+ergy: ELLL =
+�
+p2
+z + (M + F3 + F8
+√
+3)2� 1
+2, while the AMM effect diminishes the LLL energy:
+ELLL =
+�
+p2
+z + (M − κ |qf| B)2�1/2 therefore, the TSP and the AMM take almost opposite
+effects on magnetized quark matter. When the AMM and TSP contributions are not con-
+sidered, the corresponding phase transition temperature increases with the magnetic field,
+showing the characteristics of magnetic catalysis. When considering only the contribution of
+TSP, the phase transition temperature also increases with the magnetic field, showing the
+characteristics of magnetic catalysis. On the other hand, when AMM are introduced, the
+phase transition temperature decreases with the magnetic field, showing the characteristics
+of inverse magnetic catalysis.
+It is found that the square of sound-velocity shows a sudden rapid rise inflection near
+the phase transition after adding AMM sets, and this rapid rise is more obvious with the
+magnetic field increases, showing a obviously first-order phase characteristic. On the other
+hands, after adding TSP, the change of square of sound-velocity with temperature near the
+phase transition is relatively smooth inflection, showing a obviously cross-over transition
+characteristic.
+The result obtained by using the square of sound velocity is completely
+consistent with the result of entropy analysis.
+The (2 + 1)-flavor spin polarization is different from that of two flavor because of an ad-
+ditional F8 = −2Gt
+� ¯ψΣ3λ8ψ
+�
+associated with the λ8 flavor generator. The spin condensates
+affect the dynamical quark masses, chiral phase transition,and quark dispersion relation. It
+is found that the nonzero values of the two spin condensates F3 and F8 exist in the restored
+chiral symmetry phase with high temperature and large magnetic field, but F3 and F8 are
+almost zero in the chiral symmetry broken phase.
+ACKNOWLEDGMENTS
+This work was supported by National Natural Science Foundation of China (Grants No.
+11875178, No. 11475068, No. 11747115).
+17
+
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+

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+An Empirical Investigation into the Reproduction of
+Bug Reports for Android Apps
+Jack Johnson∗, Junayed Mahmud†, Tyler Wendland∗, Kevin Moran†, Julia Rubin‡, Mattia Fazzini∗
+∗University of Minnesota, MN, USA; [email protected], [email protected], [email protected]
+†George Mason University, VA, USA; [email protected], [email protected]
+‡University of British Columbia, BC, Canada; [email protected]
+Abstract—One of the key tasks related to ensuring mobile
+app quality is the reporting, management, and resolution of
+bug reports. As such, researchers have committed considerable
+resources toward automating various tasks of the bug man-
+agement process for mobile apps, such as reproduction and
+triaging. However, the success of these automated approaches is
+largely dictated by the characteristics and properties of the bug
+reports they operate upon. As such, understanding mobile app
+bug reports is imperative to drive the continued advancement
+of report management techniques. While prior studies have
+examined high-level statistics of large sets of reports, we currently
+lack an in-depth investigation of how the information typically
+reported in mobile app issue trackers relates to the specific details
+generally required to reproduce the underlying failures.
+In this paper, we perform an in-depth analysis of 180 re-
+producible bug reports systematically mined from Android apps
+on GitHub and investigate how the information contained in
+the reports relates to the task of reproducing the described
+bugs. In our analysis, we focus on three pieces of information:
+the environment needed to reproduce the bug report, the steps
+to reproduce (S2Rs), and the observed behavior. Focusing on
+this information, we characterize failure types, identify the
+modality used to report the information, and characterize the
+quality of the information within the reports. We find that bugs
+are reported in a multi-modal fashion, the environment is not
+always provided, and S2Rs often contain missing or non-specific
+enough information. These findings carry with them important
+implications on automated bug reproduction techniques as well as
+automated bug report management approaches more generally.
+I. INTRODUCTION
+The importance of the quality of mobile applications (collo-
+quially referred to as apps) has grown in recent years as smart-
+phones and tablets have become deeply integrated into users’
+daily lives. Once an application has been released to users, its
+quality is largely ensured by continuing maintenance activities,
+which have been shown to consume considerable amounts of
+engineering effort [1]. These important maintenance activities
+are typically centered around bug report management and
+include activities related to understanding, reproducing, and
+resolving bug reports.
+A number of unique development constraints related to
+mobile apps, such as pressure for frequent releases [2], [3],
+the need to cope with constantly evolving platform APIs [4],
+[5], a large volume of user feedback [6], [7], [8], [9],
+[10], and testing challenges [11] complicate the bug report
+management process. Software engineering researchers have
+recognized these domain-specific challenges and have worked
+toward providing automated solutions across several bug report
+management activities for mobile apps, including bug report
+quality assessment [12], reproduction [13], [14], triaging [15],
+and bug localization [16], [17].
+One common thread among these various automated solu-
+tions is that they operate directly upon the information con-
+tained within bug reports and, as such, are directly affected by
+the characteristics and quality of various report components,
+such as environmental information (e.g., device, software ver-
+sion), reproduction steps (S2Rs), and observed behavior (OB).
+Thus, researchers and practitioners require a solid empirical
+foundation that delineates common characteristics of mobile
+app bug reports to build effective automated techniques.
+In prior work, researchers have examined high-level statis-
+tics (e.g., number and type of report, fix rates, fix time)
+of large sets of bug reports. For example, Battacharya et
+al. [18] performed an empirical study on bugs submitted to
+the Android platform on 24 widely-used open source apps.
+Others have compared high-level bug characteristics between
+mobile apps and desktop apps [19]. However, to the best
+of our knowledge, no study has yet provided an in-depth
+characterization of how the information contained in mobile
+bug reports might impact the task of bug reproduction. One
+likely reason that past studies have not examined this relation
+is that as it requires manually reproducing real bug reports,
+which is a time-consuming and difficult task. Despite the
+difficulty of this analysis, understanding this information is
+critical as both developers and automated bug analysis tech-
+niques may need to (i) understand the type of reported failure,
+(ii) understand multiple modalities of information, such as
+text, images, or screen-recordings, and (iii) identify or infer
+information that is either vague or missing from the reports.
+In short, empirically analyzing both the characteristics and
+quality of the information reported in mobile app bugs is
+critical for both the practical and scientific advancement bug
+report management for mobile apps.
+In this paper, we conduct and in-depth characterization of
+reproducible bug reports for Android apps. To this end, we
+significantly extend ANDROR2 [20] – a dataset of reproducible
+bug reports for Android apps which contains bugs representing
+a range of failure types. We augmented the dataset with addi-
+tional, manually verified and fully reproduced bug reports from
+open source Android apps hosted on GitHub [21] and available
+on the Google Play store [22], obtaining a dataset of 180 bug
+reports. In this work, we focus on bug reports for Android
+arXiv:2301.01235v1  [cs.SE]  3 Jan 2023
+
+apps as Android is the most widely used operating system for
+mobile apps [23]. To the best of our knowledge, ours is the
+largest dataset of (i) fully reproduced bug reports for Android
+apps, which (ii) contains both user-submitted and developer-
+submitted reports, and (iii) in contrast to related work, focuses
+on different types of failures beyond app crashes. Given this
+dataset, we focused our in-depth analysis on three sources
+of information: the description of the environment needed
+to reproduce the bug report, the steps to reproduce, and the
+observed behavior.
+Leveraging the fact that our studied reports are considered
+fully reproducible, we perform an in-depth analysis of both the
+report characteristics—including the failure types and modal-
+ities of reported information—and the quality of reported
+information. In relation to the quality of reported information,
+we focus on three aspects: the types and prevalence of missing
+information, whether report discussion threads contain helpful
+information for reproducing the reports, and the specificity
+of reported information (which investigates whether reported
+information can be directly used for reproducing the reports).
+Although these aspects are only some of ones that describe the
+quality of reported information, we believe that the analysis
+of these aspects provides useful insights into the reproduction
+of bug reports and hence focus on them.
+Our analysis shows that (i) reported failures can be grouped
+into four types, three of which are not yet considered by
+existing automated reproduction techniques, (ii) different in-
+formation modalities are used to report the details related to
+the environment, steps to reproduce, and observed behavior,
+(iii) a large number of reports (74%) have at least one step
+to reproduce that requires multiple operations in the app
+indicating that the information provided for the step is not
+always specific enough, (iv) the great majority of reports
+(92%) have at least one missing reproduction step, illustrating
+that the operations required to reproduce the reports must
+often be inferred, and (v) bug report discussions can, in some
+cases (19%), provide additional information useful for the
+reproduction of the reports. Finally, we discuss implications
+of our findings, which can help guide future research on
+automated reproduction of bug reports and, more generally,
+bug report management activities.
+In summary, the main contributions of this paper are:
+• A large set of 180 manually mined and reproduced
+bug reports for Android apps that contains user- and
+developer-submitted bug reports of multiple failure types.
+• A study that examines bug characteristics and information
+quality in reproducible mobile app bug reports. This
+advances upon prior studies which do not manually verify
+and collect reproducible bug reports.
+• A discussion on the implications of our findings, which
+illustrates the need for future research on non-crashing
+oracles, multi-modal understanding of report information,
+mocking environments, and missing and non-specific
+reproduction steps.
+• A replication package [24] that contains our dataset of
+bug reports, data analysis reports, and scripts to perform
+Bug Report
+Title:
+Bug: Long pressing the amount input brings up QWERTY keyboard
+Content:
+Software specifications:
+• GnuCash Android version: 2.2.0
+• System Android version: 6.0
+• Device type: Motorola Moto G (2nd Generation)
+Steps to reproduce the behaviour:
+1. Navigate to Transactions screen
+2. Tap the Add button
+3. Enter Description (optional)
+4. Focus the Amount input
+5. Long press to bring up the context menu
+Expected behaviour:
+See the context menu
+Actual behaviour:
+Fig. 1: Bug report for the GNUCASH app.
+the study analyses, which can facilitate future replications
+and extensions of this work.
+II. BACKGROUND AND TERMINOLOGY
+Given a bug report that describes a failure in an app, we
+use the term reporter to identify the person submitting the
+bug report. A reporter can be either a user or a developer. In
+this study, we consider a person who never contributed to the
+source code of an app to be a user and all other reporters to
+be developers.
+We conceptually group the information contained in a bug
+report into multiple parts, each of which detail a particular
+aspect of the report. The parts and aspects of interest in this
+study are the ones providing details on how to reproduce the
+failure described in a report. These aspects are: the environ-
+ment, the steps to reproduce (S2Rs), and the observed behavior
+(OB). The environment includes information on the software
+and hardware necessary to reproduce the failure described in
+a report. This part can contain information such as the app
+version, the operating system (OS) version, and the device
+where the failure occurred. The S2Rs provide details on the
+operations that should be performed on a device in order
+to reproduce the failure. We use the terms GUI action (or
+simply action) and GUI interaction (or simply interaction)
+interchangeably to indicate the operations performed on the
+GUI of a device. An S2R (which are the unit of information
+composing the S2Rs) can be mapped to one or more GUI
+actions. The OB describes the failure and can be used to check
+that the failure was successfully reproduced. In practice, the
+information from these conceptual parts can be interleaved
+across the paragraphs and sections of a bug report. Bug
+reports can also have a discussion thread. A discussion thread
+contains discussion messages and these messages can provide
+2
+
+12:22
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+New transaction
+SAVE
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+口additional information on the environment, the S2Rs, and the
+OB associated with the report.
+Figure 1 provides an example of a user-submitted bug re-
+port [25]. This bug report is taken from the report management
+system of GNUCASH, an app for finance tracking, and is
+slightly modified for presentation purposes. The bug report
+contains information related to the environment, the S2Rs, and
+the OB, which are located in the Software specifications, Steps
+to reproduce the behaviour, and Actual behaviour sections of
+the report, respectively.
+To exercise the bug, the user navigated to the transactions
+screen, started adding a new transaction, and long-clicked on
+the GUI element representing the amount of the transaction.
+The failure manifests as a wrong screen being displayed to the
+user: screen with a keyboard view instead of the context menu.
+The OB describing the failure is reported using text (in the
+title) and using an image (in the Actual behaviour section). We
+refer to the way in which a piece of information is reported as
+the reporting modality (or modality in short) and reporters can
+provide the same information multiple times using different
+modalities. Because the user did not reach the desired screen,
+we identify this failure as a navigation failure. We use the
+terms failure type and failure category interchangeably to refer
+to the categorization of the failure.
+The report has five S2Rs (numbered items under the Steps
+to reproduce the behaviour section) and 13 GUI actions are
+necessary to reproduce the failure. An example of GUI action
+is performing a click on the add button in the transaction
+screen of the app as indicated by 2. Tap the Add button. An
+S2R can map to one or more GUI actions. In this example,
+the first S2R (1. Navigate to Transactions screen) maps to
+three GUI actions. We refer to S2Rs that map to multiple
+GUI actions as non-specific S2Rs. Of the remaining four S2Rs,
+three map to one GUI action and one S2R is optional (3. Enter
+Description (optional).) This optional S2R is not included in
+13 GUI actions necessary to reproduce the failure. Seven (13-
+3-3) of the GUI actions in this example are not described by
+any of the S2Rs. We refer to such GUI actions as unmapped
+GUI actions and say that they correspond to missing S2Rs.
+We refer to the remaining actions as mapped GUI actions.
+If an unmapped GUI action occurs before the first mapped
+GUI action, we call the missing S2R that corresponds to
+the unmapped action a missing context S2R, indicating that
+some contextual information is missing from the bug report.
+Otherwise, if a missing S2R is associated with a GUI action
+occurring after the first mapped GUI action, we refer to the
+S2R as a missing inline S2R.
+III. METHODOLOGY
+To characterize reproducible bug reports, inform research on
+automated bug reproduction, and, more generally, provide in-
+sights for research on bug report management, we formulated
+and answered the following research questions (RQs):
+• RQ1: What are the failure types associated with
+reproducible bug reports? In this RQ, we analyzed
+and categorized failures associated with reproducible bug
+reports. With the findings from this RQ we aim to inform
+research on automatic failure recognition.
+• RQ2: What information modalities are used to report
+the information contained in reproducible bug re-
+ports? This RQ categorizes the modalities used to report
+environment, S2Rs, and OB information. The findings
+from this RQ aim to inform research in bug triaging,
+report reproduction, and report quality assessment.
+• RQ3: Do reproducible bug reports have missing in-
+formation? We answer this question by analyzing the
+information contained in reproducible bug reports w.r.t.
+operations required to reproduce the failures described in
+the reports. This RQ aims to direct efforts on research
+for identifying and inferring missing information in bug
+reports, necessary for bug report reproduction.
+• RQ4: Do discussion threads of reproducible bug re-
+ports contain helpful information for reproducing the
+reports? In this RQ, we analyzed the information gain
+obtained by interpreting the bug report discussions. This
+RQ aims to evaluate the need for approaches that combine
+content from bug reports and their discussions.
+• RQ5: How specific is the information reported in
+reproducible bug reports? In this RQ, we investigated
+whether the information contained in reproducible bug
+reports can be directly mapped onto the operations need
+to reproduce the reports. This RQ aims to provide insights
+on how to leverage the information in bug reports for
+reproducing the failures.
+Figure 2 provides a high-level outline of the methodology
+we used to answer the RQs. In a nutshell, we first assembled
+a dataset of reproducible bug reports and then analyzed the
+characteristics of the bug reports through qualitative and
+quantitative analyses. We describe these steps in detail next.
+A. Dataset Creation
+The Dataset Creation component of Figure 2 provides an
+overview of our data collection workflow, which consisted of
+two phases: bug reports filtering and failure reproduction.
+1) Bug Reports Filtering: The objective of this phase was
+to identify a set of bug reports that we could try to reproduce
+and ultimately include in our dataset. In this study, we are
+interested in both user-submitted and developer-submitted bug
+reports that are reproducible and describe different types of
+failures. To the best of our knowledge, ANDROR2 [20] is
+the largest dataset of reproducible bug reports for Android
+apps that does not exclusively focus on crashes. This dataset
+contains 90 user-submitted bug reports, which are associated
+with apps available on the Google Play store [22] and hosted
+on GitHub [21]. The 90 bug reports are GitHub issues [26]
+and are associated with reproduction scripts created by the
+ANDROR2’s authors. This set of 90 bug reports was extracted
+from a larger set of 6,365 issues that was systematically
+mined from GitHub. The set of 6,365 issues contains issues
+that: (i) are part of repositories that use Java, (ii) have
+the label “bug”, (iii) are in repositories that contain an
+AndroidManifest.xml file (as Android apps require this
+3
+
+Bug Reports
+Filtering
+AndroR2
+Filtered
+Bug Reports
+Failure
+Reproduction
+Reproduced
+Bug Reports
+RQ1: Failure Type
+RQ2: Reporting Modality
+RQ3: Missing Information 
+Dataset Creation
+Bug Reports Analysis
+RQ4: Discussion Information
+RQ5: Information Specificity
+Reproduction
+Scripts
+Bug Reports
+Preparation
+Annotated
+Bug Reports
+E,OB,S2Rs
+Fig. 2: Overview on the methodology used in the study.
+file to properly compile [27]), (iv) contain the word “step”
+in them, and (v) are associated with apps also available on the
+Google Play store.
+Because we are also interested in developer-submitted bug
+reports, we started from the set of 6,365 GitHub issues pro-
+vided by ANDROR2 and identified 90 reproducible, developer-
+submitted bug reports (to match the number of already avail-
+able user-submitted bug reports). To identify the 90 developer-
+submitted bug reports, we used a methodology similar to that
+of ANDROR2. Specifically, we first refined the set of 6,365
+issues to only contain those created by GitHub users that
+had contributed to the repositories associated with the issues,
+resulting in 2,523 issues. Second, we selected issues that were
+closed at the time the issues were mined (November 2020) so
+that we could more easily identify whether the issues were also
+originally reproduced by the developers. This filtering resulted
+in 2,045 reports. Third, after analyzing the set of issues, we
+found that some repositories had a much larger number of
+issues compared to others. To avoid overfitting the bug report
+dataset to a specific app, we considered at most ten issues
+per repository. When a repository had more than ten issues,
+we randomly selected ten from this set resulting in 645 bug
+reports for 164 apps.
+2) Failure Reproduction Phase: In the second phase of our
+dataset creation process, we randomly selected bug reports
+from the set of 645 developer-submitted bug reports until we
+reproduced 90 of them. In this process, we disregarded trivially
+reproducible bug reports, i.e., those we could reproduce by
+simply opening the app.
+Two authors tried to reproduce the failures described in the
+bug reports. To reproduce a failure, the authors followed the
+S2Rs contained in the bug report by mapping the steps to GUI
+actions on the screen of the device running the app associated
+with the report. If a report had missing S2Rs, the authors
+manually explored the functionality of the app to identify the
+minimal sequence of GUI actions that would account for those
+missing steps, using a trial-and-error approach. When a bug
+report could be successfully reproduced by one of the two
+authors, the other author also tried to reproduced the same
+report to ensure that the reproduced failure was the same as
+the one described in the report. For all 90 bug reports, the
+authors also encoded the GUI actions in reproduction scripts
+using the UIAutomator framework [28].
+To validate whether user-submitted bug reports were still
+reproducible, we ran the scripts associated with these reports
+in the ANDROR2 dataset. Four reports were not reproducible
+as the servers associated with the apps were no longer running.
+To replace these bug reports, we identified and reproduced four
+additional user-submitted reports from the set of 6,365 GitHub
+issues provided by ANDROR2. At the end of this process,
+we obtained a set of 90 user-submitted and 90 developer-
+submitted reproducible bug reports, which we considered for
+the rest of the study.
+B. Bug Reports Analysis
+In this section, we present the analyses we performed to
+characterize aspects related to the reproducibility of Android
+bug reports. The Bug Reports Analysis Creation part of
+Figure 2 provides a summary of the analyses we performed.
+The analyses were driven by two of the paper’s authors and
+were performed one at a time to reduce cognitive load.
+1) Bug Reports Preparation: Before performing the analy-
+ses associated with the RQs, we annotated the information
+contained in the bug reports and their discussion threads,
+to identify the portions of each report that provide infor-
+mation about the environment, S2Rs, and OB. This step
+was performed by the two authors together and in multiple
+sessions; the authors associated each sentence in the report’s
+textual description, as well as each link, image, recording,
+and execution logs, with it designated purpose: to describe
+environment, S2Rs, and OB. Some elements received multiple
+annotations, e.g., a sentence can provide both S2Rs and OB.
+2) Analysis for RQ1 (What are the failure types associated
+with reproducible bug reports?): To answer RQ1, we per-
+formed a qualitative analysis that combines inductive and axial
+coding [29], [30]. Inductive coding is a systematic approach
+for categorizing data by manually coding (i.e., labeling) the
+data. Axial coding relates codes to one another and finds
+higher-level codes that represent abstractions of the original
+codes. In our analysis, a code is a label that categorizes the
+type of a failure and we assigned the code to the bug report
+describing the failure.
+The analysis was performed by two raters, who analyzed
+the description of the failure in the bug report and used the
+reproduction scripts to observe how the failure manifested.
+The analysis was divided into two parts. In the first part, the
+two raters analyzed a sample of the bug reports to define
+the analysis codebook – a document detailing the rules for
+assigning a specific code to a failure. For each code, the set
+4
+
+D</Vof rules specified the characteristics required for assigning a
+code to a failure.
+This part of the analysis was performed in six iterations. In
+each iteration, the raters independently analyzed 18 bug reports
+(10% of the report considered in the study). The set contained
+the same bug reports for both raters and was selected randomly
+from the set of not-yet-analyzed bug reports. At the end of each
+iteration, the raters used negotiated agreement [31] to resolve
+inconsistencies among created and assigned codes, and to in-
+sure the reliability of the coding process. We used this method
+due to it is advantages in research like ours, where generating
+new insights is the primary concern [32]. Because we used
+negotiated agreement, measures such as inter-rater agreement
+are not applicable in our context. To resolve disagreements,
+the raters reproduced the failures together and then decided
+on the final classification. For example, for one of the reports
+considered in the study [33], one of the raters categorized the
+failure as a crash and the other rater categorized the failure as
+a navigation issue. When the two raters met, they discussed
+the disagreement and decided to classify the failure as a crash
+because the app displayed an exception before bringing the
+user back to a different screen.
+At the sixth iteration, the raters did not create new codes and
+had assigned the same codes to all reports. From that point, the
+raters split the remaining 72 bug reports equally and coded the
+bug reports independently. At the end of the coding process,
+the raters also performed axial coding. This step led to four
+main categories of failures, which we present in Section IV.
+3) Analysis for RQ2 (What information modalities are used
+to report the details contained in reproducible bug reports?):
+The analysis to answer RQ2 was also based on inductive and
+axial coding. Two raters analyzed the environment, S2Rs, and
+OB information annotated during the bug reports preparation
+step. The raters created the analysis codebook in two iterations,
+analyzing in each iteration a sample of 18 bug reports (10%
+of all bug reports). The raters used negotiated agreement to
+address the reliability of the coding process. After finalizing
+the codebook, the authors split the remaining 144 bug reports
+equally and coded them independently.
+The raters performed axial coding at the end of the coding
+process. This process led to six main reporting modalities,
+detailed in Section IV.
+4) Analysis for RQ3 (Do reproducible bug reports have
+missing information?): To answer RQ3, we performed two
+types of analysis. First, we leveraged the annotations created
+in the bug reports preparation step to identify whether environ-
+ment, S2Rs, and OB information was completely missing from
+the reports. Second, when the S2Rs information was provided,
+we performed an in-depth analysis of S2Rs. Specifically, for
+each bug report, we compared the S2Rs information from
+the bug report with the GUI actions in our reproduction
+scripts, in order to identify missing S2Rs. Once we identified
+missing S2Rs, we categorized them into missing context S2Rs
+and missing inline S2Rs (see definitions in Section II). Two
+authors analyzed each bug report independently and then met
+to discuss and finalize the classification.
+5) Analysis for RQ4 (Do discussion threads of reproducible
+bug reports contain helpful information for reproducing the
+reports?): In RQ4, two authors manually analyzed the mes-
+sages in the bug report discussions, to identify whether they
+added information relevant to understanding and reproducing
+the bug reports. The authors leveraged the annotations from the
+bug reports preparation step to focus on messages providing
+environment, S2Rs, and OB information. The authors analyzed
+each bug report independently and labeled with the word
+additional the data from discussion messages that provided
+additional information. The two authors met and discussed
+the final classification also in this case.
+6) Analysis for RQ5 (How specific is the information re-
+ported in reproducible bug reports?): To answer RQ5, we
+analyzed whether the information provided in the bug reports
+could be directly used for reproducing the bug reports. For the
+environment-related information, two authors checked whether
+the provided information was sufficient to define the environ-
+ment where to reproduce the failure. If no additional infor-
+mation was needed, we considered the provided information
+to be of specific (and non-specific otherwise). For S2Rs, two
+authors mapped each of the S2Rs defined in a bug report
+to corresponding GUI actions from the reproduction script.
+If an S2R mapped to multiple GUI actions, we labeled that
+S2R as a non-specific S2R. We considered the other S2Rs
+to be specific. For the OB information, the authors checked
+whether the information was sufficient to verify the failure. If
+no additional information was needed (i.e., no need to check
+discussion messages), we considered the provided information
+to be specific (and non-specific otherwise).
+IV. RESULTS
+In this section, we present the results of our study on ana-
+lyzing and characterizing reproducible Android bug reports.
+A. RQ1: What are the failure types associated with repro-
+ducible bug reports?
+Our analysis identified four failures types: output, cosmetic,
+navigation, and crash. Output failures reveal issues in the
+output provided by the app. Cosmetic failures identify issues
+in the app that do not affect the functionality of the app.
+Navigation failures display the wrong screen to the user.
+Crashes abruptly terminate the execution of the app. Across the
+bug reports considered, we identify 33% of reports reporting
+output failures, 31% reporting cosmetic failures, 8% reporting
+navigation failures, and 28% reporting crashes. This finding is
+notable, as many current bug report analysis techniques focus
+solely on crashes. We discuss the implications of these findings
+further in Section V.
+This distribution reveals a comparable amount of failures
+between the output, cosmetic, and crash categories and a sig-
+nificantly lower number of navigation failures. The distribution
+is similar across both developer- and user-submitted bug re-
+ports. Specifically, among the user-submitted bug reports, there
+are 33% output failures, 31% cosmetic failures, 7% navigation
+failures, and 28% crashes. Among developer-submitted bug
+5
+
+(a) Example of output failure on the left and fix on the right.
+(b) Example of cosmetic failure on the left and fix on the right.
+(c) Example of navigation failure on the left and fix on the right.
+(d) Example of crash failure on the left and fix on the right.
+Fig. 3: Screenshot examples for the four failure types identified in the bug reports considered.
+reports, there are 32% output failures, 29% cosmetic failures,
+9% navigation failures, and 29% crashes.
+Our analysis categorized the 60 output failures into two
+subcategories: incorrect output (32) and missing output (28).
+Incorrect output identifies failures in which some computation
+of the app is displayed incorrectly or improperly saved to a
+file, and missing output describes failures where the result of
+some computation is not displayed or saved to a file. A vast
+majority of these cases affect the GUI of the app (56 cases)
+whereas a smaller number impact generated files (4 cases).
+The screenshot on the left of Figure 3a shows an example of
+a failure under the incorrect output subcategory. The example
+is taken from a bug report [34] of OMNI NOTES, a note-taking
+app. The app has a failure as it does not display the right values
+for the tags associated with the notes in the app.
+As part of our analysis, we further classified the 55 cosmetic
+failures into eight subcategories: incorrect color (10), incorrect
+cursor placement (3), content cut (3), image rendering issue
+(4), missing GUI element (9), incorrect orientation (2), incor-
+rect placement (4), and incorrect text (18). We provide details
+for each of these subcategories in our online appendix [24].
+The screenshot on the left of Figure 3b illustrates an example
+of a cosmetic failure from the incorrect placement subcategory.
+This example is taken from a report [35] submitted for
+FIREFOX FOCUS, a browser app. In this example, the text
+Show home screen tips has additional padding w.r.t other
+text elements (e.g., About Firefox Focus) on the screen.
+Our analysis of the navigation failures did not produce any
+further subcategories. The screenshot on the left of Figure 3c
+reports an example of a navigation failure. This failure was
+reported [36] for K-9 MAIL, an email client app. In this
+example, the user started setting up a new email account, went
+into the manual configuration settings, and, upon pressing the
+back button, the user was brought out of the app instead of
+the previous app screen. The screenshot in the right part of
+Figure 3c illustrates the correct app behavior where the user
+navigates to the sign-up screen after pressing the back button.
+For the 50 failures leading to a crash, we identified two
+main subcategories, immediate crash (46) and app freeze (4).
+Immediate crash identifies failures in which the app crashes
+as soon an operation is performed in the app. App freeze
+includes failures in which the app first becomes unresponsive
+after an operation is performed in the app, and then the crash
+appears after a certain amount of time. The screenshot in the
+left portion of Figure 3d reports an example of an immediate
+crash failure reported [37] for FAMILY FINANCE, a household
+finance app. The right part of the Figure 3d reports the screen
+of the app after the bug in the app was fixed.
+RQ1 answer: Our categorization identified four failure
+types: output (33%), cosmetic (31%), navigation (8%), and
+crash (28%). We also identified subcategories for output (2),
+cosmetic (8), and crash (2). Finally, the failure distribution
+does not differ dramatically when user- and developer-
+submitted reports are considered individually.
+B. RQ2: What information modalities are used to report the
+details contained in reproducible bug reports?
+In our analysis of RQ2, we identified six modalities used
+to report bug information: text, annotated text, image, anno-
+tated image, recording, and log. Text identifies information
+reported in plain text. Annotated text is a sentence containing
+text within quotes or text with casing or capitalization [38],
+which represent either app inputs or GUI elements. Image
+identifies device screenshots. Annotated image is associated
+with device screenshots that have been edited to highlight parts
+of their content. Recording refers to any animated image or
+video providing a recording of the device screen. Finally, log
+identifies reporter-provided stack traces extracted from either
+app or system logs. Figure 4 reports the distribution of the
+modalities, for reports as a whole (Figure 4-a), the environment
+(Figure 4-b), S2Rs (Figure 4-c), and OB (Figure 4-d).
+As expected, text is the most commonly used modality,
+with all 180 bug reports using text to convey some piece
+of information. Annotated text is the second most recurring
+modality and appeared in 100 bug reports. In our analysis,
+we also further categorized the annotated text modality into
+annotated GUI text and annotated input text. Annotated GUI
+text identifies bug reports in which the reporter used text within
+quotes or latter casing to identify an element in the GUI of the
+6
+
+11:26
+Reports
+NSES/INCOMES
+EXPENSES BY ARTICLES
+INCOMES BY AR
+No chart data available3:17
+★
+Test
+ab
++
+Add reminder
+Created: moments ago
+Updated: moments ago3:10
+test
+#testa #testb
+Add reminder
+Created: 4 hours ago
+Updated: 3 hours ago12:45
+Mozilla
+Show home screen tips
+About Firefox Focus
+Help
+Your Rights
+Privacy Notice12:51
+Mozilla
+Show home screen tips
+About Firefox Focus
+Help
+Your Rights
+Privacy Notice5:08
+Q Search apps
++
+Calculator
+Calendar
+Camera
+Clock
+Contacts
+@
+Custom L.
+Dev Tools
+Email
+Family Fi..
+Files
+Gallery
+K-9 Mail
+Messagi..
+Music
+Omni Not..
+Phimp.me
+Phone
+Search
+Settings
+Transistor
+UIAutom..
+Weather
+WebView.
+口A
+9:18
+Set up a new account
+[email protected]
+I Show password
+Advanced options
+MANUAL SETUP
+NEXT
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+r
+t
+y
+u
+:
+q
+W
+e
+0
+p
+d
+h
+K
+a
+g
+v
+b
+Z
+X
+C
+n
+m
+?123
+@
+V
+口LIE 4:53
+Reports
+ARTICLES (PIE CHART)
+INCOMES BY ARTICLES (PIE CHART)
+Display of Incomes by Articles (Pie
+Chart)
+Group by:
+View:
+Unfortunately, Family Finance has stopped.
+OK
+With limited group
+CANCEL
+OK200
+175
+150
+125
+100
+75
+50
+25
+0
+Text
+Annotated Text
+Image
+Annotated Image
+Recording
+Log
+180
+100
+30
+6
+18
+19
+133
+   2
+179
+89
+  4
+  1
+14
+172
+36
+29
+  5
+16
+19
+200
+175
+150
+125
+100
+75
+50
+25
+0
+160
+140
+120
+100
+80
+60
+40
+20
+0
+200
+175
+150
+125
+100
+75
+50
+25
+0
+a) Modalities for bug reports.
+b) Modalities for environment.
+c) Modalities for S2Rs.
+d) Modalities for OB.
+Fig. 4: Reporting modalities for bug reports and bug report components.
+relevant app. An example of this case appears in the bug report
+associated with Figure 3b in which the user wrote “Show
+homescreen tips” is indented in the report to describe the
+report’s OB. The annotated input text subcategory contains
+cases in which the reporter provided a textual app input using
+text within quotes. An example of this case appears in a
+bug report [39] for K-9 MAIL, where the reporter mentioned
+Add new email account with “[email protected]” as one of the S2Rs
+in the report. In total, we identified 100 bug reports with
+annotated text (85 annotated GUI text, six annotated input text,
+and nine in which both categories appeared). The remaining
+modalities, while less common, were still present and, in a
+large number of cases, provided information that would have
+been more cumbersome to convey otherwise. Among the bug
+reports considered, reporters used the image, annotated image,
+recording, and log modalities in 30, 6, 18, and 19 bug reports,
+respectively. Furthermore, image-based modalities (i.e., image,
+annotated image, and recording) appeared more frequently
+in user-submitted (35) than developer-submitted bug reports
+(14). Finally, we noticed a slight trend of increasing use of
+image data over the years, with image-based information being
+present in only 14% of reports in 2016 to 36% in 2019.
+Figures 4-b, 4-c, and 4-d report the modalities used for
+specific sections of the bug reports. Figure 4-b reports the
+modalities used for the environment sections. The great ma-
+jority of the reports (133) use the text modality to report
+environment information, and only a few use the image
+modality (2). Figure 4-c provides the modalities used for the
+S2Rs. Text is the most commonly used modality (present in
+179 bug reports). Annotated text also appears in a considerable
+number of bug reports (89). The remaining modalities are less
+common but provide relevant information for reproducing the
+bug reports. Nineteen bug reports had multiple S2R modalities
+other than text or annotated text, 16 of these bug reports
+were user-submitted and 3 were developer-submitted. These
+19 bug reports also used the recording (14), the image (4),
+and annotated image (1) modalities. Finally, Figure 4-d report
+the modalities used for the OB sections. Once more, the text
+modality is the most recurring one (172 cases). However,
+for OB, the image and recording modalities were used more
+frequently (29 and 16 cases, respectively) as compared to
+environment and S2Rs. Sixty bug reports had multiple OB
+modalities other than text or annotated text, 38 of these bug
+reports were user-submitted and 22 were developer-submitted.
+These 60 bug reports also used the image (29), the annotated
+image (5), recording (16), and log (19) modalities (with
+some bug reports having multiple modalities). Overall, user-
+submitted bug reports used reporting modalities other than text
+more frequently that developer-submitted bug reports.
+Examining the relationship between reporting modalities
+and failure types, we found that bug reports with cosmetic and
+navigation failures have a higher proportion of cases in which
+the information is reported using image-based modalities as
+compared to output and crash failures. Specifically, 45% of
+the bug reports describing cosmetic failures and 43% of the
+bug reports discussing navigation failures use image-based
+modalities, while these modalities appear in only 16% and
+18% of the bug reports describing crash and output failures,
+respectively. Focusing on specific bug reports sections, we
+find a similar result for OB descriptions. Additionally, the
+log modality was used exclusively to report the OB of bug
+reports describing crashes. These results highlight how certain
+modalities might be preferable particular failure types.
+RQ2 answer: Our categorization identified six main re-
+porting modalities. Overall, text and annotated text are the
+most recurring modalities. Certain modalities occur more
+frequently when considering specific failure types, e.g.,
+images for cosmetic and navigation failures.
+C. RQ3: Do reproducible bug reports have missing information?
+Our analysis identified that 54 bug reports did not contain
+any environment information, one bug report did not have any
+S2Rs, and four bug reports did not contain OB information.
+(Missing information is computed with respect to the bug re-
+ports initially submitted and does not consider the information
+contained in their discussions, as that is the focus of RQ4.)
+Although only one bug report did not have any S2Rs, 92.2%
+of the bug reports had at least one missing S2R. As mentioned
+in Section II, missing S2Rs include missing context S2Rs
+and missing inline S2Rs. 88.3% of bug reports had at least
+one missing context S2R and 37.7% of bug reports had at
+least one missing inline S2R. Figure 5 associates missing
+S2Rs to unmapped GUI actions. More precisely, for each bug
+report, the figure reports the percentage of unmapped GUI
+actions with respect to the number of GUI actions necessary
+to reproduce the report. The figure reports the percentage for
+missing S2Rs, missing context S2Rs, and missing inline S2Rs.
+The figure reveals that 75% of the bug reports have at least
+20% unmapped GUI actions due to missing S2Rs. Across
+7
+
+Missing S2Rs
+Missing Context S2Rs
+Missing Inline S2Rs
+100%
+80%
+60%
+40%
+20%
+0%
+% of Unmapped GUI Actions
+Fig. 5: Pct. of unmapped GUI actions due to missing S2Rs.
+all bug reports, missing S2Rs led to 43.2% of GUI actions
+being unmapped. 33.4% of unmapped GUI actions are due
+to missing context S2Rs and 9.8% are due to missing inline
+S2Rs. These results illustrate that reproducing bug reports also
+requires inferring a large number of GUI actions that are not
+specified in the description of the bug reports.
+Comparing missing S2Rs from user-submitted bug reports
+with respect to missing S2Rs from developer-submitted bug
+reports, users submitted reports that have a lower percentage of
+unmatched GUI actions due to missing context S2Rs (22.5%)
+with respect to developer-submitted reports (43.5%). This
+difference does not appear for unmatched GUI actions due
+to missing inline S2Rs (9.5% for user-submitted and 10%
+for developer-submitted bug reports). We did not observe a
+difference in missing information across failure types.
+RQ3 answer: The environment section of a bug report is the
+most likely to be missing from submitted bug reports among
+the sections considered. A large percentage of bug reports
+(92%) had at least one missing S2R. Missing S2Rs equate
+to 43.2% unmapped GUI actions necessary to reproduce the
+failures described in the reports.
+D. RQ4: Do discussion threads of reproducible bug reports
+contain helpful information for reproducing the reports?
+To answer this RQ, we analyzed the discussions associated
+with the bug reports in our dataset and identified information
+added as part of the conversations that was relevant for repro-
+ducing the bugs. In total, 35 of the bug reports contained addi-
+tional information detailing either the environment, the S2Rs,
+or the OB of the bug reports. Among these 35 bug reports,
+25 were user-submitted and 10 were developer-submitted.
+Additionally, in 22 of the 35 bug reports, a developer explicitly
+requested for the information to be added to the discussion.
+In the discussions, there were 20 instances of environment
+information added to the report, 11 instances of S2Rs, and 9
+instances of OB. The sum of these numbers is higher than
+the total number of bug reports with additional information
+because some discussions (five in total, four with two mes-
+sages and one with three) contained multiple messages that
+provided additional information. Although added information
+does not appear in a large number of cases, these results show
+0%
+20%
+40%
+60%
+80%
+100%
+Fig. 6: Percentage of non-specific S2Rs by bug report.
+that follow up conversations can be leveraged to reproduce
+reported bugs. Furthermore, considering the high number of
+reports with missing environment information and unmatched
+GUI actions identified in RQ3, automated techniques can try
+to identify and automatically seek this information through
+iterative or interactive bug reproduction approaches.
+Looking at different failure types, bug reports describing
+output failures were the ones with the highest number of
+added information in their discussions. Among the 35 bug
+reports with added information, 17 described output failures,
+15 reported crashes, 2 described cosmetic failures, and 1
+discussed a navigation failure.
+RQ4 answer: Among the bug reports considered, 35 had
+additional information relevant for reproducing the reports
+derived from follow-up, message-based discussions. In 22
+reports, the information was explicitly requested by a devel-
+oper. Finally, of the 35 reports, 20 had added environment
+info, 11 had added S2Rs, and 10 had added OB.
+E. RQ5: How specific is the information reported in repro-
+ducible bug reports?
+When the environment was reported, the information could
+be directly mapped into actions for reproducing the failure.
+That is, it was possible to select the right app version, Android
+version, and device for reproducing the failure. In the case of
+OB, we had to look at the bug report discussion of six reports
+to better understand the problem associated with the reported
+failures, meaning that, in our analysis, the OB described in
+those bug reports was not specific enough for reproducing the
+failures. Considering S2Rs, 73.9% of the bug reports had at
+least one reported S2Rs that could not be directly mapped
+into a single GUI action but, instead, required multiple GUI
+actions. Based on the terminology defined in Section II, this
+means that those bug reports had at least one non-specific S2R.
+Figure 6 reports the percentage of non-specific S2Rs in each
+bug report of our dataset. Across all reports, the S2Rs section
+had an average of 36% of S2Rs that were non-specific. This
+results shows that there is the need to fill a gap to map S2Rs
+into corresponding GUI actions when reproducing reports.
+Considering failure types, bug reports describing navigation
+failures had the highest average percentage of non-specific
+S2Rs (40%), while output failures had the lowest (34%). This
+result shows a minor difference in the specificity of S2Rs
+between reported failure types. There was also little difference
+in the average percentage of non-specific S2Rs reported by
+users (34.6%) and developers (35.8%).
+8
+
+RQ5 answer: Environment and OB information was spe-
+cific enough to reproduce reported failures in the great
+majority of cases. A large percentage of reports (73.9%) had
+at least one non-specific S2R, and the average percentage
+of non-specific S2Rs across all reports was 36%.
+V. DISCUSSION AND IMPLICATIONS
+1) New automated techniques are needed for understanding
+non-crashing oracles. Most existing automated bug repro-
+duction approaches for mobile apps focus on reproducing
+bugs leading to a crash [13], [14]. This is likely because
+failures related to crashes are easier to recognize, for example
+through detection of a crash dialog, and thus detect when a
+a crashing bug has been reproduced. However, our analysis
+shows that more than 70% of the bug reports describe failures
+other than crashes and thus require more sophisticated oracle
+definitions and detection. For example, automated techniques
+for bug report reproduction might benefit from techniques
+that can define visual oracles using computer vision, such as
+detecting an incorrect color theme through color histogram
+analysis. Similarly, navigation failures might require analysis
+of statically computed program state graphs, to determine
+feasibility of navigation paths. Extending recent work on
+defining oracles through the derivation of program invariants
+(e.g., [40]) could further aid in oracle construction.
+2) There is a need for automated multi-modal understand-
+ing of bug report information. Our analysis has illustrated
+that bug reports can mix multiple modalities of information
+together in form of text, images, and recordings, which capture
+disparate pieces of information about a given bug. However,
+most recent work on automated bug report reproduction and
+analysis only considers the textual modality [12], [13], [14].
+Given the amount of prevalence of missing information, even
+in reproducible reports, revealed through our analysis of RQ3,
+automated report analysis should strive to analyze all types of
+reported information for a more robust and complete analysis.
+As such, new techniques for multi-modal understanding of
+bugs is needed. For example, deep learning techniques that
+connect images and natural language (e.g., dense image cap-
+tioning [41]) could be used to link textual information to visual
+information for more complete report analysis. Furthermore,
+in the case of S2Rs, automated techniques would also need to
+identify how to suitably order the information and this could
+be achieved by leveraging window transition graphs computed
+statically or dynamically from the apps [42], [43].
+3) Techniques for inferring and mocking app environments
+are essential. Historically, Android app developers have strug-
+gled to reign-in issues related to the fragmented platform and
+device ecosystem. These issues also surface in bug reporting.
+As identified while analyzing the bug reports considered, it
+is possible for bugs to manifest under specific combinations
+of device and platform versions. Considering, that this in-
+formation is not always present in submitted bug reports
+(missing in 30% of the cases), techniques that are able to infer,
+prioritize environmental settings (e.g., device and platform
+versions) are needed to help drive research on more advanced
+automated mobile bug report analysis techniques. Furthermore,
+considering that apps are released frequently [44], [45] and
+bug reports do not always contain the associated app version,
+it would be beneficial to automatically infer the version of
+the app associated with a bug report. This task could be done
+by automatically by mapping bug report information into GUI
+components or code entities in the app.
+3) Reasoning about missing S2Rs is required. Our analysis
+illustrated that a large majority (92%) of our studied bug
+reports have at least one missing S2R. This represents a
+notable challenge for automated report analysis techniques
+which will likely need to infer this missing information in
+order to provide robust analyses. Current techniques do offer
+advanced solutions (i.e., they are based on random exploration)
+to help fill in certain missing gaps [13], [14], [12]. However,
+additional techniques are likely needed that allow for fine-
+grained inference of missing steps. For instance, future tech-
+niques could examine existing corpora of bug reports (such
+the artifacts associated with this research) and attempt to infer
+missing steps via patterns learned from a corpus of complete
+bug reports.
+4) Handling non-specific S2Rs in bug report data is a
+major challenge. In addition to a high prevalence of missing
+S2Rs, our analysis also revealed that 36% of the S2Rs were
+mapped to multiple GUI actions. These S2Rs identified “high-
+level” operations, in which the actions or target GUI elements
+were not explicitly delineated. This situation represents a
+challenging reasoning problem for automated reproduction
+and report analysis techniques. Current techniques attempt
+to overcome such ambiguities through the use of ontological
+matching [13] or neural representations of text [12] in addition
+to random exploration. However, additional techniques for
+performing mapping of non-specific actions or targets are
+likely needed. For example future techniques may benefit
+from inferring descriptions of app controls or functionality
+through multi-modal image captioning models that allow for
+better mapping of text to runtime app information. Automated
+“repair” of ambiguous bug report steps based on patterns
+learned form well-formed sets of reproduction steps may also
+be a worthwhile direction of exploration. Additionally, S2R
+descriptions could be extracted from sequences of GUI actions
+in existing test cases and be mapped to S2Rs in bug reports
+to facilitate their reproduction.
+In summary, the analysis performed in this paper has
+revealed several notable implications that impact future work
+on automated bug report reproduction, reporting, analysis, and
+management. We believe that future work will benefit from
+these findings and the potential new directions of research that
+they point towards.
+VI. THREATS TO VALIDITY
+While we follow a systematic methodology in collecting,
+analyzing, and reporting our results, it is important to discuss
+the threats to validity of our study to provide a comprehensive
+view of our findings. In terms of external validity, our results
+9
+
+may not generalize to bugs for other Android apps. However,
+given the number, diversity, and popularity of our subject
+applications and reports, we believe our studied reports should
+be reasonably representative of Android bug reports as a
+whole. We considered the most recent dataset of reproducible
+bug reports (with non-crashing bugs) and extended the dataset
+to also include developer-submitted bug reports. This dataset
+includes apps that vary in terms of their size and category. An
+additional threat could be posed by the fact that we only used
+open source apps. However, the evaluation includes apps such
+as FIREFOX FOCUS and SIMPLENOTE, which have complex
+functionality and millions of installs. In terms of construct
+validity, our results might be affected by errors in the tools
+we used to perform our analyses. To mitigate this threat, we
+extensively tested our tools and multiple authors manually
+inspected the results. Finally, we also performed qualitative
+analyses, which could be impacted by divergent understanding
+among evaluators. To mitigate this threat, we used open coding
+based on negotiated agreement [31].
+VII. RELATED WORK
+A. General Studies on Bug Reports
+Related work investigated bug report properties to better un-
+derstand multiple activities characterizing the bug report man-
+agement process [46], [47], [48], [49], [50], [51], [52], [53],
+[54]. Among different topics, this line of research analyzed
+bug report content, developers’ and users’ participation in
+bug report discussions, triaging, and bug fixing. A prominent
+study carried out by Bettenburg et al. [46] identified desired
+aspects that should be contained in a bug report. In follow-up
+work, Bettenburg et al. [47] also showed that duplicated bug
+reports contain some additional helpful information that could
+be used for bug triaging. Sahoo et al. [48] identified the main
+components necessary for bug reproduction by performing
+an empirical study. Some prior studies focused primarily on
+user-submitted bug reports. This line of research investigated
+how users typically communicate software problems [51], the
+usefulness of the provided information by power users [52],
+and user communitys’ expectations [55]. In this paper, we
+investigated key aspects related to both user-submitted and
+developer-submitted Android bug reports. Furthermore, we fo-
+cused on the aspects related to the reproduction of bug reports
+and specifically investigated how the bug report information
+relates to the information needed to reproduce the reports.
+B. Bug Report Studies for Mobile Apps
+Most of the initial studies on bug reports focused on
+desktop applications. However, because of smartphone apps’
+availability, usability, and popularity in the last decade, re-
+searchers have also started focusing on studying characteristics
+of bug reports for mobile apps. Zhou et al. [19] performed
+a study to understand the bug management between desktop
+and mobile software. Bhattacharya et al. [18] studied mobile
+bug reports and the bug-fixing process. Aljedaani et al. [56]
+compared the bug reports between Android and iOS. Zhang et
+al. [57] studied mobile apps bug reports, labeled those reports,
+and computed similarities with the previously labeled ones.
+In our study we reproduced bug reports, characterized the
+failures associated with the reports, analyzed the usefulness
+of the information provided in the reports, and categorized the
+reporting modalities. Previous studies also produced datasets
+of Android bugs with associated bug reports. Wendland et
+al. [20] created a dataset of reproducible, user-submitted bug
+reports. Su et al. [58] created a dataset of crashing bugs based
+on GitHub issues. Fazzini et al. [13] and Zhao et al. [14]
+also assembled a dataset of crashing bugs for their research
+on automated reproduction of bug reports. Compared to these
+datasets, to the best of our knowledge, this paper is the first
+to create and consider in its study a dataset of non-crashing
+and reproducible bug reports that contains both user-submitted
+and developer-submitted reports.
+VIII. CONCLUSION
+We presented an empirical study that characterized re-
+producible Android bug reports. Specifically, we manually
+reproduced 180 bug reports systematically mined from An-
+droid apps on GitHub and investigated how the information
+contained in the bug report relates to the task of reproducing
+the reports. Our analysis identified that reported failures can
+be grouped into four categories, three of which are not yet
+considered by existing automated reproduction techniques,
+reporters use different modalities to report the information
+relevant for reproducing failures, a large number of reports
+(74%) have at least one non-specific S2R (i.e., multiple GUI
+action are necessary to perform the operation described by the
+S2R), the great majority of reports (92%) do not provide all the
+S2Rs that are necessary to reproduce the reports, and bug re-
+port discussions can, in some cases (19%), provide additional
+information useful for the reproduction of the reports.
+In future work, we first plan to present our findings to
+Android developers and then develop techniques to aid au-
+tomated reproduction of bug reports. To support automated
+reproduction of bug reports, we first plan to define an approach
+that leverages natural language processing and computer vi-
+sion techniques to automatically encode OB information into
+oracles and so aid reproduction of output, cosmetic, and
+navigation failures. Second, we plan to define a technique that
+combines S2Rs information reported using different modali-
+ties. Third, we plan to define a technique that leverages the
+information contained in existing test cases to help mapping
+non-specific S2Rs to corresponding GUI actions. Finally, we
+believe that additional studies into the reproduction of bug
+reports for software in other domains are needed and those
+studies could inform techniques for bug report management
+in those domains.
+ACKNOWLEDGMENT
+This work was partially supported by a gift from Facebook
+and the NSF CCF-2007246 & CCF-1955853 grants. Any
+opinions, findings, and conclusions expressed herein are the
+authors’ and do not necessarily reflect those of the sponsors.
+10
+
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+

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+AJB-23-1
+BARI-TH/23-743
+331 Model Predictions for Rare B and K Decays,
+and ∆F = 2 Processes: an Update
+Andrzej J. Burasa,b and Fulvia De Fazioc
+aTUM Institute for Advanced Study, Lichtenbergstr. 2a, D-85747 Garching, Germany
+bPhysik Department, Technische Universit¨at M¨unchen, James-Franck-Straße,
+D-85747 Garching, Germany
+cIstituto Nazionale di Fisica Nucleare, Sezione di Bari, Via Orabona 4, I-70126 Bari, Italy
+Abstract
+Motivated by the improved results from the HPQCD lattice collaboration on the hadronic
+matrix elements entering ∆Ms,d in B0
+s,d − ¯B0
+s,d mixings and the increase of the ex-
+perimental branching ratio for Bs → µ+µ−, we update our 2016 analysis of various
+flavour observables in four 331 models, M1, M3, M13 and M16 based on the gauge group
+SU(3)C ×SU(3)L ×U(1)X. These four models, which are distinguished by the quantum
+numbers, are selected among 24 331 models through their consistency with the elec-
+troweak precision tests and simultaneously by the relation CNP
+9
+= −b CNP
+10
+with b ≥ 2,
+which after new result on Bs → µ+µ− from CMS is favoured over the popular relation
+CNP
+9
+= −CNP
+10 predicted by several leptoquark models. In this context we investigate in
+particular the dependence of various observables on |Vcb|, varying it in the broad range
+[0.0386, 0.043], that encompasses both its inclusive and exclusive determinations. Im-
+posing the experimental constraints from εK, ∆Ms, ∆Md and the mixing induced CP
+asymmetries SψKS and SψKS, we investigate for which values of |Vcb| the four models
+can be made compatible with these data and what is the impact on B and K branching
+ratios. In particular we analyse NP contributions to the Wilson coefficients C9 and C10
+and the decays Bs,d → µ+µ−, K+ → π+ν¯ν and KL → π0ν¯ν. This allows us to illustrate
+how the value of |Vcb| determined together with other parameters of these models is
+infected by NP contributions and compare it with the one obtained recently under the
+assumption of the absence of NP in εK, ∆Ms, ∆Md and SψKS.
+arXiv:2301.02649v1  [hep-ph]  6 Jan 2023
+
+1
+Introduction
+1
+Contents
+1
+Introduction
+1
+2
+Flavour Structure of 331 Models
+4
+3
+Selecting the 331 Models
+5
+4
+Numerical Analysis
+6
+4.1
+Determining the parameter space . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+4.2
+CNP
+9
+and CNP
+10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+4.3
+¯B(Bs → µ+µ−) and B(Bd → µ+µ−) . . . . . . . . . . . . . . . . . . . . . . . .
+8
+4.4
+Rare Kaon decays
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+5
+Summary
+10
+1
+Introduction
+The Standard Model (SM) describes globally the existing data on quark-flavour violating
+processes rather well [1] but with the reduction of experimental errors and increased precision
+in non-perturbative and perturbative QCD and electroweak calculations a number of tensions
+at the level of 2 − 5 σ seem to emerge in various seemingly unrelated observables.
+While
+some of these tensions could turn out to be the result of statistical fluctuations, underestimate
+of systematical and theoretical errors, it is not excluded that eventually they all signal the
+presence of some kind of new physics (NP). Therefore, it is interesting to investigate what this
+NP could be.
+In the present paper we will address some of these tensions in four particular 331 models
+based on the gauge group SU(3)C × SU(3)L × U(1)X [2, 3] 1. As these models have much
+smaller number of new parameters than supersymmetric models, Randall-Sundrum scenar-
+ios and Littlest Higgs models, it is not evident that they can remove all present tensions
+simultaneously.
+Our paper has been motivated by the following recent facts.
+• As demonstrated in [5] most recent lattice QCD results from HPQCD collaboration [6],
+based on 2 + 1 + 1 simulations, imply simultaneous agreement of
+|εK|,
+∆Ms,
+∆Md,
+SψKS
+Sψφ
+(1)
+within the SM with the data for rather precise values of |Vcb|, |Vub| and γ. This should
+be contrasted with the situation at the time of our previous analysis 2016 [7], when
+significant tensions between εK and ∆Ms,d within the SM have been found [8] and the
+room for NP in the quark mixing sector was much larger than it is now.
+1A recent critical reanalysis of 331 models and a collection of references can be found in [4].
+
+1
+Introduction
+2
+• The most recent data on Bs → µ+µ− from CMS imply that in the case of the dominance
+of left-handed quark currents, as is the case of the 331 models,
+CNP
+9
+= −b CNP
+10 ,
+b ≥ 2,
+(2)
+where CNP
+9 , CNP
+10 represent the shifts in the Wilson coefficients C9, C10 of the b → sℓ+ℓ−
+effective Hamiltonian in the presence of NP. The relation (2) is in contrast to the pre-
+viously favoured case b = 1 found in several leptoquark models, in particular in the U1
+model.
+• Recent messages from the LHCb [9, 10], that the lepton flavour universality violation
+(LFUV) in b → s���+ℓ−, which for many years dominated the B-physics anomalies, prac-
+tically disappeared. This is good news for 331 models for which LFUV anomalies were
+problematic, although these models could provide some shifts in the Wilson coefficients
+C9 and C10. Such shifts, in particular in C9, are still required to describe suppressed
+branching ratios in b → sµ+µ− transitions.
+• The most recent value for γ obtained by the LHCb collaboration from tree-level decays
+that reads [11]
+γ = (63.8+3.5
+−3.7)◦ .
+(3)
+It is significantly more precise than the LHCb values of γ in 2016 that could be as large
+as 75◦.
+The question then arises how 331 models face this new situation relative to the 2016
+input and what are the implications for many flavour observables, in particular for the decays
+Bd → K(K∗)µ+µ−, B+ → K+µ+µ− and Bs → φµ+µ− related to the B physics anomalies
+that imply the need for significant NP contributions to the Wilson coefficient C9 and smaller
+to C10. But it is also of interest to see what are the implications for rare decays Bs,d → µ+µ−,
+K+ → π+ν¯ν and KL → π0ν¯ν.
+It is known from many analyses, and stressed recently in particular in [5, 12] that the
+tensions between inclusive and exclusive determinations of |Vcb| and |Vub| preclude precise
+predictions for rare decay observables in the SM. However, eliminating these parameters with
+the help of εK, ∆Ms,d and SψKS and setting the latter observables to their experimental
+values allowed to obtain SM predictions for many flavour observables that are most precise to
+date [5,12]. The motivation for this strategy has been strengthened recently by one of us [13] as
+the one which could minimize the impact of NP on the determination of the CKM parameters.
+Indeed, as demonstrated in [5], presently no NP is required to describe precise experimental
+data on ∆F = 2 observables.
+This allows in turn to determine the CKM parameters on
+the basis of ∆F = 2 observables alone without being involved in the issue of |Vcb| and |Vub|
+tensions and minimizing possible impact of NP on their values that otherwise would infect
+SM predictions for rare decay branching ratios.
+The resulting values of the CKM parameters read [5]
+|Vcb| = 42.6(4) × 10−3,
+|Vub| = 3.72(11) × 10−3,
+γ = 64.6(16)◦.
+(4)
+While in this manner one can obtain rather precise SM predictions for numerous branching
+ratios [5, 12, 13], the absence of NP in the ∆F = 2 observables, if confirmed with higher
+
+1
+Introduction
+3
+Decay
+EXCLUSIVE
+HYBRID
+DATA
+B(K+ → π+ν¯ν) × 1011
+6.88(38)
+8.44(41)
+10.9(38)
+[15]
+B(KL → π0ν¯ν) × 1011
+2.37(15)
+2.74(14)
+< 300
+[16]
+B(KS → µ+µ−) × 1013
+1.49(10)
+1.72(8)
+104
+[17]
+B(Bs → µ+µ−) × 109
+3.18(12)
+3.67(12)
+3.45(29)[18–21]
+B(Bd → µ+µ−) × 1010
+0.864(34)
+0.999(34)
+< 2.05
+[18]
+|εK| × 103
+1.78(11)
+2.14(12)
+2.228(11)
+[22]
+SψKS
+0.731(24)
+0.688(22)
+0.699(17)
+[22]
+∆Ms ps−1
+15.02(87)
+17.35(94)
+17.749(20) [22]
+∆Md ps−1
+0.434(28)
+0.502(31)
+0.5065(19) [22]
+Table 1:
+Predictions (second column) for selected observables within the SM obtained in [5] using
+the EXCLUSIVE strategy for |Vcb| and |Vub| and γ = 65.4◦. In the third column we show the results
+for the HYBRID choice of |Vcb| and |Vub| as given in (6) and in the fourth the experimental data.
+precision, would be a nightmare scenario for many NP models that attempt to explain the
+B physics anomalies.
+While the ones related to lepton flavour universality violation have
+been dwarfed recently through new LHCb data [9,10], sizable anomalies remained in several
+branching ratios. In particular using the strategy of [5,12] large anomalies in the low q2 bin
+in B+ → K+µ+µ− (5.1σ) and Bs → φµ+µ− (4.8σ) have been found [13].
+Explaining such anomalies without practically no NP contributions to ∆F = 2 processes
+is in principle possible but would require significant tuning of NP parameters. Now, the value
+of γ in (4) agrees very well with the most recent value from LHCb in (3) and experimental
+value of β from SψKS is already used in obtaining the CKM parameters in (4). It is evident
+then that the most efficient and transparent strategy to allow NP to enter the ∆F = 2 sector
+is to modify the value of |Vcb|.
+In this context in [5], two scenarios for the parameters |Vcb| and |Vub| have been analysed
+within the SM. The EXCLUSIVE one based on determinations of these parameters in exclusive
+decays
+|Vcb| = 39.21(62) × 10−3,
+|Vub| = 3.61(13) × 10−3,
+(EXCLUSIVE),
+(5)
+and the HYBRID scenario in which the value for |Vcb| is the inclusive one from [14] and the
+exclusive one for |Vub| as above:
+|Vcb| = 42.16(50) × 10−3,
+|Vub| = 3.61(13) × 10−3,
+(HYBRID).
+(6)
+In Table 1 we show selected results obtained in [5] in these two scenarios. The results
+obtained in the HYBRID scenario do not differ by much from those obtained using the CKM
+
+2
+Flavour Structure of 331 Models
+4
+parameters in (4) [5, 13]. With exclusive values of |Vcb| that are much lower than given in
+(4), anomalies in ∆Ms (3σ), ∆Md (4σ) and εK (5σ) are generated. But in [5] no analysis
+of a NP scenario has been presented which would explain these anomalies and whether a
+model explaining them would also be able to explain anomalies in semi-leptonic B decays.
+In the present paper we investigate whether the 331 models could provide some insight in
+these issues and what would be the implications for rare branching ratios. As a byproduct
+our analysis illustrates in simple settings how the determination of |Vcb| in a global fit that
+includes observables exposing anomalies can be infected by NP contributions [13]. It is a
+concrete illustration of the points made in section 2 of the latter paper.
+Our paper is organized as follows. In Section 2 we recall briefly the flavour structure of
+the 331 models. In Section 3 we select four 331 models that perform best on the basis of
+electroweak precision tests and the present experimental values of the ratio CNP
+9 /CNP
+10 in (2).
+In fact these are the only models among the 24 ones considered in [23], that can successfuly
+face the new relation (2) when other contraints like electroweak precision tests are taken into
+account [7]. In Section 4 we present numerical analysis of these models addressing the issues
+mentioned above. We conclude in Section 5.
+2
+Flavour Structure of 331 Models
+Let us recall that in the 331 models new flavour-violating effects are governed by tree-level
+Z′ exchanges with a subdominant but non-negligible role played by tree-level Z exchanges
+generated through Z − Z′ mixing. All the formulae for flavour observables in these models
+can be found in [23–26] and will not be repeated here. In particular the collection of formulae
+for Z′ couplings to quarks and leptons are given in [25].
+New sources of flavour and CP violation in 331 models are parametrized by new mixing
+parameters and phases
+˜s13,
+˜s23,
+δ1,
+δ2
+(7)
+with ˜s13 and ˜s23 positive definite and smaller than unity and 0 ≤ δ1,2 ≤ 2π. They can be
+constrained by flavour observables as demonstrated in detail in [24]. The non-diagonal Z′
+couplings relevant for K, Bd and Bs meson systems can be then parametrized respectively
+within an excellent approximation through
+v∗
+32v31 = ˜s13˜s23ei(δ2−δ1),
+v∗
+33v31 = −˜s13e−iδ1,
+v∗
+33v32 = −˜s23e−iδ2 .
+(8)
+˜s13 and δ1 can be determined from ∆Md and CP-asymmetry SψKS while ˜s23 and δ2 from ∆Ms
+and CP-asymmetry Sψφ. Then the parameters in the K system are fixed. It is a remarkable
+feature of 331 models that also FCNC processes in the charm sector can be described without
+introducing no new free parameters beyond those already present in the beauty and kaon
+meson systems [27,28]. These correlations constitute important tests of these models.
+The remaining two parameters, except for MZ′ mass, are β and tan ¯β defined through2
+Q = T3 + Y
+2 = T3 + βT8 + X,
+tan ¯β = vρ
+vη
+.
+(9)
+2The parameter β should not be confused with the angle β in the unitarity triangle.
+
+3
+Selecting the 331 Models
+5
+MI
+scen.
+β
+tan ¯β
+MI
+scen.
+β
+tan ¯β
+MI
+scen.
+β
+tan ¯β
+M1
+F1
+−2/
+√
+3
+1
+M9
+F2
+−2/
+√
+3
+1
+M17
+F1
+−2/
+√
+3
+0.2
+M2
+F1
+−2/
+√
+3
+5
+M10
+F2
+−2/
+√
+3
+5
+M18
+F2
+−2/
+√
+3
+0.2
+M3
+F1
+−1/
+√
+3
+1
+M11
+F2
+−1/
+√
+3
+1
+M19
+F1
+−1/
+√
+3
+0.2
+M4
+F1
+−1/
+√
+3
+5
+M12
+F2
+−1/
+√
+3
+5
+M20
+F2
+−1/
+√
+3
+0.2
+M5
+F1
+1/
+√
+3
+1
+M13
+F2
+1/
+√
+3
+1
+M21
+F1
+1/
+√
+3
+0.2
+M6
+F1
+1/
+√
+3
+5
+M14
+F2
+1/
+√
+3
+5
+M22
+F2
+1/
+√
+3
+0.2
+M7
+F1
+2/
+√
+3
+1
+M15
+F2
+2/
+√
+3
+1
+M23
+F1
+2/
+√
+3
+0.2
+M8
+F1
+2/
+√
+3
+5
+M16
+F2
+2/
+√
+3
+5
+M24
+F2
+2/
+√
+3
+0.2
+Table 2: Definition of the various 331 models.
+Here T3,8 and X are the diagonal generators of SU(3)L and U(1)X, respectively. Y represents
+U(1)Y and vi are the vacuum expectation values of scalar triplets responsible for the generation
+of down- and up-quark masses in these models.
+Different 331 models can also be distinguished by the way quarks transform under SU(3)L.
+In [23] two classes of such models have been analyzed to be denoted by F1 and F2. F1 stands
+for the case in which the first two generations of quarks belong to triplets of SU(3)L, while
+the third generation of quarks to antitriplet. F2 stands for the case in which the first two
+generations of quarks belong to antitriplets of SU(3)L, while the third generation of quarks
+to triplet.
+A detailed analysis of 24 331 models corresponding to different values of β and tan ¯β for
+the representations F1 and F2 has been presented in [23]. They are collected in Table 2. With
+the values of β and tan ¯β being fixed, flavour phenomenology depends only on the parameters
+in (7), MZ′ and the CKM parameters which distinguish EXCLUSIVE and HYBRID scenarios.
+3
+Selecting the 331 Models
+A detailed analysis of electroweak precision tests in the 24 models in Table 2 has been per-
+formed in [23]. Interested readers are asked to look at Section 5 of that paper. Here we just
+summarize the main outcome of that study.
+Requiring that the 24 models in question perform well in these tests and are simultaneously
+consistent with the ratio C9/C10 in (2) selects, as shown in Table 3, the following models
+M1,
+M3,
+M13,
+M16,
+(favoured).
+(10)
+Note that the Z − Z′ mixing plays in some cases an important role and that the two favoured
+models M8 and M9 analysed by us in [7] are ruled out by (2).
+
+4
+Numerical Analysis
+6
+MI
+Full
+no Mixing
+MI
+Full
+no Mixing
+MI
+Full
+no Mixing
+M1
+−3.25
+−8.87
+M9
+0.42
+0.60
+M17
+−175.6
+−8.87
+M2
+−1.68
+−8.87
+M10
+0.28
+0.60
+M18
+0.75
+0.60
+M3
+−2.07
+−2.98
+M11
+−0.02
+−0.004
+M19
+−63.48
+−2.98
+M4
+−1.09
+−2.98
+M12
+−0.04
+−0.004
+M20
+0.06
+−0.004
+M5
+0.02
+−0.004
+M13
+−5.47
+−2.98
+M21
+1.15
+−0.004
+M6
+−0.03
+−0.004
+M14
+−1.56
+−2.98
+M22
+3.25
+−2.98
+M7
+0.97
+0.60
+M15
+11.3
+−8.87
+M23
+7.50
+0.60
+M8
+0.49
+0.60
+M16
+−4.59
+−8.87
+M24
+2.44
+−8.87
+Table 3: CNP
+9 /CNP
+10 in various 331 models with and without Z − Z′ mixing for MZ′ = 3 TeV.
+mBs = 5366.8(2) MeV
+[22]
+mBd = 5279.58(17) MeV [22]
+∆Ms = 17.749(20) ps−1
+[22]
+∆Md = 0.5065(19) ps−1
+[22]
+∆MK = 0.005292(9) ps−1 [22]
+mK0 = 497.61(1) MeV
+[22]
+SψKS = 0.699(17)
+[22]
+FK = 155.7(3) MeV
+[29]
+|Vus| = 0.2253(8)
+[22]
+|ϵK| = 2.228(11) · 10−3
+[22]
+FBs = 230.3(1.3) MeV
+[30]
+FBd = 190.0(1.3) MeV
+[30]
+FBs
+� ˆBs = 256.1(5.7) MeV [6]
+FBd
+� ˆBd = 210.6(5.5) MeV[6]
+ˆBs = 1.232(53)
+[6]
+ˆBd = 1.222(61)
+[6]
+mt(mt) = 162.83(67) GeV[31]
+mc(mc) = 1.279(13) GeV
+Stt(xt) = 2.303
+Sut(xc, xt) = −1.983 × 10−3
+ηtt = 0.55(2)
+[32]
+ηut = 0.402(5)
+[32]
+κε = 0.94(2)
+[33]
+ηB = 0.55(1)
+[34,35]
+τBs = 1.515(4) ps
+[36]
+τBd = 1.519(4) ps
+[36]
+Table 4: Values of the experimental and theoretical quantities used as input parameters. For
+future updates see FLAG [30], PDG [22] and HFLAV [29].
+4
+Numerical Analysis
+4.1
+Determining the parameter space
+Despite the fact that NP is not required to obtain within the SM simultaneous agreement with
+data for the ∆F = 2 observables in (1) [5], the present uncertainties in hadronic parameters
+still allow for some NP contributions, whose size depends strongly on the value of |Vcb| [5,12].
+Therefore in order to constrain the parameters in (7) and subsequently obtain predictions for
+various observables, we will proceed in each of the four considered 331 models as follows:
+• We will vary ∆Md, SψKs, ∆Ms, Sψφ, ϵK within 5% of the central value of their experi-
+mental datum.
+
+4
+Numerical Analysis
+7
+• Concerning CKM parameters, we adopt here a different strategy with respect to our
+previous analyses. We vary |Vub| as in (4), while |Vcb| is varied in such a way to encompass
+both its inclusive and exclusive determinations, i.e. |Vcb| ∈ [0.0386, 0.043].
+• For each of the four 331 models considered in this paper we then determine the allowed
+values of the 331 parameters ˜s13, δ1, ˜s23, δ2 as well as a range for |Vcb| for which a given
+model satisfies the constraints from ∆F = 2 observables in (1) within 5% as stated
+above.
+• We predict several observables in each model and discuss their dependence on |Vcb|. We
+compare the outcome in the four cases.
+The remaining parameters used in our analysis are collected in Table 4.
+Among the parameters that define the various scenarios, ∆F = 2 observables depend only
+on |β|, so that the resulting parameter space will be the same for M1 and M16 as well as for M3
+and M13. In the two cases we have constructed the tables of the allowed parameters in the form
+of 6-vectors of the kind (˜s13, δ1, ˜s23, δ2, |Vcb|, |Vub|). Of course it is not possible to display the
+space of all the variables simultaneously and therefore we do not show these plots. Instead, in
+Fig. 1 we show the allowed (|Vcb|, |Vub|) ranges in the two resulting parameter spaces. It should
+be understood that each point corresponds to a set of 331 parameters. In these figures the
+green points are obtained after imposing the constraints on ∆Md, SψKs, ∆Ms, Sψφ and show
+that even though such observables select the 331 parameters ˜s13, δ1, ˜s23, δ2 they do not have
+an impact on the allowed ranges for |Vub| and |Vcb|. On the contrary, when the constraint on εK
+is imposed, a limitation is found for |Vcb| that is the consequence of the stronger dependence
+of εK on this parameter than in the case of ∆Ms and ∆Md. However, we can observe that,
+while in the case of M1 and M16, |Vcb| cannot be smaller than ≃ 0.0405, no similar constraint
+is found in the case of M3, M13.
+4.2
+CNP
+9
+and CNP
+10
+We have already remarked the nice feature of 331 models that the ratio CNP
+9 /CNP
+10 depends only
+on the considered scenario but not on the parameters ˜s13, δ1, ˜s23, δ2. However, the separate
+values of CNP
+9
+and CNP
+10 depend on them. In Fig. 2 we show the correlation between their real
+parts in the four scenarios, while in Fig. 3 the correlation between their imaginary parts is
+displayed.
+In order to understand which values of |Vcb| correspond to the largest deviations
+in CNP
+9
+we consider Max
+��Re[CNP
+9 ]
+�� setting |Vub| at its central value. The result is shown in
+Fig. 4.
+These plots display that, consistently with the result in Fig. 1 in the case of M1
+and M16 only the values |Vcb| ≥ 0.0405 are allowed. Moreover, the deviation in |Re[C9]| is a
+decreasing function of |Vcb|, as shown in Fig. 4, together with the plots for the imaginary part.
+The situation for |Re[CNP
+10 ]| and |Im[CNP
+10 ]| is displayed in Figs. 5 and 6. It can be noticed
+that CNP
+9
+is to an excellent approximation the same in M1 and M16 on the one hand and in
+M3 and M13 on the other; for this reason we have shown the corresponding plots in a single
+figure. CNP
+10
+is instead different in all the four considered cases.
+We observe that while the pattern of NP contributions signalled by the data is correctly
+described by these models, the absolute values of CNP
+9
+are likely to turn out to be too small to
+explain the observed suppression of the branching ratios for B+ → K+µ+µ− and Bs → φµ+µ−,
+
+4
+Numerical Analysis
+8
+Figure 1: Allowed (|Vcb|, |Vub|) ranges in the parameter space of M1 and M16 (upper plot) and in
+that of M3 and M13 (lower plot). Each point corresponds to a set of 331 parameters. The green
+points are obtained after imposing the constraints on ∆Md, SψKs, ∆Ms, Sψφ, while the light blue
+points derive from imposing the constraint on εK.
+in particular if the final value for |Vcb| from tree-level decays will turn out to be in the ballpark
+of its inclusive determinations.
+4.3
+¯B(Bs → µ+µ−) and B(Bd → µ+µ−)
+In Fig. 7 we plot the correlation between the rare decays ¯B(Bs → µ+µ−) and B(Bd → µ+µ−)
+in the four considered 331 models. In these plots, the gray region is obtained considering
+all the allowed parameter space in each scenario, while the red region corresponds to |Vcb| ∈
+[0.0386, 0.0398] and the cyan region to |Vcb| ∈ [0.0422, 0.043]. The SM results for |Vcb| =
+0.03921 and |Vcb| = 0.0426 are also displayed. Comparing the four models, we can observe
+that if |Vcb| is fixed consistenlty with the exclusive determinations, a possible suppression of
+both branching ratios with respect to their SM values, that is not yet excluded in view of large
+experimental errors, could be explained only in M3 and M13. On the other hand, inclusive
+
+M1&M16
+0.00380
+0.00375
+AMd,SuKs,AMs,Su
+0.00370
+EK
+0.00365
+0.00360
+0.039
+0.040
+0.041
+0.042
+0.043
+IVcblM3 & M13
+0.00380
+0.00375
+AMd , Suk., AMs, Sud
+0.00370
+EK
+0.00365
+0.00360
+0.039
+0.040
+0.041
+0.042
+0.043
+IVcbl4
+Numerical Analysis
+9
+Figure 2: Correlation between the real parts of CNP
+9
+and CNP
+10
+in the four considered 331 models.
+values of |Vcb| do not define a clear situation in any of the four models: other correlations should
+be explored in order to discriminate among these scenarios. We detail the dependence of the
+considered branching fractions on the CKM elements in the contour plots in Fig. 8 for M1
+and M16 and in Fig. 9 for M3 and M13. Since in each scenario the parameter space involves
+6 variables it is possible that fixing (|Vcb|, |Vub|) different values for the considered branching
+ratios are obtained, because these depend also on the other four parameters of the 331 model.
+Therefore, what is plotted in Fig. 8 and in Fig. 9 is the value of the branching ratios that,
+for a given pair (|Vcb|, |Vub|), mostly deviates from the corresponding SM prediction. The
+resulting value of the branching fractions can be read from the legenda on the right of each
+plot. The benefit of these plots with respect to those already shown is that it is possible to
+relate a given value of the branching fractions to the entries for (|Vcb|, |Vub|), an information
+that is hidden in Fig. 7. The SM result as function of (|Vcb|, |Vub|) can be read from Fig. 10:
+comparison between these plots and the corresponding one in a given 331 model would give
+an idea of the possible deviation as a function of (|Vcb|, |Vub|). In particular, one can observe
+that M3 and M13 perform rather similarly to the SM, with values of the branching fractions
+that increase with |Vcb| almost independently on |Vub|. On the other hand, this pattern is not
+followed in M1 and M16.
+4.4
+Rare Kaon decays
+In Fig. 11 we display the correlation between B(K+ → π+ν¯ν) and B(KL → π0ν¯ν). The gray
+points span all the allowed parameter space in each scenario, while the red region corresponds
+
+M1
+M16
+0.2
+0.2
+0.1 
+0.1
+Re[CP]
+Re[CND]
+0.0
+0.0
+0.1
+-0.1
+0.2
+0.2
+0.5
+0.0
+0.5
+0.5
+0.0
+0.5
+Re[CgP]
+Re[CgP]
+M3
+M13
+0.2
+0.2
+0.1
+0.1
+Re[CN]
+0.0
+0.0
+0.1
+-0.1
+0.2
+0.2
+0.5
+0.0
+0.5
+0.5
+0.0
+0.5
+Re[C)P]5
+Summary
+10
+Figure 3: Correlation between the imaginary parts of CNP
+9
+and CNP
+10
+in the four considered 331
+models.
+to |Vcb| ∈ [0.0386, 0.0398] and the cyan region to |Vcb| ∈ [0.0422, 0.043]. The SM results for
+|Vcb| = 3.921 10−2 and |Vcb| = 4.26 10−2 are also displayed. In all the four models, the largest
+deviation from SM is possible in the case of B(KL → π0ν¯ν). Contour plots analogous to
+those presented for Bs, Bd decays are shown in Figs. 12 and 13, to be compared with the
+corresponding SM case in Fig. 14. We observe again that M3 and M13 behave similarly to
+the SM, while M1 and M16 show a differnt pattern.
+Correlation between B(K+ → π+ν¯ν) and ¯B(Bs → µ+µ−) is shown in Fig. 15. It can be
+observed that in all the four cases the inclusive values of |Vcb| correspond to points that can be
+compatible with the experimental result for ¯B(Bs → µ+µ−) performing slightly better than
+the SM; such points correspond to B(K+ → π+ν¯ν) ≤ 1010. Exclusive values of |Vcb| that are
+not allowed in M1 and M16, can produce in M3 and M13 also values of ¯B(Bs → µ+µ−) and
+B(K+ → π+ν¯ν) simultaneously smaller than the experimental range.
+5
+Summary
+Motivated by several changes both on experimental and theoretical frontiers we updated
+our 2016 analysis of various flavour observables in the 331 model based on the gauge group
+SU(3)C × SU(3)L × U(1)X for MZ′ = 3 TeV, that is still in the LHC reach.
+Among 24 331 models considered in our 2016 analysis only four, namely M1, M3, M13
+and M16 are simultaneously consistent with the electroweak precision tests and the relation
+between CNP
+9
+and CNP
+10 signalled by the most recent data on the B → µ+µ− decay from the
+
+M1
+M16
+0.2
+0.2
+0.1 
+0.1
+[ab]u]
+0.0
+0.0
+-0.1
+-0.1
+0.2
+0.2
+0.5
+0.0
+0.5
+0.5
+0.0
+0.5
+Im[CgP]
+Im[CgP]
+M3
+M13
+0.2
+0.2
+0.1
+0.1
+0.0
+0.0
+0.1
+0.1
+0.2
+0.2
+0.5
+0.0
+0.5
+0.5
+0.0
+0.5
+Im[C)P]5
+Summary
+11
+Figure 4: Maximal deviation of
+��Re[CNP
+9
+]
+�� and
+��Im[CNP
+9
+]
+�� in the four considered 331 models.
+Figure 5: Maximal deviation of
+��Re[CNP
+10 ]
+�� in the four considered 331 models.
+
+& M16
+M3 & M13
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.0
+0.0
+0.0405
+0.0410
+0.0415
+0.0420
+0.0425
+0.0430
+0.039
+0.040
+0.041
+0.042
+0.043
+IVcbl
+IVcbl
+M1 & M16
+M3 & M13
+0.5
+0.5
+0.4
+0.4
+0.3
+0.3
+0.2
+0.2
+0.1
+0.1
+0.0
+0.0
+0.0405
+0.0410
+0.0415
+0.0420
+0.0425
+0.0430
+0.039
+0.040
+0.041
+0.042
+0.043
+IVcbl
+IVcblM1
+M16
+0.25
+0.25
+0.20
+0.20
+0.15
+0.15
+Max
+0.10
+Max
+0.10
+0.05
+0.05
+0.00
+0.00
+0.039
+0.040
+0.041
+0.042
+0.043
+0.039
+0.040
+0.041
+0.042
+0.043
+IVcbl
+IVcbl
+M3
+M13
+0.25
+0.25
+0.20
+0.20
+[Re[CN 1
+0.15
+0.15
+Max
+0.10
+Max
+0.10
+0.05
+0.05
+0.00
+0.00
+0.039
+0.040
+0.041
+0.042
+0.043
+0.039
+0.040
+0.041
+0.042
+0.043
+IVcbl
+IVcbl5
+Summary
+12
+Figure 6: Maximal deviation of
+��Im[CNP
+10 ]
+�� in the four considered 331 models.
+CMS.
+The lessons from this analysis are as follows:
+• The 331 models allow for the values of the ratio CNP
+9 /CNP
+10 that are consistent with the
+most recent data. M13 and M16 are performing best but this can only be decided when
+new overall fits will be performed.
+• However, only models M1 and M16 can reach the values Re[CNP
+9 ] = −0.7, which although
+likely not quite sufficient to explain properly the the suppression of b → sµ+µ− branching
+ratios, they reproduce a significant portion of it. For M3 and M13 models only the
+corresponding values of −0.5 can be reached.
+• Moreover, we notice that while in the case M1 and M16 models the maximal negative
+shifts of Re[C9] can still be obtained for inclusive values in the ballpark of |Vcb| = 0.0415,
+in the case of M3 and M13 the shift of −0.5 can only be obtained for exclusive values of
+|Vcb| as low as 0.039. We conclude then that models M1 and M16 perform best in this
+context but as seen in Fig. 4 for the case of the HYBRID scenario for CKM parameters
+none of the models can provide suppression of Re[C9] by more than −0.2 which appears
+too small from present perspective.
+• Concerning Re[CNP
+10 ] all models show only a small shift which is consisten with the data.
+This is also the case of of the imaginary parts of both CNP
+9
+and CNP
+10 .
+• As seen in Fig 11, NP effects in K+ → π+ν¯ν turn out to be small but could be signifi-
+cantly larger in KL → π0ν¯ν.
+
+M1
+M16
+0.20
+0.20
+0.15
+0.15
+0.10
+0.10
+0.05
+0.05
+0.00
+0.00
+0.039
+0.040
+0.041
+0.042
+0.043
+0.039
+0.040
+0.041
+0.042
+0.043
+IVcbl
+IVcbl
+M3
+M13
+0.20
+0.20
+0.15
+0.15
+0.10
+0.10
+Max
+0.05
+0.05
+0.00
+e...
+0.00
+0.039
+0.040
+0.041
+0.042
+0.043
+0.039
+0.040
+0.041
+0.042
+0.043
+IVebl
+IVcblREFERENCES
+13
+Figure 7: Correlation between ¯B(Bs → µ+µ−) and B(Bd → µ+µ−). The gray points span all the
+allowed parameter space in each scenario. The red region corresponds to |Vcb| ∈ [0.0386, 0.0398]
+while the cyan region corresponds to |Vcb| ∈ [0.0422, 0.043]. The SM results in correspondence of
+two values of |Vcb| are displayed, as specified in the legenda.
+We are looking forward to improved data on all observables to be able to judge better the
+ability of the 331 models in explaining signs of NP.
+Acknowledgements
+A.J.B would like to thank Andreas Crivellin for the discussion on the present status of lepto-
+quark models after new LHCb and CMS data. This research was done in the context of the
+Excellence Cluster ORIGINS, funded by the Deutsche Forschungsgemeinschaft (DFG, German
+Research Foundation), Excellence Strategy, EXC-2094, 390783311. It has also been carried
+out within the INFN project (Iniziativa Specifica) QFT-HEP.
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+T
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+1.3
+μ)x 1010M16
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+ SM: IVebl=3.921 10-2
+T
+ SM: IVebl=4.26 10-2
+B
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+B
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+μ-)x 1010REFERENCES
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+3.8
+Vub
+3.6
+0.00370
+3.4
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+0.00375
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+3.6
+3.5
+0.00370
+3.4
+3.3
+3.2
+0.00365
+0.0405
+0.0410
+0.0415
+0.0420
+0.0425
+IVcblB(Bd → μ+ μ-)x 10l0, M16
+0.00380
+1.075
+1.050
+1.025
+0.00375
+1.000
+0.975
+0.950
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+0.925
+0.900
+0.875
+0.850
+0.00365
+0.0405
+0.0410
+0.0415
+0.0420
+0.0425
+IVcb!REFERENCES
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+B(Bs → μ+ μ-)x 10°, M3
+0.00380
+4.0
+3.8
+0.00375
+3.6
+ub
+3.4
+0.00370
+3.2
+3.0
+2.8
+0.00365
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+IVcblB(Ba → μ+ μ-)x 1010 , M3
+0.00380
+1.15
+1.10
+0.00375
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+1.00
+Vubl
+0.95
+0.00370
+0.90
+0.85
+0.80
+0.00365
+0.75
+0.039
+0.040
+0.041
+0.042
+0.043
+IVcblB(Bs →μ+ μ-)x 10°, M13
+0.00380
+3.8
+3.7
+0.00375
+3.6
+3.5
+ub
+3.4
+0.00370
+3.3
+3.2
+3.1
+3.0
+0.00365
+0.039
+0.040
+0.041
+0.042
+0.043
+IVcblB(Ba → μ+ μ-)x 101° , M13
+0.00380
+1.05
+0.00375
+1.00
+qA
+0.95
+0.90
+0.00370
+0.85
+0.80
+0.00365
+0.039
+0.040
+0.041
+0.042
+0.043
+IVeblREFERENCES
+16
+Figure 10: Contour Plots of ¯B(Bs → µ+µ−) (left column) and B(Bd → µ+µ−) (right column) versus
+|Vcb| and |Vub| in the SM .
+Figure 11: Correlation between B(K+ → π+ν¯ν) and B(KL → π0ν¯ν). The gray points span all
+the allowed parameter space in each scenario. The red region corresponds to |Vcb| ∈ [0.0386, 0.0398]
+while the cyan region corresponds to |Vcb| ∈ [0.0422, 0.043]. The SM results in correspondence of
+two values of |Vcb| are displayed, as specified in the legenda. The light gray region corresponds to
+the experimental range for B(K+ → π+ν¯ν) reported in Table 1.
+
+B(Bs →μt μ)x 10°, SM
+0.00380
+3.7
+3.6
+0.00375
+3.5
+3.4
+0.00370
+3.3
+3.2
+3.1
+0.00365
+0.039
+0.040
+0.041
+0.042
+IVcblB(Bd → μ+ μ-)x 10l0 , SM
+0.00380
+1.02
+1.00
+0.98
+0.00375
+0.96
+0.94
+0.92
+0.00370
+0.90
+0.88
+0.86
+0.84
+0.00365
+0.039
+0.040
+0.041
+0.042
+IVcblM1
+14
+12
+10
+8
+↑
+B(K+
+9
+1
+2
+3
+4
+5
+B(KL →°) × 1011M16
+14
+12
+x(4
+10
+SM: IVebl=3.921 10-2
+↑
+8
+SM: IVebl=4.26 10-2
+B(K+
+6
+2
+3
+4
+B(K →°) × 10l1M3
+14
+12
+10
+8
+B(K+
+9
+1
+2
+3
+4
+5
+B(KL →°) × 1011M13
+14
+12
+×(4
+10
+SM: IVebl=3.921 10-2
+↑
+8
+ SM: IVcbl=4.26 10-2
+B(K+
+6
+2
+3
+4
+B(KL →°) × 1011REFERENCES
+17
+Figure 12: Contour Plots of B(K+ → π+ν¯ν) (left column) and B(KL → π0ν¯ν) (right column) versus
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+9
+.8
+0.00370
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+6
+5
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+IVcblB(KL -→元°  ) × 10ll , M1
+0.00380
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+4.0
+3.5
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+2.5
+2.0
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+0.0410
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+IVcblB(K+ -→ +v) × 10ll, M16
+0.00380
+12
+0.00375
+10
+8
+0.00370
+6
+.4
+0.00365
+0.0405
+0.0410
+0.04150.0420
+0.0425
+0.0430
+IVeblB(KL →°  ) × 10l1 , M16
+0.00380
+5.0
+4.5
+0.00375
+4.0
+3.5
+3.0
+0.00370
+2.5
+2.0
+1.5
+0.00365
+0.0405
+0.0410
+0.0415
+0.0420
+0.0425
+0.0430
+IVecblREFERENCES
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+0.00380
+9.0
+0.00375
+8.5
+8.0
+7.5
+0.00370
+7.0
+6.5
+0.00365
+0.039
+0.040
+0.041
+0.042
+IVcblB(KL -→元°  v) × 10ll , M3
+0.00380
+4.00
+3.75
+0.00375
+3.50
+3.25
+3.00
+0.00370
+2.75
+2.50
+2.25
+2.00
+0.00365
+0.039
+0.040
+0.041
+0.042
+IVcblB(K+ -→π+vv) × 1011 , M13
+0.00380
+9.5
+9.0
+0.00375
+8.5
+8.0
+-7.5
+0.00370
+-7.0
+6.5
+6.0
+0.00365
+0.039
+0.040
+0.041
+0.042
+IVcblB(KL -→元° v) × 10ll , M13
+0.00380
+4.5
+4.0
+0.00375
+3.5
+qn
+3.0
+0.00370
+2.5
+2.0
+1.5
+0.00365
+0.039
+0.040
+0.041
+0.042
+IVcb!REFERENCES
+19
+Figure 14: Contour Plots of B(K+ → π+ν¯ν) (left column) and B(KL → π0ν¯ν) (right column) versus
+|Vcb| and |Vub| in the SM .
+Figure 15: Correlation between B(K+ → π+ν¯ν) and ¯B(Bs → µ+µ−). The gray points span all
+the allowed parameter space in each scenario. The red region corresponds to |Vcb| ∈ [0.0386, 0.0398]
+while the cyan region corresponds to |Vcb| ∈ [0.0422, 0.043]. The SM results in correspondence of two
+values of |Vcb| are displayed, as specified in the legenda. The light gray region and the blue range
+correspond to the experimental range for B(K+ → π+ν¯ν) and ¯B(Bs → µ+µ−), respectively, reported
+in Table 1.
+
+B(K+ -→元+ ) × 10l1 , SM
+0.00380
+8.75
+8.50
+0.00375
+8.25
+8.00
+7.75
+7.50
+0.00370
+7.25
+7.00
+6.75
+0.00365
+0.039
+0.040
+0.041
+0.042
+IVcblB(KL -→元  ) × 1011 , SM
+0.00380
+3.1
+3.0
+0.00375
+2.9
+2.8
+2.7
+0.00370
+2.6
+2.5
+2.4
+0.00365
+0.039
+0.040
+0.041
+0.042
+IVcblM1
+14
+12
+10
+8
+B(K+
+9
+3.0
+3.5
+4.0M16
+14
+12
+10
+SM: IVcbl=3.921 10-2
+8
+SM: IVcbl=4.26 10-2
+B(K+
+6
+3.0
+3.5
+4.0
+B(Bs → μ+ μ)x 109M3
+14
+12
+10
+8
+B(K+
+6
+3.0
+3.5
+4.0M13
+14
+12
+10
+SM: IVebl=3.921 10-2
+8
+ SM: IVebl=4.26 10-2
+B(K+
+6
+3.0
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+

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@@ -0,0 +1,2295 @@
+DOES COMPRESSING ACTIVATIONS HELP MODEL PARALLEL TRAINING?
+Song Bian * 1 Dacheng Li * 2 Hongyi Wang 2 Eric P. Xing 2 3 4 Shivaram Venkataraman 1
+ABSTRACT
+Large-scale Transformer models are known for their exceptional performance in a range of tasks, but training
+them can be difficult due to the requirement for communication-intensive model parallelism. One way to improve
+training speed is to compress the message size in communication. Previous approaches have primarily focused on
+compressing gradients in a data parallelism setting, but compression in a model-parallel setting is an understudied
+area. We have discovered that model parallelism has fundamentally different characteristics than data parallelism.
+In this work, we present the first empirical study on the effectiveness of compression methods for model parallelism.
+We implement and evaluate three common classes of compression algorithms - pruning-based, learning-based,
+and quantization-based - using a popular Transformer training framework. We evaluate these methods across
+more than 160 settings and 8 popular datasets, taking into account different hyperparameters, hardware, and both
+fine-tuning and pre-training stages. We also provide analysis when the model is scaled up. Finally, we provide
+insights for future development of model parallelism compression algorithms.
+1
+INTRODUCTION
+Transformer models have become the dominant model for
+many machine learning tasks (Devlin et al., 2018; Radford
+et al., 2018; Yang et al., 2019; Dosovitskiy et al., 2020; Gong
+et al., 2021; Sharir et al., 2021; Gong et al., 2021). However,
+state-of-the-art Transformer models have a large number
+of parameters, making it difficult for a single GPU to hold
+the entire model. As a result, training large Transformer
+models often requires partitioning the model parameters
+among multiple GPUs, a technique known as model paral-
+lelism (Shoeybi et al., 2019; Rasley et al., 2020). Model
+parallelism strategies often introduce significant commu-
+nication overhead, as demonstrated in Figure 1 (Li et al.,
+2022). For instance, the most commonly used tensor model
+parallelism strategy requires two all-reduce operations over
+a large tensor in each Transformer encoder block per iter-
+ation. This can greatly increase the overall computational
+cost of training the model (Shoeybi et al., 2019).
+To address the issue of high communication overhead in
+model parallelism, one approach is to compress the mes-
+sages communicated among GPUs, such as activation val-
+ues. In the data-parallel setting, several prior works have
+explored compressing gradients to reduce the communica-
+tion cost of training (Seide et al., 2014; Bernstein et al.,
+2018; Dettmers, 2015; Lin et al., 2017; Wang et al., 2018b;
+*Equal contribution
+1Department of Computer Science, Uni-
+versity of Wisconsin-Madison 2Machine Learning Department,
+Carnegie Mellon University 3MBZUAI 4Petuum Inc.. Correspon-
+dence to: Song Bian <[email protected]>.
+(8, 128)
+(32, 128)
+(32, 512)
+Hyper-parameters
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Comm. Overhead 
+ (% of total time)
+TP=1, PP=4
+TP=2, PP=2
+TP=4, PP=1
+Figure 1. Communication overhead of model parallelism with dif-
+ferent batch sizes and sequence lengths on BERTLarge using Py-
+torch 1.12, NCCL, fp16 and 4 GPUs. The x-axis is (batch size,
+sequence length)
+Vogels et al., 2019). However, there has been limited ex-
+ploration of compression methods specifically designed for
+model parallelism. Furthermore, it is important to note that
+compression in model parallelism is fundamentally different
+from compression in data parallelism for two main reasons.
+Firstly, as shown in Figure 2, gradients tend to be low-rank,
+while activations do not. Therefore, low-rank gradient com-
+pression methods, which have been shown to provide state-
+of-the-art end-to-end speedup in communication-efficient
+data-parallel training, may not directly apply to model paral-
+lelism (Vogels et al., 2019). Secondly, the performance ben-
+efits of gradient compression methods can be significantly
+affected by system optimizations in data parallelism (Agar-
+wal et al., 2022). However, model parallelism has a different
+arXiv:2301.02654v1  [cs.LG]  6 Jan 2023
+
+Does compressing activations help model parallel training?
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Dimension Percentage
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Sigma Value Percentage
+Activation
+Gradient
+Figure 2. Low-Rank analysis: Curves are drawn by ordering the
+singular values of the SVD decomposition. The result shows that
+the gradient is low-rank but the activation is not. The activation is
+the output of the 12th transformer layer in BERTLarge model.
+set of system optimization techniques than data parallelism,
+so it is unclear how these optimizations would impact the
+performance of compression methods in model parallelism.
+In this paper, we present the first systematic study of model
+parallelism compression for large Transformer models. We
+evaluate the impact of different compression methods in
+terms of both throughput and accuracy. We conduct ex-
+periments for both pre-training and fine-tuning tasks. (De-
+vlin et al., 2018; Gururangan et al., 2020). In particular,
+we implement and evaluate popular gradient compression
+methods, e.g., Top-K and Random-K as well as a learning-
+based compression method, i.e., auto-encoders (Hinton &
+Zemel, 1993), which can not directly be applied to gradi-
+ent compression but is compatible with activation compres-
+sion. To assist researchers and practitioners training new
+Transformer-based models (Liu et al., 2019; Izsak et al.,
+2021), we study compression methods using different train-
+ing hyper-parameters and hardware setups. We also develop
+a performance model that can be conveniently used to under-
+stand how compression methods would affect throughput
+at larger scales. In total, we evaluate compression methods
+across over 160 different settings with various compression
+algorithms, training stages, hyper-parameters, and hardware,
+and over 8 datasets (Wang et al., 2018a). Our findings in-
+clude the following takeaways.
+Our takeaways. 1. Learning-based compression meth-
+ods are most suitable for model-parallelism. On the fine-
+tuning stage(§4.2, §4.3), only auto-encoders (AEs) can pro-
+vide end-to-end speedup (upto 18%) while preserving the
+model’s accuracy (within ∼3 GLUE score (Wang et al.,
+2018a)). Top-K, Random-K, and quantization methods
+slow down training because their message encoding and de-
+coding overhead is larger than the communication time they
+reduce. Top-K and Random-K also hurt model’s accuracy.
+For the pre-training stage (§4.4), only AE provides speedup
+(upto 16%) while preserving the model’s accuracy (similar
+GLUE score). Top-K marginally improves training time,
+but degrades the accuracy. Quantization slows down the
+training time, and degrades the accuracy.
+2. Training hyper-parameters affect the performance
+benefits of compression methods. None of the compres-
+sion methods can improve performance when the batch size
+and sequence length are small because the cost of message
+encoding and decoding becomes relatively higher (as dis-
+cussed in section §4.6). In practice, we have found that
+the batch size and sequence length need to be at least 32
+and 512, respectively, for the compression methods to pro-
+vide throughput gains. The same is true when fine-tuning is
+performed on a machine with high-bandwidth NVLink con-
+nections between all GPUs (as described in section §4.2).
+3. Early model layers are more sensitive to compression.
+Our observations show that compressing the early layers or
+too many layers significantly decreases the model’s accuracy
+(as discussed in section §4.5), which is consistent with the
+findings of previous research (Wang et al., 2021). In practice,
+we have found that compressing the final 12 layers of a 24-
+layer Transformer model is an effective approach.
+Contributions. We make the following contributions:
+• We conduct the first empirical study on model paral-
+lelism compression methods for Transformer models,
+considering different compression methods, training
+stages, hyper-parameters, and hardware configurations.
+• We implement several popular compression algorithms,
+including Top-K, Random-K, quantization, and auto-
+encoders (AEs), and integrate them into an existing
+distributed training system.
+• We extensively evaluate these algorithms across over
+160 different settings and eight popular datasets. Based
+on our experimental results, we provide several take-
+aways for future model parallelism compression stud-
+ies. We also analyze the speedup when the model size
+and cluster size are scaled up.
+2
+BACKGROUND AND CHALLENGES
+In this section, we first introduce data parallelism and model
+parallelism (§2.1). Then we introduce the challenges in
+model parallelism compression (§2.2).
+2.1
+Data Parallelism and Model Parallelism
+Data Parallelism (DP).
+DP divides the training examples
+among multiple workers (Li et al., 2014; Ho et al., 2013) and
+replicates the model at each worker. During each iteration,
+
+Does compressing activations help model parallel training?
+each worker calculates the model gradient based on its as-
+signed examples and then synchronizes the gradient with the
+other workers (Sergeev & Del Balso, 2018). However, DP
+requires each worker to compute and synchronize gradients
+for the entire model, which can become challenging as the
+model size increases. One issue is that the large gradients
+can create a communication bottleneck, and several previous
+studies have proposed gradient compression methods (Seide
+et al., 2014; Bernstein et al., 2018; Dettmers, 2015; Lin
+et al., 2017; Wang et al., 2018b) to address this. Addition-
+ally, the worker may not have enough memory to train with
+the entire model using even one example, in which case
+model parallelism may be necessary.
+Model Parallelism (MP).
+Model parallelism (MP) di-
+vides the model among multiple workers, allowing large
+models to be trained by only requiring each worker to main-
+tain a portion of the entire model in memory. There are two
+main paradigms for MP: inter-layer pipeline model paral-
+lelism (PP) and intra-layer tensor model parallelism (TP). PP
+divides the layers among workers, with each worker execut-
+ing the forward and backward computations in a pipelined
+fashion across different training examples (Narayanan et al.,
+2019; Li et al., 2021).
+For example, a mini-batch of
+training examples can be partitioned into smaller micro-
+batches (Huang et al., 2019), with the forward computation
+of the first micro-batch taking place on one worker while
+the forward computation of the second micro-batch hap-
+pens on another worker in parallel. TP (Lu et al., 2017;
+Shazeer et al., 2018; Kim et al., 2016) divides the tensor
+computations among workers. In particular, we consider
+a specialized strategy developed for Transformer models
+that divides the two GEMM layers in the attention module
+column-wise and then row-wise, with the same partitioning
+applied to the MLP module (Shoeybi et al., 2019). However,
+TP still involves a communication bottleneck due to the
+need for two all-to-all collective operations in each layer,
+motivating the use of compression to reduce the communi-
+cation overhead of MP (Shoeybi et al., 2019). two all-to-all
+collective operations in each layer (Shoeybi et al., 2019).
+This bottleneck motivates our study to use compression for
+reducing the communication of model parallelism.
+2.2
+Challenges in Model Parallelism Compression
+In data parallelism, synchronizing gradients in large models
+is a major bottleneck, and several gradient compression al-
+gorithms have been proposed (Seide et al., 2014; Bernstein
+et al., 2018; Dettmers, 2015; Lin et al., 2017; Wang et al.,
+2018b) to reduce the communication volume. These algo-
+rithms often rely on the observation that the gradient matrix
+is low-rank. In model parallelism, we have observed that
+communicating activations becomes the bottleneck. How-
+ever, we have identified three challenges when adapting
+gradient compression algorithms for use in model paral-
+lelism.
+First, the low-rank observation for gradient matrices does
+not hold for activation matrices, as shown in Figure 2. The
+sigma value percentage for activation matrices increases
+nearly linearly with the dimension percentage, indicating
+that the activation matrix is not low-rank. Therefore, ap-
+plying gradient compression techniques to activations is
+likely to result in a significant loss of accuracy. Second, the
+performance of compression methods is heavily influenced
+by system optimizations (Li et al., 2020), and many gradi-
+ent compression methods do not lead to speed-ups for data
+parallelism (Zhang et al., 2017; Agarwal et al., 2022) due
+to competition for GPU resources between gradient encod-
+ing computation and backward computation. However, the
+impact of these optimizations on compression methods in
+model parallelism has not been studied. Third, model par-
+allelism introduces the possibility of using learning-based
+compression methods, such as autoencoders (AE) (Hinton &
+Zemel, 1993), which have not been examined in the gradient
+compression literature because they require gradient com-
+putations and raise new considerations. Given these three
+challenges, there is a need for a thorough study of the effects
+of different compression methods in model parallelism.
+3
+IMPLEMENTATION
+In this section, we first introduce the compression algo-
+rithms we evaluate in this work (§ 3.1). Then, we discuss
+implementation details in Sections 3.2 and 3.3.
+3.1
+Compression Algorithms
+In this work, we evaluate a range of popular compres-
+sion algorithms, including sparsification-based approaches,
+learning-based approaches, and quantization-based ap-
+proaches (as illustrated in Figure 3). We use Top-K and
+Random-K as sparsification-based approaches, as they have
+been well-studied in gradient compression (Stich et al.,
+2018). We also implement AEs, which compress messages
+using a small neural network (Hinton & Zemel, 1993). For
+quantization, we use the same scheme as in previous re-
+search (Wang et al., 2022), but compare its performance to
+other compression algorithms in the context of model paral-
+lelism, as the prior work only considered pipeline compres-
+sion over slow networks. Since the activation matrices for
+models are not low-rank (as shown in Figure 2), low-rank
+based compression algorithms (such as PowerSGD (Vo-
+gels et al., 2019)) are not suitable for model parallelism
+compression, and we do not evaluate any low-rank based
+compression algorithms in this work.
+
+Does compressing activations help model parallel training?
+3.2
+Tensor Parallelism Compression
+We base our implementation on Megatron-LM (Shoeybi
+et al., 2019), a popular Transformer models training system
+that supports tensor and pipeline model parallelism. To
+integrate the compression algorithms into Megatron-LM,
+we make the following modifications. For AE, we compress
+the activation before the all-reduce step and invoke the
+all-reduce function as usual. The implementation of AE
+is shown here: for each layer, we have a learnable matrix
+w ∈ Rh×c, and the activation X ∈ Rb×s×h, where b is the
+batch size, s is the sequence length, h is the hidden size,
+and c < h is the compressed size. By using the matrix
+w, we output the compressed activation Xw ∈ Rb×s×c.
+Then, we use a similar technique(a decoder as opposed to
+an encoder) to decompress the compressed activation and
+propagate it to the next layer. However, since the Top-K,
+Random-K, and quantization can output two independent
+tensors with different types (e.g., for Top-K values and
+their indices), we cannot use torch.distributed.all-reduce
+to sum the tensors up directly.
+In light of this, we
+replace the all-reduce step with the all-gather function:
+gather-from-tensor-model-parallel-region,
+which
+is
+implemented
+by
+Megatron-LM.
+We
+use
+torch.topk function to select the k largest absolute
+values of the activation and random.sample function to
+randomly select k values from the activation. Finally, our
+implementation of quantization is based on code released
+by (Wang et al., 2022).
+3.3
+Pipeline Parallelism Compression
+Megatron-LM can only send one tensor to the next pipeline
+stage per round, so we modify its communication functions
+to allow for the transmission of multiple tensors per round
+in order to integrate Top-K, Random-K, and quantization.
+Since we compress the activation in the forward step, us-
+ing compression also reduces the size of the gradient for
+activation and thus the communication cost in the backward
+step. However, this is not the case when using quantization
+to compress the activation for models. This is because, as
+previously noted (Wang et al., 2022), the Pytorch backward
+engine only supports gradients for floating point tensors,
+and therefore the size of the gradient is the same as the size
+of the decompressed activation. Our implementation also
+allows the integration of error-feedback compression algo-
+rithms by retaining the error information from the previous
+compression step.
+4
+EXPERIMENTS
+We next perform experiments using our implementation to
+answer the following questions:
+• What is the impact of activation compression on system
+throughput and which compression method achieves
+the best throughput?
+• What is the impact on the model’s accuracy?
+• How different network bandwidths affect the best com-
+pression method?
+• How do hyper-parameters such as the batch size and
+sequence length affect the benefits of compression?
+We answer these questions in the context of two commonly
+used scenarios in language modeling: fine-tuning on the
+GLUE benchmark (Wang et al., 2018a), and pre-training
+on the Wikipedia (Devlin et al., 2018) dataset and the
+BooksCorpus (Zhu et al., 2015) dataset.
+4.1
+Experimental Setup
+In this section, we briefly describe the hardware, model, and
+other experiment settings.
+System Configuration. To measure the performance of
+compression algorithms over different hardware, our ex-
+periments are conducted on two different setups.
+Our
+first setup uses AWS p3.8xlarge machines which have 4
+Tesla V100 GPUs with all GPUs connected by NVLink.
+AWS p3.8xlarge instances have 10 Gbps network band-
+width across instances. Our second setup uses a local ma-
+chine which also has 4 Tesla V100 GPUs but does not have
+NVLink. All the GPUs are connected by a single PCIe
+bridge. The local server runs Ubuntu 18.04 LTS and the
+server has 125GB of memory.
+Model.
+We use the BERTLARGE model provided by
+Megatron-LM (Shoeybi et al., 2019) which has 345M pa-
+rameters. We configure the model to have 24 layers with
+each layer having a hidden size of 1024 and 16 attention
+heads. We use fp16 training to train the BERTLARGE model.
+Experimental Settings. For fine-tuning, we follow the
+previous work (Devlin et al., 2018; Liu et al., 2019), and
+use micro-batch size 32 and sequence length 512 by de-
+fault. We use one machine with 4 V100 GPUs and vary
+the tensor model-parallel size and the pipeline model-
+parallel size across the following three parallelism degrees:
+{(1, 4), (2, 2), (4, 1)}, where the first number of the tuple
+represents the tensor model-parallel degree and the second
+number of the tuple stands for the pipeline model-parallel
+degree. To investigate the impact of hyper-parameters, we
+conduct experiments that vary the batch size from {8, 32},
+and sequence length from {128, 512} on fine-tuning.
+For pre-training, we use micro-batch size 128, global batch
+size 1024, and sequence length 128. To study the impact of
+the distributed settings, we use the following three different
+parallelism degrees: {(2, 8), (4, 4), (8, 2)}, where the first
+
+Does compressing activations help model parallel training?
+g
+g
+C
+C
+C
+C
+Machine 1
+Machine 2
+C
+Transformer Layer
+Transformer Layer
+ Activation
+Transformer Layer
+Machine 1,2
+Machine 3,4
+DC
+DC
+DC
+DC
+Micro-batch
+C
+DC
+Transformer Layer
+Transformer Layer
+Transformer Layer
+ Activation
+DC
+Figure 3. Illustration of compression on a 6-Layer Transformer model with 4 machines. Machine 1 and Machine 2 maintain the first three
+layers according to the TP strategy (pipeline stage 1). g stands for an all-reduce operation in the forward pass. A compression method C
+is used to reduce the message size for the all-reduce operation to reduce TP communication time. Correspondingly, a de-compression
+method DC is used after the communication. For instance, if AE are used, then C is an encoder, and DC is a decoder. Machine 3 and
+Machine 4 are responsible for the last three layers (pipeline stage 2). A compression method is used before Machine 1 sends the activation
+to Machine 3, and before Machine 2 sends the activation to Machine 4 to reduce PP communication time. The goal of this paper is to
+study the effect of different pairs of C and DC.
+number of the tuple represents the tensor model-parallel
+degree and the second number of the tuple represents the
+pipeline model-parallel degree.
+We also evaluate compression algorithms with different
+parameters. For AE, we use different dimension after com-
+pression from {50, 100}. For Top-K and Random-K algo-
+rithms, we use two comparable settings: (1) Keep the same
+compression ratio as AE (i.e., we compress the activation
+around 10 and 20 times.) (2) Keep the same communica-
+tion cost as AE. Finally, we also tune the parameters for
+quantization and compress the activation to {2, 4, 8} bits.
+By default, we perform experiments on BERTLarge model
+with 24 layers and compress the activation for the last 12
+layers. For instance, when the pipeline model-parallel de-
+gree is 2 and tensor model-parallel degree is 2, we compress
+the activation between two pipeline stages and the communi-
+cation cost over tensor parallelism in the last 12 layers. We
+also vary the number of layers compressed in Section 4.5.
+4.2
+Throughput Benefits for Fine-Tuning
+Takeaway 1 Using non-learning-based compression tech-
+niques to compress activations only slightly improves system
+throughput (by 1% or less) due to the large overhead of these
+methods. However, we see end-to-end speedups of up to
+Notation
+Description
+A1
+AE with encoder output dimension 50
+A2
+AE with encoder output dimension 100
+T1/R1
+Top/Rand-K: same comm. cost as A1
+T2/R2
+Top/Rand-K: same comm. cost as A2
+T3/R3
+Top/Rand-K: same comp. ratio as A1
+T4/R4
+Top/Rand-K: same comp. ratio as A2
+Q1
+Quantization: reduce the precision to 2 bits
+Q2
+Quantization: reduce the precision to 4 bits
+TP
+Tensor model-parallelism degree
+PP
+Pipeline model-parallelism degree
+Table 1. Notation Table. For ease of notation, we use TP/PP to
+denote the degree of tensor/pipeline model parallelism. ‘comm’
+and ‘comp’ are short for ‘communication’ and ‘compression’.
+17.8% when using learning-based compression methods on
+a machine without NVLink.
+When running fine-tune experiments on a p3.8xlarge in-
+stance on Amazon EC2, we cannot improve system through-
+put by using non-learning-based compression algorithms
+(Table 2). Comparing Tables 2 and 3, we can see that the net-
+
+Does compressing activations help model parallel training?
+Distributed Setting
+w/o
+A1
+A2
+T1
+T2
+T3
+T4
+TP=1, PP=4
+591.96
+591.36
+591.47
+594.81
+595.53
+599.65
+605.05
+TP=2, PP=2
+440.71
+437.98
+444.02
+465.73
+473.64
+493.16
+528.93
+TP=4, PP=1
+261.48
+270.22
+275.54
+314.37
+323.90
+356.57
+409.23
+Distributed Setting
+w/o
+R1
+R2
+R3
+R4
+Q1
+Q2
+TP=1, PP=4
+591.96
+749.56
+1,008.64
+1,824.36
+5,572.87
+595.29
+595.45
+TP=2, PP=2
+440.71
+3,377.59
+6,616.30
+17,117.01
+71,058.64
+489.27
+486.54
+TP=4, PP=1
+261.48
+3,254.01
+6,561.22
+16,990.88
+65,121.79
+347.68
+350.50
+Table 2. The average iteration time (ms) for fine-tuning with various compression techniques by varying the distributed setting. The
+results are collected from the AWS p3.8xlarge machine with NVLink by using batch size 32, and sequence length 512. The best setting is
+bolded in the table. And the settings which see benefits compared with the baseline, are underlined.
+With NVLink
+w/o
+A1
+A2
+TP=1, PP=4
+591.96
+591.36
+591.47
+TP=2, PP=2
+440.71
+437.98
+444.02
+TP=4, PP=1
+261.48
+270.22
+275.54
+Without NVLink
+w/o
+A1
+A2
+TP=1, PP=4
+633.17
+620.10
+620.44
+TP=2, PP=2
+646.14
+586.65
+595.25
+TP=4, PP=1
+736.01
+624.62
+636.15
+Table 3. The
+average
+iteration
+time
+(ms)
+for
+fine-tuning
+with/without NVLink. We compare time without compression
+and with AE on different distributed settings, with batch size 32,
+and sequence length 512. The best setting on each machine is
+bolded. And the settings, under which we can gain benefits com-
+pared with the baseline, are underlined.
+work bandwidth across the GPUs can affect the performance
+benefits from compression. In other words, we can improve
+system throughput by at most 17.8% when compressing
+activation for fine-tuning tasks on a 4-GPU machine without
+NVLink. That’s because, without NVLink, the communica-
+tion time for model parallelism is much longer. Thus, while
+the message encoding and decoding time remain unchanged,
+compression methods can provide more throughput benefits
+across lower bandwidth links.
+Furthermore, from Tables 2 and 3, we observe that AE out-
+performs other compression methods. In Table 4, we break-
+down the time taken by each algorithm and find that Top-K,
+Random-K and quantization have large encoding/decoding
+overheads and thus cannot provide end-to-end throughput
+improvements. Although AE slightly increases the time
+taken by the backward step, the ∼ 2× reduction in commu-
+nication time and the limited encoding/decoding overhead
+lead to better overall throughput.
+4.3
+Effect of Compression on Model Accuracy while
+Fine-tuning
+Takeaway 2 Among all evaluated compression algorithms,
+only AE and quantization preserve fine-tuning accuracy.
+From Table 5, we can see that, when using AE and quan-
+tization algorithm for compression, the accuracy loss is
+within 3% except for CoLA and RTE. In Figure 2, we have
+shown that the activation for models is not low-rank. There-
+fore, sparsification-based compression algorithms (Top-
+K/Random-K) lose important information and do not pre-
+serve model accuracy. Given that there is significant accu-
+racy difference for CoLA and RTE, we study the impact
+of varying the number and range of layers compressed for
+these two datasets in Section 4.5.
+4.4
+Throughput Benefits for Pre-training
+Takeaway 3 Only AE and Top-K algorithms improve
+throughput when performing distributed pre-training.
+First, we recap the experimental environment here. For pre-
+training, we use 4 p3.8xlarge instances on Amazon EC2
+and each instance has 4 GPUs with NVLink. From Table 6,
+we can see that using Top-K and AE can speed up pre-
+training by 7% and 16% respectively. Among the three
+distributed settings, TP=4, PP=4 is the best setting for
+pre-training. That is because the communication cost of
+tensor parallelism is larger than that of pipeline parallelism
+and with TP=4, tensor parallel communication happens over
+faster NVLinks.
+Takeaway 4 Compressing activation for models can im-
+prove throughput for pre-training by 16%.
+From Table 7, we notice that using AE and Top-K can
+reduce the waiting time and pipeline communication time
+of pre-training. This is because the inter-node bandwidth
+(10Gbps) is smaller than the intra-node bandwidth (40GB/s
+
+Does compressing activations help model parallel training?
+Compression
+Algorithm
+Forward
+Backward
+Optimizer
+Waiting &
+Pipeline Comm.
+Total Time
+Tensor Enc.
+Tensor Dec.
+Tensor
+Comm.
+w/o
+276.34
+354.16
+5.80
+9.83
+646.14
+\
+\
+150.72
+A1
+213.83
+362.61
+6.16
+4.06
+586.65
+2.16
+3.12
+80.88
+A2
+219.01
+366.51
+5.67
+4.07
+595.25
+3.12
+4.56
+84.48
+T1
+298.93
+355.71
+6.79
+4.38
+665.81
+70.08
+13.68
+85.20
+T2
+305.47
+355.51
+6.36
+3.91
+671.24
+70.32
+16.80
+87.84
+T3
+331.70
+356.80
+5.78
+5.00
+699.27
+72.24
+27.36
+100.80
+T4
+376.72
+359.19
+5.89
+6.60
+748.41
+74.88
+45.36
+124.56
+R1
+2,408.68
+357.02
+6.10
+7.68
+2,779.49
+2,040.24
+15.84
+104.16
+R2
+4,696.99
+356.33
+6.28
+6.20
+5,065.80
+4,244.64
+19.44
+135.84
+R3
+12,603.79
+362.13
+6.81
+25.28
+12,998.01
+11,499.12
+29.76
+139.92
+R4
+46,968.21
+365.36
+7.61
+22.81
+47,363.98
+44,038.56
+47.52
+567.36
+Q1
+274.03
+354.56
+5.88
+7.98
+642.46
+20.64
+32.16
+91.68
+Q2
+282.64
+354.55
+5.58
+7.58
+650.36
+19.92
+30.24
+104.64
+Table 4. We breakdown the average iteration time (ms) for fine-tuning with various compression techniques when using TP=2 and PP=2,
+batch size 32, and sequence length 512. The results are collected from the local machine without NVLink. The total time (ms) is divided
+into following parts: forward step, backward step, optimizer, and waiting & pipeline communication. The last three columns further
+breakdown the tensor encoder/decoder and communication times which are considered part of the forward step.
+Compression
+Algorithm
+MNLI-(m/mm)
+QQP
+SST-2
+MRPC
+CoLA
+QNLI
+RTE
+STS-B
+Avg.
+w/o
+88.07/88.70
+92.02
+95.07
+88.46
+62.22
+93.39
+82.67
+89.16
+86.64
+A1
+85.42/85.43
+91.07
+92.09
+86.14
+54.18
+91.31
+70.04
+87.61
+82.59
+A2
+85.53/85.65
+91.24
+93.23
+85.86
+55.93
+91.01
+65.34
+87.76
+82.40
+T1
+32.05/32.18
+74.31
+83.60
+70.78
+0.00
+58.37
+51.99
+0.00
+44.81
+T2
+44.12/45.67
+39.68
+90.83
+78.09
+0.00
+84.42
+49.82
+62.70
+55.04
+T3
+36.12/36.08
+74.75
+90.25
+81.51
+0.00
+85.41
+54.15
+0.00
+50.92
+T4
+83.85/84.41
+56.39
+93.69
+83.65
+0.00
+90.54
+59.21
+86.02
+70.86
+Q1
+87.25/87.81
+91.71
+93.46
+87.01
+55.99
+61.38
+67.51
+88.02
+80.02
+Q2
+87.85/88.47
+91.93
+93.23
+87.42
+57.67
+93.01
+78.34
+87.43
+85.04
+Table 5. Fine-tuning results over GLUE dataset under the setting that the tensor model-parallel size is 2 and pipeline model-parallel size is
+2. F1 scores are reported for QQP and MRPC, Matthews correlation coefficients are reported for CoLA, and Spearman correlations are
+reported for STS-B, and accuracy scores are reported for the other tasks.
+with NVLink), so compression is effective at reducing the
+communication time between two pipeline stages. From
+Table 9, we can observe that, by using A2 to compress
+the activation over the last 12 layers, we can reduce the
+communication cost between two pipeline stages effectively.
+Takeaway 5 Among all evaluated methods, AE is the
+best strategy to compress activation over pre-training. It
+achieves higher pre-training throughput and preserves the
+model’s accuracy.
+From Table 8, compared with the baseline (without com-
+pression), we can observe that using AE is able to keep
+the accuracy when compared to the uncompressed model.
+In addition, we observe that we can use the AE at the pre-
+training phase and remove it during the fine-tuning phase.
+In other words, we only need to load the parameter of the
+BERTLarge model to do fine-tuning, and the parameters of
+the AE can be ignored. Furthermore, Table 8 shows that pre-
+trained models suffer significant accuracy loss when using
+Top-K for compression. Finally, quantization can preserve
+the model’s accuracy, but we cannot achieve end-to-end
+speedup by using quantization as strategy to compress ac-
+
+Does compressing activations help model parallel training?
+Distributed Setting
+w/o
+A1
+A2
+T1
+T2
+T3
+T4
+TP=2, PP=8
+1,625.16
+1,550.18
+1,579.70
+1,508.34
+1,503.54
+1,593.37
+1,682.87
+TP=4, PP=4
+1,422.40
+1,242.97
+1,223.20
+1,360.37
+1,352.61
+1,410.47
+1,721.87
+TP=8, PP=2
+15,642.30
+14,577.29
+14,073.45
+14,308.12
+14,543.81
+18,919.92
+27,152.07
+Distributed Setting
+w/o
+R1
+R2
+R3
+R4
+Q1
+Q2
+TP=2, PP=8
+1,625.16
+10,308.03
+20,814.20
+55,925.28
+>100,000
+1,759.27
+1,752.24
+TP=4, PP=4
+1,422.40
+15,433.12
+31,565.19
+87,421.46
+>100,000
+2,435.03
+2,594.94
+TP=8, PP=2
+15,642.30
+32,522.47
+61,049.87
+>100,000
+>100,000
+16,414.57
+16,517.44
+Table 6. The average iteration time (ms) for pre-training with various compression techniques by varying the distributed setting. The
+results are collected from 4 AWS p3.8xlarge machines with NVLink by using micro-batch size 128, global batch size 1024, and sequence
+length 128. The best setting is bolded in the table. And the settings, under which we can gain benefits compared with the baseline, are
+underlined.
+Compression
+Algorithm
+Forward
+Backward
+Optimizer
+Waiting &
+Pipeline Comm.
+Total Time
+Tensor Enc.
+Tensor Dec.
+Tensor
+Comm.
+w/o
+467.73
+419.26
+7.42
+527.99
+1,422.40
+\
+\
+91.08
+A1
+546.95
+455.26
+7.29
+233.47
+1,242.97
+8.64
+16.20
+32.76
+A2
+459.26
+467.51
+9.64
+286.78
+1,223.20
+12.96
+20.52
+43.56
+T1
+712.22
+423.91
+7.21
+217.03
+1,360.37
+73.44
+140.4
+80.28
+T2
+671.19
+424.27
+7.35
+249.80
+1,352.61
+81.00
+170.64
+81.36
+T3
+813.03
+433.42
+7.35
+156.67
+1,410.47
+108.00
+268.92
+115.92
+T4
+1,068.38
+444.26
+6.75
+202.48
+1,721.87
+153.36
+427.68
+151.56
+R1
+14,199.56
+421.40
+4.23
+807.93
+15,433.12
+13,185.72
+181.44
+193.68
+R2
+29,344.85
+427.18
+3.91
+1,789.25
+31,565.19
+27,975.24
+181.44
+187.20
+R3
+78,906.91
+444.88
+6.08
+3,707.37
+83,065.23
+73,847.16
+279.72
+649.44
+Q1
+803.63
+417.33
+8.61
+1,205.46
+2,435.03
+90.72
+304.56
+193.68
+Q2
+805.33
+417.74
+7.55
+1,364.32
+2,594.94
+85.32
+271.08
+111.60
+Table 7. We breakdown the average iteration time (ms) for pre-training with various compression techniques when using tensor model-
+parallel size 4, pipeline model-parallel size 4, micro batch size 128, global batch size 1024, and sequence length 128. The results are
+collected from 4 AWS p3.8xlarge machines with NVLink.
+Compression
+Algorithm
+MNLI-(m/mm)
+QQP
+SST-2
+MRPC
+CoLA
+QNLI
+RTE
+STS-B
+Avg.
+w/o
+84.87/84.79
+91.25
+92.43
+86.84
+56.36
+92.26
+70.40
+86.83
+82.89
+A2
+83.77/84.32
+91.14
+91.63
+86.55
+58.61
+91.96
+71.48
+87.16
+82.96
+T2
+61.06/60.93
+80.74
+80.16
+63.83
+10.01
+59.55
+47.29
+0.37
+51.55
+Q2
+84.47/85.32
+91.36
+93.23
+85.10
+58.84
+91.69
+71.84
+86.39
+83.14
+Table 8. Fine-tuning results over GLUE dataset by using the checkpoint obtained by pre-training. F1 scores are reported for QQP and
+MRPC, Matthews correlation coefficient is reported for CoLA, and Spearman correlations are reported for STS-B, and accuracy scores
+are reported for the other tasks.
+tivation over pre-training. In conclusion, it is not a good
+choice to compress the activation by using quantization or
+Top-K.
+4.5
+Varying Compression Layers and Location
+Takeaway 6 When the number of compressed layers in-
+creases, the model accuracy decreases.
+From Figure 4(a), we can observe that the accuracy for RTE
+
+Does compressing activations help model parallel training?
+Pipeline Stages
+Comm. (w/o)
+Comm. (A2)
+0 ↔ 1
+77.82
+76.13
+1 ↔ 2
+88.69
+13.19
+2 ↔ 3
+97.67
+14.09
+Table 9. The average communication time (ms) per iteration be-
+tween two pipeline stages. The first column indicates the pipeline
+stage. And the second column shows the communication time
+per iteration without compression. Moreover, the third column
+presents the communication time with A2. We only compress
+the activation in the last 12 layers and thus the time for the first
+pipeline stage is unchanged.
+w/o
+6
+8
+10
+12
+14
+16
+18
+Number of Layers Compressed
+0
+20
+40
+60
+80
+100
+Metrics (%)
+CoLA
+RTE
+(a) Vary Number of Layers Compressed
+1-12
+4-15
+7-18
+10-21
+13-24
+w/o
+Compression Location
+0
+20
+40
+60
+80
+100
+Metrics (%)
+CoLA
+RTE
+(b) Vary Compression Location
+Figure 4. Fine-tuning results over CoLA and RTE datasets by vary-
+ing the compression location and number of layers compressed.
+The above figure shows that model performance vs the number
+of layers compressed. The below figure shows that model per-
+formance versus the compression location. We use tensor model-
+parallel degree 2, pipeline model-parallel degree 2, batch size 32,
+and sequence length 512.
+and the matthews correlation coefficient for CoLA decreases
+as we increase the number of layers compressed. This is
+because as we increase number of layers compressed, we
+lose more information in the activations leading to a loss in
+accuracy. From Figure 4(a), we observe that compressing
+activations of the last 8 layers is the best strategy to keep
+the accuracy loss within 3% for both datasets.
+Takeaway 7 Compressing the activation for the initial lay-
+ers harms the accuracy of the model.
+We keep the number of layers compressed constant and
+vary the location where we apply compression (Figure 4(b)).
+The results indicate that compressing activations of the first
+few layers of the model significantly harms the model’s
+accuracy. This is because compressing activations generates
+error and the error in the early layers can be accumulated
+and propagated to later layers.
+4.6
+Impact of Model Hyper-parameters
+Takeaway 8 Using a smaller batch size or sequence length
+for fine-tuning negates the throughput benefits from com-
+pression because of the smaller communication cost.
+We vary the batch size from {8, 32} and sequence length
+from {128, 512}, and report the results in Table 11-14. We
+provide more detailed experimental results in Appendix A.
+We notice that when the communication cost over model par-
+allelism is small, the overhead of the compression methods
+can become the bottleneck. Therefore, we cannot improve
+system throughput when using compression algorithms with
+batch size 8 and sequence length 128.
+4.7
+Performance Analysis
+In this section, we develop an analytical cost model to an-
+swer the question:
+What will happen if we scale up the model size and the
+cluster size?
+Given that prior works (Li et al., 2022) have analyzed the
+complexity of various model parallelism strategies, we only
+consider a fixed strategy of using tensor model parallelism
+here. Concretely, we use tensor model parallelism in the
+same node, and pipeline model parallelism across the node,
+a suggested strategy according to (Narayanan et al., 2021).
+In particular, we build the performance analysis for real-
+world settings similar to (Narayanan et al., 2019) in two
+steps. First, we develop our own model on a single-node
+scenario, and we scale up the model size on a single node.
+Second, we increase the cluster size and, according to the
+model-parallelism strategy we choose, assign additional
+GPUs to pipeline parallelism, and use off-the-shelf pipeline
+parallelism cost models to predict the performance (Li et al.,
+2022; Zheng et al., 2022).
+Denote the vocabulary size as V , hidden size as h, sequence
+length as s, and batch size as B. From (Narayanan et al.,
+2021), we know that the number of floating points opera-
+tions (FLOPs) and all-reduce message size in a Transformer
+layer is 96Bsh2 + 16Bs2h, and Bsh respectively.
+If we do not use compression methods, the total time of a
+Transformer layer can be modeled as a sum of the all-reduce
+communication step and the computation time step. We note
+
+Does compressing activations help model parallel training?
+2500
+5000
+7500
+10000 12500
+Hidden size
+0
+50
+100
+150
+200
+Run-time (ms)
+bs16(pred)
+bs32(pred)
+bs64(pred)
+bs128(pred)
+bs16
+bs32
+bs64
+bs128
+(a) Tcomp
+2500
+5000
+7500
+10000
+12500
+Hidden size
+0
+20
+40
+60
+Run-time (ms)
+bs16(pred)
+bs32(pred)
+bs64(pred)
+bs128(pred)
+bs16
+bs32
+bs64
+bs128
+(b) Tcomm
+2500
+5000
+7500
+10000
+12500
+Hidden size
+0
+1
+2
+3
+4
+5
+Run-time (ms)
+bs16(pred)
+bs32(pred)
+bs64(pred)
+bs128(pred)
+bs16
+bs32
+bs64
+bs128
+(c) Toverhead
+2500
+5000
+7500
+10000 12500
+Hidden Size
+1
+2
+3
+4
+5
+Speedup
+bs16(pred)
+bs32(pred)
+bs64(pred)
+bs128(pred)
+bs16
+bs32
+bs64
+bs128
+(d) Speedup
+Figure 5. We vary the batch size and the hidden size to show that our prediction model is accurate compared with the real experimental
+results. The model we use here has only one transformer layer and the tensor model-parallel size is 4. In specific, Figure (a) shows the
+real and predicted computation time with the increase of the hidden size. Figure (b) presents the real and predicted communication time
+between tensor parallelism by varying the hidden size. As for the Figure (c), it presents the computation time of AE with the increase of
+hidden size. In the end, Figure (d) show the total speedup when we use AE to compress activations over tensor parallelism.
+that these two steps can hardly overlap because , the reason
+behind it is that the all-reduce communication depends on
+the previous computational results:
+T = Tcomp(96Bsh2 + 16Bs2h) + Tcomm(Bsh)
+(1)
+Modeling Tcomp
+We model Tcomp as a linear function of
+FLOPs with the coefficient α that corresponds to the peak
+performance of the GPU. In particular, we estimate α using
+ground truth wall clock time of the largest hidden size we
+can fit, where the GPU is more likely to be of the peak
+utilization (Williams et al., 2009). During experiments, we
+found that fitting α using time of smaller hidden sizes can
+result in a 30x higher prediction time when we scale up the
+hidden size because of low GPU utilization. Our prediction
+versus the ground truth time is plotted in Figure 5(a).
+Modeling Tcomm
+we model Tcomm as a piece-wise func-
+tion of the message size (Agarwal et al., 2022). Formally,
+Tcomm(Bsh) =
+�
+c
+if Bsh < d
+βBsh
+if Bsh ≥ d
+If the message size is smaller than a threshold d, then
+Tcomm(Bsh) is a constant c because the worker needs to
+launch one communication round (Li et al., 2020). Other-
+wise, the number of communication rounds is proportional
+to the message size. The fitting result is in Figure 5(b).
+Using AE as the compression method and a fixed encoder
+dimension e (we set e to 100 in this section), the total time
+of a single Transformer layer is:
+TAE = Tcomp(96Bsh2 + 16Bs2h) + Tcomm(Bse)
++ Toverhead
+Compared with the setting without compression, the compu-
+tation time remains unchanged. In addition, Tcomm(Bse)
+is roughly equal to c because Bse is usually smaller than
+the threshold d. In our experiments, the threshold d =
+16 × 128 × 100 = 409600 and c ≈ 0.2.
+Modeling Toverhead
+In AE, Toverhead is the encoder and
+decoder computation time. It is a batched matrix multiplica-
+tion with input dimension B × s × h and h × e. Assuming
+e is kept constant, it can be modeled as Toverhead = γBsh.
+The fitting result is shown in 5(c).
+Since each Transformer layer has identical configurations in
+popular Transformer models (Devlin et al., 2018; Radford
+et al., 2018), the overall speedup ratio is the same as we vary
+the number of layers. Thus, we can estimate the speedup of
+different hidden sizes of any number of Transformer layers
+using
+T
+TAE . We provide the fitting result in Figure 5(d).
+Understanding the trend
+We consider the asymptotic
+behavior of large hidden size h:
+T
+TAE
+≈
+α(96Bsh2 + 16Bs2h) + βBsh
+α(96Bsh2 + 16Bs2h) + γBsh + c
+(2)
+Thus, we can see that as hidden layer size increases, the
+benefits from compression diminish.
+Scaling up the cluster size
+Next we analyze the speedup
+when scaling up the cluster size by combining the pipeline
+parallelism cost model developed in (Li et al., 2022; Zheng
+et al., 2022). Formally, the running time is modeled as a
+sum of per-micro-batch pipeline communication time, per-
+micro-batch of non-straggler pipeline execution time, and
+the per-mini-batch straggler pipeline execution time. To use
+the cost model, we denote the micro-batch size as m, the
+number of nodes n, the number of layers L, the pipeline
+communication time p or pAE.
+We use the default pipeline layer assignment strategy
+in (Shoeybi et al., 2019), which balances the number of
+transformer layers. Thus, every stage takes the same time in
+our scenario: L
+nT or L
+nTAE. We use the pipeline communi-
+cation model in (Jia et al., 2019; Li et al., 2022), p = Bsh
+w ,
+pAE = Bse
+w , where w is the bandwidth. Thus the overall
+speedup can be written as:
+( m−1
+n
++ 1) × LT + (n − 1) × Bsh
+w
+( m−1
+n
++ 1) × LTAE + (n − 1) × Bse
+w
+(3)
+
+Does compressing activations help model parallel training?
+From the Table 10, we see that we can maintain a ∼1.5x
+speedup as we scale the hidden size to 25600. This shows
+that if we increase the number of nodes when we increase in
+hidden size, AE compression retains its benefits. However,
+it is possible to avoid the diminishing speedup by properly
+scaling up the number of nodes n, where the speedup will
+asymptotically converge to h
+e .
+In summary, compression in model parallelism has dimin-
+ishing returns if we only scale up the model on a fixed
+cluster. To gain benefits from compression methods, one
+needs to also properly manage other parameters in the
+cost model, e.g. also scaling up the number of nodes and
+use the pipeline parallelism.
+5
+RELATED WORK
+In this section, we first introduce work related to the de-
+velopment of large Transformer models. Then, we discuss
+strategies to train these models at scale. In the end, we
+discuss prior work that accelerates distributed ML models
+training by using compression techniques.
+Transformer Models. Transformer models were first intro-
+duced by Vaswani et al. (2017) in the machine translation
+context. It has been shown to be effective in various other
+language understanding tasks such as text generation, text
+classification and question answering (Devlin et al., 2018;
+Radford et al., 2018; Wang et al., 2018a; Rajpurkar et al.,
+2016). Recent research has also successfully applied Trans-
+former models to images (Dosovitskiy et al., 2020; Touvron
+et al., 2021), audio (Gong et al., 2021) and beyond (Sharir
+et al., 2021). An N-layers transformer model is composed
+of three major components: (1) An embedding layer that
+maps an input token to a hidden state, (2) A stack of N
+transformer layers, and (3) a prediction layer that maps the
+hidden state proceeded by transformer layers to the task
+output. A transformer layer is composed of an attention
+module (Bahdanau et al., 2014) and several matrix multipli-
+cations. Several optimizations have been proposed to speed
+up Transformer model training such as carefully managing
+the I/O (Dao et al., 2022) and reducing the complexity of the
+attention module (Wang et al., 2020). In this work, we speed
+up the Transformer model training in the distributed setting,
+where we reduce the communication between workers.
+Training Large Transformer models. Several parallelism
+strategies have been proposed to train Transformer mod-
+els. Megatron (Shoeybi et al., 2019) proposes tensor model
+parallelism, which parallelizes the computation in attention
+layers and in the following matrix multiplications. Deep-
+Speed (Rasley et al., 2020) uses a specialized form of
+pipeline parallelism (Huang et al., 2019; Narayanan et al.,
+2019) that treats a transformer layer as the smallest unit
+in pipeline stages. It further combines the tensor model
+parallelism developed in Megatron and data parallelism to
+train Transformer models at the scale of trillion parame-
+ters. (Li et al., 2022) considers a more sophisticated model
+parallelism strategy space for Transformer models and uses
+a cost model to automatically search for the optimal one.
+Our work is orthogonal to the direction of developing new
+parallel training strategies. In this work, we study how to
+compress communication on existing parallel strategies.
+Distributed training with Compression. Distributed ML
+model training requires frequent and heavy synchronization
+between workers. Several directions have been proposed
+to reduce the communication bottleneck by compressing
+the message size. One direction is developed on the data
+parallelism setting, where workers communicate model gra-
+dients (Wang et al., 2021; Agarwal et al., 2022) during
+backward propagation. Common techniques to reduce the
+gradient communication include low-rank updates (Wang
+et al., 2018b), sparsification (Lin et al., 2017), and quanti-
+zation (Seide et al., 2014; Bernstein et al., 2018; Dettmers,
+2015). A more recent direction find that the activation pro-
+duced during the forward propagation in neural networks is
+large, and thus compressing them is beneficial (Wang et al.,
+2022). In particular, they use quantization to compress the
+activation volume between pipeline parallelism workers.
+However, they focus on the geo-distributed setting where
+the network bandwidth is very low. In this paper, we study
+the effect of a rich set of popular compression techniques
+on tensor and pipeline parallelism, and in a typical cloud
+computing setting.
+6
+CONCLUSION
+In this work, we studied the impact of compressing acti-
+vations for models trained using model parallelism. We
+implemented and integrated several popular compression
+algorithms into an existing distributed training framework
+(Megatron-LM) and evaluated their performance in terms
+of throughput and accuracy under various settings. Our re-
+sults show that learning-based compression algorithms are
+the most effective approach for compressing activations in
+model parallelism. We also developed a performance model
+to analyze the speedup when scaling up the model. Our ex-
+periments provide valuable insights for the development of
+improved activation compression algorithms in the future.
+Acknowledgments
+Shivaram Venkataraman is supported by the Office of the
+Vice Chancellor for Research and Graduate Education at
+UW-Madison with funding from the Wisconsin Alumni
+Research Foundation.
+Eric Xing is supported by NSF
+IIS1563887, NSF CCF1629559, NSF IIS1617583, NGA
+HM04762010002, NSF IIS1955532, NSF CNS2008248,
+NSF IIS2123952, and NSF BCS2040381.
+
+Does compressing activations help model parallel training?
+hidden size
+number of layers
+number of nodes
+batch size
+speedup
+6144
+40
+1
+1024
+1.91×
+8192
+48
+2
+1536
+1.75×
+10240
+60
+4
+1792
+1.63×
+12288
+80
+8
+2304
+1.55×
+16384
+96
+16
+2176
+1.46×
+20480
+105
+35
+2528
+1.46×
+25600
+128
+64
+3072
+1.47×
+Table 10. Weak-scaling speedup for the Transformer models. The number of tensor model parallelism is 4, and the micro-batch size is 16.
+As for the change of the hidden size, the number of layers, and the batch size, we follow the setting of Table 1 in (Narayanan et al., 2021).
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+Does compressing activations help model parallel training?
+A
+MORE EXPERIMENTAL RESULTS
+We provide more experimental results in this section.
+Distributed Setting
+w/o
+A1
+A2
+T1
+T2
+T3
+T4
+TP=1, PP=4
+151.82
+154.62
+155.03
+155.78
+155.12
+156.84
+158.58
+TP=2, PP=2
+145.58
+157.49
+163.63
+175.67
+177.39
+186.71
+178.91
+TP=4, PP=1
+136.66
+155.43
+145.97
+170.04
+176.88
+186.06
+190.01
+Distributed Setting
+R1
+R2
+R3
+R4
+Q1
+Q2
+Q3
+TP=1, PP=4
+206.89
+273.49
+449.70
+1,292.15
+154.30
+153.65
+152.33
+TP=2, PP=2
+844.66
+1,589.66
+3,915.32
+15,732.57
+178.09
+175.23
+172.93
+TP=4, PP=1
+820.37
+1,588.59
+3,915.52
+15,469.87
+188.10
+168.90
+167.90
+Table 11. The total time (ms) for fine-tuning with various compression techniques by varying the distributed setting. The results are
+collected from the AWS p3.8xlarge machine with NVLink by using batch size 32, and sequence length 128.
+Distributed Setting
+w/o
+A1
+A2
+T1
+T2
+T3
+T4
+TP=1, PP=4
+106.04
+113.67
+106.35
+109.58
+109.10
+109.18
+110.57
+TP=2, PP=2
+121.26
+142.41
+140.05
+152.91
+154.60
+162.00
+157.12
+TP=4, PP=1
+122.22
+142.33
+139.47
+171.24
+165.77
+172.69
+170.61
+Distributed Setting
+R1
+R2
+R3
+R4
+Q1
+Q2
+Q3
+TP=1, PP=4
+124.39
+137.51
+187.59
+333.61
+108.18
+109.56
+109.49
+TP=2, PP=2
+314.51
+507.00
+998.51
+3,197.42
+163.18
+155.48
+150.31
+TP=4, PP=1
+329.33
+513.89
+1,007.65
+3,406.20
+171.06
+163.96
+152.82
+Table 12. The total time (ms) for fine-tuning with various compression techniques by varying the distributed setting. The results are
+collected from the AWS p3.8xlarge machine with NVLink by using batch size 8, and sequence length 128.
+Distributed Setting
+w/o
+A1
+A2
+T1
+T2
+T3
+T4
+TP=1, PP=4
+154.82
+152.50
+153.47
+155.56
+156.01
+156.81
+158.37
+TP=2, PP=2
+184.48
+175.29
+180.35
+206.56
+204.48
+207.66
+214.30
+TP=4, PP=1
+212.76
+201.39
+200.31
+234.16
+240.42
+242.62
+261.39
+Distributed Setting
+R1
+R2
+R3
+R4
+Q1
+Q2
+Q3
+TP=1, PP=4
+185.83
+231.78
+368.95
+963.62
+155.33
+154.85
+154.82
+TP=2, PP=2
+684.28
+1,228.36
+2,900.86
+10,499.14
+188.82
+189.14
+194.25
+TP=4, PP=1
+722.87
+1,275.57
+2,973.04
+10,891.70
+225.42
+230.69
+242.42
+Table 13. The total time (ms) for fine-tuning with various compression techniques by varying the distributed setting. The results are
+collected from the local machine without NVLink by using batch size 32, and sequence length 128.
+Distributed Setting
+w/o
+A1
+A2
+T1
+T2
+T3
+T4
+TP=1, PP=4
+73.19
+72.94
+72.58
+75.98
+74.15
+73.62
+74.86
+TP=2, PP=2
+100.86
+107.73
+100.54
+113.59
+117.36
+114.86
+112.11
+TP=4, PP=1
+100.73
+107.90
+115.18
+129.31
+124.94
+136.18
+133.91
+Distributed Setting
+R1
+R2
+R3
+R4
+Q1
+Q2
+Q3
+TP=1, PP=4
+82.45
+94.84
+123.78
+239.81
+73.33
+74.41
+71.80
+TP=2, PP=2
+235.02
+366.59
+769.47
+2,183.39
+111.61
+106.75
+101.25
+TP=4, PP=1
+238.28
+368.45
+733.03
+2,509.73
+120.14
+114.73
+118.98
+Table 14. The total time (ms) for fine-tuning with various compression techniques by varying the distributed setting. The results are
+collected from the local machine without NVLink by using batch size 8, and sequence length 128.
+
+Does compressing activations help model parallel training?
+Compression
+Algorithm
+MNLI-(m/mm)
+QQP
+SST-2
+MRPC
+CoLA
+QNLI
+RTE
+STS-B
+w/o
+87.87/88.02
+91.96
+95.18
+87.71
+59.40
+92.99
+76.90
+88.43
+A1
+85.30/85.33
+91.28
+92.32
+84.58
+55.18
+90.87
+59.93
+87.92
+A2
+85.25/85.19
+91.41
+93.23
+86.72
+57.02
+90.92
+64.26
+87.74
+T1
+34.38/34.01
+72.29
+49.54
+70.38
+36.64
+59.89
+53.43
+70.81
+T2
+40.10/38.97
+58.91
+79.24
+66.49
+0.00
+80.40
+45.49
+11.32
+T3
+68.76/69.23
+64.58
+91.40
+80.93
+0.00
+67.34
+66.43
+69.24
+T4
+84.24/85.23
+89.17
+92.09
+81.68
+51.54
+91.71
+63.54
+84.80
+Q1
+86.85/87.58
+91.50
+93.58
+86.96
+59.20
+92.24
+59.57
+86.89
+Q2
+87.46/88.02
+91.82
+94.95
+87.48
+57.02
+93.36
+68.95
+87.84
+Table 15. Fintune results over GLUE dataset under the setting using tensor parallelism size 2, pipeline parallelism size 2, batch size
+32, and sequence length 128. F1 scores are reported for QQP and MRPC, Matthews correlation coefficient is reported for CoLA, and
+Spearman correlations are reported for STS-B, and accuracy scores are reported for the other tasks.
+Compression
+Algorithm
+MNLI-(m/mm)
+QQP
+SST-2
+MRPC
+CoLA
+QNLI
+RTE
+STS-B
+w/o
+86.23/86.07
+91.22
+91.74
+88.17
+59.02
+92.09
+78.70
+88.40
+A1
+82.49/82.41
+89.93
+91.85
+82.43
+43.56
+89.84
+47.29
+87.03
+A2
+82.18/82.23
+90.45
+90.52
+83.54
+0.00
+89.02
+62.82
+87.66
+T1
+36.69/38.13
+66.85
+55.32
+68.93
+0.00
+59.13
+52.71
+1.97
+T2
+43.92/43.66
+73.63
+51.26
+62.26
+0.00
+60.13
+49.82
+0.00
+T3
+49.07/47.96
+72.02
+83.57
+69.33
+12.04
+83.60
+55.60
+84.96
+T4
+83.99/84.37
+35.78
+68.30
+83.54
+47.33
+60.52
+64.62
+86.72
+Q1
+84.91/85.18
+90.54
+92.43
+85.91
+53.25
+60.68
+57.04
+87.91
+Q2
+85.66/86.09
+90.99
+91.74
+86.84
+53.92
+91.31
+75.81
+88.19
+Table 16. Fintune results over GLUE dataset under the setting using tensor parallelism size 2, pipeline parallelism size 2, batch size 8, and
+sequence length 128. F1 scores are reported for QQP and MRPC, Matthews correlation coefficient is reported for CoLA, and Spearman
+correlations are reported for STS-B, and accuracy scores are reported for the other tasks.
+

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+Highlights
+Active Learning-based Domain Adaptive Localized Polynomial Chaos Expansion
+Lukáš Novák, Michael D. Shields, Václav Sadílek, Miroslav Voˇrechovský
+• Effective construction of a general purpose surrogate model based on polynomial chaos expansion.
+• Novel method for sequential decomposition of the input random space and construction of local approxi-
+mations.
+• Sequential domain decomposition and sample size extension based on an active learning methodology.
+• Active learning is represented by variance-based Θ criterion developed for polynomial chaos expansion.
+arXiv:2301.13635v1  [cs.LG]  31 Jan 2023
+
+Active Learning-based Domain Adaptive Localized Polynomial Chaos
+Expansion
+Lukáš Novák∗
+Brno University of Technology, Brno, Czech Republic
+Michael D. Shields
+Johns Hopkins University, Baltimore, USA
+Václav Sadílek, Miroslav Voˇrechovský
+Brno University of Technology, Brno, Czech Republic
+Abstract
+The paper presents a novel methodology to build surrogate models of complicated functions by an active learning-
+based sequential decomposition of the input random space and construction of localized polynomial chaos expan-
+sions, referred to as domain adaptive localized polynomial chaos expansion (DAL-PCE). The approach utilizes
+sequential decomposition of the input random space into smaller sub-domains approximated by low-order poly-
+nomial expansions. This allows approximation of functions with strong nonlinearties, discontinuities, and/or sin-
+gularities. Decomposition of the input random space and local approximations alleviates the Gibbs phenomenon
+for these types of problems and confines error to a very small vicinity near the non-linearity. The global behavior
+of the surrogate model is therefore significantly better than existing methods as shown in numerical examples.
+The whole process is driven by an active learning routine that uses the recently proposed Θ criterion to assess
+local variance contributions [1]. The proposed approach balances both exploitation of the surrogate model and
+exploration of the input random space and thus leads to efficient and accurate approximation of the original
+mathematical model. The numerical results show the superiority of the DAL-PCE in comparison to (i) a single
+global polynomial chaos expansion and (ii) the recently proposed stochastic spectral embedding (SSE) method
+[2] developed as an accurate surrogate model and which is based on a similar domain decomposition process.
+This method represents general framework upon which further extensions and refinements can be based, and
+which can be combined with any technique for non-intrusive polynomial chaos expansion construction.
+Keywords: Polynomial Chaos Expansion, Adaptive Sampling, Sequential Sampling, Local Approximations,
+Active Learning, Stochastic Spectral Embedding
+1. Introduction
+The Polynomial Chaos Expansion (PCE), originally proposed by Norbert Wiener [3] and further investigated
+in the context of engineering problems by many researchers, e.g. [4, 5], is a preferred method for uncertainty
+quantification (UQ) and surrogate modeling in industrial applications [6, 7] thanks to its efficiency and powerful
+post-processing. Once a PCE is available for a given problem, the constructed explicit function can be exploited
+∗Corresponding author
+Email addresses: [email protected] (Lukáš Novák), [email protected] (Michael D. Shields),
+[email protected] (Václav Sadílek), [email protected] (Miroslav Voˇrechovský)
+Preprint submitted to Computer Methods in Applied Mechanics and Engineering
+February 1, 2023
+
+to directly estimate important properties of the original problem including its statistical moments, response prob-
+ability distribution or sensitivity indices (without additional sampling [8]), which brings significant efficiency for
+surrogate modeling, sensitivity analysis, uncertainty quantification and reliability analysis [9].
+The PCE, in its non-intrusive form, offers a convenient way to perform probabilistic analysis of any black-box
+model, e.g. finite element models representing complex physical systems in engineering. There are generally
+two types of non-intrusive methods to calculate the deterministic PCE coefficients: spectral projection and lin-
+ear regression. The spectral projection approach utilizes the orthogonality of the multivariate polynomials and
+calculates the coefficients using inner products. The spectral projection leads to an explosion of computational
+complexity referred to as the curse of dimensionality. Therefore, the non-intrusive approach based on linear re-
+gression is often preferred. Although it is typically less expensive than the spectral projection (the number of
+samples should be at least � (P ln(P)), where P is the number of terms in the PCE [10, 11]), it suffers from the
+curse of dimensionality as well, since the number of PCE terms grows rapidly with both dimension and maximum
+polynomial order. Therefore, it becomes necessary to employ advanced adaptive techniques to construct sparse
+PCEs that yield efficient solutions for real-world physical systems.
+Regression-based PCE can be significantly affected by the selected sampling scheme, as was recently shown
+in an extensive review paper [12] comparing several general statistical sampling techniques. However, PCE
+construction as a linear regression model is a very problem specific task and it can be highly beneficial to use
+methods that exploit information from the given mathematical model and sequentially update the surrogate
+model – referred to as active learning. Active learning is a common approach for surrogate-based reliability
+analysis, wherein an initial experimental design is iteratively updated based on the current estimate of the limit-
+state surface [13, 14, 15]. Active learning for reliability analysis with PCE was used e.g. in [16, 17, 18]. For
+general UQ studies, some recent studies have focused on general sequential sampling for PCE based on space-
+filling criteria or alphabetical optimality [19, 20]. However, it is beneficial to use both exploitation (leveraging
+model behavior) criteria and exploration (space filling) criteria to define an optimally balanced criterion [21].
+Such sequential sampling for sparse Bayesian learning PCE combining both aspects – epistemic uncertainty of
+the statistical inference (exploration) together with quadratic loss function (local exploitation) – was recently
+proposed in [22]. However, its application is limited to PCE built by sparse Bayesian learning only.
+The authors of this paper recently proposed a general active learning method based on sequential adaptive
+variance-based sampling [1], which is an efficient tool for accurate surrogate modeling that is sufficiently general
+for further extension [23]. Although this approach leads to superior results in comparison to standard approaches
+without active learning, it is limited by the inherently smooth nature of the PCE. More specifically, polynomial
+basis functions are not able to approximate functions with discontinuities or singularities. Moreover, it is nec-
+essary to use high-order polynomials to approximate functions with local non-linearities, even when the rest of
+the input random space could be easily approximated by a low-order PCE. This can lead to spurious oscillations
+in the approximation and over-fitting. To overcome this limitation, we propose a method to construct localized
+PCEs based on the concept of divide-and-conquer, i.e. decomposition of the input random space to sub-domains
+approximated by many low-order PCEs instead of a single high-order global PCE. Although this concept is not
+entirely new in stochastic finite elements [24] and stochastic collocation [25, 26], there is no such approach
+for non-intrusive PCE. However there are two primary techniques based on similar concepts as described in the
+following section.
+1.1. Related Developments
+Stochastic Spectral Embedding (SSE) [2] is a general approximation technique based on a decomposition of
+the input random space and the construction of embedded local approximations. Although it is generally possible
+to use any spectral approximation technique, it is beneficially coupled with PCE. SSE is based on a novel idea of
+embedding – instead of constructing local approximations of the original mathematical model, local surrogates
+are constructed to approximate the residuals between the model and approximation from the previous level of
+the decomposed space. Although such an approach can lead to significant improvement in comparison to a single
+global approximation [2], it is not a sequential approach based on active learning and thus it does not iteratively
+reflect new information obtained from the previous steps of the algorithm. Active learning is crucial in analysis of
+functions with discontinuity or singularity because it allows for the aforementioned exploration and exploitation
+necessary to find and resolve these features. For the sake of completeness, active learning for SSE has been
+2
+
+proposed for reliability analysis [27], but it does not lead to an accurate approximation over the entire input
+random space. Its accuracy is limited to regions around the limit surface, which are important for an estimation
+of failure probability.
+The second related technique is Multi-element generalized Polynomial Chaos Expansion (ME-gPC) [28]. ME-
+gPC was developed as an extension of generalized PCE based on Wiener-Askey scheme [29] allowing analysis of
+models with arbitrary distribution of input random vector. The ME-gPC method consists of three main parts: de-
+composition of the input random space, numerical construction of locally orthogonal polynomials and an adaptive
+procedure based on the decay rate of local error in estimated variance derived from local PCE. ME-PCE applies
+an h-type mesh refinement procedure akin to mesh refinement in finite element methods. By doing so, they
+introduce a structured grid of uniform points in each new element and solve for the PCE coefficients. This can
+be cumbersome and does not afford the flexibility to adaptively select sparse and near-optimal training points.
+Moreover, we note that the ME-gPC was created mainly for uncertainty propagation in models with arbitrary
+input distributions, and thus in contrast to SSE, its objective is not necessarily to construct the best possible sur-
+rogate model using adaptive algorithms, but rather to minimize errors in response statistics. This is a subtle, but
+important difference that distinguishes its use as a predictive tool from that of a tool for statistical estimation.
+1.2. Contributions of this paper
+This paper describes a novel method, termed Domain Adaptive Localized PCE (DAL-PCE) that applies adap-
+tive sequential decomposition of the input random space and adaptive sequential sampling within the sub-
+domains. Both of these features are based on recently a proposed criterion for variance-based sequential sta-
+tistical sampling, developed specifically for PCE in [30]. In the context of previously described methods SSE and
+ME-gPC, the proposed novel approach can be though to lie between them. Like SSE, it is developed specifically
+for the construction of accurate surrogate models, especially for functions with high non-linearity or disconti-
+nuity. But the decomposition of the input random space is rather similar to ME-gPC. The uniqueness of our
+proposal lies in the combination of active learning, sequential sampling, sequential decomposition of the input
+space and regression-based PCE using sparse solvers such as Least Angle Regression (LARS) allowing adaptivity
+and learning in each iteration of the proposed algorithm.
+2. Polynomial Chaos Expansion
+Assume a probability space (Ω,F,P), where Ω is an event space, F is a σ-algebra on Ω and P is a probability
+measure on F. If the input variable of a mathematical model, Y = f (X), is a random variable X(ω),ω ∈ Ω, the
+model response Y (ω) is also a random variable. Assuming that Y has a finite variance, PCE represents the output
+variable Y as a function of an another random variable ξ called the germ with a known distribution
+Y = f (X) = f PCE(ξ),
+(1)
+and represents the function f (X) via infinite polynomial expansion. A set of polynomials, orthogonal with respect
+to the distribution of the germ, are used as a basis of the Hilbert space L2 (Ω,F,P) of all real-valued random
+variables of finite variance, where P takes over the meaning of the probability distribution. The orthogonality
+condition is given by the inner product of L2 (Ω,F,P) defined for any two functions ψj and ψk for all j ̸= k
+with respect to the weight function pξ (probability density function of ξ) as:
+〈ψj,ψk〉 =
+�
+ψj(ξ)ψk(ξ)pξ(ξ) dξ = 0.
+(2)
+This means that there are specific orthogonal polynomials associated with the corresponding distribution of
+the germ via its weighting function. For example, Hermite polynomials orthogonal to the Gaussian measure are
+associated with normally distributed germs. Orthogonal polynomials corresponding to other distributions can
+be chosen according to Wiener-Askey scheme [29] or constructed numerically [31]. For further processing, it is
+beneficial to use normalized polynomials (orthonormal), where the inner product of ith and jth polynomials is
+equal to the Kronecker delta δjk, i.e. δjk = 1 if and only if j = k, and δjk = 0 otherwise.
+3
+
+In the case of XXX and ξ being vectors containing M independent random variables, the polynomial Ψ(ξ) is
+multivariate and it is built up as a tensor product of univariate orthonormal polynomials, i.e.
+Ψααα(ξ) =
+M
+�
+i=1
+ψαi(ξi),
+(3)
+where ααα ∈ �M is a set of integers called the multi-index reflecting polynomial degrees associated to each ξi. The
+quantity of interest (QoI), i.e. the response of the mathematical model Y = f (XXX), can then be represented as [5]
+Y = f (XXX) =
+�
+ααα∈�M
+βαααΨααα(ξ),
+(4)
+where βααα are deterministic coefficients and Ψααα are multivariate orthonormal polynomials.
+2.1. Non-intrusive computation of PCE coefficients
+For practical computation, the PCE expressed in Eq. (4) must be truncated to a finite number of terms P.
+One can generally choose any truncation rule (e.g. tensor product of polynomials up to the selected order p),
+but the most common truncation is achieved by retaining only terms whose total degree |ααα| is less than or equal
+to a given p, in which case the truncated set of PCE terms is then defined as
+AM,p =
+�
+ααα ∈ �M : |ααα| =
+M
+�
+i=1
+αi ≤ p
+�
+.
+(5)
+The cardinality of the truncated index set AM,p is given by
+card AM,p = (M + p)!
+M! p!
+≡ P .
+(6)
+When the PCE is truncated to a finite number of terms, there is an error ϵ in the approximation such that
+Y = f (XXX) =
+�
+ααα∈A
+βαααΨααα(ξ) + ϵ .
+From a statistical point of view, PCE is a simple linear regression model with intercept. Therefore, it is possible
+to use ordinary least squares (OLS) regression to minimize the error ϵ.
+Knowledge of vector βββ fully characterizes the approximation via PCE. To solve for βββ, first it is necessary to
+create Nsim realizations of the input random vector XXX and the corresponding results of the original mathematical
+model Y, together called the experimental design (ED). Then, the vector of P deterministic coefficients βββ can be
+determined by OLS as
+βββ = (Ψ TΨ)−1 Ψ TY,
+(7)
+where Ψ is the data matrix
+Ψ =
+�
+Ψi j = Ψj(ξ(i)), i = 1,..., Nsim, j = 0,..., P − 1
+�
+.
+(8)
+A well-known problem, the curse of dimensionality, states that P is highly dependent on the number of input
+random variables M and the maximum total degree of polynomials p, which is clear from Eq. (6). Considering
+that estimation of βββ by regression requires at least � (P ln(P)) number of samples for stable solution [10, 11],
+the problem can become computationally highly demanding in case of a large or strongly non-linear stochastic
+models. Although one can use advanced model selection algorithms such as Least Angle Regression (LAR) [32, 4],
+orthogonal matching pursuit [33] or Bayesian compressive sensing [34] to find an optimal set of PCE terms, and
+thus reduce the number of samples needed to compute the unknown coefficients, the benefit of these techniques
+is significant only if the true coefficient vector is sparse or compressible. The sparse set of basis functions obtained
+by any adaptive algorithm is further denoted by A for the sake of clarity.
+4
+
+2.2. Approximation Error Estimation
+Once the PCE is constructed, it is crucial to estimate its accuracy. Further, the PCE accuracy can be used
+to directly compare several PCEs to choose the best surrogate model. Ideally the ED should be divided into
+validation and training sets, but this might be extremely computationally demanding in engineering applications
+with complex numerical models. Therefore in the field of uncertainty quantification (UQ) of engineering models,
+it is preferred to estimate the approximation error directly from the training set, without any additional sampling
+of the original model. A common choice is the coefficient of determination R2, which is well-known from machine
+learning or statistics. However, R2 may lead to over-fitting and thus advanced methods should be used. One of
+the most widely-used methods is the leave-one-out cross-validation (LOO-CV) error Q2. The LOO-CV is based on
+residuals between the original surrogate model and the surrogate model built with the ED while excluding one
+realization. This approach is repeated for all realizations in the ED and the average error is estimated. Although
+the calculation of Q2 is typically highly time-consuming, it is possible to obtain results analytically from a single
+PCE as follows [35]:
+Q2 =
+1
+Nsim
+Nsim
+�
+i=1
+�
+g
+�
+x (i)�
+− gPCE �
+x (i)�
+1 − hi
+�2
+σ2
+Y,ED
+,
+(9)
+where σ2
+Y,ED is the variance of the ED calculated using the original mathematical model and hi represents the ith
+diagonal term of matrix H = Ψ
+�
+Ψ TΨ
+�−1 Ψ T.
+2.3. Statistical Moments Derived from PCE
+The form of PCE as a linear summation over orthonormal polynomials allows for powerful and efficient
+post-processing. In particular, once a PCE approximation is created, it is possible to directly estimate statistical
+moments of the output from the expansion.
+The first statistical moment (the mean value) is simply the first deterministic coefficient of the expansion
+µY =
+�
+Y 1�
+= β000. The second raw statistical moment,
+�
+Y 2�
+, can be estimated by
+�
+Y 2�
+=
+� ��
+ααα∈A
+βαααΨααα (ξ)
+�2
+pξ (ξ) dξ =
+�
+ααα1∈A
+�
+ααα2∈A
+βααα1βααα2
+�
+Ψααα1 (ξ)Ψααα2 (ξ) pξ (ξ) dξ
+(10)
+=
+�
+ααα∈A
+β2
+ααα
+�
+Ψααα (ξ)2pξ (ξ) dξ =
+�
+ααα∈A
+β2
+ααα 〈Ψααα,Ψααα〉.
+Considering the orthonormality of the polynomials, it is possible to obtain the variance σ2
+Y =
+�
+Y 2�
+− µ2
+Y as the
+sum of all squared deterministic coefficients except the intercept (which represents the mean value), i.e.
+σ2
+Y =
+�
+ααα∈A
+ααα̸=000
+β2
+ααα.
+(11)
+Note that the computation of higher statistical central moments, specifically skewness γY (3rd moment) and
+kurtosis κY (4th moment), are more complicated since they require triple and quad products. These can be
+obtained analytically only for certain polynomial families, e.g. formulas for Hermite and Legendre polynomials
+(and their combination) can be found in [30].
+3. Active Learning-based Domain Adaptive Localized PCE (DAL-PCE)
+In this section, we propose a novel methodology to constructed localized PCEs designed for highly non-linear
+functions, termed Domain Adaptive Localized PCE (DAL-PCE). Instead of increasing the maximum polynomial
+order p (p-adaptivity), which brings high computational requirements due to the curse of dimensionality, we
+5
+
+propose to decompose the input random space into several sub-domains approximated by low-order PCEs (h-
+adaptivity). Although this idea is not entirely new, we use this approach in combination with novel active learning
+methods to identify domains for refinement and for sequential sample selection and regression-based PCEs. This
+allows us to use any sparse adaptive solver (e.g. LAR) and thus it can be easily implemented into the existing
+software packages [36, 37]. In the following sections, we define the requisite components of the proposed method
+and provide an algorithm (Algorithm 1) for its implementation.
+3.1. Variance-based Adaptive Sequential Sampling
+The decomposition of the input random space is a sequential process coupled with adaptive sampling assuring
+optimal coverage of the sub-domains of interest. The whole process thus consists of two steps: (i) identification of
+an important sub-domain, that is, a domain that is either large compared to other sub-domains or that is associated
+with a high local variance; and (ii) identification of the best positions for additional samples extending the current
+ED in the selected sub-domain. Each of these steps must be based on a criterion that balances exploration of the
+input random space with exploitation of the surrogate model, which in our case is in the form of a PCE. The
+Θ-criterion for adaptive sequential sampling, which is driven by the output variance and its approximation via
+local variance using PCE [1], is employed for both steps. We will first discuss the process for adaptive sequential
+sampling within a specified sub-domain in this section. This will be followed by the process for refinement of the
+domain in the subsequent sections.
+Consider a pool of candidate samples containing realizations of the random vector ξ generated by an arbitrary
+sampling technique, e.g., Latin Hypercube Sampling (LHS) [38, 39] or Coherence sampling [40, 41, 10]. From
+this pool of candidates, we select the best sample using a method inspired by the sequential sampling proposed in
+[21] and based on Koksma-Hlawka inequality [42]. The Θ-criterion for PCE, which accounts for both variation
+of the function and discrepancy of the samples, was proposed as follows [1]:
+Θ(ξ(c)) ≡ Θc =
+�
+σ2
+A(ξ(c)) · σ2
+A(ξ(s))
+ave variance density
+l M
+c,s
+vol.
+≡
+�
+σ2
+c · σ2
+s l M
+c,s.
+(12)
+The criterion is a product of two terms – the exploitation term (denoted as “ave variance density”) and the
+exploration part (the distance term lc,s raised to the domain dimension) – which are multiplied to maintain an
+optimal balance between exploration and exploitation [1].
+The exploration aspect is maintained by accounting for the distance lc,s between a candidate ξ(c) and its nearest
+neighboring realization from the existing ED, ξ(s) as
+lc,s =
+�
+�
+�
+M
+�
+i=1
+|ξ(c)
+i
+− ξ(s)
+i |2.
+(13)
+If the criterion was reduced to this term only, sequential filling of the greatest empty regions would occur, con-
+verging to uniform space coverage in the spirit of the space-filling “miniMax criterion” [43, 44, 45].
+The exploitation component is motivated by the desire to sample points in regions with the greatest contribu-
+tions to the total variance of the QoI σ2
+Y , i.e. at points with the highest variance density. Once the PCE has been
+established at any given stage of the algorithm, the variance density is computationally cheap to evaluate for any
+location ξ as
+σ2
+A(ξ) =
+� �
+ααα∈A
+ααα̸=000
+βαααΨααα (ξ)
+�2pξ (ξ).
+(14)
+The local variance is therefore estimated directly using the basis functions and coefficients β of the PCE. When
+considering a candidate “c”, an estimate of the variance contribution of the region between the candidate and its
+nearest neighbor “s” may be obtained by averaging the local variance densities between the two. Therefore, we
+can say that the candidate with the greatest Θc criterion is the one that represents the largest amount of total
+variance to be refined by its selection.
+A significant advantage of this method is the ability to add candidates into an existing ED one-by-one. Thus,
+it can be employed at any moment of the PCE construction process. Moreover, this learning function can be
+6
+
+combined with any sampling algorithm for the construction of the initial ED and candidates for extension. The
+ideas behind the Θ criterion will now be used in the proposed domain decomposition and ED extension algorithm.
+3.2. Decomposition of Input Random Space
+The core of the proposed approach is a sequential decomposition of the input random space � for the construc-
+tion of local approximations. This approach assumes that the original mathematical model can be approximated
+by piecewise low-order PCEs that are valid only in individual sub-domains of �. Therefore, in the proposed ap-
+proach, the input random space is sequentially decomposed into n� smaller non-overlapping sub-domains �i ⊂ �
+that collectively fill the full input random space �, i.e.
+n�
+�
+i=1
+�i = �
+such that
+�i ∩ �j = �
+∀i, j
+(15)
+In each iteration of the algorithm, a single sub-domain �i (referred to as the parent) is identified for refinement
+and divided by a plane perpendicular to the direction of one selected input random variable. Specifically, �i is
+divided into a refinement-child �i, which is further processed, and an inheriting-child �⋆
+i adopting the PCE from
+the parent as illustrated for a one-dimensional function in Fig. 1. In this case, we see that the space is divided
+into two subdomains. In the left (refinement child) a new PCE is constructed. In the right (inheriting child), the
+original PCE is retained. Such process assures an exhaustive decomposition into disjoint subsets i.e. �i = �i⊕�⋆
+i .
+This sequential domain decomposition is illustrated in Fig. 2, which depicts the original input random space and
+the first four iterations of the decomposition process.
+Figure 1: The first iteration of the algorithm: the original sub-domain is split and the new local PCE is constructed in �i (red background),
+while the second part in �⋆
+i inherits the PCE approximation from the original domain.
+In contrast to SSE [2], the selection of a single sub-domain for refinement in each iteration is based on an active
+learning approach, the details of which are provided in subsequent sections. Importantly, actively integrating
+information from the original mathematical model leads to a significantly more effective decomposition of the
+space and thus assures accurate approximations, even for small-size EDs. On the other hand, the identified
+decomposition and the associated ED are directly connected to the given mathematical model and therefore
+might be inefficient for general statistical analysis.
+The complete surrogate model is assembled from the n� local PCEs associated with all sub-domains �i as:
+Y ≈
+n�
+�
+i=0
+�
+αααi∈Ai
+βαααiΨαααi(ξ)��i(ξ),
+(16)
+where ��i(ξ) represents indicator function, i.e. ��i(ξ) = 1 only if ξ ∈ �i and ��i(ξ) = 0 otherwise. In other
+words, to approximate the original model at any point, it suffices to determine the one relevant sub-domain and
+use the corresponding local PCE. Each such local PCE has its own set of basis functions Ai and corresponding co-
+efficients βαααi, which can be obtained by any model-selection algorithm. In this paper the OLS and LAR algorithms
+are employed, but generally any non-intrusive technique can be used.
+7
+
+Figure 2: The first four steps of the decomposition of a 3D space of input random variables. The thick black lines outline the parent domain
+selected for division. The red and green boxes inside it represent the two newly created refinement-child �i (red) and inheriting-child �⋆
+i
+(green) sub-domains created by splitting the parent domain �i (bold boundaries), selected via Eq. (17), by the cutting plane (blue). The
+cutting plane is perpendicular to the variable selected for splitting (blue arrow).
+3.3. Domain Selection via Modified Variance-based Criterion
+The selection process to identify the “best” subdomain for possible division is governed by extending the
+Θ-criterion from Eq. (12) as follows:
+Θi = Wi · exp(Q2
+i )
+weight of subdomain
+·
+�
+σ2
+Ai(ξ(c)) · σ2
+Ai(ξ(s)) l M
+c,s
+Θc in ith subdomain
+.
+(17)
+This extended criterion aims to identify sub-domains of the input random space associated with the maximum
+value of Θc, while simultaneously accounting for the size of each subdomain and the accuracy of the existing
+local PCE. The former is calculated using Eq. (12) calculated for a rich pool of screening global candidates,
+while the latter are measured by incorporating the volume of each sub-domain Wi and the LOO-CV error Q2
+i ,
+respectively. The LOO-CV term, exp(Q2
+i ), can be thought to artificially inflate the domain volume as a penalization
+for inaccurate approximation. When the approximation is perfect (Q2
+i = 0) the true volume of the sub-domain is
+used. Meanwhile, a poor approximation with Q2
+i = 1 leads to roughly 2.72 times increased volume.
+The three terms featured in Eq. (17) aim at different aspects affecting the accuracy of the final surrogate
+model: large sub-domains are preferred by Wi, sub-domains containing poor PCE approximation are promoted
+via exp(Q2
+i ) and finally, Θc prefers sub-domains with high concentration of variance. Note that Θc is calculated
+for a rich pool of screening candidates, and Wi and exp(Q2
+i ) are calculated directly from the geometry of existing
+sub-domain and the local PCE model, respectively. The product of all three terms in the extended criterion
+therefore maintains the desired balance and assures the selection of the sub-domain, �i, that currently seems to
+be the most important for increasing the accuracy of the PCE surrogate model.
+Sub-domain � with the greatest Θi is selected and one of the operations described in detail in Sec. 3.6 is
+performed, depending on whether �i contains a critical number of ED points. Two scenarios can occur:
+• �i contains a sufficient number of ED points (ni ≥ nsim) to ensure accuracy of a PCE on the domain.
+Therefore, it becomes a parent �i (bold boundaries in Fig. 2) and is divided into two parts by a selected
+rule. The child domain containing the decisive candidate with the greatest Θc becomes the refinement-child
+�i (see the red subdomains in steps 1 − 4 in Fig. 2). The remaining volume becomes an inheriting-child
+denoted �⋆
+i (see the green subdomains in Fig. 2), which retains the PCE from the parent. Division occurs
+by a cutting plane, oriented perpendicular to the selected direction (blue arrows in Fig. 2) and naturally, the
+coordinates of the cutting plane are restricted to the bounding box of the selected parent �i, see Sec. 3.6.
+If needed, the refinement-child domain �i is sequentially filled with additional ED points (according to Θc)
+to reach ni = nsim needed to construct a new PCE approximation.
+• �i does not contain a sufficient number of ED points (ni < nsim). The domain is not divided because the
+suggestion for division is based on insufficient information. Instead, new ED points are sequentially added
+to �i, again using the Θc criterion. Note that this scenario practically arises when the selected domain was
+an inheriting-child in the previous iteration. In this case, the selected domain has inherited a PCE model
+that was constructed over a larger domain. When that domain was divided, it was left with an insufficient
+number of points from which to construct a new PCE.
+8
+
+3.4. PCE Basis Functions
+Without loss of generality, the proposed method operates on the M-dimensional unit hypercube with uniform
+distributions of input random variables, i.e. XXX ∼ U[0,1]M. In the case of a general joint probability distribution
+of XXX, it is always possible to transform input random vector to the unit hypercube by Rosenblatt transformation
+[46], Nataf transformation [47] or various methods based on copulas [48]. Standard normalized Legendre
+polynomials, orthonormal to the uniform distribution, can thus be used as basis functions for the PCE. However,
+due to the decomposition of the input random space to smaller sub-domains, each with lower bound ai and upper
+bound bi, it is necessary to use univariate scaled orthonormal Legendre polynomials of nth order ˜
+ψn(ξ) defined
+as follows:
+˜
+ψn(ξ) = ψn
+�2ξ − ai − bi
+bi − ai
+�
+,
+(18)
+where ψn represents standard orthonormal Legendre polynomials. Naturally, the transformation of the original
+input random vector to the unit hypercube might bring additional non-linearity, and thus one might prefer the
+direct construction of polynomials locally orthonormal to the given original probability measure as proposed
+in the Me-gPC [28]. While certainly possible, this brings additional computational demands and thus it is not
+employed here.
+3.5. Local and Global Statistical Estimates from DAL-PCE
+The significant advantage of PCE is that analytically post-processing of the expansion yields highly efficient
+estimates of statistical moments [30], sensitivity indices [8] and LOO-CV [4]. In the proposed DAL-PCE, since
+the original domain � is decomposed into a set of sub-domains (see Eq. (15)), standard analytical post-processing
+can be applied locally and global characteristics can be obtained by simple weighted summations that converge
+to the true values as n� increases. Specifically, the global mean value and variance of a QoI are obtained from
+localized PCEs (denoted by subscript �i) as follows:
+µY =
+n�
+�
+i=1
+Wiβ0i =
+n�
+�
+i=1
+Wiµ�i,
+(19)
+σ2
+Y =
+n�
+�
+i=1
+Wi
+�
+αααi∈Ai
+αααi̸=000
+β2
+αααi =
+n�
+�
+i=1
+Wiσ2
+�i.
+(20)
+where the local mean µ�i and variance σ2
+�i are obtained as described in Section 2.3.
+Local Sobol’ indices, S�i, of any order can be derived directly from localized PCEs and their first-order (main
+effect) estimates are given by
+S
+X j
+�i =
+1
+σ2
+�i
+�
+αααi∈A
+X j
+i
+β2
+αααi
+A
+X j
+i
+=
+�
+αααi ∈ Ai : αj
+i > 0,αk̸=j
+i
+= 0
+�
+.
+(21)
+These local Sobol’ indices are used in the DAL-PCE to determine the cut direction (see Section 3.6). Likewise,
+global Sobol’ indices can be obtained easily from weighted summation of local contributions to partial variances
+normalized by σ2
+Y as follows:
+SX j =
+�n�
+i=1 Wi
+�
+αααi∈A
+X j
+i
+β2
+αααi
+σ2
+Y
+.
+(22)
+Similarly, global LOO-CV, Q2, of a QoI can be approximated by the weighted summation of the local contributions
+as
+Q2 =
+n�
+�
+i=1
+WiQ2
+�i,
+(23)
+where Q2
+�i are obtained from each local PCE using Eq. (9).
+These estimates are used throughout the proposed DAL-PCE, as described in detail next.
+9
+
+3.6. Numerical Algorithm
+Based on the presented theoretical background, we now present the numerical algorithm for the domain
+adaptive localized PCE. As mentioned above, the whole process can be divided to two iterative tasks: (i) decom-
+position of the input random space and (ii) construction of localized PCEs. Both of these tasks are described in
+the following paragraphs with specific reference to the steps in Algorithm 1.
+Algorithm 1 DAL-PCE: Active Domain Decomposition and Construction of Localized PCEs
+Input: maximum local polynomial order p, number of screening global candidates nc,g, number of local
+candidates nc,l, number of iterations niter
+1: set the minimum number of realizations for local PCE construction nsim ∈ 〈P,2P〉
+2: generate a rich pool of nc,g screening candidates
+3: generate the initial ED (size nsim) and construct the initial global PCE
+4: for 1 to niter do
+5:
+identify the sub-domain �i with the highest Θi based on screening candidates
+6:
+ni ← number of ED samples existing in �i
+7:
+if ni ≥ nsim then
+8:
+the identified sub-domain �i becomes a parent �i
+9:
+identify the direction of the highest first-order Sobol’ index S�i of the parent �i
+10:
+restrict coordinates of �i → �i and create �⋆
+i
+11:
+ni ← number of ED samples existing in �i
+12:
+end if
+13:
+generate nc,l local candidates in �i
+14:
+while ni < nsim do
+15:
+extend size of local ED ni using the local Θc criterion
+16:
+end while
+17:
+reconstruct local PCEs in the �i
+18: end for
+Output: list of subdomains and corresponding PCEs
+The first task identifies the important sub-domain �i that should be divided and over which low-order local
+PCE should be constructed. The sub-domain �i is specifically identified using the Θi criterion from Eq. (17),
+which again incorporates three important characteristics for accurate surrogate modeling – the size of the sub-
+domain Wi, the accuracy of the existing local PCE measured by Q2
+�i, and the original Θc criterion measuring the
+variance contribution in �i. While Wi and Q2
+�i are computed for the whole sub-domain, Θc is computed at specific
+realizations of input random vector. Therefore, it is necessary to cover the sub-domains by a sufficiently large
+number of screening candidates, such that the total global number of screening candidates is given by nc,g. Based
+on numerical experiments, we recommend nc,g ≥ 1000 M to ensure that each sub-domain contains a sufficient
+number of screening candidates. Note that the screening candidates are used only to identify �i [step 5]. They
+are not used for the ED, and thus even high nc,g does not bring any additional computational demand.
+Once �i is identified, it is necessary to check whether there are enough samples to construct a PCE inside the
+sub-domain. We start with finding out how many points belong to the selected domain �i [step 6]. If the number
+of samples in the identified sub-domain, ni, is greater than (or equal to) nsim [step 7], a local PCE already exists
+for �i. The subdomain is then assigned as a parent �i for division [step 8] and the first-order Sobol’ indices
+are estimated by Eq. (22) [step 9]. This identified parent �i is divided in the direction of the highest first-order
+Sobol’ index S
+X j
+�i . The new restricted coordinates of refinement-child �i are identified and the inheriting-child �⋆
+i
+is created [step 10]. Further, the number of ED samples ni in the refinement-child �i is determined [step 11]. On
+the other hand, if the identified sub-domain �i does not contain enough samples (i.e. ni < nsim), the inherited
+PCE from the previous iteration is not sufficiently local (it was trained over a domain that has since been divided)
+and it is necessary to add new samples to �i before constructing a new local PCE.
+The second task of the proposed algorithm is sequential sampling and adaptive PCE construction in sub-
+domain �i. Recall that this domain may be either
+10
+
+(i) a refinement-child that was just divided but does not contain a sufficient number of points (ni < nsim) or,
+(ii) an inheriting-child that now does not contain at least nsim ED samples.
+Next, a set of local candidates is generated in region �i [step 13]. To ensure sufficient assessment of the coverage
+of the domain, the number of local candidates is empirically recommended as nc,l ∈ 〈3P,5P〉 [1]. From these
+candidates, the standard Θc criterion in Eq. (12) is used to iteratively select the best candidates until there are
+nsim samples in �i [step 14-16]. This sequential extension of the sample in �i is adaptive in the sense that the
+pairwise distances in Eq. (12) between candidates and existing ED points are updated after the addition of each
+new point. However, because ni < nsim the local variance densities are estimated from the previously existing
+PCE, which cannot be updated until a sufficient number of samples are available in �i.
+The last step of each iteration is to construct the local PCE using scaled Legendre polynomials as basis func-
+tions (see Eq. (18)) [step 17]. Any non-intrusive technique can be used to estimate the coefficients βββ; we use LARS
+and OLS for an adaptive construction of the local PCEs in this paper. At the end of the iteration, all sub-domains
+are re-numbered and a list of sub-domains with corresponding PCEs can be exported or the next iteration can be
+started.
+3.7. Adaptivity in PCE Construction and Domain Decomposition
+Adaptivity is central to the proposed DAL-PCE. In the proposed algorithm, there are two types of adaptivity
+employed:
+(i) adaptivity in PCE construction (selection of the optimal set of basis functions), and
+(ii) adaptivity in domain decomposition
+Since the PCE can be constructed by any regression technique in each sub-domain, PCE adaptivity is incorporated
+by sparse solvers and best model selection algorithms, e.g. Least Angle Regression [32], orthogonal matching
+pursuit [33] or Bayesian compressive sensing [34]. Although sparse solvers are often used for PCE with high p,
+this adaptivity is also important for reducing the number of basis functions (and thus the minimum number of ED
+samples) for high-dimensional examples or, in our case, for very low-size ED in each �i approximated by low-p
+local PCE.
+The second type of adaptivity is the proposed adaptivity in the domain decomposition. At any point in the
+iterative process, the existing ED samples can be used to construct local PCEs or a single global PCE. The DAL-
+PCE is not guaranteed to provide a better approximation than the global PCE. This can be measured via Q2,
+specifically by computing Q2
+local from Eq. (23) and Q2
+global from a single global PCE according to Eq. (9). If
+Q2
+local > Q2
+global at a given iteration, the domain decomposition is deemed to be poor and the whole decomposition
+process is re-started. That is, the complete geometrical decomposition is forgotten and all existing ED points
+are taken as an initial ED for a brand new run of the algorithm. This is illustrated in Fig. 3 which shows the
+decomposition (top) and the associated error (bottom) right before the restart a) at Nsim = 181, b) the new
+decomposition and error right after the restart, and c) the final decomposition/error which shows significant
+improvement over the global PCE. These histories show the standard R2 error defined in Eq. (24). It is not
+necessary to check this criterion at every iteration, but it is suggested to check it periodically, every nr steps, to
+ensure adequate local refinement.
+3.8. Stopping Criteria
+The proposed DAL-PCE algorithm can be fully automated by adding an adequate stopping criterion. A simple
+but practical stopping criterion is based on computational budget, i.e. once the total number of model evaluations
+Nsim or number of iterations niter have reached a critical level/budget. One may also use a stopping criterion
+based on decomposition pattern, e.g. the smallest or the largest volumes of any subdomain, to ensure a desired
+resolution. Valuable stopping criterion can be also obtained directly from Q2, corresponding to a target/threshold
+level of achieved accuracy. Regardless of the selected stopping criteria, it can easily be applied before step 5 of
+the proposed algorithm (start of each iteration).
+11
+
+Figure 3: Illustration of domain decomposition restart. a) decomposition and error evolution prior to restart, b) rebuilt decomposition and
+error drop right after the restart, c) final decomposition and error showing that the restart unlocks a dramatic decrease in approximation
+error.
+4. Numerical Experiments
+The proposed DAL-PCE is presented on four numerical examples of increasing complexity and which illus-
+trated different aspects of the approach. The obtained results are compared (a) to the standard global PCE
+approach with adaptive maximum order p ∈ [5,25] and (b) to SSE [2], as current state-of-the-art non-intrusive
+surrogate modeling technique based on the domain decomposition. The PCE is constructed using the UQPy pack-
+age [36] and the original implementation of SSE is used from the UQLab package [37]. To compare methods,
+the relative mean squared errors ε are calculated for all three approximations ˜f on a validation set containing
+a large pool of 106 integration points generated by crude Monte Carlo according to:
+ε(XXX) :=
+�
+��
+f (XXX) − ˜f (XXX)
+�2�
+�
+�
+f (XXX)
+�
+,
+(24)
+where �[] and �[] are the mean value and variance operators, respectively.
+To show representative results of the proposed DAL-PCE algorithm, the calculations were repeated 100 times,
+and the same settings of the algorithm for all examples were selected as follows: maximum local polynomial
+degree p = 2, number of global candidates nc,g = 1000 M, number of local candidates nc,l = 5P, minimum
+number of samples for local PCE construction nsim = 1.5P, minimum number of iterations before checking for
+restart nr = 20, and βββ are obtained by LARS and OLS algorithm. Minimum number of samples in sub-domains
+required to justify an expansions for SSE was set identically to DAL-PCE and polynomial order is adaptively
+selected in the range p ∈ [2,6]. Since the SSE is not a sequential approach, the presented results were obtained
+for 10 discrete sample sets of increasing size to compare convergence of the method. Note that all samples and
+candidates are generated by LHS for all compared approaches, though it was shown [1] that for the variance-
+based sequential sampling, it is significantly better to use advanced techniques such as Coherence D-optimal
+sampling [41].
+12
+
+4.1. One-dimensional Toy Example
+The first example involves a simple 1D function [2] that is extremely difficult to approximate with PCE due
+to the third, highly nonlinear “exp” term:
+f (X) = −X + 0.1sin(30X) + exp(−(50(X − 0.65))2),
+X ∼ U[0,1].
+(25)
+The poor performance of a single global PCE learned from 200 samples is depicted by the blue line in Fig. 4c
+where it is clear that a single global PCE is not able to accurately approximate the function even for a high
+number of samples and high maximum polynomial order p ∈ [5,25]. This function was originally developed to
+demonstrate the efficiency of SSE based on domain decomposition and thus it is a natural choice for comparison
+of the proposed DAL-PCE and SSE.
+Fig. 4a-b show a typical realization of the DAL-PCE where the algorithm sequentially decomposes the domain
+and adds additional samples to the ED. Specifically shown are the 4th and 11th iterations. The boundaries of
+sub-domains are represented by blue vertical lines and red dots show the positions of samples in the ED. Once
+the algorithm discovers the highly nonlinear region (the steep peak caused by exp), it progressively refines this
+region and adds more samples there as a result of the high variance density. Of course, these figures show only
+one realization of the algorithm and the decomposition is dependent on the initial ED. Therefore, it is necessary
+to repeat the algorithm many times with random initial ED to assess convergence. Fig. 4d shows convergence
+Figure 4: (a), (b) The adapted domain and ED before (iteration 4) and after (iteration 11) exploration and discovery of the exponential part
+of the mathematical model. (c) Final surrogate models from global PCE and DAL-PCE. (d) Convergence plot comparing the mean square
+error for global PCE SSE, and DAL-PCE. The convergence plots for Global PCE and DAL-PCE show continuous mean value ±σ intervals
+from 100 repeated trials, while those for SSE are plotted for several discrete ED sizes.
+of the error ε from 100 repeated trials. The single global PCE is unable to accurately approximate the original
+function even when using high p and thus the ε does not converge, as expected. Both methods based on domain
+decomposition (DAL-PCE and SSE) achieve great accuracy already for 200 samples. However, the DAL-PCE
+consistently has 1–2 orders of magnitude higher accuracy than SSE for the given number of samples. Moreover,
+increase in variance of ε is, in general, slower in DAL-PCE than in SSE. Fast increment in variance of SSE can
+be seen also in the original paper [2]. Finally, we again observe that convergence is continuous with DAL-PCE,
+where convergence can only be assessed at discrete sample sizes with SSE through a new analysis. All of these
+13
+
+Figure 5: Results for the 2-dimensional Singularity function: a) original mathematical model, b) approximation via DAL-PCE (background
+color), current domain division and the corresponding ED, c) local LOO-CV Q2
+�i and Θi value for each sub-domain, d) convergence plots for
+DAL-PCE, Global PCE, and SSE showing the mean value and ±σ interval. Convergence plots for SSE show the mean ±σ at discrete sample
+sizes.
+advantages of the DAL-PCE can be attributed to the active learning, which both explores the space and exploits
+the behavior of the function to decompose the domain and add samples. Although active learning might lead to
+lower accuracy (higher ε) initially (for small nsim = 10–20) as it is dominated by exploration, it rapidly improves
+once it identifies important features and begins to favor exploitation.
+4.2. Two-dimensional Singularity
+The second example involves a 2D function with mirrored quarter-circle arc line singularities [1]. The form
+of the function is give by:
+f (XXX) =
+1
+|0.3 − X 2
+1 − X 2
+2| + δ −
+1
+|0.3 − (1 − X1)2 − (1 − X2)2| + δ,
+XXX ∼ U[0,1]2,
+(26)
+where the strength of the singularities is controlled by the parameter δ, which we set as δ = 0.1. The singularities
+in this example represent a challenging task for a global PCE even with high order, due to the well-known Gibbs
+phenomenon [49]. It is thus beneficial to identify the location of the singularity, locally decompose the domain,
+and construct low-order local PCEs.
+Fig. 5 illustrates the decomposition and DAL-PCE approximation at a given stage of the computation. Panel
+a) visualizes the true values of the function via a background color. The same coloring scheme is used in panel b)
+for the pointwise information available in the current ED (small circles) and for the function approximation via
+DAL-PCE by the background color. Panels b) and c) show also the final domain decomposition. The symmetry
+14
+
+Figure 6: Results for the 2-dimensional discontinuiy function: a) original mathematical model, b) approximation via DAL-PCE and ED, c)
+local LOO-CV Q2
+�i and Θi value for each sub-domain, d) convergence plots for DAL-PCE, Global PCE, and SEE showing the mean value and
+±σ interval. Convergence plots for SSE show the mean ±σ at discrete sample sizes.
+in the decomposition documents the great convergence of the DAL-PCE thanks to an adaptive decomposition
+described in the previous section. Plot c) shows the local Q2
+�i error in each individual sub-domain (darker color
+corresponds to higher local error). These local errors clearly show localization of the prediction error to very
+small areas near singularities, which are continually being refined. The color of the small solid squares in the
+center of each sub-domains shows the Θi value for that sub-domain.
+Finally, the convergence plot in Fig. 5d) shows that both DAL-PCE and SSE outperform the global PCE, as
+expected. The SSE performs comparable to or slightly better than DAL-PCE for small Nsim, but the DAL-PCE
+begins to outperform SSE as Nsim grows thanks to the active learning approach that targets samples in the vicinity
+of the singularities. Note that the error converges for both SSE and DAL-PCE as we approach 1000 samples and
+does not seem to substantially reduce after this. This is due to the fundamental limitation of trying to approximate
+this singularity, even locally, with low-order polynomials.
+4.3. M-dimensional Discontinuity
+The third example investigates the role of dimensionality on the performance of the proposed DAL-PCE. The
+following discontinuous function is defined for an arbitrary number of input random variables M [26]:
+f (XXX) =
+�
+sin(X1π)sin(X2π)
+if x1 ≤ 0.5 and x2 ≤ 0.5
+�M
+i=3 Xi
+otherwise
+,
+XXX ∼ U[0,1]M.
+(27)
+This function has a discontinuity in the first two input random variables, which can be seen in Fig. 6a. A single
+global PCE cannot accurately approximate the function because of the discontinuity, although the function f (XXX)
+15
+
+Figure 7: Convergence plots for the M-dimensional function: a) 3-dimensional version, b) 5-dimensional version, c) 6-dimensional version,
+and d) 8-dimensional version. Convergence plots for the DAL-PCE and global PCE show the mean value ±σ interval. Convergence plots
+for SSE also show the mean ±σ, but at discrete sample sizes.
+can be easily approximated by two separate PCEs in the two regions for which the definitions differ. But, this
+requires a priori knowledge of the discontinuity location. Since the location of the discontinuity is assumed to be
+unknown, this function is a good example for domain adaptation using DAL-PCE.
+The detailed results for a 2D version of this problem are depicted in Fig. 6 in identical form as in the previous
+example. Note that the local Q2
+i errors Fig. 6c show perfect accuracy in the part of the input random space where
+f (XXX) = 0 and thus the associated sub-domains are not preferred for further decomposition. The convergence
+plot in Fig. 6d confirms that a single global PCE is not able to create an accurate approximation and adding
+more points to ED does not lead to significant improvements in the approximation. The mean values of errors
+ε associated to the proposed DAL-PCE approach are significantly lower in comparison to SSE (1–2 orders of
+magnitude) similarly as in the first example, though the convergence trend is similar for both methods. SSE,
+however, uses a random splitting routine. This can lead to very high variance of results, since the accuracy is
+highly dependent on the pattern of the decomposed input random space. This clearly shows the advantage of an
+active learning approach.
+The influence of dimensionality M on convergence of the DAL-PCE, SSE, and global PCE is studied in Fig. 7
+for a) 3, b) 5, c) 6, and d) 8 input random variables. As the domain dimension increases, the linear part of the
+function f (XXX) occupies an increasing proportion of the domain while the discontinuity remain low-dimensional.
+The proposed DAL-PCE greatly improves the convergence because it is able to identify an ideal decomposition
+and local samples to resolve the discontinuity. For low-dimensions (M = 2,3), SSE error ε shows a decreasing
+trend that is better than global PCE but has an extremely high variance. This is caused by a lack of control in
+sample placement. The domain decomposition in SSE is a product of sample location and without active learning
+to guide sample placement, SSE will sometimes produce a very good decomposition and sometimes a very poor
+decomposition. Meanwhile, the proposed DAL-PCE errors have comparably low variance for low-dimensions
+and consistently have accuracy comparable to, or better than, the best SSE realizations.
+As the dimension, M, increases the DAL-PCE is able to maintain a very high level of accuracy, while the
+accuracy degrades completely for the SSE such that it is comparable to the global PCE. The DAL-PCE is able
+to maintain its low error because the discontinuity remains low-dimensional and the active learning process is
+able to target this region for domain re��nement and sampling. This means that the DAL-PCE remains largely
+independent of the problem dimension, and instead depends predominantly on the intrinsic dimension of the
+16
+
+Figure 8: Convergence plots for the modified M-dimensional function: a) 3-dimensional version, b) 5-dimensional version, c) 6-dimensional
+version, and d) 8-dimensional version. Convergence plots for the DAL-PCE and global PCE show the mean value ±σ interval. Convergence
+plots for SSE also show the mean ±σ, but at discrete sample sizes.
+discontinuous/nonlinear features of the model. The performance of SSE, on the other hand, degrades with
+dimension because its domain decomposition depends only on a set of a priori specified points that are not selected
+in a way that is aware of the important features of the model. Consequently, as the dimension increases the
+algorithm becomes less likely to refine the domain appropriately around an embedded low-dimensional feature.
+We remark that this desirable scalable convergence trend of the DAL-PCE is not likely a universal property, as
+the trend may break down in problems where the intrinsic dimension of the discontinuity/nonlinearity is high or
+where the discontinuity occupies a very small proportion of the domain – in which case exploration of the space
+to find the important feature may take a very large number of samples.
+In the present example, the discontinuity in the function given in Eq. (27) lies at x1 = 0.5 and x2 = 0.5, which
+corresponds to the exact location where the domain will be split for both SSE and during the early iterations of
+the DAL-PCE. One might argue that this presents an unreasonable advantage for the proposed algorithm. We
+therefore modified the function such that the discontinuity lies at x1 = 0.61 and x2 = 0.61. Fig. 8 shows the
+convergence for the DAL-PCE and SSE for this modified function with varying dimension, M. The absolute errors
+ε exhibit slower decrease, especially for dimensions M = 3 and M = 5. However, the proposed active learning
+still leads to superior results (especially for higher dimensions as in the previous case). Note that there are visible
+spikes in the DAL-PCE convergence graph for the 3-dimensional example. Although the results were statistically
+processed, these spikes are caused by the restart adaptivity occurring at the same Nsim in each replication. In
+this case, the optimal decomposition pattern is very complicated and therefore the algorithm activates the restart
+adaptivity frequently (after multiples of nr steps), until it finds a suitable pattern to continue convergence. SSE
+in the 3- and 5-dimensional cases has higher mean error and significantly lower variance in comparison to the
+previous example. This is caused by the fact that the modified discontinuity location no longer lies along the
+boundary of the domain decomposition. In the previous example, some SSE realizations achieved near-perfect
+accuracy because the domain was coincidentally divided along the discontinuity.
+This phenomenon is investigated more closely in Fig. 9, which compares number of outliers in both versions
+of 3D examples. In addition to the mean ±σ seen previously, the figure also shows standard boxplots for SSE
+(median along with lower and upper quartiles) and the corresponding number of “extreme” realizations produc-
+ing very high accuracy (top axis) for a) the original position of discontinuity; and b) discontinuity at x1 = 0.61
+17
+
+Figure 9: Convergence plots for DAL-PCE and SSE with additional boxplots for SSE showing the median, lower and upper quartiles and
+outliers for: a) the 3D example with discontinuity at x1 = 0.5 and x2 = 0.5, b) the 3D example with discontinuity at x1 = 0.61 and x2 = 0.61.
+and x2 = 0.61. As can be seen, in panel a) there are many outliers producing ε < −7, which effectively decreases
+µ relative to the median while also significantly increasing the variance. In contrast DAL-PCE has no outliers and
+it leads to very consistent results. In panel b), there are no outliers for either SSE or DAL-PCE and the results
+are thus consistent with low variance for both methods.
+4.4. Asymmetric shallow von Mises truss
+In this section, we demonstrate the relevance of the proposed method for a representative engineering exam-
+ple exhibiting discontinuous response. Consider the shallow two-bar planar truss subjected to a vertical load at
+its top joint, as presented in [50] and illustrated in Fig. 10a.
+The truss is formed by two prismatic bars made of a hard wood (density 800 kg/m3, modulus of elasticity
+E = 12 GPa). There are two variables in the studied von Mises truss: (i) the loading vertical force F, and (ii)
+a half sine-wave imperfection of the left bar having magnitude δ, see the sketch in Fig. 10a. The load is applied
+dynamically as a step function at time zero for an unlimited duration. The structure is modeled, as illustrated
+in Fig. 10b. In particular, the mass of the bar is concentrated in 21 mass points, including the supports and
+the loading point. These mass points are connected via 10 + 10 translational springs representing the normal
+stiffness of the true bars. The pairs of the axial members are connected via rotational spring having zero moment
+for a zero angle between adjacent bars. The only exceptions are the loading ans support points where there
+are no rotational springs attached (hinges). The damping is associated with the mass points via linear viscous
+damping coefficient set to 11 N · s/(kg · m) approximating the relative damping of about 3%. Explicit dynamics
+solver FyDiK [51, 52] was used to solve the equations of equilibrium at the mass points. The numerical solution
+lasts to up to two seconds, which is the time needed for almost complete stabilization of the solution (kinetic
+energy drops below a negligible threshold).
+Since the structure is very shallow, sudden application of the vertical force can cause snap-through buckling,
+wherein the loading point drops down between the supports and the members switch from a state of compression
+to tensile stresses in the final stable state. We specifically study the horizontal coordinate yF of the loading point
+after the dynamic response stabilizes to the final deformed shape. The force F ∈ (31.6,772.6) kN and initial
+imperfection δ ∈ (−0.4,0.4) m are treated as uniform random variables mapped to the unit square such that
+the model input X ∼ U[0,1]2. Because of the potential snap-through buckling, the solution is discontinuous as
+illustrated in Fig. 10c. On each side of the discontinuity, the solution yF is smooth and slowly-varying having
+values near +1 m and -1 m, respectively. Note that the output is not symmetric with respect to δ = 0 because the
+dynamical response evolves differently for concave and convex initial displacements.
+The sharp boundary between the buckled and unbuckled regions, shown in Fig. 11a cause global PCE to
+produce poor approximations that are vulnerable to the Gibbs phenomenon, similar to the example in subsection
+18
+
+b)
+a)
+c)
+Figure 10: Asymmetric shallow von Mises truss. a) Initial geometry with two random variables F and δ; b) illustrative sketch of the discrete
+dynamical model and the meaning of output variable yF, c) illustration of the discontinuous response function of the two input variables.
+Figure 11: Results for the von Misses truss example: a) original mathematical model (numerical solution), b) approximation via DAL-PCE
+and ED, c) local LOO-CV Q2
+�i and Θi value for each sub-domain, d) convergence plots for DAL-PCE, Global PCE, and SSE showing the
+mean value and ±σ interval; convergence plots for SSE show the mean ±σ at discrete sample sizes.
+4.2. This is shown by the convergence plots in Fig. 11d comparing global PCE, DAL-PCE, and SSE. Clearly, the
+complexity of this example and the complicated shape of the discontinuity limits the accuracy of all the surrogate
+models. The proposed DAL-PCE achieves low accuracy for small sample sizes because the corresponding small
+number of sub-domains and low-order PCEs are unable to sufficiently approximate the boundary. Therefore, the
+global PCE and SSE (with a low number of embedding levels) are initially better. With increasing number of
+samples, the proposed DAL-PCE approach leads to superior results because the active learning is able to resolve
+the discontinuity as illustrated in Fig. 11b, which shows the domain decomposition and approximation after
+2000 samples. Fig. 11c shows the corresponding LOO-CV errors for each subdomain, demonstrating the errors
+are confined to small, localized regions near the boundary.
+19
+
+5. Discussion & Future Work
+The proposed DAL-PCE approach is a general methodology for the decomposition of the input random space
+and construction of localized PCEs using active learning. The proposed active learning is based on a novel Θ
+criterion that optimally balances global exploration with local exploitation of the model. Although this paper
+presents one specific learning algorithm, the methodology is general and amenable to modifications to reflect the
+specific user’s needs. The whole process can be divided into two tasks: A) decomposition of the input random
+space and B) construction of localized PCEs; and both can be easily modified as discussed further:
+A) The most important sub-domain �i is identified by extended Θ according to Eq. (17) evaluated for a large
+number of global candidates. In this paper, we use standard LHS for candidate generation, but it may
+be beneficial to use different sampling methods that produce more uniform coverage of the whole input
+random space (see e.g. [53, 54, 45]). Although it is generally possible to generate a large number of
+candidates, it might be challenging to uniformly cover the entire input random space, especially in high
+dimensions. Thus, one can use any sampling technique suitable for a specific example, e.g. [55].
+Once the �i is identified via Eq. (17), it is either divided (providing it contains enough ED points) or
+the sample is extended inside it, to achieve a better PCE approximation. The simplest division occurs by
+splitting the volume into two parts of identical hypervolume in the direction of the highest first-order Sobol’
+index. However, the algorithm can accommodate various different approaches. For example, it is possible
+to divide the �i into a higher number of sub-domains, not just two. Moreover, instead of splitting the
+domain into parts of equal hypervolume, other criteria can be used. For example, the cutting plane can be
+positioned so to split the domain variance into equal parts.
+B) The user can choose to employ any existing method to construct the non-intrusive PCEs, including various
+sparse solvers or adaptive algorithms, which may be preferable for certain applications [12]. For example,
+we use LARS with OLS. However, it is generally more efficient to use active learning based on the Θ criterion
+for PCE as shown in [1], which employs variance-based sequential sampling. This improvement can be
+integrated within the DAL-PCE to make local PCE more efficient in each subdomain, and thereby improving
+the overall convergence. The can be compounded by the use of advanced sampling techniques within the
+subdomains such as Coherence D-optimal sampling [40, 41].
+As seen from the previous paragraphs, the whole algorithm can be adapted for specific needs reflecting the
+characteristics of a given mathematical model, such as dimensionality, sparsity, non-linearity etc., by simply ex-
+changing components of the proposed algorithm for suitable existing (or new) techniques. Note that even after
+the modification, the whole methodology based on Θ criterion is still valid and can be used for uncertainty
+quantification and surrogate modelling as described in this paper. Moreover, in comparison to SSE, the DAL-
+PCE sequentially adds points and divides the sub-domains one-by-one based on information obtained from the
+previous iteration.
+Another significant advantage of the DAL-PCE is that it provides estimates of the local errors, Q�i, associ-
+ated with each sub-domain. Since localized PCEs are constructed independently, local errors estimate the local
+accuracy of the surrogate model directly, and can be assembled to provide global error measures. Naturally, local
+accuracy is very important information that can be used for further probabilistic analysis and active learning.
+Although this paper does not propose any specific approach for further processing of this information, it could
+serve as a main ingredient for various active learning algorithms. For example, it could be directly used to predict
+uncertainty in industrial applications and possibly extend the ED in a sub-domain of interest.
+Finally, an important topic of further research is to study the behavior of the proposed criterion in higher
+dimensions. In particular, the geometrical terms l M
+c,s and �i likely cause poor convergence in high dimensions.
+Although some preliminary results focused on investigating of l M
+c,s in high dimensions was previously performed in
+the paper [1] proposing the original Θ criterion, it is still necessary to perform an extensive study of its behavior
+as well as investigating the influence of �i, which may need to be reformulated for high dimensions.
+20
+
+6. Conclusion
+The paper presented a novel approach, domain adaptively localzed PCE, for the adaptive sequential con-
+struction of localized PCEs based on active learning and decomposition of the input random space. It combines
+adaptive sequential sampling based on the recently proposed Θ criterion to maintain the balance between ex-
+ploration of the input random space and exploitation of the current characteristics of the PCE together with the
+adaptive sequential decomposition of the input random space creating sub-domains approximated by local sur-
+rogate models. The methodology offers a general technique that can be easily adapted or modified for specific
+functions extending its applicability. The performance of the proposed methodology was validated on several nu-
+merical examples of increasing complexity investigating different aspects of the algorithm and leading to superior
+results in comparison to a single global PCE and the recently proposed SSE.
+Acknowledgments
+The first author acknowledge financial support provided by the Czech Science Foundation under project num-
+ber 22-00774S. Additionally, the major part of this research was conducted during the research stay of the first
+author at Johns Hopkins University supported by the project International Mobility of Researchers of Brno Uni-
+versity of Technology, Czechia under project No. EF18_053/0016962.
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+23
+

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+ 
+ 
+ 
+ 
+ Số chuyên san (11/2022): 1 – 11 
+  1 
+ 
+ 
+MỐI TƯƠNG QUAN CỦA CÁC NHÂN TỐ ẢNH HƯỞNG                             
+TỚI VIỆC SỬ DỤNG ỨNG DỤNG BLUEZONE 
+Nguyễn Thế Vịnh1*, Nguyễn Tuấn Anh1, Nguyễn Hồng Tân1, Lương Khắc Định2  
+1Khoa Công nghệ thông tin, Trường ĐH Công nghệ thông tin và Truyền thông, ĐH Thái Nguyên 
+2Khoa Công nghệ thông tin, Trường ĐH Hạ Long  
+* Email: [email protected] 
+Ngày nhận bài: 11/6/2022 
+Ngày nhận bài sửa sau phản biện: 09/11/2022 
+Ngày chấp nhận đăng: DD/MM/YYYY 
+TÓM TẮT 
+Sự xuất hiện của đại dịch Covid-19 đã gây ra nhiều tác động tiêu cực đến mọi mặt của đời 
+sống. Chính phủ đã áp dụng nhiều biện pháp để giảm thiểu sự ảnh hưởng và lây truyền của dịch 
+bệnh. Trong số đó có việc áp dụng chuyển đổi số đối với việc quản lý và truy vết người bị nhiễm 
+Covid thông qua phần mềm Bluezone (nay là PC-Covid). Tuy nhiên, việc cài đặt và sử dụng 
+Bluezone lại không được như kỳ vọng. Vì vậy, nghiên cứu này tìm hiểu những nhân tố chính và 
+sự ảnh hưởng của chúng tới ý định hành vi của người dùng về việc sử dụng phần mềm truy vết 
+Bluezone. Phiếu khảo sát được gửi tới người dùng thông qua công cụ Google Form. Kết quả 
+phân tích các nhân tố khám phá trên 224 đối tượng khảo sát cho thấy, có bốn nhân tố chính ảnh 
+hưởng tới hành vi của người dùng, trong đó: sự tin tưởng và kỳ vọng hiệu quả, kỳ vọng nỗ lực, 
+ảnh hưởng xã hội có tác động tích cực đến ý định hành vi của việc sử dụng phần mềm truy vết 
+Bluezone; trong khi rủi ro về quyền riêng tư có ảnh hưởng tiêu cực đến hành vi này.   
+Từ khóa: EFA, SEM, UTAUT, tin tưởng, quyền riêng tư, Covid-19.   
+FACTORS INFLUENCING TO USE OF BLUEZONE 
+ABSTRACT 
+The emergence of the Covid-19 pandemic has been causing many negative impacts on all 
+aspects of life. The government has taken many measures to minimize the impact and 
+transmission of the disease. Among them is the application of digital transformation to the 
+management and tracing of people infected with Covid through the Bluezone app (now PC-
+Covid). However, using and installing Bluezone is not as expected. Therefore, this study aims 
+to understand the main factors and their influence on the behavioral intention of users about 
+using Bluezone. Surveys are sent to users through the Google Form tool. Experimental results 
+through analysis of exploratory factors on 224 survey subjects show that there are 4 main factors 
+affecting user behavior. Structural equation modeling indicates that trust, performance 
+expectations, effort expectations, and social influence have a positive impact on behavioral 
+intention of using Bluezone. Meanwhile, privacy risks have a negative effect on this behavior.   
+Keywords: EFA, SEM, UTAUT, trust, privacy, Covid-19.
+
+ap chi khoa hoc
+DAI HOC HA LONGTAP CHI KHOAHOC DAI HOCHALONG
+Scientific JournalofHa Long Vniversity
+KHOAHOC
+DAIHOCHALONG
+http://uhl.edu.vnl
+Hac de thanh cong 
+ 
+ 
+ 
+2 Số 01(2021): 1 – 11  
+ 
+ 
+ 
+KHOA HỌC TỰ NHIÊN 
+1.  ĐẶT VẤN ĐỀ 
+Đại dịch Covid-19 xuất hiện vào cuối năm 
+2019 và bùng phát mạnh mẽ trong thời gian 
+qua đã có những ảnh hưởng tiêu cực tới tất cả 
+các quốc gia trên toàn thế giới (Whitelaw và 
+c.s., 2020). Đứng trước vấn đề đó, chính phủ 
+các quốc gia trên thế giới đã tiến hành nhiều 
+biện pháp cấp bách nhằm hạn chế tầm ảnh 
+hưởng, lây lan của dịch bệnh (Nguyen và c.s., 
+2021). Song song với các biện pháp tuyên 
+truyền đến người dân về ý thức phòng chống 
+dịch thông qua các phương tiện truyền thông, 
+chính phủ Việt Nam cũng tiến hành nhiều 
+biện pháp hỗ trợ nhằm truy vết tiếp xúc và 
+cảnh báo người nhiễm Covid-19 (Le và c.s., 
+2021). Cụ thể, Bộ Y tế và Bộ Thông tin và 
+Truyền thông đã phối hợp tạo ra ứng dụng 
+Bluezone. Bluezone được coi là “cần thiết 
+trong quá trình sinh hoạt hàng ngày, khi mọi 
+người có tiếp xúc, ứng dụng trên điện thoại 
+của họ sẽ tự “nói chuyện” với nhau” 
+(baochinhphu.vn, 2020). Ứng dụng Bluezone 
+được kỳ vọng là sẽ giúp ích cho các cơ quan 
+nhà nước có thể nhanh chóng truy vết và quản 
+lý được các ca nhiễm trong cộng đồng, người 
+dân có thể nắm bắt được thông tin kịp thời để 
+phòng dịch (Nguyen và c.s., 2021). 
+Mặc dù Bluezone được kỳ vọng sẽ mang 
+lại hiệu quả tích cực cao và nhiều người sẽ sử 
+d���ng, nhưng số liệu thống kê thực tế lại 
+không được như mong muốn (Nguyen và c.s., 
+2021). Tính đến 27 tháng 5 năm 2021, cả 
+nước chỉ ghi nhận 33,48 triệu lượt tải (khoảng 
+34,7% so với tổng dân số), trong đó tập trung 
+chủ yếu ở hai địa phương lớn là Hà Nội (3,1 
+triệu lượt cài đặt) và Thành phố Hồ Chí Minh 
+(2,83 triệu lượt cài đặt). Ở chiều ngược lại, 
+các tỉnh khác như Điện Biên, Kon Tum, Lai 
+Châu, Bắc Kạn lại ghi nhận số lượng người 
+tải ứng dụng Bluezone thấp nhất. Vì vậy, câu 
+hỏi đặt ra là: Những yếu tố nào ảnh hưởng tới 
+việc sử dụng phần mềm Bluezone? 
+Trả lời được câu hỏi nghiên cứu trên đóng 
+vai trò quan trọng trong việc khuyến khích 
+người dân tham gia, hỗ trợ phòng chống dịch 
+trên môi trường số (Nguyen & Nguyen, 2022; 
+Whitelaw và c.s., 2020). Có nhiều nghiên cứu 
+trên thế giới tìm hiểu các yếu tố ảnh hưởng 
+tới việc sử dụng phần mềm truy vết nói chung 
+(Mbunge, 2020; Whitelaw và c.s., 2020), 
+nhưng chưa có nghiên cứu nào được thực 
+hiện ở Việt Nam trả lời cho câu hỏi trên một 
+cách đầy đủ. Vì vậy nghiên cứu này có vị trí 
+riêng biệt và cần thiết trong bối cảnh hiện 
+nay, đặc biệt khi đại dịch Covid-19 vẫn chưa 
+có dấu hiệu kết thúc do sự xuất hiện của các 
+biến chủng mới. Nghiên cứu của Nguyen và 
+c.s. (2021) mới chỉ dừng lại ở việc trích xuất 
+được các nhân tố mà chưa xem xét đến mối 
+tương quan giữa các nhân tố đó tới ý định sử 
+dụng phần mềm Bluezone như thế nào. Chính 
+vì vậy, nghiên cứu này được mở rộng bằng 
+cách áp dụng mô hình phương trình cấu trúc 
+nhằm đánh giá mối quan hệ giữa các yếu tố 
+tới ý định sử dụng phần mềm Bluezone. Kết 
+quả của bài báo được kỳ vọng sẽ có những 
+đóng góp tích cực trong lĩnh vực nghiên cứu 
+bao gồm: 1) việc khám phá ra các nhân tố 
+chính ảnh hưởng tới ý định sử dụng phần 
+mềm Bluezone, 2) đánh giá mối quan hệ giữa 
+các yếu tố tới ý định sử dụng phần mềm 
+Bluezone. Kết quả nghiên cứu sẽ là tài liệu 
+tham khảo cho các nghiên cứu tương tự và là 
+một trong các chỉ báo giúp các nhà quản lý 
+điều chỉnh chính sách phù hợp nhằm nâng cao 
+hiệu quả của ứng dụng truy vết. 
+2.  MÔ HÌNH NGHIÊN CỨU VÀ CƠ SỞ 
+LÝ THUYẾT 
+2.1.  Tổng quan về mô hình nghiên cứu  
+Sự phát triển không ngừng của các thiết bị 
+mới và phần mềm mới đã giúp cho người 
+dùng trải nghiệm và giải quyết các vấn đề 
+trong cuộc sống dễ dàng hơn. Tuy nhiên, 
+không phải mọi công nghệ mới đều được 
+người dùng chấp nhận và sử dụng. Để giảm 
+thiểu các rủi ro trên, nhiều mô hình chấp nhận 
+công nghệ được phát triển và áp dụng rộng rãi 
+như: mô hình SOR – stimulus (kích thích), 
+organism (chủ thể), response (phản hồi) – mô 
+tả cách mà sinh vật, con người phản ứng, đáp 
+lại với kích thích từ môi trường (Mehrabian 
+& Russell, 1974), mô hình chấp nhận công 
+nghệ – Technology Acceptance Model 
+(TAM) (Davis, 1985), mô hình lý thuyết chấp 
+nhận công nghệ hợp nhất (UTAUT). UTAUT 
+được phát triển bằng việc kết hợp và tinh 
+chỉnh tám mô hình trước đây thành một mô 
+hình duy nhất để mô tả hành vi của người 
+
+ 
+ 
+ 
+ 
+ Số 02 (2022): 1 – 11 
+  3 
+ 
+KHOA HỌC TỰ NHIÊN 
+dùng với một hệ thống công nghệ thông tin 
+(Venkatesh và c.s., 2003). Mô hình UTAUT 
+chỉ ra có 4 yếu tố chính ảnh hưởng đến hành vi 
+của người dùng bao gồm: kỳ vọng  hiệu quả 
+(performance expectancy), kì vọng nỗ lực 
+(effort expectancy), ảnh hưởng xã hội (social 
+influence), và các điều kiện thuận lợi 
+(facilitating conditions). Ngoài ra còn có các 
+yếu tố khác điều chỉnh đến ý định sử dụng như 
+giới tính, độ tuổi, sự tự nguyện và kinh nghiệm. 
+UTAUT được áp dụng rộng rãi trong nhiều lĩnh 
+vực khác nhau (Jung và c.s., 2020, 2021; 
+Nguyen, 2022). Trong nghiên cứu này, chúng 
+tôi mở rộng mô hình UTAUT với hai nhân tố 
+mới là sự riêng tư (privacy) và độ tin cậy (trust) 
+được tham khảo từ những nghiên cứu tương tự 
+(Arfi và c.s., 2021; Chopdar, 2022). 
+2.2.  Cơ sở lý thuyết 
+Kỳ 
+vọng 
+hiệu 
+quả 
+(Performance 
+Expectancy) được định nghĩa là mức độ mà 
+một cá nhân tin rằng vi���c sử dụng hệ thống sẽ 
+giúp họ đạt được hiệu quả trong công việc 
+(Venkatesh và c.s., 2003). Năm yếu tố từ các 
+mô hình khác nhau liên quan đến kỳ vọng 
+hiệu quả là nhận thức phần mềm hữu ích, 
+động lực bên ngoài, sự phù hợp với công việc, 
+lợi thế tương đối và kỳ vọng kết quả.  
+Kỳ vọng nỗ lực (Effort Expectancy) được 
+định nghĩa là mức độ dễ dàng liên quan đến 
+việc sử dụng hệ thống (Venkatesh và c.s., 
+2003). Ba yếu tố từ các mô hình khác nhau 
+liên quan đến kỳ vọng nỗ lực là nhận thức dễ 
+sử dụng, độ phức tạp (mô hình sử dụng máy 
+tính) và tính dễ dùng (mô hình khuếch tán 
+đổi mới).  
+Ảnh hưởng xã hội (Social Influence) được 
+định nghĩa là mức độ mà một cá nhân nhận 
+thấy rằng những người khác quan trọng tin 
+rằng họ nên sử dụng hệ thống mới (Venkatesh 
+và c.s., 2003). Ba yếu tố từ các mô hình khác 
+nhau liên quan đến ảnh hưởng xã hội là chuẩn 
+chủ quan, yếu tố xã hội và hình ảnh. 
+Các điều kiện thuận lợi (Facilitating 
+Conditions) được định nghĩa là “Mức độ mà 
+một cá nhân tin rằng có sẵn cơ sở hạ tầng kỹ 
+thuật và tổ chức để hỗ trợ việc sử dụng hệ 
+thống” (Venkatesh và c.s., 2003). Venkatesh 
+cho rằng các điều kiện thuận lợi không ảnh 
+hưởng đến ý định hành vi, nhưng ảnh hưởng 
+đến hành vi sử dụng. Các điều kiện thuận lợi 
+liên quan đến sự sẵn có của nguồn lực và hỗ 
+trợ cho các cá nhân sử dụng công nghệ. 
+Rủi ro về quyền riêng tư (Privacy Risk) 
+được hiểu là mối quan ngại của người dùng 
+về việc tiết lộ thông tin cá nhân (Arfi và c.s., 
+2021; Chopdar, 2022; Li, 2011). Nhiều 
+nghiên cứu đã chỉ ra rằng rủi ro về quyền 
+riêng tư có ảnh hưởng tới độ tin cậy của người 
+dùng và gián tiếp ảnh hưởng đến ý định sử 
+dụng hệ thống (Arfi và c.s., 2021; Bansal và 
+c.s., 2010; Chopdar, 2022).  
+Sự tin tưởng (Trust) phản ánh sự sẵn sàng 
+ở trong tình trạng dễ bị tổn thương dựa trên 
+kỳ vọng tích cực đối với hành vi trong tương 
+lai của yếu tố ngoại vi (Arfi và c.s., 2021; 
+Chopdar, 2022). Nhiều nghiên cứu đã chỉ ra 
+rằng sự tin tưởng có ảnh hưởng tới ý định 
+hành vi và nhận thức rủi ro (Arfi và c.s., 2021; 
+Chopdar, 2022).  
+3.  PHƯƠNG PHÁP NGHIÊN CỨU 
+3.1.  Đối tượng nghiên cứu 
+Phiếu khảo sát được tạo ra và gửi đến người 
+dùng thông qua ứng dụng Zalo và mạng xã hội 
+Facebook trong khoảng thời gian từ ngày 
+18/6/2021 đến ngày 21/6/2021. Số lượng ước 
+lượng người dùng tham gia khảo sát là 400 
+người, tỷ lệ phản hồi là 73,75% (295 phản 
+hồi), nhóm nghiên cứu loại bỏ 25 phản hồi do 
+người dùng không cài đặt ứng dụng Bluezone, 
+41 câu trả lời không hợp lệ do chỉ chọn một 
+lựa chọn duy nhất, 5 phản hồi không hoàn 
+thành khảo sát. Tổng số dữ liệu cuối cùng để 
+đưa vào phân tích là 224 (75,93%). Bảng 1 
+tổng hợp dữ liệu từ phiếu khảo sát, tỷ lệ nam 
+chiếm 16,07%, trong khi đó tỷ lệ nữ chiếm 
+83,48%. Hơn một nửa đối tượng tham gia điều 
+tra là sinh viên, học sinh trong độ tuổi từ 10 – 
+20 (52,68%), 27,23% nằm trong độ tuổi từ 21 
+– 30, 11,16% nằm trong độ tuổi 31 – 40%, số 
+còn lại trên 41 tuổi chiếm 8,93%. Khu vực sinh 
+sống của người dùng ứng dụng Bluezone chủ 
+yếu tập trung ở khu vực thị xã, nông thôn và 
+miền núi (52,23%), còn lại là ở các khu vực 
+thành phố (28,57%) và quận /huyện (19,20%). 
+Kết quả của phiếu khảo sát này cũng phù hợp 
+với đặc tính vùng miền của tỉnh Thái Nguyên 
+– là tỉnh miền núi. 
+
+ap chi khoa hoc
+DAI HOC HA LONG 
+ 
+ 
+ 
+4 Số 01(2021): 1 – 11  
+ 
+ 
+ 
+KHOA HỌC TỰ NHIÊN 
+3.2.  Công cụ khảo sát   
+Sau khi nghiên cứu các câu hỏi dùng cho 
+việc khảo sát dựa trên mô hình nghiên cứu 
+(Arfi và c.s., 2021; Chopdar, 2022), 18 câu 
+hỏi được nhóm tác giả lựa chọn và đưa vào 
+nghiên cứu (xem  
+Bảng 2). Thang điểm Likert năm điểm (1 
+= Hoàn toàn không đồng ý, 2 = Không đồng 
+ý, 3 = Trung lập, 4 =  Đồng ý, 5 = Hoàn toàn 
+đồng ý) được sử dụng cho mỗi câu hỏi. 
+Bảng 1. Thông tin chung về đối tượng khảo sát 
+Thông tin chung 
+Số lượng 
+% 
+ 
+Giới tính 
+ 
+ 
+ 
+Nam 
+36 
+16,07  
+Nữ 
+187 
+83,48  
+Không xác định 
+1 
+0,45  
+��ộ tuổi 
+ 
+  
+10 – 20 
+118 
+52,68  
+21 – 30 
+61 
+27,23  
+31 – 40 
+25 
+11,16  
+Trên 40 tuổi 
+20 
+8,93  
+Khu vực sinh sống 
+ 
+  
+Thành phố 
+64 
+28,57  
+Quận/huyện 
+43 
+19,20  
+Thị xã, nông thôn 
+117 
+52,23  
+Tổng 
+224 
+100  
+3.3.  Phân tích các nhân tố khám phá 
+Phân tích nhân tố khám phá (Explatory 
+Factor Analysis - EFA) là một phương pháp 
+thống kê dùng để rút gọn nhiều biến đo lường 
+phụ thuộc lẫn nhau (đo được) thành một tập 
+biến ít hơn (gọi là các nhân tố – không đo 
+được trực tiếp) mà vẫn chứa đựng hầu hết nội 
+dung thông tin của tập biến ban đầu (Hair Jr 
+và c.s., 2009). EFA giả định rằng mỗi chỉ số 
+trong một tập hợp các chỉ số là một hàm tuyến 
+tính của một hoặc nhiều nhân tố chung và một 
+nhân tố duy nhất. Các nhân tố chung là các 
+yếu tố tiềm ẩn không thể quan sát được có ảnh 
+hưởng đến nhiều hơn một chỉ số trong một 
+tập hợp các chỉ số (Fabrigar & Wegener, 
+2012). Các nhân tố duy nhất là các biến tiềm 
+ẩn được giả định chỉ ảnh hưởng đến một chỉ 
+số từ một tập hợp các chỉ số và không tính 
+đến mối tương quan giữa các chỉ số.  Mục tiêu 
+của mô hình nhân tố chung là tìm hiểu cấu 
+trúc mối tương quan giữa các chỉ số bằng 
+cách ước tính các mô hình mối quan hệ giữa 
+các chỉ số và các nhân tố tiềm ẩn được lập chỉ 
+mục gọi là tải nhân tố.  
+Bảng 2. Bảng câu hỏi sử dụng khảo sát 
+Mã Câu hỏi 
+ 
+Kỳ vọng hiệu quả (Venkatesh và c.s., 2003)  
+ 
+PE1   Sử dụng phần mềm Bluezone giúp tôi nắm 
+bắt thông tin về Covid nhanh hơn. 
+ 
+PE2   Sử dụng phần mềm Bluezone giúp tôi 
+nâng cao hiệu quả về phòng tránh Covid. 
+ 
+PE3 Sử dụng phần mềm Bluezone giúp tôi nắm 
+bắt kịp thời các thông tin cần thiết nơi tôi 
+sinh sống. 
+ 
+Kỳ vọng nỗ lực  (Venkatesh và c.s., 2003) 
+ 
+EE1 Học cách sử dụng phần mềm Bluezone là 
+tương đối dễ với tôi. 
+ 
+EE2 Các chức năng và thao tác của Bluezone là 
+rõ ràng và dễ hiểu. 
+ 
+EE3 Phần mềm Bluezone là dễ sử dụng.  
+ 
+EE4 Tôi dễ dàng sử dụng thành thạo phần mềm 
+Bluezone. 
+ 
+Ảnh hưởng xã hội  (Venkatesh và c.s., 2003) 
+ 
+SI1 Người thân trong gia đình tôi cho rằng tôi 
+nên sử dụng phần mềm Bluezone. 
+ 
+SI2 Bạn bè và đồng nghiệp tôi cho rằng tôi 
+nên sử dụng phần mềm Bluezone.  
+ 
+SI3 Tôi sử dụng phần mềm Bluezone là do 
+được tuyên truyền từ các phương tiện 
+truyền thông. 
+ 
+Các điều kiện thuận lợi (Venkatesh và c.s., 2003) 
+ 
+FC1 Tôi có thiết bị để cài đặt phần mềm 
+Bluezone (ví dụ: điện thoại, máy tính 
+bảng). 
+ 
+FC2 Phần mềm Bluezone tương thích với các 
+thiết bị của tôi.  
+ 
+FC3 Tôi có sự hỗ trợ khi gặp trục trặc với phần 
+mềm Bluezone. 
+ 
+Rủi ro về quyền riêng tư (Arfi và c.s., 2021; 
+Chopdar, 2022)  
+ 
+PR1 Tôi nghĩ rằng việc sử dụng Bluezone sẽ 
+khiến quyền riêng tư của tôi gặp rủi ro. 
+ 
+PR2 Dữ liệu cá nhân của tôi có thể bị rò rỉ khi 
+sử dụng phần mềm Bluezone. 
+ 
+Sự tin tưởng (Trust) (Arfi và c.s., 2021; 
+Chopdar, 2022) 
+ 
+T1 
+Tôi tin rằng thông tin mà Bluezone cung 
+cấp là đáng tin cậy. 
+ 
+T2 
+Tôi tin tưởng việc sử dụng phần mềm 
+Bluezone. 
+ 
+T3 
+Bluezone cung cấp các chức năng mà 
+người dùng cần. 
+ 
+Nếu giá trị trung bình của một câu được tìm 
+thấy là gần với 1 hoặc 5 thì nhóm nghiên cứu 
+
+ 
+ 
+ 
+ 
+ Số 02 (2022): 1 – 11 
+  5 
+ 
+KHOA HỌC TỰ NHIÊN 
+loại bỏ câu trả lời đó ra khỏi bảng số liệu vì nó 
+có thể làm giảm tiêu chuẩn tương quan giữa các 
+mục còn lại (J. J. Kim, 2011). Sau bước này, 
+tính chuẩn mực trong phân phối đã được kiểm 
+tra bằng cách kiểm tra độ lệch (skewness) và độ 
+nhọn (kurtosis) trước khi tiến hành phân tích 
+nhân tố khám phá. Vì tính chuẩn mực của phân 
+phối đã được xác nhận, nên việc phân tích nhân 
+tố khám phá được tiến hành thông qua việc sử 
+dụng phần mềm SPSS 26 (Statistical Package 
+for the Social Sciences). 
+Tiến trình phân tích nhân tố khám phá được 
+bắt đầu bằng việc thu thập các giá trị riêng 
+(eigenvalues) cho mỗi nhân tố. Tiếp theo, thang 
+đo Kaiser-Meyer-Olkin (KMO) được sử dụng 
+để đo về mức độ phù hợp của dữ liệu cho việc 
+phân tích nhân tố (Goretzko và c.s., 2021). Giá 
+trị của KMO thay đổi giữa 0 và 1 và các giá trị 
+trên 0,5 thường được coi là đủ cho EFA 
+(Goretzko và c.s., 2021; Schneeweiss & 
+Mathes, 1995). Mức độ tương quan giữa các 
+câu hỏi có đủ lớn để phân tích nhân tố có ý 
+nghĩa thống kê hay không được kiểm tra thông 
+qua phương pháp Bartlett. Chỉ khi kiểm định 
+Bartlett có ý nghĩa thống kê (sig. < 0,05) thì các 
+phân tích tiếp theo mới được tiến hành. 
+3.4.  Mô hình phương trình cấu trúc 
+Sau khi có kết quả từ phân tích nhân tố 
+khám phá, các nhân tố tìm được sẽ được sử 
+dụng để tìm hiểu sự tác động của chúng đối 
+với ý định hành vi của việc sử dụng phần 
+mềm Bluezone. Mô hình phương trình cấu 
+trúc (Structural Equation Modeling – SEM) 
+được sử dụng để tìm hiểu sự tác động của các 
+biến độc lập (nhân tố) đối với biến phụ thuộc 
+(ý định hành vi) (Kline, 2015). SEM là một 
+mô hình cấu trúc tuyến tính bao gồm các mô 
+hình thống kê nhằm tìm lời giải thích mối 
+quan hệ giữa các biến số (Kline, 2015). SEM 
+được ứng dụng rộng rãi trong nhiều lĩnh vực 
+với các tên gọi khác nhau như phân tích cấu 
+trúc hiệp phương sai, phân tích biến ẩn, hoặc 
+mô hình nhân quả. Mục đích của SEM là 
+kiểm tra lý thuyết bằng cách chỉ định một mô 
+hình đại diện cho các dự đoán của lý thuyết 
+đó trong số các cấu trúc hợp lý được đo bằng 
+các biến quan sát thích hợp. 
+4.  KẾT QUẢ NGHIÊN CỨU 
+4.1.  Phân tích nhân tố khám phá 
+EFA được thực hiện trên 18 câu hỏi với 
+vòng quay Varimax. Kết quả phân tích từ 
+phần mềm SPSS cho phép nhóm nghiên cứu 
+trích xuất được giá trị đặc trưng cho từng 
+nhân tố. Phép đo KMO đã xác minh tính thích 
+hợp của việc lấy mẫu cho phép phân tích với 
+giá trị là 0,889 (xem Bảng 3), cao hơn đề xuất 
+của J. O. Kim & Mueller (1978) là 0,6. 
+Bảng 3. Kiểm định KMO và Barlett 
+Kaiser-Meyer-Olkin 
+0,889  
+Kiểm định 
+Bartlett 
+Chi-Square 
+2825,528  
+df 
+153  
+Sig. 
+0,000  
+Kiểm định Bartlett (Bartlett's test of 
+sphericity) cho kết quả χ2 (153) = 2825,528, 
+ρ < 0,000, chỉ ra rằng mối tương quan giữa 
+các hạng mục câu hỏi là đủ lớn để tiến hành 
+phân tích nhân tố khám phá. 
+Số liệu từ 
+Bảng 4 cho thấy có bốn nhân tố chính 
+được hình thành từ tập 18 câu hỏi với giá trị 
+đặc trưng lớn hơn 1. Nói cách khác, 18 câu 
+hỏi này đóng góp 70,269% tầm quan trọng 
+của các yếu tố tác động đến việc sử dụng ứng 
+dụng Bluezone, 29,731% còn lại là do các 
+yếu tố khác. Tỷ lệ phần trăm được giải thích 
+theo từng nhân tố là: nhân tố 1 (46,749%), 
+nhân tố 2 (10,563%), nhân tố 3 (6,587%) và 
+nhân tố 4 (3,369%). 
+Dữ liệu trong Bảng 5 cho thấy có sự dịch 
+chuyển về hạng mục câu hỏi giữa các nhân tố 
+chính. Trong mô hình ban đầu, chúng tôi giả 
+định rằng có sáu nhân tố chính ảnh hưởng tới 
+việc sử dụng phần mềm Bluezone, tuy nhiên 
+kết quả phân tích chỉ ra bốn nhân tố cơ bản 
+phản ánh mối tương quan giữa các câu hỏi. Có 
+một điểm đáng chú ý trong kết quả phân tích 
+đó là nhóm nhân tố chính thứ hai và thứ tư vẫn 
+giữ nguyên theo giả định ban đầu của nhóm 
+tác giả, trong khi nhóm nhân tố chính thứ nhất 
+được hình thành bằng việc kết hợp giữa hai 
+yếu tố sự tin cậy (trust) và kỳ vọng hiệu quả 
+(Performance Expectancy) – đặt lại tên là Hiệu 
+
+ap chi khoa hoc
+DAI HOC HA LONG 
+ 
+ 
+ 
+6 Số 01(2021): 1 – 11  
+ 
+ 
+ 
+KHOA HỌC TỰ NHIÊN 
+quả tin cậy; Nhóm nhân tố thứ 3 được hình 
+thành bằng việc kết hợp giữa ảnh hưởng xã hội 
+và các điều kiện thuận lợi – đặt lại tên là Xã 
+hội và Kỳ vọng hiệu quả. Hạng mục FC3 (Tôi 
+có sự hỗ trợ khi gặp trục trặc với phần mềm 
+Bluezone) bị loại bỏ sau quá trình phân tích. 
+Bảng 4. Các nhân tố chính 
+Bảng 5. Ma trận nhân tố xoay 
+ 
+1 
+2 
+3 
+4 
+T3 
+0,721 
+ 
+ 
+ 
+PE2 
+0,712 
+ 
+ 
+ 
+PE3 
+0,690 
+ 
+ 
+ 
+T2 
+0,649 
+ 
+ 
+ 
+PE1 
+0,575 
+ 
+ 
+ 
+T1 
+0,481 
+ 
+ 
+ 
+FC3 
+ 
+ 
+ 
+ 
+EE1 
+ 
+0,769 
+ 
+ 
+EE2 
+ 
+0,739 
+ 
+ 
+EE3 
+ 
+0,688 
+ 
+ 
+EE4 
+ 
+0,664 
+ 
+ 
+SI2 
+ 
+ 
+0,796 
+ 
+SI1 
+ 
+ 
+0,671 
+ 
+FC1 
+ 
+ 
+0,614 
+ 
+FC2 
+ 
+ 
+0,566 
+ 
+SI3 
+ 
+ 
+0,375 
+ 
+PR1 
+ 
+ 
+ 
+0,905 
+PR2 
+ 
+ 
+ 
+0,872 
+4.2.  Mô hình phương trình cấu trúc 
+Dựa vào kết quả của phân tích nhân t��� 
+khám phá, nhóm nghiên cứu đưa ra các giả 
+thiết sau: 
+H1. Sự tin tưởng và kỳ vọng hiệu quả có 
+ảnh hưởng tích cực tới ý định hành vi của việc 
+sử dụng phần mềm Bluezone. 
+H2. Kỳ vọng nỗ lực có ảnh hưởng tích cực tới ý 
+định hành vi của việc sử dụng phần mềm Bluezone. 
+H3. Ảnh hưởng xã hội có tác động tích cực tới ý 
+định hành vi của việc sử dụng phần mềm Bluezone. 
+H4. Rủi ro về quyền riêng tư có ảnh hưởng 
+tiêu cực tới ý định hành vi của việc sử dụng 
+phần mềm Bluezone. 
+Chỉ có các hạng mục có hệ số trong Bảng 
+5 lớn hơn 0,6 được giữ lại trong phân tích. Số 
+mẫu tối thiểu cần thiết để phân tích có ý nghĩa 
+thống kê theo công cụ tính toán của (Soper, 
+2022) là 166 (với 4 biến tiềm ẩn và 13 biến 
+quan sát được). Số mẫu trong nghiên cứu là 
+224 lớn hơn so với số mẫu tối thiểu. Kỹ thuật 
+phân tích thành phần có cấu trúc tổng quát 
+(GSCA) được sử dụng để phân tích mô hình 
+nghiên cứu được đề xuất do khả năng xử lý 
+với kích thước mẫu nhỏ trong khi cần phân 
+phối chuẩn nghiêm ngặt (Hwang & Takane, 
+2014). GSCA là một thành phần dựa trên mô 
+hình phương trình cấu trúc có thể được sử 
+dụng để mô phỏng các đường dẫn Bình 
+phương tối thiểu một phần (PLS). Nghiên cứu 
+này sử dụng phần mềm GSCA Pro trong việc 
+ước lượng các tham số (Hwang và c.s., 2021). 
+ 
+Tính nhất quán của dữ liệu và các phép 
+đo giá trị hội tụ cho mỗi nhân tố được thể hiện 
+trong Bảng 6. Dillon-Goldstein’s rho được sử 
+dụng để đánh giá cho các yêu cầu về tính nhất 
+quán và độ tin cậy bên trong của mỗi nhân tố 
+(Hwang & Takane, 2014).  
+Nhân tố 
+Giá trị đặc trưng khởi tạo 
+Tổng bình phương của                     
+hệ số tải nhân tố 
+Tổng bình phương 
+của hệ số tải nhân 
+tố xoay 
+Tổng 
+% Phương 
+sai 
+% Tích lũy 
+Tổng 
+% Phương 
+sai 
+% Tích lũy 
+Tổng 
+1 
+8,415 
+46,749 
+46,749 
+8,068 
+44,823 
+44,823 
+3,765 
+2 
+1,901 
+10,563 
+57,312 
+1,682 
+9,343 
+54,165 
+3,326 
+3 
+1,186 
+6,587 
+63,900 
+0,911 
+5,062 
+59,228 
+2,652 
+4 
+1,146 
+3,369 
+70,269 
+0,760 
+4,220 
+63,447 
+1,677 
+5 
+0,973 
+5,405 
+75,674 
+ 
+ 
+ 
+ 
+6 
+0,728 
+4,043 
+79,717 
+ 
+ 
+ 
+ 
+
+ 
+ 
+ 
+ 
+ Số 02 (2022): 1 – 11 
+  7 
+ 
+KHOA HỌC TỰ NHIÊN 
+Bảng 6. Độ tin cậy của thang đo 
+Nhân tố 
+Rho 
+AVE 
+Sự tin tưởng và kỳ 
+vọng hiệu quả 
+0,912 
+0,722 
+Kỳ vọng nỗ lực 
+0,940 
+0,796 
+Ảnh hưởng xã hội 
+0,897 
+0,746 
+Rủi ro về quyền 
+riêng tư 
+0,947 
+0,899 
+Ý định hành vi 
+0,090 
+0,750 
+Hầu hết tất cả các giá trị, nằm trong 
+khoảng từ 0,897 đến 0,947, đều lớn hơn 0,7, 
+trên mức ước tính độ tin cậy có thể chấp nhận 
+được (Hwang & Takane, 2014). Chúng tôi 
+cũng đã xem xét giá trị phương sai trung bình 
+được trích xuất (Average Variance Extracted 
+– AVE) của mỗi biến tiềm ẩn để xác định xem 
+biến có hội tụ hay không. Tất cả các giá trị 
+AVE đều lớn hơn 0,5 (Hwang & Takane, 
+2014), nằm trong khoảng từ 0,722 đến 0,899, 
+cho thấy độ tin cậy hội tụ.
+Bảng 7. Ước lượng hệ số tải (loadings) 
+ 
+Ước lượng 
+SE 
+95%CI_LB 
+95%CI_UB 
+PE2  
+0,876  
+0,022  
+0,826  
+0,911  
+PE3  
+0,850  
+0,031  
+0,786  
+0,904  
+T2  
+0,833  
+0,031  
+0,782  
+0,893  
+T3  
+0,839  
+0,027  
+0,763  
+0,887  
+EE1  
+0,849  
+0,032  
+0,789  
+0,907  
+EE2  
+0,912  
+0,017  
+0,873  
+0,939  
+EE3  
+0,915  
+0,020  
+0,867  
+0,947  
+EE4  
+0,890  
+0,019  
+0,851  
+0,932  
+SI1  
+0,896  
+0,014  
+0,869  
+0,921  
+SI2  
+0,943  
+0,008  
+0,924  
+0,955  
+FC1  
+0,739  
+0,050  
+0,621  
+0,822  
+PR1  
+0,949  
+0,008  
+0,934  
+0,968  
+PR2  
+0,948  
+0,009  
+0,931  
+0,967  
+BI1  
+0,893  
+0,029  
+0,836  
+0,939  
+BI2  
+0,869  
+0,029  
+0,808  
+0,917  
+BI3  
+0,835  
+0,043  
+0,716  
+0,889  
+Bảng 7 cho thấy hệ số tải của các hạng 
+mục cùng với các tham số khác như sai số 
+chuẩn (SE), khoảng tin cậy dưới (CI_LB) và 
+khoảng tin cậy trên (CI_UB). Phương pháp 
+Boostrap thực hiện với số mẫu lặp lại là 100 
+lần, giá trị trung bình của 100 lần lặp này 
+được dùng để ước lượng giá trị gần đúng của 
+tổng thể. Ở mức 0,05 alpha, ước tính tham số 
+được coi là có ý nghĩa thống kê nếu 95% 
+khoảng tin cậy không bao gồm giá trị 0. Kết 
+quả Bảng 7 cho thấy tất cả các hạng mục đều 
+đáng tin cậy và các ước lượng tải đều có ý 
+nghĩa thống kê. 
+ 
+Kết quả phân tích từ phần mềm GSCA 
+Pro cho các kết quả như: độ phù hợp của mô 
+hình (Model FIT) là 0,59; độ phù hợp điều 
+chỉnh của mô hình (Adjusted FIT - AFIT) là 
+0,586. Cả FIT và FIT điều chỉnh (AFIT) đều 
+được sử dụng để điều tra sự khác biệt trong 
+dữ liệu được giải thích bởi một cấu hình mô 
+hình nhất định. Các giá trị FIT nằm trong 
+khoảng từ 0 đến 1. Các đặc điểm và ý nghĩa 
+của FIT và AFIT tương đương với R2 và R2 
+điều chỉnh trong hồi quy tuyến tính. Kết quả 
+thực nghiệm của FIT và AFIT cho thấy mô 
+hình lần lượt chiếm khoảng 59% và 58,6% 
+tổng phương sai của tất cả các biến. 
+
+ap chi khoa hoc
+DAI HOC HA LONG 
+ 
+ 
+ 
+8 Số 01(2021): 1 – 11  
+ 
+ 
+ 
+KHOA HỌC TỰ NHIÊN 
+Bảng 8. Ước tính hệ số đường dẫn 
+ 
+Ước lượng 
+SE 
+95%CI_LB 95%CI_UB 
+Sự tin tưởng và kỳ vọng hiệu quả  
+→ Ý định hành vi sử dụng Bluezone (H1)   
+0,218* 
+0,105 
+0,029 
+0,049 
+Kỳ vọng nỗ lực  
+→ Ý định hành vi sử dụng Bluezone (H2) 
+0,116* 
+0,084 
+0,05 
+0,290 
+Ảnh hưởng xã hội  
+→ Ý định hành vi sử dụng Bluezone (H3) 
+0,137* 
+0,097 
+0,043 
+0,320 
+Rủi ro về quyền riêng tư  
+→ Ý định hành vi sử dụng Bluezone (H4)  
+-0,06* 
+0.63 
+0.06 
+0.185 
+                                                                                                * có ý nghĩa thống kê ở mức 0,05
+Bảng 8 trình bày các ước tính của hệ số 
+đường dẫn của mô hình phương trình cấu 
+trúc, cùng với sai số chuẩn và khoảng tin cậy 
+95% cận dưới và cận trên. Kết quả thực 
+nghiệm cho thấy sự tin tưởng và kỳ vọng hiệu 
+quả có ảnh hưởng tích cực tới ý định hành vi 
+sử dụng phần mềm Bluezone (H1 = 0,218; SE 
+= 0,105; CIs = 0,029 – 0,049). Tương tự, kỳ 
+vọng nỗ lực có ảnh hưởng tích cực tới ý định 
+hành vi (H2 = 0,016; SE = 0,084; CIs = 0,05 
+– 0,29). Ảnh hưởng xã hội cũng đóng góp vào 
+ý định hành vi một cách tích cực (H3 = 0,137; 
+SE = 0,097; CIs = 0,043 – 0,32) và cuối cùng 
+rủi ro về quyền riêng tư có ảnh hưởng tiêu cực 
+tới ý định hành vi sử dụng Bluezone của 
+người dùng (H4 = -0,06; SE = 0,63; CIs = 
+0,06 – 0,185).  
+5.  THẢO LUẬN 
+Đã hơn hai năm kể từ khi đại dịch Covid-
+19 xuất hiện, mặc dù số lượng ca nhiễm và tử 
+vong đã giảm đáng kể so với giai đoạn đầu 
+nhưng vẫn chưa có dấu hiệu nào cho thấy sự 
+kết thúc của đại dịch này. Cùng với các biện 
+pháp cách ly xã hội, tiêm vắc xin, khai báo 
+trực tiếp bằng văn bản, việc ứng dụng công 
+nghệ thông tin trong hỗ trợ đại dịch cũng đã 
+và đang đem lại nhiều lợi ích nhất định. Hầu 
+như các hoạt động xã hội đều đã được số hóa 
+như họp trực tuyến, đặt hàng trực tuyến, 
+thanh toán trực tuyến, giảng dạy trực tuyến... 
+đến truy vết bằng công nghệ số. Các hoạt 
+động này, cho dù không có đại dịch xảy ra, 
+cũng là xu hướng tất yếu trong chuyển đổi số, 
+nhưng sự xuất hiện của đại dịch khiến cho 
+quá trình này được chuyển đổi nhanh hơn. 
+Phần mềm Bluezone là sản phẩm kịp thời để 
+ứng phó nhanh với đại dịch. Tuy nhiên, số 
+lượng người dùng sử dụng liên tục lại không 
+được như kỳ vọng. Điều đó dẫn đến ứng dụng 
+công nghệ thông tin này chưa phát huy được 
+hết sức mạnh. Do đó, ý thức về việc tích cực 
+tham gia vào việc sử dụng ứng dụng 
+Bluezone (hay PC-Covid) vẫn cần phải được 
+nâng cao để giúp các nhà chức trách nhanh 
+chóng tìm ra các giải pháp kịp thời. 
+Kết quả thực nghiệm từ mô hình phương 
+trình cấu trúc cho thấy, cả bốn nhân tố thu 
+được từ phân tích các nhân tố khám phá đều 
+có ảnh hưởng tích cực hoặc tiêu cực tới ý định 
+hành vi của người dùng đối với phần mềm 
+Bluezone. Cụ thể, sự tin tưởng và kỳ vọng 
+hiệu quả, kỳ vọng nỗ lực, ảnh hưởng xã hội 
+có tác động tích cực đến ý định hành vi của 
+việc sử dụng phần mềm truy vết Bluezone. 
+Trong khi đó, rủi ro về quyền riêng tư có ảnh 
+hưởng tiêu cực đến hành vi này. 
+Về mặt lý thuyết, kết quả của nghiên cứu 
+này một lần nữa xác thực các mối quan hệ 
+nguyên nhân – hậu quả đã được nghiên cứu 
+và xác định ở trong mô hình phương trình cấu 
+trúc, qua đó tạo thêm nhiều minh chứng cho 
+sự tồn tại và ảnh hưởng của các nhân tố này. 
+Những độc giả quan tâm hoặc các nhà nghiên 
+cứu khác có thể tham khảo  kết quả trên cho 
+các nghiên cứu tương tự.  
+Về mặt thực tiễn, kết quả nghiên cứu là cơ 
+sở để các nhà phát triển phần mềm, người 
+quản lý đưa ra các chiến lược và giải pháp phù 
+hợp để tăng cường ý định hành vi sử dụng 
+
+ 
+ 
+ 
+ 
+ Số 02 (2022): 1 – 11 
+  9 
+ 
+KHOA HỌC TỰ NHIÊN 
+phần mềm truy vết Bluezone. Cụ thể, đối với 
+các nhân tố có ảnh hưởng tích cực, cần phải 
+liên tục và cập nhật phần mềm sao cho nó 
+thực sự mang lại hiệu quả hay nói cách khác, 
+dữ liệu có được sử dụng tối ưu cho các nhà 
+quản lý hay không. Hơn nữa, phần mềm phải 
+nên thiết kế dễ sử dụng để bất kỳ ai cũng có 
+thể tự thao tác. Ảnh hưởng xã hội cho thấy 
+phương tiện truyền thông, gia đình, bạn bè và 
+đồng nghiệp đóng vai trò quan trọng tới ý 
+định hành vi, do đó việc tuyên truyền cũng 
+nên tiếp tục được duy trì thông qua các 
+phương tiện truyền thông khác nhau. Vì rủi ro 
+về quyền riêng tư cũng đóng vai trò quyết 
+định tới ý định, hành vi của người sử dụng, 
+do đó các nhà quản lý, các nhà phát triển phần 
+mềm, an ninh mạng cũng phải có các kỹ 
+thuật, cơ chế, chính sách sử dụng và bảo vệ 
+một cách phù hợp để giúp người dùng yên 
+tâm hơn về dữ liệu cá nhân của mình. 
+Ngoài các yếu tố tích cực, nghiên cứu này 
+cũng tồn tại một số giới hạn. Thứ nhất, việc 
+lấy mẫu là không hoàn toàn ngẫu nhiên vì đối 
+tượng tham gia nghiên cứu nằm trong mạng 
+lưới của tác giả. Do đó việc khái quát hóa đến 
+một số lượng người dùng lớn hơn cần phải 
+được xem xét một cách kỹ lưỡng. Thứ hai, 
+việc khảo sát chỉ được thực hiện trong một 
+khoảng thời gian nhất định nên hành vi của 
+đối tượng tham gia nghiên cứu có thể không 
+nhất quán trong tương lai. Thứ ba, chỉ có một 
+số các nhân tố được đưa vào phân tích trong 
+mô hình phương trình cấu trúc, có thể tồn tại 
+nhiều nhân tố khác cũng có tầm ảnh hưởng 
+tới việc sử dụng Bluezone, do đó chúng tôi 
+khuyến nghị các nhà nghiên cứu quan tâm tìm 
+hiểu thêm các nhân tố mới này. 
+6.  KẾT LUẬN 
+Nghiên cứu này khám phá các nhân tố 
+và đánh giá sự ảnh hưởng của các nhân tố 
+đó tới ý định hành vi của người dùng trong 
+việc sử dụng phần mềm truy vết Bluezone. 
+Mô hình lý thuyết thống nhất về chấp nhận 
+và sử dụng công nghệ được mở rộng thêm 
+hai nhân tố mới bao gồm sự tin tưởng và rủi 
+ro về quyền riêng tư. Kết quả khảo sát từ 
+224 người dùng cho thấy có bốn nhân tố 
+chính ảnh hưởng tới việc sử dụng phần 
+mềm truy vết, trong đó có 3 nhân tố ảnh 
+hưởng tích cực tới ý định hành vi, trong khi 
+đó nhân tố rủi ro về quyền riêng tư có ảnh 
+hưởng theo chiều ngược lại. Kết quả nghiên 
+cứu đóng góp về mặt lý thuyết bằng cách 
+giải thích sự ảnh hưởng của các nhân tố đối 
+với ý định hành vi một cách tinh gọn hơn 
+(giảm chiều từ 6 nhân tố xuống còn 4 nhân 
+tố trong EFA) và xác thực các mối quan hệ 
+nguyên nhân – hậu quả thông qua mô hình 
+phương trình cấu trúc. Đồng thời, kết quả 
+nghiên cứu cũng có thể được sử dụng trong 
+thực tiễn giúp các nhà quản lý, nhà phát 
+triển phần mềm, an ninh môi trường mạng 
+có thêm cơ sở để tiếp tục hoàn thiện phần 
+mềm truy vết Covid-19. 
+LỜI CẢM ƠN  
+ 
+Nhóm tác giả trân trọng cảm ơn các 
+bạn bè, đồng nghiệp trong việc tham gia khảo 
+sát. Cảm ơn TS. Nguyễn Hải Minh (ICTU) vì 
+đã phổ biến phiếu khảo sát đến các sinh viên 
+trong trường. 
+TÀI LIỆU THAM KHẢO 
+Arfi, W. B., Nasr, I. B., Kondrateva, G., & 
+Hikkerova, L. (2021). The role of trust in 
+intention to use the IoT in eHealth: 
+Application of the modified UTAUT in a 
+consumer 
+context. 
+Technological 
+Forecasting and Social Change, 167, 
+120688. 
+https://doi.org/10.1016/j.techfore.2021.120688 
+Bansal, G., Zahedi, F. “Mariam”, & Gefen, 
+D. (2010). The impact of personal 
+dispositions on information sensitivity, 
+privacy concern and trust in disclosing 
+health information online. 
+Decision 
+Support 
+Systems, 
+49(2), 
+138–150. 
+https://doi.org/10.1016/j.dss.2010.01.010 
+baochinhphu.vn. (2020, Tháng Tư 18). Thủ 
+tướng dự khai trương 2 sản phẩm công 
+nghệ giúp phòng chống COVID-19. 
+
+ap chi khoa hoc
+DAI HOC HA LONG 
+ 
+ 
+ 
+10 Số 02 (2022): 1 – 11  
+ 
+ 
+ 
+baochinhphu.vn. 
+https://baochinhphu.vn/thu-tuong-du-
+khai-truong-2-san-pham-cong-nghe-
+giup-phong-chong-covid-19-
+102271400.htm 
+Chopdar, P. K. (2022). Adoption of Covid-19 
+contact tracing app by extending UTAUT 
+theory: Perceived disease threat as moderator. 
+Health Policy and Technology, 11(3), 
+100651. 
+https://doi.org/10.1016/j.hlpt.2022.100651 
+Davis, F. D. (1985). A technology acceptance 
+model for empirically testing new end-
+user information systems: Theory and 
+results [Thesis, Massachusetts Institute of 
+Technology]. 
+https://dspace.mit.edu/handle/1721.1/15192 
+Fabrigar, L. R., & Wegener, D. T. (2012). 
+Exploratory Factor Analysis. Oxford 
+University Press. 
+Goretzko, D., Pham, T. T. H., & Bühner, M. 
+(2021). Exploratory factor analysis: 
+Current 
+use, 
+methodological 
+developments and recommendations for 
+good practice. Current Psychology, 40(7), 
+3510–3521. https://doi.org/10.1007/s12144-
+019-00300-2 
+Hair Jr, J. F., Black, W. C., Babin, B. J., & 
+Anderson, R. E. (2009). Multivariate 
+Data Analysis (7th edition). Pearson. 
+Hwang, H., Cho, G., & Choo, H. (2021). 
+GSCA Pro 1.0 User’s Manual. 
+Hwang, 
+H., 
+& 
+Takane, 
+Y. 
+(2014). 
+Generalized 
+Structured 
+Component 
+Analysis: A Component-Based Approach 
+to 
+Structural 
+Equation 
+Modeling. 
+Chapman 
+and 
+Hall/CRC. 
+https://doi.org/10.1201/b17872 
+Jung, K., Nguyen, T. V., Piscarac, D., & 
+Yoo, S.-C. (2020). Meet the Virtual Jeju 
+Dol Harubang—The Mixed VR/AR 
+Application for Cultural Immersion in 
+Korea’s 
+Main 
+Heritage. 
+ISPRS 
+International 
+Journal 
+of 
+Geo-
+Information, 
+9(6), 
+Art. 
+6. 
+https://doi.org/10.3390/ijgi9060367 
+Jung, K., Nguyen, V. T., & Lee, J. (2021). 
+BlocklyXR: An Interactive Extended 
+Reality Toolkit for Digital Storytelling. 
+Applied 
+Sciences, 
+11(3), 
+Art. 
+3. 
+https://doi.org/10.3390/app11031073 
+Kim, J. J. (2011). Developing an instrument 
+to measure social presence in distance 
+higher education. British Journal of 
+Educational Technology, 42(5), 763–777. 
+https://doi.org/10.1111/j.1467-
+8535.2010.01107.x 
+Kim, J. O., & Mueller, C. W. (1978). Factor 
+Analysis: 
+Statistical 
+Methods 
+and 
+Practical Issues. SAGE Publications, Inc. 
+Kline, R. B. (2015). Principles and Practice 
+of Structural Equation Modeling (Fourth 
+edition). The Guilford Press. 
+Le, T.-A. T., Vodden, K., Wu, J., & Atiwesh, 
+G. (2021). Policy Responses to the 
+COVID-19 
+Pandemic 
+in 
+Vietnam. 
+International Journal of Environmental 
+Research and Public Health, 18(2), Art. 2. 
+https://doi.org/10.3390/ijerph18020559 
+Li, Y. (2011). Empirical Studies on Online 
+Information Privacy Concerns: Literature 
+Review and an Integrative Framework. 
+Communications of the Association for 
+Information 
+Systems, 
+28(1). 
+https://doi.org/10.17705/1CAIS.02828 
+Mbunge, E. (2020). Integrating emerging 
+technologies into COVID-19 contact 
+tracing: Opportunities, challenges and 
+pitfalls. Diabetes & Metabolic Syndrome: 
+Clinical Research & Reviews, 14(6), 
+1631–1636. 
+https://doi.org/10.1016/j.dsx.2020.08.029 
+Mehrabian, A., & Russell, J. A. (James A. 
+(1974). An approach to environmental 
+psychology. Cambridge, M.I.T. Press. 
+http://archive.org/details/approachtoenvir
+o00albe 
+Nguyen, T. V. (2022). The perceptions of 
+social media users of digital detox apps 
+considering personality traits. Education 
+and Information Technologies, 27(7), 
+9293–9316. 
+https://doi.org/10.1007/s10639-022-
+11022-7 
+Nguyen, T. V., Anh, N., Tan, N., & Dinh, L. 
+(2021). Tìm hiểu các yếu tố ảnh hưởng tới 
+
+ap chi khoa hoc
+DAI HOC HA LONG 
+ 
+ 
+ 
+ Số 02 (2022): 1 – 11 
+  11 
+ 
+KHOA HỌC TỰ NHIÊN 
+việc sử dụng ứng dụng Bluezone tại Việt 
+Nam. Hội thảo quốc gia lần thứ XXIV: 
+Một số vấn đề chọn lọc của Công nghệ 
+thông tin và truyền thông, Thái Nguyên. 
+Nguyen, T. V., & Nguyen, T. H. C. (2022). 
+Factors Influencing Intention to use the 
+COVID-19 Contact Tracing Application. 
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+462. 
+https://doi.org/10.3844/jcssp.2022.453.462 
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+Factor 
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+55(1), 
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+Calculator for Structural Equation Models. 
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+Spall, H. G. C. V. (2020). Applications of 
+digital 
+technology 
+in 
+COVID-19 
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+Lancet Digital Health, 2(8), e435–e440. 
+https://doi.org/10.1016/S2589-
+7500(20)30142-4 
+
+ap chi khoa hoc
+DAI HOC HA LONG

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+arXiv:2301.03340v1  [physics.atom-ph]  9 Jan 2023
+Formation of strongly shifted EIT resonances using "forbidden" transitions of Cesium
+Armen Sargsyan,1 Ara Tonoyan,1 Rodolphe Momier,1, 2, ∗ Claude Leroy,2 and David Sarkisyan1
+1Institute for Physical Research, NAS of Armenia, Ashtarak-2, 0203 Armenia
+2Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS 6303,
+Université Bourgogne Franche-Comté, 21000 Dijon, France
+(Dated: January 10, 2023)
+Atomic transitions satisfying Fe − Fg = ∆F = ±2 (where Fe stands for excited and Fg stands
+for ground state) of alkali atoms have zero probability in zero magnetic field (they are so-called
+"forbidden" transitions) but experience a large probabilty increase in an external magnetic field.
+These transitions are called magnetically induced (MI) transitions. In this paper, we use for the first
+time the σ+ (∆mF =
++ 1) MI transitions Fg = 3 → Fe = 5 of Cesium as probe radiation to form
+EIT resonances in strong magnetic fields (1 - 3 kG) while the coupling radiation frequency is resonant
+with Fg = 4 → Fe = 5 σ+ transitions. The experiment is performed using a nanometric-thin cell
+filled with Cs vapor and a strong permanent magnet. The thickness of the vapor column is 852 nm,
+corresponding to the Cs D2 line transition wavelength. Due to the large frequency shift slope of the
+MI transitions (∼ 4 MHz/G), it is possible to form contrasted and strongly frequency-shifted EIT
+resonances. Particularly, a strong 12 GHz frequency shift is observed when applying an external
+magnetic field of ∼ 3 kG. Preliminary calculations performed considering Doppler-broadened three
+level systems in a nanocell are in reasonable agreement with the experimental measurements.
+I.
+INTRODUCTION
+Optical processes occurring in Rubidium, Cesium,
+Potassium and Sodium vapors confined in optical cells
+have important applications such as optical atomic
+clocks, optical atomic magnetometers, atomic gyro-
+scopes, markers of atomic transition frequencies, as de-
+scribed for example in [1–6]. Therefore, the study of the
+peculiarities of atomic transitions (in particular Zeeman
+transitions in an external magnetic field) of alkali atoms
+is of utmost importance. It is well known that the appli-
+cation of a strong magnetic field can significantly change
+the probabilities (intensities) of the Zeeman transitions,
+as shown in [7–13]. High interest has recently been fo-
+cused on atomic transitions between ground and excited
+levels that satisfy the condition Fe − Fg = ∆F = ±2
+(these transitions are so-called forbidden by the selection
+rules, thus their probability is zero when no external mag-
+netic field is applied). However, the probabilities of these
+transitions in a magnetic field increase significantly. For
+this reason, we refer to these transitions as Magnetically
+Induced (MI) transitions [8, 11, 12].
+This giant increase in the probabilities of the MI transi-
+tions is due to the “mixing” of magnetic sublevels |F, mF ⟩
+of the ground (Fg) or excited (Fe) levels with sublevels
+having the same magnetic quantum number mF . This
+mixing is the strongest for D2 lines of alkali atoms, as
+up to four states |Fe, 0⟩ can experience mixing, thus re-
+sulting in a 4×4 block in the magnetic Hamiltonian, as
+described in [7, 8, 11, 12].
+Magnetically-induced transitions are of great interest
+because, over a wide range of magnetic field, their proba-
+bilities can be much higher than the probabilities of usual
+∗ [email protected]
+("allowed", satisfying the selection rule on F) transitions.
+It is important to note that the slope of the frequency
+shifts (obtained by diagonalizing the magnetic Hamilto-
+nian [7]) as a function of the magnetic field B in strong
+magnetic fields can reach up to around 4 MHz/G, which
+is 3 times larger than in the case of ordinary transitions.
+Thus, the frequency shift of MI transitions in strong mag-
+netic fields can reach several tens of GHz, which can be
+useful for working in higher frequency ranges, for exam-
+ple for the frequency stabilisation of lasers on strongly
+shifted frequencies [14, 15].
+In [11, 12], we established the following rule for the
+probabilities of MI transitions:
+the probabilities and
+number of MI transitions with ∆F = +2 are maximal
+for σ+ radiation, whereas the probabilities and number
+of MI transitions with ∆F = −2 are maximal for σ−
+radiation. The difference between the intensities of MI
+transitions for the σ+ and σ−-polarized radiation beams
+can reach several orders of magnitude.
+It
+has
+been
+recently
+demonstrated
+that
+electromagnetically-induced
+transparency
+(EIT)
+resonances can be formed using Λ-system made of
+∆F
+=
++2 MI transitions only if both probe and
+coupling beam are σ+-polarized.
+This statement was
+experimentally and theoretically verified for 87Rb (MI
+transitions Fg = 1 → Fe = 3) and 85Rb (MI transitions
+Fg = 2 → Fe = 4) [16, 17]. However, if the Λ-system
+is formed by MI transitions satisfying ∆F = −2, then
+both probe and coupling radiation must be σ−-polarized
+in order to form EIT resonances.
+This statement
+was experimentally and theoretically verified for Cs
+(MI transitions Fg = 4 → Fe = 2).
+This is a direct
+consequence of magnetically-induced circular dichroism
+[18].
+In this work, we consider seven σ+ MI transitions of
+Cs (Fg = 3 → Fe = 5, see Fig. 1). The probabilities of
+these transitions increase highly in the range 0.3 - 3 kG
+
+2
+0
++1
++2
++3
+-1
+-2
+-3
+0
++1
++2
++3
+-1
+-2
+1
+2
+3
+4
+5
+7
+6
++4
+0
++1
++2
++3
+-1
+-2
+-3
+FIG. 1. Scheme of Cs D2 line σ+ transitions between Fg = 3, 4
+and Fe = 5. The probe frequency νp is scanned across the
+MI transitions labelled 1-7 (Fg = 3 → Fe = 5). The coupling
+frequencies νcn are resonant with Fg = 4 → Fe = 5 transi-
+tions, forming seven Λ-systems. Only the states involved in
+the process under consideration are shown. Note that |F, mF ⟩
+is just a notation for visualization, as the atomic states are
+better described in the uncoupled basis |J, mJ, I, mI⟩ in high
+magnetic fields.
+and we used these transitions to form EIT resonances
+in strong B-fields.
+A nanometric-thin cell (NC) filled
+with Cs vapor (thickness L ≈ 850 nm, approximately
+the resonant wavelength of Cs D2 line [19]) has been
+used. The advantages of using thin cells, including strong
+reduction of Doppler broadening, are noted in [12, 17, 20].
+A.
+Probabilities and frequency shifts of the MI
+transitions of Cs D2 line
+The curves in Fig. 2 were calculated using a known the-
+oretical model depicting the changes of transition proba-
+bilities as a function of the external magnetic field. The
+block-diagonal (each block corresponding to a given value
+of the magnetic quantum number) magnetic Hamiltonian
+is built for each value of the magnetic field and then diag-
+onalized in order to calculate the probability coefficients.
+This model was presented in a number of papers, e.g.
+[7, 11, 13].
+The evolution of the probabilities of MI transitions (la-
+belled 1 to 7, see Fig. 1) with respect to the magnetic field
+B is shown in Fig. 2a). Note that in the range 0.3 - 2
+kG the probabilities of the MI transitions labeled 5, 6
+and 7 are the strongest among all transitions occurring
+from Fg = 3 [8, 12]. The frequency shift slope of the
+MI transitions, obtained through the eigenvalues of the
+Hamiltonian, is quite large (∼ 4 MHz/G) while for usual
+transitions the slope is 3 times smaller. Despite the fact
+that the probabilities of the MI transitions decrease as
+B increases, they can still be recorded easily at 7 kG.
+As noted below, this is due to the fact that these tran-
+sitions are formed far on the high-frequency wing where
+there are no intersections with other transitions (spec-
+tra are presented for Na in [21], but Cs behaves almost
+identically).
+The evolution of the probabilities of the corresponding
+seven coupling transitions Fg = 4 → Fg = 5 (Ac1 to Ac7)
+that are used to form seven Λ-systems (see Fig. 1) with
+respect to the magnetic field are shown in Fig. 2b). In the
+case of σ− polarization, the probability of the strongest
+Fg = 4 → Fe = 5 σ− transition already tends to zero for
+B > 300 G, as shown in Fig. 2c). Thus, both the probe
+and the coupling beams must be σ+-polarized in order
+to form EIT resonances.
+B.
+Qualitative description of the EIT process
+For a qualitative description of the EIT process, we
+present a formula from [3, 22]. The ratio of absorption at
+the probe radiation frequency νp at which EIT resonance
+is observed (in the presence of νc radiation) to absorp-
+tion (when there is no coupling radiation), assuming low
+radiation intensity νp and zero frequency detuning of the
+coupling radiation, is described by the expression:
+α(Ωc)
+α(0) =
+K
+1 + Ω2c/4Γ21γN
+,
+(1)
+where K is a constant including the Doppler width,
+γN is the natural width of the level (γN/2π ≃ 5.2 MHz
+for the 62P3/2 level of the Cs atom), Ωc is the Rabi fre-
+quency for the coupling radiation and Γ21 is the dephas-
+ing rate of the coherence between the two ground states
+of the Λ-system, which is caused in particular by colli-
+sions of atoms with the windows of the nanocell. The
+case α(Ωc) = 0 corresponds to complete transparency
+(the contrast of the EIT resonance reaches 100%) and a
+large amplitude of the EIT resonance, which decreases
+with an increase in Γ21. The spectral width of the EIT
+resonance can be described by the simple expression [3]:
+γEIT ≃ 2Γ21 + Ω2
+c/γN .
+(2)
+It follows from formula (1) that in order to obtain small
+value of α(Ωc) (which means high electromagnetically in-
+duced transparency of the medium), it is necessary to in-
+crease Ωc, however, an increase in Ωc leads to an increase
+in the spectral width of the EIT resonance. Therefore,
+it is necessary to find a compromise for Ωc. Estimates
+can be obtained from Ωc/2π = aγN(I/8)1/2 where I is
+the laser intensity in mW/cm2, γN ∼ 5 MHz, and a
+is a fit parameter (for our case a is of ∼ 0.5) [23] and
+Ωc ∼ 15 MHz.
+II.
+EXPERIMENT
+A.
+Experimental setup
+The layout of the experimental setup is shown on
+Fig. 3. Two extended cavity diode lasers are tuned in the
+vicinity of the Cs D2 line, with a wavelength λ ≃ 852 nm.
+The Λ-systems shown in Fig. 1 are formed by scanning
+the frequency νp of a VitaWave laser (δνp ∼ 1 MHz) [24]
+
+3
+1
+2
+3
+4
+5
+6
+7
+a)
+b)
+c)
+FIG. 2. Magnetic field dependence of the Zeeman transition intensities of the D2 line of Cs. a) Fg = 3 → Fe = 5 σ+ MI
+transitions. b) Fg = 4 → Fe = 5 σ+ transitions. c) Transition |4, −1⟩ → |5, −2⟩ (σ−). This transition forms a Λ-system with
+transition 7 as shown in panel a) and in the inset (see Fig. 1). Its probability tends to 0 as the magnetic field increases, thus
+forming EIT resonances at high magnetic fields requires both probe and coupling beams to be σ+-polarized.
+FI
+FI
+SO
+BS
+PD
+ECDL 1
+ECDL 2
+probe
+coupling
+C
+Ref. channel
+Meas. channel
+PBS2
+PBS1
+PBS3
+PBS4
+M
+IF PD
+NC
+PM
+FIG. 3.
+Scheme of the experimental setup.
+ECDL: CW
+narrow-band external-cavity diode lasers with λ = 852 nm
+(resonant with Cs D2 line). FI: Faraday insulators. PBSi:
+polarizing beam splitters. BS: beam splitter. IF: interference
+filter. C: saturated absorption spectroscopy unit for frequency
+reference. NC: nanocell placed in oven. PM: permanent mag-
+net. PD: photodiodes. SO: 4-channel digital oscilloscope.
+in the vicinity of the MI transitions Fg = 3 → Fe = 5,
+while keeping the frequency νc from a MOGLabs “cat-
+eye” laser (δνp ≃ 0.1 MHz) on resonance with one of the
+4 → 5 transitions. A fraction about 10% of the coupling
+radiation power was sent to a frequency stabilization unit
+based on the DAVLL method [25]. Probe radiation has
+vertical polarization, while the coupling radiation has
+horizontal polarization. In the case of a longitudinal B-
+field, linearly polarized laser radiation can be considered
+as consisting of σ+ and σ− radiations. The use of mutu-
+ally perpendicular polarizations allows by using PBS4 to
+direct only probe radiation to the photo-receiver, while
+cutting off the coupling radiation. As noted above, in
+the case of MI transitions with ∆F = +2 for the for-
+mation of the EIT resonance, both probe and coupling
+radiations must have σ+ polarization. A photograph of
+the Cs nanocell is shown in Fig. 3. Interference fringes
+are formed by the reflection of light on the inner surfaces
+of windows (made of sapphire). The region correspond-
+Coupling off
+7
+6
+5
+4
+3
+EIT 7
+EIT 6
+EIT 5
+EIT 4
+EIT 3
+(1)
+(2)
+(3)
+(4)
+(5)
+(6)
+FIG. 4. Probe transmission spectra of the Cs nanocell (L =
+λ = 852 nm). Spectra labelled 1 to 5 show five EIT reso-
+nances, labelled EIT 3 to EIT 7, while the probe frequency is
+scanned across transitions 3 to 7 (see Fig. 1). The coupling
+and probe powers are respectively 10 and 0.05 mW and the ex-
+ternal longitudinal magnetic field is B = 1400 G. Spectrum n°
+6 corresponds to the case where coupling is off. Small VSOP
+peaks are visible on each atomic resonance. Zero frequency
+corresponds to the transition frequency of Cs D2 line.
+ing to a thickness L ≈ λ ∼ 850 nm is outlined by an
+oval. The design of the Cs-filled NC used in our experi-
+ments is similar to that of extremely thin cell described
+in [26]. Earlier it was demonstrated in [16, 17, 27] that
+the use of a nanocell (NC) with thickness L = λ makes it
+easy to record contrasted EIT resonances, which is due
+to the low absorption of the NC, while the disadvan-
+tage is broadening of the EIT resonance caused by fre-
+quent inelastic collisions of atoms with the windows of
+the NC. Studies of the EIT resonances were done using
+a strong neodymium–iron–boron alloy ring-shaped per-
+manent magnet (PM). Due to the small thickness of the
+vapor column, the high-gradient field produced by magne
+can be considered uniform across the interaction region.
+The PM was placed after the rear window of the NC,
+with the axis aligned along the probe beam propagation
+
+4
+direction. The magnetic field in the NC was simply var-
+ied by longitudinal displacement of the PM, calibrated
+using a Teslameter HT201 magnetometer.
+B.
+Experimental results: using MI transitions to
+form EIT resonances
+Curves 1 to 5 in Fig. 4 show the experimental trans-
+mission spectra of the probe radiation which contain the
+resonances EIT 3 to EIT 7 (numbers 3-7 means that MI
+transitions with numbers 3-7 are involved, respectively)
+in a longitudinal magnetic field B = 1400 G. The NC
+thickness is L = λ = 852 nm and the temperature of
+the reservoir is 100 ◦C (to prevent Cs vapor condensa-
+tion on the windows, the temperature of the windows is
+slightly higher). The coupling and the probe powers are
+20 mW and 0.1 mW, respectively. Note that since only
+σ+ radiations participate to the formation of the EIT
+resonances (see Fig. 1), only half of the power of these
+radiations must be considered, meaning 10 mW and 50
+µW, respectively. Curve n° 6 is a probe spectrum when
+the coupling is blocked. Since the cell thickness is L = λ,
+small peaks formed by velocity selective optical pump-
+ing (VSOP) resonances are located exactly at the atomic
+transitions frequencies, as described in [9].
+The amplitude of the EIT resonance is a factor ∼10
+larger than the amplitude of the VSOP resonance,
+whereas the spectral width of the EIT resonance is a
+factor of 1.5 smaller, which is characteristic of the coher-
+ent EIT process [17]. Note that the contrast of the EIT
+resonance defined as the ratio of the EIT resonance am-
+plitude divided by the peak absorption of the Cs vapor
+when the coupling is blocked reached 40-50 % which is
+typical when a nanocell is used [27].
+In Fig. 6, curves 1 to 4 are probe transmission spectra
+which contain EIT 6, EIT 5, EIT 4 and EIT 3 resonances
+for B = 1770 G. Curve n° 5 shows only the probe spec-
+trum when the coupling is blocked. In Fig. 7, lines 1 to 3
+show the probe transmission spectra which contain EIT
+6, EIT 4 and EIT 3 resonances for B = 2880 G. Line
+n° 4 shows only the probe spectrum when the coupling
+is blocked. The inset shows the profile of EIT 6 reso-
+nance fitted with a Gaussian profile with a FWHM of ∼
+35 MHz. There is also a small VSOP resonance which is
+formed when the coupling is blocked. The typical FWHM
+of VSOP resonances is 40-50 MHz.
+Preliminary theoretical calculations (shown in the
+right part of the inset of Fig. 7 were obtained by solv-
+ing the Liouville equations of motion for an ensemble of
+three-level Λ-systems (as presented in Fig. 5), taking into
+account the geometry of the nanocell (coherence dephas-
+ing rate determined by the time of flight of the atoms), its
+Fabry-Perot nature (reflections of the fields on the inner
+surfaces of the cell) and Doppler broadening, following
+the procedure described in [28]. The Rabi frequencies of
+the probe and coupling lasers are respectively Ωc = 1.5γN
+and Ωp = 0.06γN. Reasonable agreement between theory
+and experiment regarding the width and depth of the EIT
+resonance is obtained and the VSOP resonance is seen.
+Small discrepancies (assymetry of the profile and ampli-
+tude of the VSOP resonance) can arise notably from the
+need of considering neighboring Zeeman sublevels (not
+shown in Fig. 1, and therefore more than three levels, to
+obtain more accurate results.
+FIG. 5. Scheme of the three-level Λ-system used in the calcu-
+lations. The total decay rate Γ33 of state |3⟩ is 1/2(γ31 + γ32)
+[29].
+The dephasing rate of coherence between the ground
+states is Γ21 = (2πt)−1 where t is the time of flight of
+the atoms through the cell (at the most probable velocity
+u =
+�
+2kBT/M where T is the vapor temperature and M the
+atomic mass).
+The amplitude of resonance n° 6 is ∼ 50 times greater
+than that of the VSOP resonance and is spectrally nar-
+rower than the latter (this is a manifestation of the co-
+herent EIT process [2, 17]). In Fig. 8 the solid lines in-
+dicate the calculated dependences of the frequency shifts
+for transitions 1–7 (Fig. 1) and Fg = 3 → Fe = 4 (marked
+with dotted oval) to the magnetic field B.
+The black
+squares represent the experimental results. As mentioned
+earlier, due to the high value of the frequency shift slope
+for B > 3 kG, the group of MI transitions 1–7 is com-
+pletely separated in frequency from Fg = 3 → Fe = 4
+transitions.
+The curves in the inset of Fig. 8 show experimental and
+theoretical spectra (calculated by combining the models
+presented in [7] and [30]) of the seven MI transitions ab-
+sorption for B = 6 kG when frequency shift reaches ∼ 30
+GHz. Note that the amplitude of transition 6 is slightly
+bigger than that of transition 7 (while for B < 5 kG the
+amplitude of transition 7 is bigger, see Fig. 2a), because
+of the “mixing” effect. Note that one of the remarkable
+features of the σ+ MI transitions 3 → 5′ is that they
+are still well recorded for a magnetic field B ≈ 8 kG.
+They are located in the high frequency wing of the spec-
+trum presented in Fig. 18 of paper [31] and for this case
+the frequency shift reaches 34 GHz. Using our theoret-
+ically calculated curves for MI transitions 3 → 5′ we
+checked the frequency position of these MI transitions
+and found good agreement with the experimental curves
+presented in Fig. 18. In paper [31] the 3 → 5′transitions
+are not identified. Therefore, it is important to inform
+
+5
+Coupling off
+EIT 6
+EIT 5
+EIT 4
+EIT 3
+(1)
+(2)
+(3)
+(4)
+(5)
+6
+5
+4
+3
+FIG. 6.
+Probe transmission spectra of the Cs nanocell
+(L = λ ≈ 850 nm). Spectra 1 to 4 exhibit four EIT reso-
+nances, labelled EIT 3 to EIT 6, while the probe frequency
+is scanned across transitions 3 to 6. The external longitudi-
+nal magnetic field is B = 1770 G. Spectrum n° 5 is a probe
+transmission spectrum when the coupling is off. Small VSOP
+peaks are visible on each atomic transition. Zero frequency
+corresponds to the transition frequency of Cs D2 line.
+Coupling off
+Coupling off
+(1)
+(2)
+(3)
+(4)
+Experiment
+Coupling o���
+EIT 6
+EIT 4
+EIT 3
+6
+5
+4
+3
+EIT 6 
+Theory
+FIG. 7. Probe transmission spectra of the Cs nanocell (L =
+λ = 852 nm).
+Lines 1 to 3 show four EIT resonances, la-
+belled EIT 4, EIT 5 and EIT 6. The external longitudinal
+magnetic field is B = 2880 G. Line 4 is a probe transmission
+spectrum when the coupling is off. The left part of the inset
+is a zoom on EIT 6, fitted with a Gaussian profile (FWHM
+35 MHz). The right curves are calculated. Red: coupling on,
+black: coupling off. Small VSOP peaks are visible on each
+atomic transitions formed by the probe radiation. Their typ-
+ical linewidth is 40-50 MHz. Zero frequency corresponds to
+the transition frequency of Cs D2 line.
+scientists working in the field of laser spectroscopy of al-
+kali metal atoms about the MI atomic transitions. The
+above-mentioned MI transitions can be exploited in such
+high B-fields as new frequency markers, for using new fre-
+quency ranges, as well as for the frequency stabilization
+of lasers at strongly shifted frequencies from the initial
+transition in unperturbed atoms [13, 14].
+Exp.
+Theory
+6
+5
+4
+3 2 1
+7
+6
+5
+4
+3
+2
+1
+7
+FIG. 8. Red solid lines: frequency shift of transitions 1 to 7
+(see figure 1) as a function of the magnetic field. The black
+squares with error bars represent experimental measurements,
+the inaccuracy is around 1 %. Black dashed lines: frequency
+shift of Fg = 3 → Fe = 4 transitions. For B > 3 kG, both
+groups are well separated in frequency. Inset: theoretical and
+experimental absorption spectra for B = 6 kG, the frequency
+shift reaches 30 GHz from the Cs D2 line transition frequency.
+III.
+CONCLUSION
+In this paper, we used for the first time forbidden
+transitions of Cs (Fg = 3 → Fe = 5, more precisely
+σ+(∆mF = +1) transitions) to create Λ-system allowing
+the formation of EIT resonances. This was done in an ex-
+ternal magnetic field, as such transitions have zero proba-
+bility in the absence of magnetic field. A nanometric-thin
+cell filled with Cs vapor was used, with a thickness corre-
+sponding to the resonant wavelength of Cs D2 line (≈ 850
+nm), and the magnetic field was varied by longitudinal
+displacement of the permanent magnet along the prop-
+agation direction (Fig. 3). As expected, when the cou-
+pling is blocked, small VSOP resonances are formed right
+at the different transitions’ frequencies, while coupling
+radiation allows for the formation of EIT resonances,
+spectrally narrower and with a bigger amplitude.
+We
+formed EIT resonances with 6 out the 7 transitions de-
+picted in Fig. 1. This was possible up to 3 kG thanks
+to the big value of the frequency shift, reaching up to 4
+MHz/G, therefore leading to EIT resonances shifted 12
+GHz apart from the Cs D2 line transition frequency [31].
+This result is of great interest, as the highly-shifted spec-
+tra can serve as frequency references [14, 15], especially
+taking into account that these transitions are still easily
+recorded up to 8 kG when the frequency shift reaches 35
+GHz. As for the theoretical description, further investi-
+gation is necessary, mainly in order to take into account
+the effect of neighbouring states, and thus including more
+levels in the model. The complexity of the manifold and
+the number of coupled equations make it a challenging
+
+6
+and computationally-intensive task. However, reasonable
+agreement was already achieved by simply considering an
+ensemble of three-level systems. To the best of our knowl-
+edge, there are no reports on obtaining EIT resonances
+in Λ-systems in such strong fields using usual transitions
+of alkali atoms. We note that much narrower EIT reso-
+nances can be attained by using cm-long cells (to lower
+the effect of inelastic collisions of atoms with the win-
+dows), and by using coherently coupled probe and cou-
+pling radiations derived from a single narrow-band laser
+beam [3].
+ACKNOWLEDGMENTS
+This work was supported by the Science Committee of
+the Republic of Armenia, in the frame of research project
+n° 21T-1C005, and by the NATO Science for Peace and
+Security Project under grant G5794.
+DATA AVAILABILITY STATEMENT
+Data underlying the results presented in this paper are
+not publicly available at this time but may be obtained
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+Precision Measurement of the Excited State Landé g-factor and Diamagnetic Shift of the Cesium D2 Line
+(2022), arXiv:2208.00077.
+

D9E1T4oBgHgl3EQfqQU2/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf,len=456
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='03340v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='atom-ph]  9 Jan 2023  Formation of strongly shifted EIT resonances using "forbidden" transitions of Cesium  Armen Sargsyan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='1 Ara Tonoyan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='1 Rodolphe Momier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' ∗ Claude Leroy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='2 and David Sarkisyan1  1Institute for Physical Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' NAS of Armenia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Ashtarak-2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 0203 Armenia  2Laboratoire Interdisciplinaire Carnot de Bourgogne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' UMR CNRS 6303,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Université Bourgogne Franche-Comté,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 21000 Dijon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' France  (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 2023)  Atomic transitions satisfying Fe − Fg = ∆F = ±2 (where Fe stands for excited and Fg stands  for ground state) of alkali atoms have zero probability in zero magnetic field (they are so-called  "forbidden" transitions) but experience a large probabilty increase in an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  These transitions are called magnetically induced (MI) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' In this paper, we use for the first  time the σ+ (∆mF =  + 1) MI transitions Fg = 3 → Fe = 5 of Cesium as probe radiation to form  EIT resonances in strong magnetic fields (1 - 3 kG) while the coupling radiation frequency is resonant  with Fg = 4 → Fe = 5 σ+ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The experiment is performed using a nanometric-thin cell  filled with Cs vapor and a strong permanent magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The thickness of the vapor column is 852 nm,  corresponding to the Cs D2 line transition wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Due to the large frequency shift slope of the  MI transitions (∼ 4 MHz/G), it is possible to form contrasted and strongly frequency-shifted EIT  resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Particularly, a strong 12 GHz frequency shift is observed when applying an external  magnetic field of ∼ 3 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Preliminary calculations performed considering Doppler-broadened three  level systems in a nanocell are in reasonable agreement with the experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  INTRODUCTION  Optical processes occurring in Rubidium, Cesium,  Potassium and Sodium vapors confined in optical cells  have important applications such as optical atomic  clocks, optical atomic magnetometers, atomic gyro-  scopes, markers of atomic transition frequencies, as de-  scribed for example in [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Therefore, the study of the  peculiarities of atomic transitions (in particular Zeeman  transitions in an external magnetic field) of alkali atoms  is of utmost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' It is well known that the appli-  cation of a strong magnetic field can significantly change  the probabilities (intensities) of the Zeeman transitions,  as shown in [7–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' High interest has recently been fo-  cused on atomic transitions between ground and excited  levels that satisfy the condition Fe − Fg = ∆F = ±2  (these transitions are so-called forbidden by the selection  rules, thus their probability is zero when no external mag-  netic field is applied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' However, the probabilities of these  transitions in a magnetic field increase significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' For  this reason, we refer to these transitions as Magnetically  Induced (MI) transitions [8, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  This giant increase in the probabilities of the MI transi-  tions is due to the “mixing” of magnetic sublevels |F, mF ⟩  of the ground (Fg) or excited (Fe) levels with sublevels  having the same magnetic quantum number mF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' This  mixing is the strongest for D2 lines of alkali atoms, as  up to four states |Fe, 0⟩ can experience mixing, thus re-  sulting in a 4×4 block in the magnetic Hamiltonian, as  described in [7, 8, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Magnetically-induced transitions are of great interest  because, over a wide range of magnetic field, their proba-  bilities can be much higher than the probabilities of usual  ∗ rodolphe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='momier@u-bourgogne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='fr  ("allowed", satisfying the selection rule on F) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  It is important to note that the slope of the frequency  shifts (obtained by diagonalizing the magnetic Hamilto-  nian [7]) as a function of the magnetic field B in strong  magnetic fields can reach up to around 4 MHz/G, which  is 3 times larger than in the case of ordinary transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Thus, the frequency shift of MI transitions in strong mag-  netic fields can reach several tens of GHz, which can be  useful for working in higher frequency ranges, for exam-  ple for the frequency stabilisation of lasers on strongly  shifted frequencies [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  In [11, 12], we established the following rule for the  probabilities of MI transitions:  the probabilities and  number of MI transitions with ∆F = +2 are maximal  for σ+ radiation, whereas the probabilities and number  of MI transitions with ∆F = −2 are maximal for σ−  radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The difference between the intensities of MI  transitions for the σ+ and σ−-polarized radiation beams  can reach several orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  It  has  been  recently  demonstrated  that  electromagnetically-induced  transparency  (EIT)  resonances can be formed using Λ-system made of  ∆F  =  +2 MI transitions only if both probe and  coupling beam are σ+-polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  This statement was  experimentally and theoretically verified for 87Rb (MI  transitions Fg = 1 → Fe = 3) and 85Rb (MI transitions  Fg = 2 → Fe = 4) [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' However, if the Λ-system  is formed by MI transitions satisfying ∆F = −2, then  both probe and coupling radiation must be σ−-polarized  in order to form EIT resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  This statement  was experimentally and theoretically verified for Cs  (MI transitions Fg = 4 → Fe = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  This is a direct  consequence of magnetically-induced circular dichroism  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  In this work, we consider seven σ+ MI transitions of  Cs (Fg = 3 → Fe = 5, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The probabilities of  these transitions increase highly in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='3 - 3 kG  2  0  +1  +2  +3  1  2  3  0  +1  +2  +3  1  2  1  2  3  4  5  7  6  +4  0  +1  +2  +3  1  2  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Scheme of Cs D2 line σ+ transitions between Fg = 3, 4  and Fe = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The probe frequency νp is scanned across the  MI transitions labelled 1-7 (Fg = 3 → Fe = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The coupling  frequencies νcn are resonant with Fg = 4 → Fe = 5 transi-  tions, forming seven Λ-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Only the states involved in  the process under consideration are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Note that |F, mF ⟩  is just a notation for visualization, as the atomic states are  better described in the uncoupled basis |J, mJ, I, mI⟩ in high  magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  and we used these transitions to form EIT resonances  in strong B-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  A nanometric-thin cell (NC) filled  with Cs vapor (thickness L ≈ 850 nm, approximately  the resonant wavelength of Cs D2 line [19]) has been  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The advantages of using thin cells, including strong  reduction of Doppler broadening, are noted in [12, 17, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Probabilities and frequency shifts of the MI  transitions of Cs D2 line  The curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 2 were calculated using a known the-  oretical model depicting the changes of transition proba-  bilities as a function of the external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The  block-diagonal (each block corresponding to a given value  of the magnetic quantum number) magnetic Hamiltonian  is built for each value of the magnetic field and then diag-  onalized in order to calculate the probability coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  This model was presented in a number of papers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  [7, 11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  The evolution of the probabilities of MI transitions (la-  belled 1 to 7, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 1) with respect to the magnetic field  B is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Note that in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='3 - 2  kG the probabilities of the MI transitions labeled 5, 6  and 7 are the strongest among all transitions occurring  from Fg = 3 [8, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The frequency shift slope of the  MI transitions, obtained through the eigenvalues of the  Hamiltonian, is quite large (∼ 4 MHz/G) while for usual  transitions the slope is 3 times smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Despite the fact  that the probabilities of the MI transitions decrease as  B increases, they can still be recorded easily at 7 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  As noted below, this is due to the fact that these tran-  sitions are formed far on the high-frequency wing where  there are no intersections with other transitions (spec-  tra are presented for Na in [21], but Cs behaves almost  identically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  The evolution of the probabilities of the corresponding  seven coupling transitions Fg = 4 → Fg = 5 (Ac1 to Ac7)  that are used to form seven Λ-systems (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 1) with  respect to the magnetic field are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' In the  case of σ− polarization, the probability of the strongest  Fg = 4 → Fe = 5 σ− transition already tends to zero for  B > 300 G, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Thus, both the probe  and the coupling beams must be σ+-polarized in order  to form EIT resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Qualitative description of the EIT process  For a qualitative description of the EIT process, we  present a formula from [3, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The ratio of absorption at  the probe radiation frequency νp at which EIT resonance  is observed (in the presence of νc radiation) to absorp-  tion (when there is no coupling radiation), assuming low  radiation intensity νp and zero frequency detuning of the  coupling radiation, is described by the expression:  α(Ωc)  α(0) =  K  1 + Ω2c/4Γ21γN  ,  (1)  where K is a constant including the Doppler width,  γN is the natural width of the level (γN/2π ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='2 MHz  for the 62P3/2 level of the Cs atom), Ωc is the Rabi fre-  quency for the coupling radiation and Γ21 is the dephas-  ing rate of the coherence between the two ground states  of the Λ-system, which is caused in particular by colli-  sions of atoms with the windows of the nanocell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The  case α(Ωc) = 0 corresponds to complete transparency  (the contrast of the EIT resonance reaches 100%) and a  large amplitude of the EIT resonance, which decreases  with an increase in Γ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The spectral width of the EIT  resonance can be described by the simple expression [3]:  γEIT ≃ 2Γ21 + Ω2  c/γN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  (2)  It follows from formula (1) that in order to obtain small  value of α(Ωc) (which means high electromagnetically in-  duced transparency of the medium), it is necessary to in-  crease Ωc, however, an increase in Ωc leads to an increase  in the spectral width of the EIT resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Therefore,  it is necessary to find a compromise for Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Estimates  can be obtained from Ωc/2π = aγN(I/8)1/2 where I is  the laser intensity in mW/cm2, γN ∼ 5 MHz, and a  is a fit parameter (for our case a is of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='5) [23] and  Ωc ∼ 15 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  EXPERIMENT  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Experimental setup  The layout of the experimental setup is shown on  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Two extended cavity diode lasers are tuned in the  vicinity of the Cs D2 line, with a wavelength λ ≃ 852 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  The Λ-systems shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 1 are formed by scanning  the frequency νp of a VitaWave laser (δνp ∼ 1 MHz) [24]  3  1  2  3  4  5  6  7  a)  b)  c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Magnetic field dependence of the Zeeman transition intensities of the D2 line of Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' a) Fg = 3 → Fe = 5 σ+ MI  transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' b) Fg = 4 → Fe = 5 σ+ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' c) Transition |4, −1⟩ → |5, −2⟩ (σ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' This transition forms a Λ-system with  transition 7 as shown in panel a) and in the inset (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Its probability tends to 0 as the magnetic field increases, thus  forming EIT resonances at high magnetic fields requires both probe and coupling beams to be σ+-polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  FI  FI  SO  BS  PD  ECDL 1  ECDL 2  probe  coupling  C  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' channel  Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' channel  PBS2  PBS1  PBS3  PBS4  M  IF PD  NC  PM  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Scheme of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  ECDL: CW  narrow-band external-cavity diode lasers with λ = 852 nm  (resonant with Cs D2 line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' FI: Faraday insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' PBSi:  polarizing beam splitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' BS: beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' IF: interference  filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' C: saturated absorption spectroscopy unit for frequency  reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' NC: nanocell placed in oven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' PM: permanent mag-  net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' PD: photodiodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' SO: 4-channel digital oscilloscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  in the vicinity of the MI transitions Fg = 3 → Fe = 5,  while keeping the frequency νc from a MOGLabs “cat-  eye” laser (δνp ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='1 MHz) on resonance with one of the  4 → 5 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' A fraction about 10% of the coupling  radiation power was sent to a frequency stabilization unit  based on the DAVLL method [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Probe radiation has  vertical polarization, while the coupling radiation has  horizontal polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' In the case of a longitudinal B-  field, linearly polarized laser radiation can be considered  as consisting of σ+ and σ− radiations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The use of mutu-  ally perpendicular polarizations allows by using PBS4 to  direct only probe radiation to the photo-receiver, while  cutting off the coupling radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' As noted above, in  the case of MI transitions with ∆F = +2 for the for-  mation of the EIT resonance, both probe and coupling  radiations must have σ+ polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' A photograph of  the Cs nanocell is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Interference fringes  are formed by the reflection of light on the inner surfaces  of windows (made of sapphire).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The region correspond-  Coupling off  7  6  5  4  3  EIT 7  EIT 6  EIT 5  EIT 4  EIT 3  (1)  (2)  (3)  (4)  (5)  (6)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Probe transmission spectra of the Cs nanocell (L =  λ = 852 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Spectra labelled 1 to 5 show five EIT reso-  nances, labelled EIT 3 to EIT 7, while the probe frequency is  scanned across transitions 3 to 7 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The coupling  and probe powers are respectively 10 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='05 mW and the ex-  ternal longitudinal magnetic field is B = 1400 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Spectrum n°  6 corresponds to the case where coupling is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Small VSOP  peaks are visible on each atomic resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Zero frequency  corresponds to the transition frequency of Cs D2 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  ing to a thickness L ≈ λ ∼ 850 nm is outlined by an  oval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The design of the Cs-filled NC used in our experi-  ments is similar to that of extremely thin cell described  in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Earlier it was demonstrated in [16, 17, 27] that  the use of a nanocell (NC) with thickness L = λ makes it  easy to record contrasted EIT resonances, which is due  to the low absorption of the NC, while the disadvan-  tage is broadening of the EIT resonance caused by fre-  quent inelastic collisions of atoms with the windows of  the NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Studies of the EIT resonances were done using  a strong neodymium–iron–boron alloy ring-shaped per-  manent magnet (PM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Due to the small thickness of the  vapor column, the high-gradient field produced by magne  can be considered uniform across the interaction region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  The PM was placed after the rear window of the NC,  with the axis aligned along the probe beam propagation  4  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The magnetic field in the NC was simply var-  ied by longitudinal displacement of the PM, calibrated  using a Teslameter HT201 magnetometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Experimental results: using MI transitions to  form EIT resonances  Curves 1 to 5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 4 show the experimental trans-  mission spectra of the probe radiation which contain the  resonances EIT 3 to EIT 7 (numbers 3-7 means that MI  transitions with numbers 3-7 are involved, respectively)  in a longitudinal magnetic field B = 1400 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The NC  thickness is L = λ = 852 nm and the temperature of  the reservoir is 100 ◦C (to prevent Cs vapor condensa-  tion on the windows, the temperature of the windows is  slightly higher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The coupling and the probe powers are  20 mW and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='1 mW, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Note that since only  σ+ radiations participate to the formation of the EIT  resonances (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 1), only half of the power of these  radiations must be considered, meaning 10 mW and 50  µW, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Curve n° 6 is a probe spectrum when  the coupling is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Since the cell thickness is L = λ,  small peaks formed by velocity selective optical pump-  ing (VSOP) resonances are located exactly at the atomic  transitions frequencies, as described in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  The amplitude of the EIT resonance is a factor ∼10  larger than the amplitude of the VSOP resonance,  whereas the spectral width of the EIT resonance is a  factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='5 smaller, which is characteristic of the coher-  ent EIT process [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Note that the contrast of the EIT  resonance defined as the ratio of the EIT resonance am-  plitude divided by the peak absorption of the Cs vapor  when the coupling is blocked reached 40-50 % which is  typical when a nanocell is used [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 6, curves 1 to 4 are probe transmission spectra  which contain EIT 6, EIT 5, EIT 4 and EIT 3 resonances  for B = 1770 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Curve n° 5 shows only the probe spec-  trum when the coupling is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 7, lines 1 to 3  show the probe transmission spectra which contain EIT  6, EIT 4 and EIT 3 resonances for B = 2880 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Line  n° 4 shows only the probe spectrum when the coupling  is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The inset shows the profile of EIT 6 reso-  nance fitted with a Gaussian profile with a FWHM of ∼  35 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' There is also a small VSOP resonance which is  formed when the coupling is blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The typical FWHM  of VSOP resonances is 40-50 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Preliminary theoretical calculations (shown in the  right part of the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 7 were obtained by solv-  ing the Liouville equations of motion for an ensemble of  three-level Λ-systems (as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 5), taking into  account the geometry of the nanocell (coherence dephas-  ing rate determined by the time of flight of the atoms), its  Fabry-Perot nature (reflections of the fields on the inner  surfaces of the cell) and Doppler broadening, following  the procedure described in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The Rabi frequencies of  the probe and coupling lasers are respectively Ωc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='5γN  and Ωp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='06γN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Reasonable agreement between theory  and experiment regarding the width and depth of the EIT  resonance is obtained and the VSOP resonance is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Small discrepancies (assymetry of the profile and ampli-  tude of the VSOP resonance) can arise notably from the  need of considering neighboring Zeeman sublevels (not  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 1, and therefore more than three levels, to  obtain more accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Scheme of the three-level Λ-system used in the calcu-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The total decay rate Γ33 of state |3⟩ is 1/2(γ31 + γ32)  [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  The dephasing rate of coherence between the ground  states is Γ21 = (2πt)−1 where t is the time of flight of  the atoms through the cell (at the most probable velocity  u =  �  2kBT/M where T is the vapor temperature and M the  atomic mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  The amplitude of resonance n° 6 is ∼ 50 times greater  than that of the VSOP resonance and is spectrally nar-  rower than the latter (this is a manifestation of the co-  herent EIT process [2, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 8 the solid lines in-  dicate the calculated dependences of the frequency shifts  for transitions 1–7 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 1) and Fg = 3 → Fe = 4 (marked  with dotted oval) to the magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  The black  squares represent the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' As mentioned  earlier, due to the high value of the frequency shift slope  for B > 3 kG, the group of MI transitions 1–7 is com-  pletely separated in frequency from Fg = 3 → Fe = 4  transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  The curves in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 8 show experimental and  theoretical spectra (calculated by combining the models  presented in [7] and [30]) of the seven MI transitions ab-  sorption for B = 6 kG when frequency shift reaches ∼ 30  GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Note that the amplitude of transition 6 is slightly  bigger than that of transition 7 (while for B < 5 kG the  amplitude of transition 7 is bigger, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 2a), because  of the “mixing” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Note that one of the remarkable  features of the σ+ MI transitions 3 → 5′ is that they  are still well recorded for a magnetic field B ≈ 8 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  They are located in the high frequency wing of the spec-  trum presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 18 of paper [31] and for this case  the frequency shift reaches 34 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Using our theoret-  ically calculated curves for MI transitions 3 → 5′ we  checked the frequency position of these MI transitions  and found good agreement with the experimental curves  presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' In paper [31] the 3 → 5′transitions  are not identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Therefore, it is important to inform  5  Coupling off  EIT 6  EIT 5  EIT 4  EIT 3  (1)  (2)  (3)  (4)  (5)  6  5  4  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Probe transmission spectra of the Cs nanocell  (L = λ ≈ 850 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Spectra 1 to 4 exhibit four EIT reso-  nances, labelled EIT 3 to EIT 6, while the probe frequency  is scanned across transitions 3 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The external longitudi-  nal magnetic field is B = 1770 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Spectrum n° 5 is a probe  transmission spectrum when the coupling is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Small VSOP  peaks are visible on each atomic transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Zero frequency  corresponds to the transition frequency of Cs D2 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Coupling off  Coupling off  (1)  (2)  (3)  (4)  Experiment  Coupling off  EIT 6  EIT 4  EIT 3  6  5  4  3  EIT 6  Theory  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Probe transmission spectra of the Cs nanocell (L =  λ = 852 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Lines 1 to 3 show four EIT resonances, la-  belled EIT 4, EIT 5 and EIT 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The external longitudinal  magnetic field is B = 2880 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Line 4 is a probe transmission  spectrum when the coupling is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The left part of the inset  is a zoom on EIT 6, fitted with a Gaussian profile (FWHM  35 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The right curves are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Red: coupling on,  black: coupling off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Small VSOP peaks are visible on each  atomic transitions formed by the probe radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Their typ-  ical linewidth is 40-50 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Zero frequency corresponds to  the transition frequency of Cs D2 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  scientists working in the field of laser spectroscopy of al-  kali metal atoms about the MI atomic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The  above-mentioned MI transitions can be exploited in such  high B-fields as new frequency markers, for using new fre-  quency ranges, as well as for the frequency stabilization  of lasers at strongly shifted frequencies from the initial  transition in unperturbed atoms [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  Theory  6  5  4  3 2 1  7  6  5  4  3  2  1  7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Red solid lines: frequency shift of transitions 1 to 7  (see figure 1) as a function of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The black  squares with error bars represent experimental measurements,  the inaccuracy is around 1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Black dashed lines: frequency  shift of Fg = 3 → Fe = 4 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' For B > 3 kG, both  groups are well separated in frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' Inset: theoretical and  experimental absorption spectra for B = 6 kG, the frequency  shift reaches 30 GHz from the Cs D2 line transition frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  CONCLUSION  In this paper, we used for the first time forbidden  transitions of Cs (Fg = 3 → Fe = 5, more precisely  σ+(∆mF = +1) transitions) to create Λ-system allowing  the formation of EIT resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' This was done in an ex-  ternal magnetic field, as such transitions have zero proba-  bility in the absence of magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' A nanometric-thin  cell filled with Cs vapor was used, with a thickness corre-  sponding to the resonant wavelength of Cs D2 line (≈ 850  nm), and the magnetic field was varied by longitudinal  displacement of the permanent magnet along the prop-  agation direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' As expected, when the cou-  pling is blocked, small VSOP resonances are formed right  at the different transitions’ frequencies, while coupling  radiation allows for the formation of EIT resonances,  spectrally narrower and with a bigger amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  We  formed EIT resonances with 6 out the 7 transitions de-  picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' This was possible up to 3 kG thanks  to the big value of the frequency shift, reaching up to 4  MHz/G, therefore leading to EIT resonances shifted 12  GHz apart from the Cs D2 line transition frequency [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  This result is of great interest, as the highly-shifted spec-  tra can serve as frequency references [14, 15], especially  taking into account that these transitions are still easily  recorded up to 8 kG when the frequency shift reaches 35  GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' As for the theoretical description, further investi-  gation is necessary, mainly in order to take into account  the effect of neighbouring states, and thus including more  levels in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' The complexity of the manifold and  the number of coupled equations make it a challenging  6  and computationally-intensive task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' However, reasonable  agreement was already achieved by simply considering an  ensemble of three-level systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' To the best of our knowl-  edge, there are no reports on obtaining EIT resonances  in Λ-systems in such strong fields using usual transitions  of alkali atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content=' We note that much narrower EIT reso-  nances can be attained by using cm-long cells (to lower  the effect of inelastic collisions of atoms with the win-  dows), and by using coherently coupled probe and cou-  pling radiations derived from a single narrow-band laser  beam [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported by the Science Committee of  the Republic of Armenia, in the frame of research project  n° 21T-1C005, and by the NATO Science for Peace and  Security Project under grant G5794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
+page_content='  DATA AVAILABILITY STATEMENT  Data underlying the results presented in this paper are  not publicly available at this time but may be obtained  from the authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfqQU2/content/2301.03340v1.pdf'}
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+Enantio-specific state transfer of chiral molecules through enantio-selective
+shortcut-to-adiabaticity paths
+Jian-Jian Cheng,1, 2 Chong Ye,3 and Yong Li1, 4, ∗
+1Center for Theoretical Physics and School of Science, Hainan University, Haikou 570228, China
+2Beijing Computational Science Research Center, Beijing 100193, China
+3Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems,
+School of Physics, Beijing Institute of Technology, 100081 Beijing, China
+4Synergetic Innovation Center for Quantum Effects and Applications,
+Hunan Normal University, Changsha 410081, China
+(Dated: January 10, 2023)
+An interesting method of fast enantio-specific state transfer is proposed for cyclic three-level
+systems of chiral molecules. We show that the fast population transfer via shortcut to adiabaticity
+can be accomplished for the cyclic three-level system of a general (chiral) molecule with invariant-
+based inverse engineering of the coupling strengths. By choosing appropriate parameters, the two
+enantiomers, which are initially prepared in their ground states in the three-level systems, will
+evolve respectively along their enantio-selective shortcut-to-adiabaticity paths to different-energy
+final states simultaneously, namely achieving the fast enantio-specific state transfer.
+I.
+INTRODUCTION
+Since Pasteur first discovered chiral molecules in
+1848, the theoretical and experimental studies of chiral
+molecules have proliferated in chemistry [1], biotechnolo-
+gies [2], and pharmaceutics [3]. Chiral molecules contain
+two species, e.g. left- and right-handed ones [4], which are
+often called enantiomers. The two enantiomers are mir-
+ror images of each other but can be superposed on each
+other via translations and rotations. The enantiodiscrim-
+ination (as well as enantioseparation and enantioconver-
+sion) [5–8] of chiral molecules remains an enormous chal-
+lenge. The traditional method of enantiodiscrimination
+is to break the mirror symmetry of the enantiomers by
+using circularly polarized light [9]. Some commonly used
+chiroptical methods of enantiodiscrimination are circular
+dichroism [10], vibrating circular dichroism [11], optical
+rotation [9], and Raman optical activity [12]. However,
+these methods rely on the interference between electric-
+dipole and weak magnetic-dipole (or electric-quadrupole)
+transitions.
+Alternatively,
+enantiodiscrimination
+methods
+that
+only use electric-dipole interactions [13, 14], have also
+been proposed.
+The left- and right-handed chiral
+molecules can be modeled as cyclic three-level systems,
+where three electromagnetic (optical or microwave) fields
+couple respectively to three transitions via electric-dipole
+interactions [15, 16]. Due to the intrinsic property of chi-
+ral molecules, the product of the corresponding three cou-
+pling strengths (Rabi frequencies) in the cyclic three-level
+systems can differ in signs for the two enantiomers [15,
+16].
+So the corresponding overall phases in the cyclic
+three-level systems differ by π with the enantiomers.
+Based on such cyclic three-level systems, one can use
+different schemes, such as enantio-selective three-wave
+∗ [email protected]
+mixing [17–21], enantio-selective absorption [22], enantio-
+selective AC stark effect [23] and enantio-selective two-
+dimensional spectra [24], to discriminate the left- and
+right-handed molecules. Moreover, some more ingenious
+sources of modern optics physics, such as frequency en-
+tangled photons [25], quantized photons [26, 27], and cor-
+related photons in cavities [28], have been introduced to
+enhance the performance of enantiodiscrimination.
+Beyond the enantiodiscrimination, the cyclic three-
+level systems of chiral molecules have also been used in
+some more ambitious issues, such as the enantio-specific
+state transfer (ESST) [15, 29–38], enantioseparation [39–
+42], and enantioconversion [16, 43–46]. The perfect ESST
+of chiral molecules can be realized by transferring the left-
+and right-handed chiral molecules from the same-energy
+initial states to different-energy final states by choosing
+suitable electromagnetic fields [15, 29–38]. Recently, the
+feasibility of ESST based on the cyclic three-level sys-
+tems has been demonstrated experimentally in gaseous
+samples by using microwave fields [47–50].
+After the
+achievement of the ESST, one can further realize the
+enantiodiscrimination and spatial enantioseparation for
+the chiral molecules [39, 40].
+In the original ESST method based on cyclic three-
+level systems of chiral molecules [15], the ESST was re-
+alized by using the adiabatic (and also diabatic) passage
+technique, which makes the ESST process slow and com-
+plicated. To overcome these defects, several theoretical
+methods of fast ESST were proposed and developed [29–
+38] based on cyclic three-level systems. Among them, an
+ingenious method [31] was proposed to achieve the fast
+ESST of chiral molecules by using the “shortcut to adi-
+abaticity” (STA) concept via adding a counterdiabatic
+field to accelerate the stimulated Raman adiabatic pas-
+sage.
+Motivated by Ref. [31], here we propose to achieve the
+ESST by a different STA with invariant-based inverse
+engineering [51–53], instead of the STA with adding the
+counterdiabatic field [31].
+The invariant-based inverse
+arXiv:2301.03341v1  [quant-ph]  9 Jan 2023
+
+2
+engineering starts by introducing a Lewis-Riesenfeld in-
+variant in a time-dependent system. The invariant can
+be used to derive a law that governs the evolution state
+for the designed Hamiltonian. By means of the invariant-
+based inverse engineering of the time-dependent Hamil-
+tonians with designing appropriate control parameters,
+the left- and right-handed chiral molecules prepared ini-
+tially in their corresponding ground states would evolve
+(approximately) along their enantio-selective shortcut-
+to-adiabaticity paths to different-energy final states.
+II.
+CYCLIC THREE-LEVEL SYSTEMS
+A general chiral molecule can be modeled as the cyclic
+three-level system by choosing appropriate three electro-
+magnetic fields to couple with three electric-dipole tran-
+sitions [15, 54]. Here, we only consider the case that all
+the three electromagnetic fields couple resonantly with
+the electric-dipole transitions respectively, as shown sim-
+ilar to Fig. 1(a). In the basis of {|1⟩, |2⟩, |3⟩}, the Hamil-
+tonian of the cyclic three-level system can be described
+in the interaction picture as (ℏ = 1) [31]
+ˆH(t) =
+�
+�
+0
+Ωx(t) Ωz(t)e−iφ
+Ωx(t)
+0
+Ωy(t)
+Ωz(t)eiφ Ωy(t)
+0
+�
+�
+(1)
+with |1⟩ = (1, 0, 0)T , |2⟩ = (0, 1, 0)T , |3⟩ = (0, 0, 1)T .
+Here Ωj(t) (j = x, y, z) are the Rabi frequencies, which
+can be controlled by varying the amplitudes of the ap-
+plied electromagnetic fields. φ is the overall phase of the
+three Rabi frequencies. Here we set φ = π/2. Without
+loss of generality, we have assumed Ωj are real. Then the
+Hamiltonian can be expressed as
+ˆH(t) = Ωx(t) ˆKx + Ωy(t) ˆKy + Ωz(t) ˆKz.
+(2)
+Here, ˆKx, ˆKy, and ˆKz are the SU(2) angular-momentum
+operators [55]
+ˆKx =
+�
+�
+0 1 0
+1 0 0
+0 0 0
+�
+� ,
+ˆKy =
+�
+�
+0 0 0
+0 0 1
+0 1 0
+�
+� ,
+ˆKz =
+�
+�
+0 0 −i
+0 0
+0
+i 0
+0
+�
+� .
+(3)
+They satisfy the commutation relations
+[ ˆKx, ˆKy] = i ˆKz, [ ˆKy, ˆKz] = i ˆKx, [ ˆKz, ˆKx] = i ˆKy.(4)
+The fact that Hamiltonian (2) is written as the sum of
+three SU(2) operators, means it addresses the SU(2) al-
+gebraic structure [53].
+For the two enantiomers of chiral molecules, the overall
+phases in the cyclic three-level systems under consider-
+ation differ by π [50]. For convenience, we specify that
+the signs before Ωx and Ωz are equal for the two enan-
+tiomers, while the sign before Ωy is opposite, as shown
+in Fig. 1.
+|3〉L
+|3〉R
+|2〉L
+|2〉R
+|1〉L
+|1〉R
+Ωzeiϕ
+Ωy
+Ωx
+Ωzeiϕ
+-Ωy
+Ωx
+(a) Left-handed
+(b) Right-handed
+FIG. 1.
+(a) Left- and (b) right-handed chiral molecules of
+cyclic three-level systems, where three electromagnetic fields
+couple resonantly to the three electric-dipole transitions, re-
+spectively, with Ωx, ±Ωy, and Ωzeiφ the corresponding Rabi
+frequencies.
+Therefore, the Hamiltonians of the cyclic three-level
+systems for the two enantiomers in the basis {|m⟩L} and
+{|m⟩R} (m = 1, 2, 3) can be described as
+ˆHL,R(t) = Ωx(t) ˆKL,R
+x
+± Ωy(t) ˆKL,R
+y
++ Ωz(t) ˆKL,R
+z
+. (5)
+Here, the indices L and R [which correspond, respec-
+tively, to the signs + and − in the right side of Eq. (5)],
+denote the left- and right-handed chiral molecules, re-
+spectively.
+ˆKQ
+j (j = x, y, z, Q = L, R) is just
+ˆKj in
+Eq. (3) for the two enantiomers. In this work, when refer-
+ring to left- or right-handed chiral molecules, we will add
+the index. When there is no index, we refer to general
+molecules.
+III.
+INVARIANT DYNAMICS
+Shortcut to adiabaticity (STA) is a fast route to ac-
+celerate a slow adiabatic process by controlling the pa-
+rameters of a system [56], while keeping the same initial
+and final states as that in the adiabatic passage. A mo-
+tivation to apply the STA technique is to manipulate the
+quantum system on timescales shorter than decoherence
+times.
+There are two main STA techniques that have
+been proposed theoretically and implemented experimen-
+tally to inversely engineer the time-dependent Hamilto-
+nian of a quantum system for accelerating slow adiabatic
+process [52]. One is the counterdiabatic driving method
+with adding an auxiliary field in a reference Hamilto-
+nian to cancel the nonadiabatic coupling, where the dy-
+namics follows exactly the adiabatic passage defined by
+the reference Hamiltonian [52, 57, 58]. The other one is
+the invariant-based inverse engineering method, which is
+based on the Lewis-Riesenfeld invariant that carries the
+eigenstates of a system from the initial state to the de-
+sired final state [52], with keeping the same initial and
+final states as those in the adiabatic passage, but without
+following the adiabatic passage at the intermediate time
+instants [51, 52]. In what follows, we focus on how to use
+
+3
+the latter STA technique to achieve the ESST of chiral
+molecules.
+Commonly a Lewis-Riesenfeld invariant for a Hamilto-
+nian ˆH(t) is a Hermitian operator ˆI(t) that satisfies [59]
+dˆI(t)
+dt
+≡ ∂ ˆI(t)
+∂t
+− i[ˆI(t), ˆH(t)] = 0,
+(6)
+so that its eigenvalues remain constant in time. Accord-
+ing to the Lewis-Riesenfeld theory [51, 52, 59], if {|φn(t)⟩}
+is a set of orthogonal eigenstates of the invariant ˆI(t),
+the solution to the time-dependent Sch¨ordinger equation
+can be constructed as |Ψ(t)⟩ = �
+n cneiαn(t)|φn(t)⟩, with
+cn being a time-independent coefficient. Here αn(t) =
+� t
+0⟨φn(t′)|[i∂t′ − ˆH(t′)]|φn(t′)⟩dt′ is the Lewis-Riesenfeld
+phase [51, 52, 59].
+In general, ˆH(t) does not commute with the invari-
+ant ˆI(t) at all time. We only require the invariant and
+the Hamiltonian to commute at the initial and final time
+instants, i.e., [ ˆH(0), ˆI(0)] = 0 and [ ˆH(τ), ˆI(τ)] = 0 [51–
+53, 56]. The eigenstates of the Hamiltonian and the in-
+variant coincide at the initial and final time instants but
+may be different at the intermediate time. This leaves
+large freedom to choose how the state evolves in the in-
+termediate time. We can use Eq. (6) to find the Hamilto-
+nian (2) that drives such a designed evolution of a given
+state in the cyclic three-level system. Moreover, we con-
+sider, respectively, the evolutions of the left- and right-
+handed chiral molecules with cyclic three-level structures
+by invariant-based inverse engineering of the Rabi fre-
+quencies (equivalently the amplitude of the electromag-
+netic fields). By choosing appropriate Rabi frequencies,
+the fast ESST can be achieved by transferring the two
+enantiomers from their ground states to different-energy
+final states through their corresponding eigenstates of in-
+variants, following their enantio-selective STA paths.
+A.
+Invariant dynamics for the left-handed chiral
+molecules
+We first consider the state transfer of the left-handed
+chiral molecules with the cyclic three-level structures by
+the invariant-based inverse engineering. Since ˆHL(t) in
+Eq. (5) possesses the SU(2) algebraic structure, the cor-
+responding invariant ˆIL(t) can be given as [53]
+ˆIL =Ω0
+2 (cos γ sin β · ˆKL
+x + cos γ cos β · ˆKL
+y + sin γ · ˆKL
+z )
+=Ω0
+2
+�
+�
+0
+cos γ sin β
+−i sin γ
+cos γ sin β
+0
+cos γ cos β
+i sin γ
+cos γ cos β
+0
+�
+�
+L
+(7)
+in the basis {|1⟩L, |2⟩L, |3⟩L}. Here, Ω0 is an arbitrary
+constant with unit of frequency, and the time-dependent
+auxiliary parameters γ and β satisfy the equations
+˙γ = Ωx cos β − Ωy sin β,
+˙β = (Ωx sin β + Ωy cos β) tan γ − Ωz.
+(8)
+The eigenstates of the invariant ˆIL(t), which satisfy
+ˆIL(t)|φn(t)⟩L = λL
+n|φn(t)⟩L (n = 0, ±), are
+|φ0⟩L =
+�
+�
+cos γ cos β
+−i sin γ
+− cos γ sin β
+�
+�
+L
+,
+(9)
+|φ±⟩L =
+1
+√
+2
+�
+�
+sin γ cos β ± i sin β
+i cos γ
+− sin γ sin β ± i cos β
+�
+�
+L
+(10)
+with the corresponding (time-independent) eigenval-
+ues
+λL
+0
+=
+0
+and
+λL
+±
+=
+±Ω0.
+In
+this
+case,
+the
+Lewis-Riesenfeld
+phases
+are
+αL
+0 (t)
+=
+0,
+and
+αL
+±(t) = ∓
+� t
+0[ ˙β(t′) sin β(t′) + Ωx(t′) sin β(t′) cos γ(t′) +
+Ωy(t′) cos β(t′) cos γ(t′) + Ωz(t′) sin γ(t′)]dt′.
+Here, we take Ωx(t) = Ωz(t) for simplicity. By using
+Eq. (8), we have
+Ωx = Ωz =
+˙β sin β + ˙γ cos β tan γ
+tan γ − sin β
+,
+Ωy =
+˙β cos β + ˙γ(1 − tan γ sin β)
+tan γ − sin β
+.
+(11)
+Once the appropriate boundary conditions for γ and β
+are fixed, one can insert a polynomial function to deter-
+mine Ωx, Ωy, and Ωz. Our task is to design the Hamilto-
+nian ˆHL(t) to drive the initial state |1⟩L to the final state
+|3⟩L (up to a phase factor) along the invariant eigenstate
+|φ0(t)⟩L in a given time τ. Therefore, based on the invari-
+ant eigenstate |φ0(0)⟩L = (1, 0, 0)T
+L = |1⟩L at the initial
+instant time and |φ0(τ)⟩L = (0, 0, −1)T
+L = −|3⟩L at the
+final instant time τ, the boundary conditions for γ and
+β can be given as
+γ(0) = 0, β(0) = 0,
+γ(τ) = 0, β(τ) = π
+2 .
+(12)
+On one hand, one needs to impose the boundary con-
+ditions to make ˆHL(t) and ˆIL(t) commute at t = 0 and
+t = τ so that they have common eigenstates at these time
+instants. On the other hand, one requires the Rabi fre-
+quencies to vanish at the initial and final time instants to
+make the electromagnetic fields turn on and off smoothly.
+These requirements further imply the additional bound-
+ary conditions
+˙γ(0) = 0, ˙β(0) = 0,
+˙γ(τ) = 0, ˙β(τ) = 0.
+(13)
+There are many interpolating functions consistent with
+the boundary conditions at the initial and final time in-
+stants. With these boundary conditions, we can simply
+choose
+γ(t) = 0, β(t) = 3π
+2τ 2 t2 − π
+τ 3 t3 + ��.
+(14)
+Here the small value η is set to avoid the infinite values
+of the Rabi frequencies at the initial time instant. Thus
+
+4
+the designed Rabi frequencies in Eq. (11) reduce to
+Ωx = Ωz = 3πt
+τ 2
+� t
+τ − 1
+�
+,
+Ωy = 3πt
+τ 2
+� t
+τ − 1
+�
+cot
+� 3π
+2τ 2 t2 − π
+τ 3 t3 + η
+�
+.
+(15)
+(a)
+Ωx (Ωz)
+Ωy
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+t /τ
+Rabi frequencies (2π /τ)
+(b)
+P1
+L
+P3
+L
+P2
+L
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t /τ
+population
+FIG. 2.
+(Color online) (a) The designed Rabi frequen-
+cies for the left-handed chiral molecules with Ωx = Ωz (red
+solid line) and Ωy (blue dashed line) given in Eq. (15). (b)
+Time evolution of corresponding populations in |1⟩L (red solid
+line), |2⟩L (black dotted line), and |3⟩L (blue dashed line) for
+the left-handed chiral molecules with the initial state |1⟩L.
+Here η = 0.02.
+Fig. 2 shows the designed Rabi frequencies for the left-
+handed chiral molecules and corresponding evolution of
+the populations in the states |m⟩L (m = 1, 2, 3) for the
+initial state |Ψ(0)⟩L = |1⟩L. In the ideal condition (i.e.
+the case of η = 0), the left-handed chiral molecules will
+evolve from the initial state |1⟩L (= |φ0(0)⟩L) to the
+final target state −|3⟩L (up to a phase factor), along
+the invariant eigenstate |φ0(t)⟩L. For the case of small
+value η = 0.02 as shown in Fig. 2(b), the initial state
+|1⟩L ≈ |φ0(0)⟩L, thus the populations in the initial state
+|1⟩L with P L
+1 (0) = 1 are finally transferred approxi-
+mately to that in the target state |3⟩L with probabil-
+ity P L
+3 (τ) = 0.9991 for the left-handed chiral molecules.
+Correspondingly, P L
+2 (0) = 0 = P L
+3 (0), P L
+1 (τ) = 0.0005,
+and P L
+2 (τ) = 0.0004.
+B.
+Invariant dynamics for the right-handed chiral
+molecules
+Then we consider the state transfer of the right-handed
+chiral molecules with the cyclic three-level structures
+by the invariant-based inverse engineering.
+Since the
+Hamiltonian ˆHR(t) in Eq. (5) of the right-handed chi-
+ral molecules has the same SU(2) algebraic structure as
+ˆHL(t) of the left-handed ones, similarly the invariant
+ˆIR(t) can be given in the basis {|1⟩R, |2⟩R, |3⟩R} as the
+form
+ˆIR= Ω0
+2 (cos ξ sin χ · ˆKR
+x + cos ξ cos χ · ˆKR
+y + sin ξ · ˆKR
+z )
+= Ω0
+2
+�
+�
+0
+cos ξ sin χ
+−i sin ξ
+cos ξ sin χ
+0
+cos ξ cos χ
+i sin ξ
+cos ξ cos χ
+0
+�
+�
+R
+.
+(16)
+Here the time-dependent auxiliary parameters ξ(t) and
+χ(t) satisfy the equations
+˙ξ = Ωx cos χ + Ωy sin χ,
+˙χ = (Ωx sin χ − Ωy cos χ) tan ξ − Ωz.
+(17)
+The eigenstates of the invariant ˆIR(t), which satisfy
+ˆIR(t)|φn(t)⟩R = λR
+n |φn(t)⟩R (n = 0, ±), are
+|φ0⟩R =
+�
+�
+cos ξ cos χ
+−i sin ξ
+− cos ξ sin χ
+�
+�
+R
+,
+(18)
+|φ±⟩R =
+1
+√
+2
+�
+�
+sin ξ cos χ ± i sin χ
+i cos ξ
+− sin ξ sin χ ± i cos χ
+�
+�
+R
+(19)
+with the corresponding eigenvalues λR
+0 = 0 and λR
+± =
+±Ω0. Here the Lewis-Riesenfeld phase is αR
+0 (t) = 0, and
+αR
+±(t) = ∓
+� t
+0[ ˙χ(t′) sin χ(t′) + Ωx(t′) sin χ(t′) cos ξ(t′) −
+Ωy(t′) cos χ(t′) cos ξ(t′) + Ωz(t′) sin ξ(t′)]dt′.
+Here we still take Ωx = Ωz for simplicity. According
+to Eq. (17), we have
+Ωx = Ωz = ˙χ sin χ + ˙ξ cos χ tan ξ
+tan ξ − sin χ
+,
+Ωy = ˙χ cos χ + ˙ξ(1 − tan ξ sin χ)
+sin χ − tan ξ
+.
+(20)
+Similar to the case of the left-handed chiral molecules
+in the above subsection, once the functions χ and ξ are
+fixed, we can construct Ωx, Ωy, and Ωz and thus the
+Hamiltonian HR(t) can be determined. Here we aim to
+design the Hamiltonian ˆHR(t) to make the system evolve
+from the initial state |1⟩R to the finial state |2⟩R (up to
+a phase factor) along the invariant eigenstate |φ0(t)⟩R
+in a given time τ.
+Therefore, based on the invariant
+eigenstate |φ0(0)⟩R = (1, 0, 0)T
+R = |1⟩R at the initial time
+instant and |φ0(τ)⟩R = (0, −i, 0)T
+R = −i|2⟩R at the final
+time instant τ, the boundary conditions for ξ and χ can
+be given as
+ξ(0) = 0, χ(0) = 0, ξ(τ) = −π
+2 .
+(21)
+
+5
+(a)
+Ωx (Ωz)
+Ωy
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+t /τ
+Rabi frequencies (2π /τ)
+(b)
+P1
+R
+P2
+R
+P3
+R
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t /τ
+population
+FIG. 3. (Color online) (a) The designed Rabi frequencies for
+the right-handed chiral molecules with Ωx = Ωz (red solid
+line) and Ωy (blue dashed line) given in Eq. (24). (b) Time
+evolution of corresponding populations in |1⟩R (red solid line),
+|2⟩R (black dotted line), and |3⟩R (blue dashed line) for the
+right-handed chiral molecules with the initial state |1⟩R. Here
+η′ = −0.02.
+Similarly, we set ˆHR(t) and ˆIR(t) commute at the ini-
+tial and final time instants (so that they have the same
+eigenstates at these time instants) and make the electro-
+magnetic fields (equivalently the Rabi frequencies) turn
+on and off smoothly for the right-handed chiral molecules.
+Thus, the additional boundary conditions for ξ(t) and
+χ(t) can be given as
+˙ξ(0) = 0, ˙χ(0) = 0,
+˙ξ(τ) = 0, ˙χ(τ) = 0.
+(22)
+Consistent with these boundary conditions, we can
+choose
+χ(t) = 0, ξ(t) = − 3π
+2τ 2 t2 + π
+τ 3 t3 + η′.
+(23)
+Here the small value η′ is set to avoid the infinite values
+of the Rabi frequencies at the initial time instant. Thus
+the designed Rabi frequencies in Eq. (20) reduce to
+Ωx = Ωz = 3πt
+τ 2
+� t
+τ − 1
+�
+,
+Ωy = 3πt
+τ 2
+� t
+τ − 1
+�
+cot
+� 3π
+2τ 2 t2 − π
+τ 3 t3 − η′
+�
+.
+(24)
+Fig. 3 shows the designed Rabi frequencies of the right-
+handed chiral molecules and corresponding evolution of
+the populations in the states |m⟩R (m = 1, 2, 3) for the
+initial state |Ψ(0)⟩R = |1⟩R.
+In the ideal condition (i.e. the case of η′ = 0), the
+right-handed chiral molecules will evolve from the initial
+state |1⟩R (= |φ0(0)⟩R) to the final target state −i|2⟩L
+(up to a phase factor), along the invariant eigenstate
+|φ0(t)⟩R. When we set the small value η′ = −0.02 as
+shown in Fig. 3(b), the initial state |1⟩R ≈ |φ0(0)⟩R, thus
+the populations in the initial state |1⟩R with P R
+1 (0) = 1
+are finally transferred approximately to that in the tar-
+get state |2⟩R with P R
+2 (τ) = 0.9991 for the right-handed
+chiral molecules. Correspondingly, P R
+2 (0) = 0 = P R
+3 (0),
+P R
+1 (τ) = 0.0005, and P R
+3 (τ) = 0.0004.
+C.
+Achieving the fast enantio-specific state transfer
+So far we have designed the desired evolution for the
+left- and right-handed chiral molecules of the cyclic three-
+level systems via the STA technique with invariant-based
+inverse engineering in the above two subsections, respec-
+tively.
+By comparing Eq. (15) with Eq. (24), it can
+be found that the two groups of designed Rabi frequen-
+cies for the two enantiomers are exactly the same when
+η = −η′. This means that the two enantiomers are driven
+by the same three electromagnetic fields indeed. In this
+case, the left-handed chiral molecule begins with |1⟩L and
+terminates approximately at −|3⟩L, almost along the in-
+variant eigenstate |φ0(t)⟩L, while the right-handed chi-
+ral molecule begins with |1⟩R and terminates approxi-
+mately at −i|2⟩R, almost along the invariant eigenstate
+|φ0(t)⟩R simultaneously.
+As also shown in Fig. 2 and
+Fig. 3, the left- and right-handed chiral molecules pre-
+pared in the same-energy initial states evolves (approx-
+imately) to the different-energy final states via the dif-
+ferent enantio-selective STA processes of invariant-based
+inverse engineering, driven by the same electromagnetic
+fields. Thus, the fast ESST via enantio-selective STA is
+achieved (approximately).
+In the above ESST method via the enantio-selective
+STA with invariant-based inverse engineering, the enan-
+tiomeric excess of the ESST can be defined as [23, 38]
+ϵ ≡
+���P L
+3 (τ) − P R
+3 (τ)
+P L
+3 (τ) + P R
+3 (τ)
+���.
+(25)
+Although the small values η and η′ (e.g.
+η = −η′ =
+0.02) have been introduced to avoid the infinite Ωy at
+the initial time instant, we still obtain a highly efficient
+ESST with enantiomeric excess ϵ = 99.92% at the final
+time instant (with most of left-chiral molecule staying
+in |3⟩L and very few of the right-chiral molecule staying
+in the same-energy state |3⟩R, as shown in Fig. 2 and
+Fig. 3).
+In general, the final populations are effected by the
+small value η (or η′) and are independent of the param-
+eter τ. As shown in Fig. 4(a), the population of the tar-
+
+6
+get state |3⟩L can be further decreased by increasing the
+small value η, while the population of the other target
+state |3⟩R would be commonly increased by increasing
+the small value η. Therefore, it is possible to achieve a
+better enantiomeric excess with relatively small value η.
+According to Eq. (15) and Eq. (24), decreasing the small
+amount η (or η′) implies the tradeoff of requiring larger
+Rabi frequencies and laser intensities [53]. Here we define
+Ωmax=Max{|Ωx(t)|, |Ωy(t)|, |Ωz(t)|} as the maximum ab-
+solute value of the Rabi frequencies during the whole evo-
+lution process. As shown in Fig. 4(b), the maximum ab-
+solute value of the Rabi frequencies increase dramatically
+when decreasing the small value η.
+(a)
+P3
+R
+P3
+L
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.80
+0.85
+0.90
+0.95
+1.00
+0
+0.01
+0.02
+0.03
+0.04
+η
+population
+(b)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+5
+10
+50
+100
+500
+1000
+η
+Ωmax (2π⨯MHz)
+FIG. 4. (Color online) (a) The corresponding populations in
+|3⟩L (red solid line) and |3⟩R (blue dashed line) at the final
+time versus the small value η. The initial states are |1⟩L,R.
+(b) The maximum absolute value of the Rabi frequencies Ωmax
+versus the small value η with τ = 0.5 µs.
+In experiments, the typical Rabi frequencies for the
+transitions of chiral molecules are about 2π×10 MHz [18,
+47, 48]. That means the evolution time can be shortened
+to be 0.5 µs for the experimentally available Rabi fre-
+quencies. Thus, the decoherence effects (typically being
+about 5 ∼ 6 µs) [17, 47] will become negligable.
+This
+is the advantage of our ESST method since it allows to
+manipulate the quantum system on the timescales much
+shorter than the typical decoherence time.
+Note that in the previous ESST method via STA [31],
+an auxiliary counterdiabatic field has been applied. It
+works as a shortcut to adiabaticity for canceling the
+nonadiabatic coupling and induces perfect population
+transfer between the states |1⟩L and |3⟩L for the left-
+handed chiral molecules.
+Simultaneously, it also acts
+oppositely for strengthening the nonadiabatic coupling
+for the right-handed chiral molecules and the population
+transfer between the states |1⟩R and |3⟩R is canceled com-
+pletely. Therefore, under such an ESST process, the left-
+handed chiral molecule begins with |1⟩L and terminates
+at −|3⟩L, following a STA path. But the right-handed
+chiral molecule is subject to a free evolution, instead
+of following the STA path.
+By contrast, in our ESST
+method via STA, the eigenstates of invariants for the two
+enantiomers define their corresponding enantio-selective
+STA paths. Thus, our ESST can be achieved with trans-
+ferring the two enantiomers from their ground states to
+different-energy final states along their enantio-selective
+STA paths simultaneously, by choosing appropriate in-
+tensities of the three electromagnetic fields (that is, the
+Rabi frequencies).
+IV.
+CONCLUSION
+In conclusion, we have proposed the fast ESST method
+of chiral molecules via the STA technique with invariant-
+based inverse engineering. Based on the cyclic three-level
+systems, the ESST of chiral molecules can be achieved
+through enantio-selective STA paths: for the left- and
+right-handed chiral molecules prepared initially in their
+ground states, they will evolve (approximately) finally to
+the different-energy states almost along the eigenstates
+of the invariants within a short operation time simulta-
+neously.
+Hence, our fast ESST method via STA with
+invariant-based inverse engineering has promising appli-
+cations in discriminating molecular chirality and control-
+ling the dynamics of chiral molecules.
+ACKNOWLEDGMENTS
+This work was supported by the Natural Science Foun-
+dation of China (Grants No. 12074030, No. 12274107,
+and No. U1930402), National Science Foundation for
+Young Scientists of China (No. 12105011), and Bei-
+jing Institute of Technology Research Fund Program for
+Young Scholars.
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