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+Boundary conditions dependence of the phase transition in the quantum
+Newman-Moore model
+Konstantinos Sfairopoulos,∗ Luke Causer, Jamie F. Mair, and Juan P. Garrahan
+School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK and
+Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems,
+University of Nottingham, Nottingham, NG7 2RD, UK
+We study the triangular plaquette model (TPM, also known as the Newman-Moore model) in
+the presence of a transverse magnetic field on a lattice with periodic boundaries in both spatial
+dimensions.
+We consider specifically the approach to the ground state phase transition of this
+quantum TPM (QTPM, or quantum Newman-Moore model) as a function of the system size and
+type of boundary conditions. Using cellular automata methods, we obtain a full characterization of
+the minimum energy configurations of the TPM for arbitrary tori sizes. For the QTPM, we use these
+cycle patterns to obtain the symmetries of the model, which we argue determine its quantum phase
+transition: we find it to be a first-order phase transition, with the addition of spontaneous symmetry
+breaking for system sizes which have degenerate classical ground states.
+For sizes accessible to
+numerics, we also find that this classification is consistent with exact diagonalization, Matrix Product
+States and Quantum Monte Carlo simulations.
+I.
+INTRODUCTION
+In this paper, we study the ground state phase transi-
+tion of the quantum Newman-Moore model, or quantum
+triangular plaquette model. The classical triangular pla-
+quette model (TPM), introduced by Newman and Moore
+[1], is a model of Ising spins interacting in triplets in (half
+of) the plaquettes of a triangular lattice.
+Despite the
+absence of quenched disorder and its trivial static prop-
+erties, the model has rich glassy dynamics [1–3].
+The
+TPM is an important model as it realises in an interact-
+ing system the paradigm of slow (super-Arrhenius) re-
+laxation at low temperatures due to effective kinetic con-
+straints. This phenomenon is central to the dynamic fa-
+cilitation picture of the glass transition [4–6]. The physics
+of the TPM can also be generalised to three dimensions,
+for example in the five-spin interaction square-pyramid
+model [7], or maintaining the triangular interactions in
+the models of Ref. [8]. A three-dimensional generalisa-
+tion of the TPM with non-commuting terms [9] actually
+started what is now the field of fractons [10, 11].
+The simplest way to transform the TPM into a quan-
+tum model is by adding a transverse field term to the
+classical Hamiltonian. Such quantum TPM (QTPM, or
+quantum Newman-Moore model) was considered in the
+context of fractons in Refs. [12, 13]. Numerics in Ref. [12]
+suggested that the ground state undergoes a first-order
+transition. A related work studying the large deviations
+of plaquette observables in the stochastic dynamics of in-
+dependent spins [14] also found numerical evidence for a
+first-order transition at the self-dual point of the model.
+In contrast, the results from Ref. [15] indicated that the
+transition is continuous, with a particular form of fractal
+symmetry breaking. The classical TPM and its connec-
+tion with fractals and topological order was also drawn in
+∗ [email protected]
+Ref. [16]. Here we aim to resolve these discrepancies by
+exploiting a general connection between D-dimensional
+cellular automata (CA) [17] and the ground states of
+(D + 1)-dimensional classical spin models [18]. By us-
+ing this method in the specific case of the QTPM with
+periodic boundary conditions, we are able to character-
+ize the approach to its quantum phase transition in the
+large size limit. Our key observation is that the nature of
+the transition depends on the specific lattice dimensions,
+and this is manifested in the finite size scaling.
+For the TPM, the relevant CA is Rule 60 [17] and
+not Rule 90 that might be assumed from comment [18]
+in Ref. [1]. For system sizes where one dimension is a
+power of two, Rule 60 has a single fixed point [19], im-
+plying a single energy minimum for the classical TPM.
+In such cases, we verify that the quantum phase tran-
+sition in the QTPM is first-order (that is, a sequence
+of such system sizes tends to a first-order transition in
+the large size limit). This also holds for other sizes for
+which Rule 60 has no non-trivial attractors. However,
+for certain sizes there can be periodic orbits on top of
+the fixed point for the CA, giving rise to classical ground
+state degeneracies in the TPM. For the quantum model,
+this translates into a mixed order quantum phase transi-
+tion. We provide evidence for this scenario by means of
+numerical simulations, namely, for small sizes using ex-
+act diagonalisation, and for large sizes using both Matrix
+Product State approximations of the ground state, and
+continuous-time Quantum Monte Carlo [20–23].
+The paper is organised as follows. In Sec. II, we review
+the classical and quantum TPM. In Sec. III, we provide
+the necessary background on CA and the connection to
+the ground states of the classical TPM. In Sec. IV, we dis-
+cuss the ground state phase transition of the QTPM in
+terms of the symmetries that follow from the properties
+of the associated CA, and support our predictions with
+numerical simulations. In Sec. V we give our conclusions.
+In the Appendix we provide further details, including a
+discussion on the case of the QTPM with open bound-
+arXiv:2301.02826v1  [cond-mat.stat-mech]  7 Jan 2023
+
+2
+aries.
+II.
+TRIANGULAR PLAQUETTE MODEL,
+CLASSICAL AND QUANTUM
+A.
+Classical
+The triangular plaquette model (or TPM or Newman-
+Moore model) [1–3] is a model of Ising spins si = ±1
+on the sites i = 1, . . . , N of a triangular lattice, with
+cubic interactions between the spins on the corners of
+downward-pointing triangles of the lattice, Fig. 1. The
+Hamiltonian of this classical model reads
+ETPM = −J
+�
+i,j,k∈▽
+sisjsk.
+(1)
+In what follows, it will be convenient to consider the
+equivalent model on a square lattice of size N = L × M,
+with classical Hamiltonian,
+ETPM = −J
+L,M
+�
+x,y=1
+sx,ysx+1,ysx+1,y+1,
+(2)
+where we assume periodic boundary conditions in both
+directions by identifying
+x + L = x
+mod L
+y + L = y
+mod L.
+In Eq.(2), we label the spins by sx,y at site with coordi-
+nates (x, y) in the square lattice.
+The classical TPM has been predominantly studied in
+the context of the glass transition. For lattices with at
+least one dimension being a power of two, the energy
+Eq.(1) reduces to that of the non-interacting plaquette
+variables [1–3],
+ETPM = −J
+�
+▽
+d▽,
+(3)
+where d▽ = sisjsk with i, j, k ∈ ▽ for every downward-
+pointing triangle in the lattice. When at least one di-
+mension is a power of two, the relation between plaque-
+ttes and spin variables is one-to-one, exactly proving the
+above. The thermodynamics of the TPM is therefore one
+of free binary excitations and, as such, it is essentially
+trivial.
+In contrast to the statics, the single spin-flip dynamics
+of the TPM is highly non-trivial, as flipping one spin
+changes three adjacent plaquettes.
+This implies that
+at low temperatures, where excited plaquettes are sup-
+pressed, cf. Eq.(3), the dynamics has effective kinetic con-
+straints [2]. These dynamical constraints lead to an ac-
+tivated relaxation similar to that of the East model [24],
+with relaxation times growing as the exponential of the
+inverse temperature squared [2] (a super-Arrhenius form
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+FIG. 1: Triangular plaquette model. The shaded
+triangles indicate the interacting triplets, Eq.(1).
+Dotted lines indicate spins that interact for periodic
+boundary conditions on an N = 4 × 4 lattice.
+known as the “parabolic law” [25]). Similar glassy be-
+haviour is seen in generalisations of the TPM with odd
+plaquette interactions [8, 26]. The TPM has also been
+considered in the presence of a (longitudinal) magnetic
+field [27] and in the related case of coupled replicas [28],
+and the TPM with open boundary conditions was studied
+in Ref. [29] through partial trace methods. The classical
+TPM was studied in the context of topological order in
+fractal models in Ref. [16]. Autoregressive neural net-
+works were applied to the classical TPM with limited
+success in [30].
+B.
+Quantum: TPM in a transverse field
+Taking Eq.(1) and adding a transverse field, we obtain
+the Hamiltonian of the quantum TPM (QTPM),
+HQTPM = −J
+�
+i,j,k∈▽
+ZiZjZk − h
+�
+i
+Xi,
+(4)
+where Zi and Xi represent Pauli operators acting non-
+trivially on site i. This quantum model was studied in
+Refs. [12, 13] and its connections to models of fractons
+were investigated.
+The Hamiltonian (4) is expected to have a quantum
+phase transition at the self-dual point J = h [14]. Nu-
+merical results from [12, 14] suggested the transition to be
+first-order. Ref. [14] used trajectory sampling in systems
+with linear size a power of two and periodic boundaries.
+In contrast, Ref. [15] found evidence for a continuous
+phase transition with fractal symmetry breaking using
+stochastic series expansion methods, also with periodic
+boundary conditions but not restricted to power of two
+sizes. The study of QTPM was connected to Rydberg
+atoms in Ref. [31]. Further studies on the QTPM and its
+generalisations from the viewpoint of fracton field theory
+were presented in Refs. [32, 33].
+
+3
+III.
+CELLULAR AUTOMATA AND GROUND
+STATES OF THE CLASSICAL TPM
+A.
+General aspects of CA
+Cellular automata (CA) consist of a D-dimensional ar-
+ray of sites evolving under discrete-time and synchronous
+dynamics given by a CA rule. Under this evolution, the
+state is updated by a deterministic rule which is local in
+time (although generalisations exist) [17, 34–36]. As a
+result, CA dynamics gives rise to, often rich, (D + 1)-
+dimensional structures.
+In what follows, we consider
+D = 1, linear CA with deterministic local transition rules
+with sites taking values from the finite field F2.
+Discrete cellular automata are fully specified by an ini-
+tial configuration of L sites and a local transition rule.
+The local update rule determines the configuration of ev-
+ery site at each timestep using the local neighbourhood
+of size r. For D = 1, the simplest CA rules are those
+defined only by r = 1.
+For a neighbourhood of three
+sites, r = 1, there are 8 possible configurations giving
+rise to 256 possible choices for update rules. These ele-
+mentary CA were classified by Wolfram [17, 34]. Figure
+2(a) shows Rule 60, which will be the relevant one for
+the TPM: if we identify the empty and occupied sites of
+the CA with the up and down spins of the TPM, then
+Rule 60 is the same as the condition that the product of
+spins in Eq.(2) is one, thus maximising the local energy.
+Figure 2(b) also shows the closely related Rule 90. CA
+like these are called elementary [34].
+Figure 3(a,b) shows the patterns generated for Rules
+60 and 90 starting from an initial single seed. Note that
+these depend on the boundary conditions (for a generic
+analysis on cellular automata with periodic boundaries,
+see Ref. [37]). For example, Fig. 3(b) shows Rule 90 for
+L = 64 and periodic boundaries: the timestep after the
+last one shown will take the CA to the trivial, empty
+configuration; in contrast for a length L = 63 the Sier-
+pinski fractal [38] will continue. Similarly, time evolution
+will bring in one timestep the pattern of Fig. 3(a) to the
+trivial one for periodic boundaries with L = 32, but the
+fractal shape would continue to be reproduced for open
+boundaries. Focusing on Rules 60 and 90 for concrete-
+ness, CA evolution can be written as polynomials (or
+generating functions, see Ref. [34]), as
+T60(x) = 1 + x
+(5)
+T90(x) = x−1 + x,
+(6)
+with the evolution of the whole lattice obeying [34]
+A(t)(x) = TA(t−1)(x)
+mod (xL − 1).
+(7)
+Given a configuration at timestep t for the evolution
+of any given rule, the configuration at timestep t + 1 will
+be its successor and the one at t − 1 its predecessor. For
+Rules 60 and 90, following Ref. [34], we have:
+• There are no predecessors for configurations with
+an odd number of sites being equal to 1 for both
+(a)
+(b)
+FIG. 2: (a) Rule 60 and (b) Rule 90 evolution rules.
+(a)
+(b)
+(c)
+FIG. 3: (a) Evolution from a single site for Rule 60. (b)
+Same for Rule 90. (c) A stable cycle generated by Rule
+60.
+rules. For Rule 90, 2L−1 configurations for a system
+of size L cannot be reached if L is odd and 3 ×
+2L−2 if L is even. For Rule 60, 2L−1 configurations
+cannot be reached for any system size. This means
+that rows with an odd number of ones can only
+occur as initial states for any given time evolution
+for Rule 60.
+• Similarly, there may exist configurations that can
+be reached in one timestep by more than one pre-
+decessors. More specifically, given a configuration
+with at least one predecessor, there are 2 predeces-
+sors if the length L is odd and 4 if it is even for
+Rule 90, while there are always 2 for Rule 60.
+Regarding the length of cycles for Rule 90, more results
+can be found in Ref. [34]. For Rule 60, which is most
+relevant for the TPM, we discuss the cycle lengths and
+their multiplicities next.
+B.
+Periodic orbits of Rule 60
+The iterated map
+Dn(x) = (|x1 − x2|, |x2 − x3|, ..., |xL − x1|)
+(8)
+for Dn : Zn → Zn and x = (x1, x2, ..., xL) is known as the
+Ducci map (also known as rule 102 in Wolfram’s nota-
+tion [17, 34], the mirror image of Rule 60, e.g. Ref. [39]).
+Ducci’s map (and therefore Rule 60) exhibits periodic
+orbits or evolves to a fixed point depending on L being
+a power of two or not.
+For L = 2k, any initial state
+
+4
+(b)
+(a)
+(c)
+FIG. 4: The fixed point and limit cycles of Rule 60. Filled circles indicate states of the CA and the arrows the flow
+under the dynamics. (a) For L = 5 there is a cycle of length C = 15 and the fixed point (which flows into itself). (b)
+For L = 6 there are two cycles of period C = 6, a cycle of period C = 3 and the fixed point. (c) For L = 8, there is
+the fixed point and no cycles.
+evolves to the trivial state of zeros [40], while for L ̸= 2k
+it evolves to a richer attractor structure [41].
+In F2 and by using periodic boundary conditions, the
+Ducci map can be brought into matrix form as follows,
+Dn x = ((x1 + x2), (x2 + x3), · · · , (xL + x1))
+mod 2,
+(9)
+where
+Dn =
+�
+�������
+1 1 0 . . . 0 0 0
+0 1 1 . . . 0 0 0
+...
+...
+... ... ...
+...
+...
+0 0 0 . . . 1 1 0
+0 0 0 . . . 0 1 1
+1 0 0 . . . 0 0 1
+�
+�������
+.
+(10)
+The matrix D can be expressed as
+D = I + SL
+(11)
+with SL the left shift map [19, 41]. It is easy to check
+that
+D2k = (I + SL)2k
+= I + S2k
+L = I + I = 0
+mod 2, (12)
+with k ∈ Z and, thus, a system that has L a power of
+2 ends up in the trivial configuration. For Rule 60 the
+corresponding matrix is D⊺.
+In order to obtain the cycles of Rule 60, we need
+the following definitions and results for Rule 102 from
+Refs. [19, 35, 36, 41–44]:
+• For any given CA, each array of sites can be
+thought of as a vector, v. For any such vector v in
+F2, its order is defined through the monic polyno-
+mial, which satisfies µv(D)v = 0.
+• The order of the minimal annihilating polynomial,
+ord µv(λ), is equal to the smallest natural number
+n, such that µv(λ)|λn − 1. If µv(0) = 0, then for
+µv(λ) = λk˜µv(λ) with ˜µv(0) ̸= 0 and k ∈ N, the
+order of µv is ord (µv) = ord (˜µv).
+• Assuming that µv(λ) = λk˜µv(λ) with k ≥ 0, then
+the k-th successor of v belongs to a cycle of length
+c = ord µv. This applies to a vector v of any pos-
+itive integer L and any linear map (or cellular au-
+tomaton rule).
+The minimal annihilating polynomial for the Ducci
+map was calculated in Refs. [19, 43], based on the char-
+acteristic polynomial of the matrix D. It was found that
+µn(λ) = pn(λ) = (1 + λ)L + 1.
+(13)
+We are now in a position to obtain the attractor struc-
+ture of Rule 60 by following Ref. [36] (specifically “Prin-
+ciple C”). We decompose the minimal annihilating poly-
+nomial into the product of its irreducible polynomials,
+πi(λ). We call their polynomial powers bi. Thus,
+µv(λ) =
+m
+�
+i=1
+πi(λ)bi,
+(14)
+and
+ord µv(λ) = rpt,
+(15)
+where r is the least common multiple of ord πi and t the
+smallest integer satisfying pt ≥ max (b1, b2, . . . , bm).
+Figure 4 illustrates the different scenarios for cycles as
+a function of L. We show three different sizes, L = 5, 6, 8,
+small enough to be able to visualise the network of states.
+Configurations of the CA are identified by blue circles,
+
+5
+1
+1
+0
+2
+1
+0
+3
+1
+3
+3
+4
+1
+0
+5
+1 15
+15
+6
+1
+3
+6
+15
+7
+1
+7
+63
+8
+1
+0
+9
+1
+3 63
+255
+10
+1 15 30
+255
+11
+1 341
+1023
+12
+1
+3
+6 12
+255
+13
+1 819
+4095
+14
+1
+7 14
+4095
+15
+1
+3
+5 15
+16383
+16
+1
+0
+17
+1 85 255
+65535
+18
+1
+3
+6 63 126
+65535
+19
+1 9709
+262143
+20
+1 15 30 60
+65535
+21
+1
+3
+7 21 63
+1048575
+22
+1 341 682
+1048575
+23
+1 2047
+4194303
+24
+1
+3
+6 12 24
+4095
+25
+1 15 25575
+16777215
+26
+1 819 1638
+16777215
+27
+1
+3 63 13797
+67108863
+28
+1
+7 14 28
+16777215
+29
+1 475107
+268435455
+30
+1
+3
+5
+6 10 15 30
+268435455
+31
+1 31
+1073741823
+32
+1
+0
+33
+1
+3 341 1023
+4294967295
+34
+1 85 170 255 510
+4294967295
+35
+1
+7 15 105 819 4095
+17179869183
+36
+1
+3
+6 12 63 126 252
+4294967295
+37
+1
+3233097
+68719476735
+38
+1 9709 19418
+68719476735
+39
+1
+3 455 819 1365 4095
+274877906943
+40
+1 15 30 60 120
+16777215
+TABLE I: Cycle lengths for Rule 60. The second
+column indicates the cycle lengths and the third the
+total number of periods for Rule 60 (each period of
+length C is counted C-times), assuming that the least
+common multiple of the periods for each given L divides
+M, lcm C|M. The last column is constructed based on
+the number of predecessors for all configurations for
+Rule 60, given a length L.
+and arrows indicate to which configurations they evolve
+to under the CA dynamics. Figure 4(a) shows L = 5:
+here there is one fixed point to which one other state
+evolves to (shown at the centre of the figure), and one
+limit cycle of period C = 15, to which all other configura-
+tions flow. Figure 4(b) shows a more general situation of
+multiple distinct cycles for the case of L = 6: there are
+two cycles of period C = 6, one cycle of period C = 3 and
+one fixed point. For the case of L a power of 2, as shown
+in Fig. 4(c) for L = 8, there is a unique fixed point and
+all states evolve towards it.
+C.
+Classical ground states of the TPM
+From the analysis above for Rule 60, we can enumerate
+all the minimum energy configurations of the TPM, cf.
+Eq.(2). The classical ground states for a system of size
+N = L × M are: (i) the state with all spins up, corre-
+sponding to the fixed point of Rule 60, for any value of
+L and M; (ii) two-dimensional spin configurations that
+correspond to periodic trajectories of Rule 60, that is,
+CA trajectories starting from any of the states of a limit
+cycle, for all limit cycles whose period is contained an in-
+teger number of times in M—this occurs only for certain
+combinations of L and M (never if L is a power of two).
+We show the relevant Rule 60 information in Table
+I for up to L = 40. Under the column C we give the
+distinct periods of the limit cycles. In the column labelled
+by M we give the corresponding degeneracy of classical
+ground states of the TPM, apart from the uniform spin-
+up state, given that lcm C|M. For example, for L = 15,
+there is one cycle of length 3, three cycles of length 5,
+and 1091 cycles of length 15, which means 1 + M =
+1+3+3×5+1091×15 = 16384 distinct two-dimensional
+spin configurations that minimise the energy in a TPM
+of size N = 15 × M as long as 15|M. In contrast, for
+L = 15, if 5|M but 3 ∤ M and 15 ∤ M, then there are
+1+15 different ground states. The symmetries of a TPM
+of size N = L × M can be similarly constructed from
+the number of ground states and their multiplicities, as
+determined by Rule 60.
+IV.
+PHASE TRANSITION IN THE QUANTUM
+TPM
+As we will now show, the set of minimum energy con-
+figurations and the associated symmetries of the classical
+TPM, as obtained from the Rule 60 CA, determine the
+properties of the ground state phase transition in the
+quantum TPM, Eq.(4).
+A.
+Symmetries of the QTPM
+Like its classical counterpart, the QTPM with periodic
+boundaries has full translational invariance. The symme-
+
+6
+(b)
+(a)
+FIG. 5: (a) Symmetry operators for the 3 × 3 QTPM
+with periodic boundaries. (b) One of the symmetry
+operators of the 9 × 9 QTPM.
+tries of the QTPM will then be deduced by the results of
+Sec. III.
+For systems with dimensions N = L × M, which can
+accommodate non-trivial cycles of Rule 60, the symme-
+tries of the corresponding QTPM easily follow. Consider
+as a simple example the case of N = 3 × 3 with periodic
+boundaries in both dimensions.
+From Table I, we see
+that L = 3 has one cycle of period 3. This means that
+for M = 3 there are three non-trivial symmetries, given
+by the operators
+G1 = X1,2X1,3X2,1X2,2X3,1X3,3
+(16)
+G2 = X1,1X1,3X2,2X2,3X3,1X3,2
+(17)
+G3 = X1,1X1,2X2,1X2,3X3,2X3,3
+(18)
+see Fig. 5(a). Note that translational invariance plays a
+crucial role in the determination of the symmetries: in
+the example above G2 = TxG1T −1
+x
+and G3 = TxG2T −1
+x
+,
+where Tx is the translation operator in the x direction,
+and these symmetries alongside the identity form the
+Klein group K4, which, in turn, is isomorphic to Z2 ⊗Z2.
+This approach generalises to other system sizes, and
+the symmetries are products of K4.
+For example, the
+symmetries of the N = 5×15 and the N = 6×6 systems
+form the group K4 ⊗ K4. Since the symmetry operators
+are products of the X Pauli matrices, cf. Eqs.(16-18),
+they commute with the transverse field term in Eq.(4),
+and, therefore, are symmetries of the QTPM for all values
+of J and h.
+B.
+The character of the quantum phase transition
+The character of the phase transition for the QTPM
+is now easy to predict based on the information of its
+symmetries for a given system size. By this we mean:
+given a sequence of increasing system sizes with periodic
+boundaries, all having the same number of symmetries,
+the progressively sharper finite size crossovers will be in-
+dicative of an eventual phase transition in the large size
+limit whose character—first-order or continuous—will be
+determined by the underlying symmetries of the given
+system sizes in the sequence. As these symmetries in a
+system of size N = L × M depend on the precise values
+of both L and M, this analysis has to be done carefully.
+In the next section we show numerical evidence for the
+general considerations we give here.
+First consider the system sizes N = L × M such that
+the underlying Rule 60 has only one fixed point, as de-
+scribed previously. Then, the QTPM has no non-trivial
+symmetries. In the limit J ≫ h, there is a single ground
+state corresponding to the all-up state, and a vanishing
+number of excited plaquettes, as the classical TPM has
+a single minimum.
+In the opposite limit, J ≪ h, the
+ground state is paramagnetic, with spins aligned in the
+x direction, with an explicit Z2 symmetry for its ground
+state, corresponding to a high density of excited plaque-
+ttes.
+In the limit of large N, we expect the quantum
+phase transition at the self-dual point J = h to be first-
+order.
+A second scenario is that of system sizes where the
+underlying Rule 60 cycles give rise to non-trivial sym-
+metries in the QTPM. In this case, in the limit J ≪ h,
+the ground state is the same paramagnetic one as be-
+fore, invariant under a global Z2 symmetry.
+However,
+for J ≫ h there exists spontaneous breaking of the sym-
+metries emerging from Rule 60.
+This case, therefore,
+has characteristics of a first-order phase transition (the
+explicit symmetry breaking of the global symmetry of
+the paramagnetic ground state) with those of a contin-
+uous transition (spontaneous symmetry breaking). This
+is similar to the first-order phase transitions observed in
+kinetically constrained models [45] and suggests that, for
+local observables, the phase transition will appear first-
+order; for example, in a discontinuous jump in the excited
+plaquette density,
+Mzzz = 1
+N
+�
+i,j,k∈▽
+ZiZjZk.
+(19)
+Appropriate operators will quantify the symmetry break-
+ing of the degenerate classical ground states.
+The choice of these operators depends on the size and
+specific lattice dimensions of the system. Consider, for
+example, the case of L = 3 and M = 3k with k ∈ N,
+where we know from Rule 60 that there is a single fixed
+point and the three non-trivial ground states. For the
+trivial ground state this operator is just the magnetisa-
+tion Mz =
+1
+N
+�N
+i Zi. To detect the symmetry breaking
+into the other three states, we can define the three op-
+erators ˜
+M m
+z =
+1
+N
+�N
+i (−1)niZi, where ni = 1 if the spin
+i is flipped for a state of the cycle m of Rule 60, and
+ni = 0 otherwise. For example, for the state associated
+
+7
+to Eq.(16), we have
+˜
+Mz = 1
+N (Z1,1 − Z1,2 − Z1,3 − Z2,1 − Z2,2 + Z2,3
+−Z3,1 + Z3,2 − Z3,3) .
+(20)
+Note that when there are multiple non-trivial ground
+states connected by translations, there will be no sin-
+gle operator taking the form of a sum of local terms for
+which it will be possible to discern between them. For
+example, in the N = 3k case, Mz will only distinguish be-
+tween the trivial ground state and the 3-fold degenerate
+ones, while the operators ˜
+M m
+z
+will be able to distinguish
+between one of the non-trivial ground states.
+C.
+Numerics
+We now provide evidence for the general observations
+above from numerical simulations.
+For small systems
+we use exact diagonalization (ED) [46–48], allowing the
+study of system sizes up to 28 sites. For larger systems
+we estimate the properties of the ground states using
+two different approaches.
+The first of these is Matrix
+Product States (MPS) [49–52], which we “snake” around
+the 2D lattice, and optimize with the 2D Density-Matrix
+Renormalization Group (DMRG) [53, 54]. By employing
+a bond dimension up to D = 1000, we are able to reli-
+ably estimate the ground state properties for system sizes
+on the square lattice for up to N = 16 × 16. Time and
+memory constraints hinder progressively the convergence
+in the paramagnetic phase, where J ∼ h. As discussed
+below, we are also able to apply MPS to cylindrical sys-
+tems, which can be considered to be quasi-1D, allowing
+us to reach much larger sizes than in the case of square
+geometries.
+To confirm the results of 2D DMRG, we also employ
+Quantum Monte Carlo (QMC) methods. In particular,
+we use the continuous-time expansion (ctQMC) [20], with
+local spin updates which re-draw the entire trajectory
+of a single spin, subject to a time-dependent environ-
+ment, where the trajectories of unmodified spins are con-
+sidered to act as a “heat bath”, e.g., see Refs. [21, 22].
+We run our simulations with an inverse temperature of
+β = 128, which we find to be large enough to converge
+to the ground state [23].
+1.
+First-order transition
+As explained above, when the underlying Rule 60 has a
+single fixed point, and the classical TPM a single energy
+minimum with all spins up, we thus expect the phase
+transition of the QTPM to be first-order.
+Figure 6 shows results for N = L×L with L = 4, 8, 16.
+We show that our MPS and ctQMC results coincide with
+the large deviations results of Ref. [14], obtained via tran-
+sition path sampling (TPS). What is plotted is the pla-
+quette density Mzzz as a function of the coupling J for
+□□□□□□□□□□□□□□□□
+▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇
+■
+▼
+▼
+◆
+◆
+0.95
+1.00
+1.05
+1.10
+0.4
+0.6
+0.8
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+□
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+□
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+▼
+▼
+▼
+▼ ▼
+▼
+▼ ▼▼
+▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼
+◆
+◆
+◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ctQMC
+□
+▽
+◇
+MPS
+■
+▼
+◆
+TPS
+△
+○
+FIG. 6: Normalized three-spin correlator, Eq.(19), in
+the QTPM as a function of J for fixed h = 1, with
+N = L × L with L a power of two. We compare results
+from MPS and ctQMC obtained here with results from
+Ref. [14]. The numerical data indicates a first-order
+transition at J = h.
+fixed transverse field h = 1. The data indicates a first-
+order transition at J = h, as expected. For J > 1.0, we
+see small deviations in the TPS results, due to the extra
+field used for the acquisition of this data in Ref. [14]. Sim-
+ilar issues are observed for ctQMC close to the J = 1.0
+point, where the single spin updates do not allow for the
+collective effects necessary to move between phases.
+In Fig. 7, we show the results for several system sizes
+in a square geometry, N = L × L. Figure
+7(a) shows
+the ground state energy as a function of J (at h = 1)
+for L = 5 to 16, obtained from MPS numerics. For the
+smallest size we also show ED results, which coincide
+with the MPS ones. The kink near J = 1 indicates a
+quantum phase transition. Note that this behaviour is
+similar in systems with a single classical ground state
+(L = 5, 8, 16) or multiple ones (L = 6, 7), cf. Table I. In
+Figs. 7(b,c) we show the average transverse magnetisa-
+tion, Mx = 1
+N
+�
+i Xi, and Mzzz, respectively, for systems
+with L a power of two. We get exactly the same results
+for different system sizes too.
+Both MPS and ctQMC
+show clear indications of a first-order transition at J = 1
+in both observables.
+Figure 8 shows similar results in a rectangular geome-
+try, N = 3×M. For such thin stripe systems we can per-
+form MPS more efficiently for larger system sizes than for
+square geometries. Once again, MPS and ctQMC results
+coincide, and indicate a first-order transition at J = 1
+(although weaker than in the square lattice case, in the
+sense that the discontinuity in the local operators shown
+is smaller). Note that these results include not only val-
+ues of M which are multiples of three, for which there
+are multiple classical ground state cycles, but also values
+of M for which a single ground state is found. What we
+see in this case is that the observables Mx and Mzzz are
+unable to detect changes related to any given classical
+ground states.
+
+8
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+0.0
+0.5
+1.0
+1.5
+2.0
+-2.0
+-1.5
+-1.0
+(a)
+△
+●
+●
+●
+●
+●
+● MPS
+△ ED
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+■ ■ ■ ■ ■ ■ ■
+■
+■
+■ ■ ■ ■ ■ ■ ■
+▮ ▮ ▮ ▮ ▮
+▮
+▮ ▮▮
+▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮
+▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b)
+ctQMC
+□
+▯
+△
+▽
+◇
+○
+MPS
+■
+▮
+▲
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+○○○○○○○○○○○○○○○○��○○○○○○○○○○○
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+■
+■
+■
+■
+■
+■
+■
+■
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+■
+■
+■
+■
+■
+■
+■
+▮
+▮
+▮
+▮ ▮
+▮
+▮ ▮▮
+▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮
+▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(c)
+ctQMC
+□
+▯
+△
+▽
+◇
+○
+MPS
+■
+▮
+▲
+FIG. 7: First-order transition in the QTPM for systems of size N = L × L. (a) The normalised by the system size
+ground state energy as a function of J at fixed h = 1. Open symbols are results from ED, filled symbols from
+numerical MPS. (b) Transverse magnetisation as a function of J. In this case the open symbols are from ctQMC. (c)
+Average three-spin interaction as a function of J.
+■
+■
+■
+■ ■ ■ ■
+■
+■
+■
+■
+■
+■
+▲
+▲
+▲
+▲ ▲ ▲ ▲
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+▼
+▼ ▼ ▼ ▼
+▼
+▼
+▼
+▼
+▼
+▼
+◆
+◆
+◆
+◆
+◆ ◆ ◆
+◆
+◆
+◆
+◆
+◆
+◆
+●
+●
+●
+●
+● ● ●
+●
+●
+●
+●
+●
+●
+■
+■
+■
+■ ■ ■ ■
+■
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+■
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+▼
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+▼ ▼ ▼ ▼
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+▼
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+▼
+▼
+▼
+◆
+◆
+◆
+◆
+◆ ◆ ◆
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+◆
+◆
+◆
+◆
+◆
+●
+●
+●
+●
+● ● ●
+●
+●
+●
+●
+●
+●
+0.0
+0.5
+1.0
+1.5
+2.0
+-2.0
+-1.5
+-1.0
+(a)
+MPS
+■
+▲
+▼
+◆
+●
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+□
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+○○○○○○○○○○○○○○○○○○○○ ○○○○○○○
+■
+■
+■
+■ ■
+■
+■ ■
+■ ■
+■
+■ ■
+▲
+▲
+▲
+▲ ▲
+▲
+▲ ▲
+▲ ▲
+▲
+▲ ▲
+▼
+▼
+▼
+▼ ▼
+▼
+▼ ▼
+▼ ▼
+▼
+▼ ▼
+◆
+◆
+◆
+◆
+◆
+◆
+◆ ◆
+◆ ◆ ◆
+◆ ◆
+●
+●
+●
+●
+●
+●
+● ●
+● ● ●
+● ●
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b)
+ctQMC
+□
+△
+▽
+◇
+○
+MPS
+■
+▲
+▼
+◆
+●
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+□
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+○○○○○○○○○○○○○○○○○○○○ ○○○○○○○
+■
+■
+■
+■ ■
+■
+■ ■
+■ ■
+■
+■ ■
+▲
+▲
+▲
+▲ ▲
+▲
+▲ ▲
+▲ ▲
+▲
+▲ ▲
+▼
+▼
+▼
+▼ ▼
+▼
+▼ ▼
+▼ ▼
+▼
+▼ ▼
+◆
+◆
+◆
+◆
+◆
+◆
+◆ ◆
+◆ ◆ ◆
+◆ ◆
+●
+●
+●
+●
+●
+●
+● ●
+● ● ●
+● ●
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(c)
+ctQMC
+□
+△
+▽
+◇
+○
+MPS
+■
+▲
+▼
+◆
+●
+FIG. 8: Same as Fig. 7 but for systems of size N = 3 × L.
+2.
+Symmetry breaking
+In Figs. 7 and 8, we show the two terms that com-
+pete in the Hamiltonian, Mx and Mzzz. For system sizes
+where there is one classical ground state and no non-
+trivial symmetries, the total longitudinal magnetisation
+Mz can also serve as an order parameter, as it picks up
+the orientation of the ground state. Figure 9(a) shows
+that the transition is also clear for this observable for
+square lattices.
+For system sizes where degeneracies are expected for
+J ≥ h, however, Mz is unable to detect the symme-
+try breaking related to the extra symmetries. For these
+cases, we need the staggered magnetisations, ˜
+M m
+z , such
+as that for N = 3 × 3 in Eq.(20). Figure 9(b) shows that
+such operators are able to detect the spontaneous break-
+ing of symmetry for these lattices. Note that Fig. 9(b)
+was obtained through the use of a small symmetry break-
+ing field. This is a standard method for the detection of
+the symmetry breaking in the ground state of a degen-
+erate quantum model [55]. As a result, the calculations
+were performed through the use of a modified Hamilto-
+nian H = HQTPM − p ˜
+Mz, where p is chosen to be small.
+The detection of the spontaneous symmetry breaking can
+be similarly preformed for any of the classical ground
+states of the given lattice size with the appropriate oper-
+ator ˜
+Mz.
+In order to more clearly understand the mechanism of
+the phase transition, in Figs. 10 and 11 we plot the low-
+lying spectrum of the QTPM from ED as a function of
+h for fixed J.
+These results support our above obser-
+vations: for system sizes where only a first-order phase
+transition is expected, there is an avoided crossing be-
+tween the ground state and the first excited state; for
+system sizes with extra symmetries from the cycles of
+Rule 60, we see both an avoided crossing (indicative of
+first-order transitions) and a merging of eigenstates in-
+dicative of spontaneous symmetry breaking. As seen in
+Fig. 11 for the case of N = 3 × M, the avoided crossing
+becomes apparent only with increasing system size.
+We now comment on how our results compare to those
+in Ref. [15]. For the numerics, Ref. [15] used a stochastic
+series expansion (SSE) approach. We in turn use MPS
+and ctQMC. Both SSE and ctQMC are Quantum Monte
+Carlo based methods, which indicates that they, in prin-
+ciple, should be able to roughly access system sizes of the
+same order of magnitude.
+Furthermore, while Ref. [15] also considered periodic
+boundaries, there was no specific restriction on system
+size, and therefore no distinction between sizes for which
+there is a single classical minimum and sizes where there
+are multiple ones, with the implications for symmetries of
+the corresponding QTPM. Ref. [15] also used a non-local
+order parameter, compared to our local ones (the stag-
+gered magnetisations) that do reflect the minima of the
+underlying TPM. In [15], the existence of a phase tran-
+sition at J = h was confirmed through the study of the
+Binder cumulant; this was done, however, with limited
+
+9
+accuracy on the location of the phase transition point. It
+is important to note that some of the local observables
+we calculate here are also studied for specific system sizes
+in the Appendix of Ref. [15]. Since the temperature used
+for those calculations varied for different system sizes,
+it is possible that the smoothness observed in Ref. [15]
+is a consequence of thermal effects. We instead used a
+fixed inverse temperature β = 128 which we verified is
+sufficient to make thermal effects negligible.
+D.
+Nature of the phase transition in the
+thermodynamic limit
+The discussion above and the numerical results indi-
+cate the existence of a quantum phase transition in the
+thermodynamic limit, N → ∞, at the self-dual point,
+J = h, of the QTPM. However, the approach to the
+thermodynamic limit is different across different system
+size geometries.
+There are three different limits to thermodynamics: (i)
+across one of the two dimensions while the other one re-
+mains constant (that is, infinite stripes), (ii) across both
+dimensions, and (iii) on making the spins continuous. We
+briefly discuss the differences between these limits and
+the complications that might arise.
+In the case (i), if the limit is taken for fixed L and with
+M such that lcm C|M (e.g. M = 3k, with k ∈ N), the
+number of classical ground states remains the same. In
+our numerics we are restricted to narrow stripes to allow
+convergence of the MPS algorithm. Fig. 8 suggests that
+in such quasi-1D systems the transition will eventually
+be slighly weaker than for square system sizes.
+Case (ii) can be more involved. The simplest situation
+is that of square lattices N = L × L with L a power of
+two, where it is guaranteed that for all sizes there will be
+a single classical ground state, and therefore the transi-
+tion is certainly first-order. For other size sequences, the
+number of relevant Rule 60 cycles, and therefore symme-
+tries of the QTPM, may grow or decline with system size.
+For some cases this growth is monotonic (as for example
+for N = 3k × 3k with k → ∞), while in others it is not
+(as for example when N = 3k × 3k with k → ∞), see
+Table I.
+In case (iii) the nature of the underlying CA is altered
+[56, 57]. In this limit, Rule 60 becomes
+f(p, q, r) = p + q − 2pq,
+(21)
+where p, q and r indicate the state of the three sites
+in the neighbourhood determining the local evolution of
+the CA, see Fig. 2. Basic arguments [56] indicate a single
+fixed point in the evolution of this fuzzy CA. We spec-
+ulate that the same behaviour will be observed in the
+quantum field theory limit for QTPM; a single ground
+state across different regions of the whole J − h space
+and thus a first-order phase transition. However, a field
+theoretic description of the QTPM might not be as obvi-
+ous and straightforward to get for the above elementary
+argument to hold.
+V.
+CONCLUSIONS
+In this work, we used the cycles of the cellular automa-
+ton Rule 60 to describe the symmetries of the quantum
+triangular plaquette model. We found that the attrac-
+tor structure of Rule 60 plays an important role in the
+characterization of the degeneracies of the ground states
+of the classical TPM, allowing in turn to construct the
+symmetry operators of the QTPM. In this way, the exis-
+tence or absence of stable cycles in Rule 60 imply whether
+it is possible or not for the QTPM to display sponta-
+neous symmetry breaking of the corresponding symme-
+tries, which in turn impacts on the nature of the quantum
+phase transition at the self-dual point. These general ob-
+servations are also consistent with the finite size trends
+from our numerical simulations. A full description of the
+QTPM phase transition would require a field theoretical
+description and a renormalization group treatment; we
+leave these tasks for future works.
+ACKNOWLEDGMENTS
+We thank L. Vasiloiu for insightful comments.
+We
+acknowledge financial support from EPSRC Grant no.
+EP/R04421X/1, the Leverhulme Trust Grant No. RPG-
+2018-181,
+and University of Nottingham grant no.
+FiF1/3. LC was supported by an EPSRC Doctoral prize
+from the University of Nottingham.
+Simulations were
+performed using the University of Nottingham Augusta
+HPC cluster, and the Sulis Tier 2 HPC platform hosted
+by the Scientific Computing Research Technology Plat-
+form at the University of Warwick (funded by EPSRC
+Grant EP/T022108/1 and the HPC Midlands+ consor-
+tium).
+[1] M. E. J. Newman and C. Moore, Glassy dynamics and
+aging in an exactly solvable spin model, Phys. Rev. E 60,
+5068 (1999).
+[2] J. P. Garrahan and M. E. J. Newman, Glassiness and
+constrained dynamics of a short-range nondisordered spin
+model, Phys. Rev. E 62, 7670 (2000).
+[3] J. P. Garrahan, Glassiness through the emergence of
+effective dynamical constraints in interacting systems,
+Journal of Physics: Condensed Matter 14, 1571 (2002).
+
+10
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+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
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+▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮ ▮
+▲
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+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀
+◀
+◀
+◀
+◀◀◀◀
+◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀
+◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀◀
+◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
+◆
+◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
+◆◆
+◆◆◆◆◆◆◆
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+■
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+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯���▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0 (a)
+ED
+◀
+◆
+MPS
+■
+▮
+▲
+ctQMC
+□
+▯
+△
+▽
+◇
+○
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0 (b)
+MPS
+●
+●
+●
+●
+●
+FIG. 9: (a) Longitudinal magnetisation for systems with no symmetries. (b) Staggered magnetisation for detecting
+symmetry breaking in systems with multiple symmetries.
+□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+-18
+-16
+-14
+-12
+(a)
+Energy Levels
+□
+1
+▯
+2
+△ 3-4
+□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+-22
+-20
+-18
+-16
+(b)
+Energy Levels
+□
+1
+▯
+2
+△ 3-4
+□ □ □ □ □ □ □ □
+□
+□
+□
+□
+□
+▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯
+△ △ △ △ △ △ △ △ △ △ △ △ △
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+-30
+-28
+-26
+-24
+-22
+(c)
+Energy Levels
+□
+1
+▯
+2
+△ 3-4
+FIG. 10: Low-lying spectrum of the QTPM as a function of h for fixed J = 1 from ED, for sizes without extra
+symmetries. (a) N = 3 × 4. (b) N = 4 × 4. (c) N = 3 × 7. The avoided crossing between the ground (black squares)
+and first excited (purple rectangles) states is indicative of a first-order transition.
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+11
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+-16
+-14
+-12
+-10
+-8
+-6
+(a)
+Energy Levels
+□
+1
+▯ 2-4
+△
+5
+□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+-26
+-24
+-22
+-20
+-18
+(b)
+Energy Levels
+□
+1
+▯ 2-4
+△
+5
+□ □ □ □ □ □ □ □ □ □ □ □ □ □ □
+▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯
+△ △ △ △ △ △ △ △ △ △ △ △ △ △ △
+0.9
+1.0
+1.1
+1.2
+-38
+-36
+-34
+-32
+-30
+-28
+(c)
+Energy Levels
+□
+1
+▯ 2-4
+△
+5
+FIG. 11: Same as Fig. 10 but for systems with multiple symmetries. (a) N = 3 × 3. (b) N = 3 × 6. (c) N = 3 × 9.
+The merging of the ground state (black squares) with three degenerate excited states (purple rectangles) is
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+Appendix A: TPM and QTPM for other boundary
+conditions
+In the main text, we show that ground state properties
+of the TPM and, consequently, the quantum phase tran-
+sition of the QTPM depends on the boundary conditions,
+but we only focused on systems with periodic boundaries.
+Here, we consider the cases of periodic boundaries in only
+the x-dimension (PBCx) and of open boundaries (OBC),
+using the same Rule 60 CA considerations as for the pe-
+riodic case.
+For PBCx, we use the update rule for Rule 60 as before,
+with the only difference that we do not need to explic-
+itly check for the periodicity across the y-direction. As
+a result, for an initial array of L sites, there will be 2L
+configurations and, thus, 2L ground states for the clas-
+sical TPM. In this case, only the number of sites in the
+x-direction matters for the number of classical ground
+states. For example, a lattice with size N = 3 × 3 and
+one with N = 3 × 80 will have the same number, 8, of
+ground states. The identification of the classical ground
+states can be worked out from the Rule 60 evolution, as
+before.
+For OBC, the update rule for Rule 60 is modified for
+the first cell of an L-length array so that it is not updated.
+This freedom on choosing two of the boundaries of the
+lattice gives an increased number of ground states for
+the classical TPM. Specifically, given a lattice of N spins,
+N = L × M, the number of the classical ground states is
+2L+M−1.
+We now perform a similar numerical analysis as in
+Sec. IV C, but only using MPS methods. Data is nor-
+malised with the system size of the given lattice. The
+system sizes accessible do not give a clear indication of
+a well-formed phase transition, but only signatures of it.
+The first-order transition found is weaker than in the case
+of the fully PBC, which we attribute to the high number
+of ground states for the classical TPM, given the system
+sizes. We note that all these states for h ̸= 0 constitute
+low-lying excited states which affect the convergence of
+the MPS algorithm.
+As seen from Figs. 12 and 13 for PBCx, the difference
+between the square lattice size scaling and the quasi-1D
+rectangular stripes is more pronounced, when compared
+to the finite-size scaling for PBC. Extra calculations on
+wider rectangular stripes verify that this difference is only
+a feature of the quasi-1D geometry of the lattice and not
+an inherent property of the system. Accuracy is lost with
+increasing size and the MPS results for the sizes studied
+are not reflective of the true thermodynamic limit.
+The above considerations for PBCx are even more no-
+ticeable for the case of OBC, Figs. 14 and 15. Rectangu-
+lar stripes are fully continuous, while they seem to have
+converged to their “thermodynamic” behaviour.
+How-
+ever, as seen from the square system sizes, the behaviour
+of the model remains the same regardless of the bound-
+ary conditions. It becomes apparent though that bigger
+system sizes soon become computationally inaccessible
+due to the exponential number of classical ground states.
+This behaviour shows an obvious discrepancy with stan-
+dard MPS methods; normally, fully periodic system sizes
+are computationally harder to access.
+Here, we stud-
+ied a model where the exponential number of low-lying
+states (ground states for h = 0) deters the convergence of
+the algorithm and also significantly increases the lattice
+size where the “thermodynamic limit” has been reached.
+QTPM belongs to this class of models, coming from clas-
+sical glasses, where the number of classical ground states
+for PBCx or OBC would progressively constitute the
+model numerically inaccessible. Only for PBC, the ther-
+modynamic limit is evidently accessible.
+Another conclusion that can be drawn from Figs. 12,
+13, 14 and 15 concerns the nature of the phase transi-
+tion. The possession of data for only OBC and Fig. 15,
+in particular, kschangewould possibly point towards a
+continuous quantum phase transition or an inconclusive
+statement. Note, however, that this conclusion would be
+inaccurate. Even if the “thermodynamic limit” has pos-
+sibly been reached, numerics for some system sizes alone
+does not provide enough evidence for the characterization
+of the phase transition for these cases; further knowledge
+of the ground states of the model is necessary, while also
+a general understanding of the behaviour (and number)
+of the low-lying states.
+The significance of these arguments is further evident
+from Figs. 16 and 17.
+For the case of PBCx, all de-
+generate ground states for the J ≫ h region are easily
+found from exact diagonalization calculations and classi-
+cally excited states are easily tractable too. However, the
+same is not true for OBC. The number of classically de-
+generate ground states increases exponentially and this
+is the reason why it would be pointless to show more
+ground states.
+Appendix B: Gap Scaling Analysis for PBC
+In this section we present a restricted and with limited
+accuracy analysis on the energy difference between the
+ground state and the first excited state. This analysis was
+conducted based on ED and MPS methods, which limits
+the validity of the conclusions that can be reached: it be-
+comes quickly obvious that MPS methods are not power-
+ful enough for the detection of the actual gap, especially
+in regions of the parameter space with high entanglement
+or with a high number of low-lying excited states, where
+MPS often converge to excited states above the lowest-
+lying ones. However, the analysis below still provides an
+
+13
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+(a)
+□□□□□□□□□□□□□□
+□
+□□□□□□□□□□□□□□□□
+△ △ △ △ △ △
+△
+△
+△ △ △ △ △ △ △ △
+□□□□□□□□□□□□□□
+□
+□□□□□□□□□□□□□□□□
+△ △ △ △ △ △
+△
+△
+△ △ △ △ △ △ △ △
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b)
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△ △ △ △ △ △
+△
+△ △ △ △ △ △ △ △ △
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△ △ △ △ △ △
+△
+△ △ △ △ △ △ △ △ △
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8 (c)
+MPS
+●
+●
+●
+●
+●
+ED
+□
+△
+FIG. 12: (a) Normalised energy, (b) Mx and (c) Mzzz of ground states from MPS and ED for square lattices with
+PBCx. Data from ED are denoted as empty squares and empty triangles.
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+(a)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0 (c)
+MPS
+●
+●
+●
+●
+●
+●
+FIG. 13: Same as Fig. 12 but for systems of size N = 3 × L.
+indication of the behaviour of the gap with system size
+when comparing systems with different symmetries.
+This limited accuracy when measuring the first excited
+state energy is evident in Fig. 18(a). Data is calculated
+for the J = h = 1.0 point. The gap seems to decrease
+with increasing the system size, but at the same time,
+the power of MPS to detect it is significantly reduced.
+The situation seems clearer for Fig. 18(b). However, it is
+equally problematic despite the monotonically decreasing
+gap. The only significance of these results are an upper
+bounds of the actual gap.
+For the first case, the gap
+seems to decrease algebraically to zero, while for the case
+of multiple classical ground states, it seems to decrease
+exponentially.
+This underlines the different behaviour
+depending on the existence or not of multiple classical
+ground states.
+The same problems are encountered close to the phase
+transition from the MPS results in Fig. 19(a). Both plots
+are normalised by the maximum value of the gap encoun-
+tered in the region of J values studied. For 0.0 < J < 2.0,
+the gap appears always to be maximum at J = 2.0 for
+Fig. 19(a) and at J = 0 for Fig. 19(b). In Fig. 19(b), for
+J > h = 1.0 the gap approaches zero, as expected from
+the existence of degenerate ground states.
+
+14
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+-2.5
+-2.0
+-1.5
+-1.0
+(a)
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△ △ △ △ △ △ △
+△
+△
+△ △ △ △ △ △ △
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△ △ △ △ △ △ △
+△
+△
+△ △ △ △ △ △ △
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b)
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8 (c)
+MPS
+●
+●
+●
+●
+●
+ED
+□
+△
+FIG. 14: (a)Normalised energy, (b) Mx and (c) Mzzz of ground states from MPS and ED for square lattices with
+OBC. Data from ED are denoted as empty squares and empty triangles.
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+-2.0
+-1.8
+-1.6
+-1.4
+-1.2
+-1.0
+(a)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6 (c)
+MPS
+●
+●
+●
+●
+●
+●
+FIG. 15: Same as Fig. 14 but for systems of size N = 3 × L.
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
+◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+○○○○○○○○○○○○○○○○○○○○○○○○○���○○○○○○○
+◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻◻
+0.2
+0.4
+0.6
+0.8
+1.0
+-18
+-16
+-14
+-12
+-10
+(a)
+Energy Levels
+□
+1
+▯ 2-7
+△
+8
+▽ 9-12
+◇ 13-14
+○
+15
+◻
+16
+17
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽
+◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+-26
+-24
+-22
+-20
+-18
+-16
+-14
+-12
+(b)
+Energy Levels
+□
+1
+▯ 2-4
+△ 5-7
+▽ 8
+◇ 9
+FIG. 16: The unnormalised state diagrams for a (a) 4 × 4 and a (b) 3 × 6 lattice with PBCx.
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△
+0.5
+1.0
+1.5
+-30
+-25
+-20
+-15
+-10
+(a)
+Energy Levels
+□
+1
+▯ 2-5
+△
+6
+□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
+▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯▯
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+the J = h = 1/0 point. Both ED and MPS methods are used (where appropriate) for the calculation of the given
+gaps. Square and rectangular sizes are equally used.
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+MPS without (a) and with (b) symmetries for the QTPM. For (a) gmax ≈ 3.43 − 3.50 and for (b) gmax = 2.0.
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+ExplainableFold: Understanding AlphaFold Prediction with Explainable AI
+Juntao Tan 1 Yongfeng Zhang 1
+Abstract
+This paper presents ExplainableFold, an explain-
+able AI framework for protein structure prediction.
+Despite the success of AI-based methods such as
+AlphaFold in this field, the underlying reasons for
+their predictions remain unclear due to the black-
+box nature of deep learning models. To address
+this, we propose a counterfactual learning frame-
+work inspired by biological principles to generate
+counterfactual explanations for protein structure
+prediction, enabling a dry-lab experimentation
+approach. Our experimental results demonstrate
+the ability of ExplainableFold to generate high-
+quality explanations for AlphaFold’s predictions,
+providing near-experimental understanding of the
+effects of amino acids on 3D protein structure.
+This framework has the potential to facilitate a
+deeper understanding of protein structures.
+1. Introduction
+The protein folding problem studies how a protein’s amino
+acid sequence determines its tertiary structure. It is crucial
+to biochemical research because a protein’s structure influ-
+ences its interaction with other molecules and thus its func-
+tion. Current machine learning models have gain increasing
+success on 3D structure prediction (AlQuraishi, 2021; Tor-
+risi et al., 2020). Among them, AlphaFold (Jumper et al.,
+2021) provides near-experimental accuracy on structure pre-
+diction, which is considered as an importance achievement
+in recent years. Nevertheless, one of the important problems
+of AlphaFold, as well as other deep models, is that they can-
+not provide explanations for their predictions. Essentially,
+the why question still remains largely unsolved: the model
+gives limited understanding of why the proteins are folded
+into the structures they are, which hinders the model’s ability
+to provide deeper insights for human scientists.
+However, it is crucial to understand the mechanism of pro-
+tein folding from both AI and scientific perspectives. From
+1Department of Computer Science, Rutgers University. Corre-
+spondence to: Juntao Tan <[email protected]>, Yongfeng
+Zhang <[email protected]>.
+Copyright by the author(s).
+the AI perspective, explainability has long been an important
+consideration. State-of-the-art protein structure prediction
+models leverage complex deep and large neural networks,
+which makes it difficult to explain their predictions or debug
+the trained model for further improvement. From the sci-
+entific perspective, scientists’ eager to conquest knowledge
+is not satisfied with just knowing the prediction results, but
+also knowing the why behind the results (Li et al., 2022).
+In particular, structural biologists not only care about the
+structure of proteins, but also need to know the underlying
+relationship between protein primary sequences and tertiary
+structures (Dill et al., 2008; Dill & MacCallum, 2012).
+It has been established that certain amino acids play signifi-
+cant roles in the protein folding process. For instance, one
+single disorder in the HBB gene can significantly change
+the structure of hemoglobin, the protein that carries oxygen
+in blood, causing the sickle-cell anaemia (Kato et al., 2018).
+Knowing the relationship between amino acids and protein
+structure helps scientists to produce synthetic proteins with
+precisely controlled structures (Tan et al., 2020) or mod-
+ify existing proteins with desired properties (Szymkowski,
+2005; Mart´ınez et al., 2020; Ackers & Smith, 1985), which
+are essential for advanced research directions such as drug
+design. Additionally, in certain research tasks, scientists
+would like to modify the amino acids without drastically
+changing the protein structure, which requires the knowl-
+edge of “safe” residue substitutions (Bordo & Argos, 1991),
+i.e., the knowledge of which amino acids are not the most
+crucial ones in the folding process.
+While currently there are few Explainable AI-based methods
+to study the mechanism of protein folding, many previous
+biochemical researches have been conducted for this pur-
+pose. One of the best known methods is via site-directed
+mutagenesis (Hutchison et al., 1978; Carter, 1986; Sarkar &
+Sommer, 1990). To test the role of certain residues in pro-
+tein folding, biologists either delete them from the sequence
+(i.e., site-directed deletion) (Arpino et al., 2014; Gl¨uck &
+Wool, 2002; Flores-Ram´ırez et al., 2007; Dominy & An-
+drews, 2003) or replace them with other types of amino acids
+(i.e., site-directed substitution) (Flores-Ram´ırez et al., 2007;
+Bordo & Argos, 1991; Betts & Russell, 2003) and then mea-
+sure their influences on the 3D structure. However, these
+approaches suffer from several limitations: 1) So far, modi-
+fication of such residues can be limited by methods for their
+arXiv:2301.11765v1  [cs.AI]  27 Jan 2023
+
+ExplainableFold: Understanding AlphaFold Prediction with Explainable AI
+Figure 1. (a) Some amino acids play crucial roles in protein folding. By removing the effects of a relative small set of these residues, the
+predicted structure will be different. (b) Some other residues are less important. Despite deleting a large set of these residues, the protein
+still folds into a similar structure. (c) Some substitutions are radical to the protein structure and even a small number of such substitutions
+can drastically change the structure. (d) Some other substitutions are conservative and have small effect on the protein structure.
+installation and the chemistry available for reaction, and
+the modification of some residues can be very challenging
+(Spicer & Davis, 2014), 2) Wet-lab methods for determin-
+ing protein structures are very difficult and time-consuming
+(Ilari & Savino, 2008), and 3) The wet lab experiments de-
+scribed above have many prerequisites and obstacles, and
+may not be completely safe for many researchers.
+Recently, AI-based dry-lab methods such as AlphaFold pro-
+vide near-experimental protein structure predictions (Jumper
+et al., 2021), which sheds light on the possibility to gen-
+erate insightful understandings of protein folding by ex-
+plaining AlphaFold’s inference process. Such (Explainable)
+AI-driven dry-lab approach will largely overcome the afore-
+mentioned limitations and can be very helpful for human
+scientists. Furtunately, we observe that the process of testing
+the effects of residues on protein structure by site-directed
+mutagenesis is fundamentally similar to counterfactual rea-
+soning, a commonly used technique for generating explana-
+tions for machine learning models (Tan et al., 2021; 2022;
+Goyal et al., 2019; Tolkachev et al., 2022; Cito et al., 2022).
+Intuitively, counterfactual reasoning perturbs parts of the
+input data, such as interaction records of a user (Tan et al.,
+2021), nodes or edges of a graph (Tan et al., 2022), pixels
+of an image (Goyal et al., 2019), or words of a sentence
+(Tolkachev et al., 2022), and then observes how the model
+output changes accordingly.
+In this paper, we propose ExplainableFold, a counterfac-
+tual explanation framework that generates explanations for
+protein structure prediction models. ExplainableFold mim-
+ics existing biochemical experiments by manipulating the
+amino acids in a protein sequence to alter the protein struc-
+ture through carefully designed optimization objectives. It
+provides insights about which residue(s) of a sequence is cru-
+cial (or indecisive) to the protein’s structure and how certain
+changes on the residue(s) will change the structure, which
+helps to understand, e.g., what are the most impactful amino
+acids on the structure, and what are the most radical (or safe)
+substitutions when modifying a protein structure. An exam-
+ple of applying our framework on CASP14 target protein
+T1030 is shown in Figure 1, which shows that deletion or
+substitution of a small number of residues can result in sig-
+nificant changes to the protein structure, while some other
+deletions or substitutions may have very small effects. We
+evaluate the framework based on both standard explainable
+AI metrics and biochemical heuristics. Experiments show
+that the proposed method produces more faithful explana-
+tions compared to previous statistical baselines. Meanwhile,
+the predicted relationship between protein amino acids and
+their structures are highly positively correlated with wet-lab
+biochemical experimental results.
+2. Related Work
+The essential idea of the proposed method is to integrate
+counterfactual reasoning and site-directed mutagenesis anal-
+ysis in a unified machine learning framework. We discuss
+the two research directions in this section.
+2.1. Residue Effect Analysis by Site-directed Mutagenesis
+Many studies in molecular biology, such as those involv-
+ing genes and proteins, rely on the use of human-induced
+mutation analysis (Stenson et al., 2017). Early metagenesis
+methods were not site-specific, resulting in entirely random
+and indiscriminate mutations (Egli et al., 2006). In 1978,
+
+a) Most Intolerant Deletion:
+c) Most Radical Substitution:
+TM-score: 0.47 Changed # of Residues: 80 / 273 (29%)
+b) Most Tolerant Deletion:
+d) Most Conservative Substitution:
+TM-score: 0.66 Changed # of Residues: 142 / 273 (52%)
+TM-score: 0.57 Changed # of Residues: 82 / 273 (30%)ExplainableFold: Understanding AlphaFold Prediction with Explainable AI
+Hutchison et al. proposed the first method that can modify
+biological sequences at desired positions with specific inten-
+tions, which is known as site-directed mutagenesis. Later,
+more precise and effective tools were constantly developed
+(Motohashi, 2015; Doering et al., 2018). Site-directed muta-
+genesis is widely utilized in biomedical research for various
+applications. In this section, we focus on the use of site-
+directed mutagenesis to study the impact of amino acid
+mutations on protein structures (Studer et al., 2013).
+Two common approaches to site-directed mutagenesis are
+amino acid deletion and substitution (Choi & Chan, 2015).
+The deletion approach deletes certain residues from the
+sequence and observes the effects on the structure. For in-
+stance, Gl¨uck & Wool (2002) identified the amino acids that
+are essential to the action of the ribotoxin restrictocin by
+systematic deletion of its amino acids. Flores-Ram´ırez et al.
+(2007) proposed a random deletion approach to measure the
+amino acids’ effects on the longest loop of GFP. Arpino et al.
+(2014) conducted experiments to measure the the protein’s
+tolerance to random single amino acid deletion. The substi-
+tution approach, on the other hand, replaces one or multiple
+residues with other types of amino acids to test their influ-
+ence. For example, Clemmons (2001) substituted a small
+domain of the IGF-binding protein to measure weather spe-
+cific domains account for specific structures and functions.
+Zhang et al. (2018) mutated a specific amino acid on the
+surface of a Pin1 sub-region, known as the WW domain, and
+observed significant structural change on the protein struc-
+ture. Guo et al. (2004) randomly replaced amino acids to
+test proteins’ tolerance to substitution at different positions.
+When developing our framework, we draw insights from the
+aforementioned biochemical methods, which were proven
+effective in wet-lab experiments. We aim to translate the
+wet-lab methods for understanding protein structures into a
+dry-lab AI-driven approach. We note that there have been ex-
+isting attempts which built models to understand the relation-
+ship between protein structures and their residues (Masso
+& Vaisman, 2008; Masso et al., 2006; Sotomayor-Vivas
+et al., 2022). However, they were mostly based on statistical
+analysis on wet-lab experimentation data. Our method is
+the first AI-driven machine learning method developed for
+understading protein structure predictions.
+2.2. Counterfactual Reasoning for Explainable AI
+Counterfactual explanation is a type of model-agnostic ex-
+plainable AI method that tries to understand the underlying
+mechanism of a model’s behavior by perturbing its input.
+The basic idea is to investigate the difference of the model’s
+prediction before and after changing the input data in spe-
+cific ways (Wachter et al., 2017). Since counterfactual ex-
+planation is well-suited for explaining black-box models,
+it has been an important explainable AI method and has
+been employed in various applications, including but not
+limited to recommender systems (Tan et al., 2021; Ghaz-
+imatin et al., 2020), computer vision (Goyal et al., 2019;
+Vermeire et al., 2022), natural language processing (Yang
+et al., 2020; Lampridis et al., 2020; Tolkachev et al., 2022),
+molecular analysis (Tan et al., 2022; Ying et al., 2019; Lin
+et al., 2021), and software engineering (Cito et al., 2022).
+In this paper, we explore counterfactual explanation to ex-
+plain the amino acids’ effects on protein folding. How-
+ever, counterfactual explanation for protein folding exhibits
+unique challenges compared with previous tasks. For exam-
+ple: 1) most of the aforementioned applications are classifi-
+cation tasks, for which the explanation goal is very clear –
+looking for a minimal change that alter the predicted label.
+However, protein structure prediction is a generation task
+in a continuous space, which requires careful design of the
+counterfactual reasoning objective; 2) protein structure pre-
+diction models such as AlphaFold take complicated input
+besides the primary sequence, e.g., the MSA and templates;
+3) it is easier to evaluate the explanations for the classifi-
+cation tasks, nevertheless, as a new explainable AI task,
+protein structure prediction poses unique challenges on the
+evaluation of explanation. We will show how to overcome
+these challenges in the following parts of the paper.
+3. Problem Formulation
+In this section, we first provide formulation of the Explain-
+ableFold problem. Since a protein tertiary (3D) structure is
+uniquely determined by its primary structure (amino acid
+sequences) (Dill et al., 2008; Wiltgen, 2009), according to
+the key idea of counterfactual explanation, we define the ex-
+planation as identifying the most crucial residues that cause
+the proteins to fold into the structures they are.
+Suppose a protein consists of a chain of l residues, where
+the i-th residue is encoded as a 21-dimensional one-hot col-
+umn vector ri. The “1” element in ri indicates the type
+of the residue, which can be one of the 20 common amino
+acids or an additional dimension for unknown residue. By
+concatenating all the residue vectors, a protein P is de-
+noted as P = [r1, r2, · · · , rl], where P ∈ {0, 1}21×l is
+called the protein embedding matrix. Many state-of-the-art
+protein structure prediction models predict the 3D struc-
+ture not only based on the residue sequence, but also uti-
+lize supplementary evolutionary information (Senior et al.,
+2020; Jumper et al., 2021) by extracting Multiple Sequence
+Alignment (MSA) (Edgar & Batzoglou, 2006) from pro-
+tein databases. Suppose m proteins are retrieved from the
+evolutionary database based on their similarity with protein
+P, the constructed MSAs can be encoded as another ma-
+trix M(P ) ∈ {0, 1}m×21×l. A protein structure prediction
+model fθ predicts the protein 3D structure S based on the
+
+ExplainableFold: Understanding AlphaFold Prediction with Explainable AI
+residue sequence and MSA embeddings:
+S = fθ
+�
+P , M(P )
+�
+(1)
+where M(P ) can be omitted if the model only takes the
+residue sequence information. Though a structure predic-
+tion model may predict the positions of all atoms, in many
+structural biology research, only the backbone of residues
+are used for comparing the similarities of protein structures
+(Zhang & Skolnick, 2004; 2005; Xu & Zhang, 2010; Zemla,
+2003). Therefore, we adopt the same idea in this paper,
+where S ∈ R3×l only contains the predicted (x, y, z)T co-
+ordinates of the α-carbon atom of each amino acid residue.
+The explanation is expected to be a subset of residues ex-
+tracted from the protein sequence, expressed as E. The
+objective of the ExplainableFold problem is to find the mini-
+mum set of E that contains the most influential information
+for the prediction of the 3D structure.
+4. The ExplainableFold Framework
+In biochemistry, the most common methods for studying
+the effects of amino acids on protein structure fall into two
+categories: amino acid deletion and substitution (Choi &
+Chan, 2015). We design the ExplainableFold framework
+from both of the two perspectives, and we introduce them
+separately in the following.
+4.1. The Residue Deletion Approach
+The deletion approach simulates the biochemical studies
+that detect essential residues for a protein by deleting one or
+more residues and measuring the protein’s tolerance to such
+deletion (Arpino et al., 2014; Gl¨uck & Wool, 2002; Flores-
+Ram´ırez et al., 2007). The key idea is to apply a residue
+mask that removes the effect of certain residues from the se-
+quence and then measure the change of the protein structure.
+From the counterfactual machine learning perspective, this
+can be considered from two complementary views (Guidotti
+et al., 2019; Tan et al., 2022): 1) Identify the minimal dele-
+tion that will alter the predicted structure and the deleted
+residues will be the necessary explanation; 2) Identify the
+maximal deletion that still keeps the predicted structure and
+the undeleted residues will be the sufficient explanation. We
+design the counterfactual explanation algorithm from these
+two views accordingly.
+4.1.1. NECESSARY EXPLANATION (MOST INTOLERANT
+DELETION)
+From the necessary perspective, we aim to find the minimal
+set of residues in the original sequence which, if deleted,
+will change the AI model’s (such as AlphaFold’s) predicted
+structure. The deleted residues thus contain the most neces-
+sary information for the model’s original prediction.
+We can express the perturbation on the original sequence
+as a multi-hot vector ∆ = {0, 1}1×l, where δi = 1 means
+that the i-th residue will be deleted and δi = 0 means it will
+be kept. Then the counterfactual protein embedding matrix
+P ∆ can be expressed as:
+P ∆ = P ⊙ (1 − ∆) + U ⊙ ∆
+(2)
+where ⊙ is the element-wise product and U ∈ {0, 1}21×l
+denotes an “unknown” matrix of the same shape with P ,
+but with all elements being 0 except for the last row being 1
+(i.e., unknown type amino acid). Thus, for δi = 0, the i-th
+residue in the original sequence will be preserved, while for
+δi = 1, the i-th residue will be treated as an unknown amino
+acid without any specific chemical property.
+Motivated by the Occam’s Razor Principle (Blumer et al.,
+1987), we aim to find simple and effective explanations. The
+simpleness can be characterized by the number of residues
+that need to be deleted, which should be as few as possi-
+ble, while effectiveness means that the predicted protein
+structure should be different before and after applying the
+deletions. We can use zero-norm ∥∆∥0 to represent the
+number of deletions (for simpleness), while using the TM-
+score between the original and the new protein structures
+TM(S, S∗) to represent the degree of change on the struc-
+ture (for effectiveness). TM-score is a standard measure-
+ment for comparing aligned protein structures, where TM-
+score > 0.5 suggests the same folding and TM-score ≤ 0.5
+suggests different foldings (Zhang & Skolnick, 2004; Xu &
+Zhang, 2010). The counterfactual expxlanation algorithm
+then learns the optimal explanation by solving the following
+constrained optimization problem:
+minimize ∥∆∥0
+s.t. TM(S, S∗) ≤ 0.5, ∆ = {0, 1}1×l
+where S∗ = fθ(P ∆, M(P ∆))
+(3)
+where the objective ∥∆∥0 aims to find the minimal dele-
+tion, while the constraint guarantees the effectiveness of the
+deletion, i.e., the deletion will change the predicted protein
+structure to be different from before.
+Due to the exponential combinations of sub-sequences for
+a given sequence, it is impractical to search for an optimal
+solution on the discrete space. To solve the problem, we use
+a continuous relaxation approach to solve the optimization
+problem by relaxing the multi-hot vector ∆ to a real-valued
+vector. We also relax the hard constraint in Eq.(3) and
+combine them into a single trainable loss function:
+L1 = max
+�
+0, TM(S, S∗) − 0.5 + α
+�
++ λ1∥σ(∆)∥1
+s.t. ∆ ∈ R1×l, where S∗ = fθ(P σ(∆), M(P σ(∆)))
+(4)
+where the sigmoid function σ(·) is applied so that σ(∆) ∈
+(0, 1)1×l approximates the probability distribution between
+
+ExplainableFold: Understanding AlphaFold Prediction with Explainable AI
+the original residues and unknown residues, and α is the
+margin value of the hinge loss whose default value is 0.1.
+This relaxation approach has been justified in several previ-
+ous studies which also learn explanation on discrete inputs
+(Ying et al., 2019; Goyal et al., 2019; Tan et al., 2022). The
+1-norm regularizer assures the learned perturbation σ(∆)
+to be sparse (Candes & Tao, 2005), i.e., the learned expla-
+nation only contains a small set of residues. λ1 is a hyper-
+parameter that controls the trade-off between the complexity
+and strength of the generated explanation. Eq.(4) can be
+easily optimized with gradient descent. After optimization,
+we convert σ(∆) to a binary vector with the threshold 0.5.
+4.1.2. SUFFICIENT EXPLANATION (MOST TOLERANT
+DELETION)
+Symmetrically, from the sufficiency perspective, we aim to
+find the maximal set of residues in the orignal sequence
+which, if deleted, will not change the AI model’s predicted
+structure. The undeleted residues thus contain the most
+sufficient information for the model’s original prediction.
+This can be formulated as a similar but reversed optimization
+process as Eq.(3), which looks for the maximal perturba-
+tion ∆ while keeping the same folding (TM-score > 0.5).
+Therefore, the optimization problem is formulated as:
+maximize ∥∆∥0
+s.t. TM(S, S∗) > 0.5, ∆ = {0, 1}1×l
+where S∗ = fθ
+�
+P ∆, M(P ∆)
+�
+(5)
+Similarly, we relax Eq.(5) to a differentiable loss function:
+L2 = max
+�
+0, 0.5 − TM(S, S∗) + α
+�
+− λ2∥σ(∆)∥1
+s.t. ∆ ∈ R1×l, where S∗ = fθ(P σ(∆), M(P σ(∆)))
+(6)
+Contrary to the necessary explanation, the sufficient explana-
+tion consists the undeleted residues. Hence, after optimiza-
+tion, we filter the residues according to
+�
+1 − σ(∆)
+�
+> 0.5
+and include them into the sufficient explanation.
+4.2. The Residue Substitution Approach
+Another popular approach in biochemistry, site-directed sub-
+stitution, studies the influence of the amino acids on protein
+folding by replacing certain residues with other known-type
+residues (Flores-Ram´ırez et al., 2007; Bordo & Argos, 1991;
+Betts & Russell, 2003). Different replacements may have
+different effects on protein structures, and they can be clas-
+sified into two types: conservative substitution and radical
+substitution (Zhang, 2000; Dagan et al., 2002). A conser-
+vative substitution is considered as a “safe” substitution for
+which the amino acid replacement usually have no or minor
+effects on the protein structure. A radical substitution is
+considered “unsafe,” which usually causes significant struc-
+tual changes. Based on the above concepts, we design the
+substitution approach from these two different perspectives.
+4.2.1. RADICAL SUBSTITUTION EXPLANATION
+From the radical substitution perspective, we aim to find the
+mimimal set of residue replacements which will lead to a
+different folding, and then the learned substitutions are the
+most radical substitutions for the protein.
+For a target protein with binary embedding matrix P , we
+learn a counterfactual binary protein embedding P ′, which
+has the same shape as the original embedding matrix. The
+number of substitutions is represented by ∥P −P ′∥0, which
+is the 0-norm of the difference between the two matrices.
+To find the minimal residue substitution that changes the
+original folding, the optimization problem is defined as:
+minimize ||P − P ′||0
+s.t. TM(S, S′) ≤ 0.5, P ′ ∈ {0, 1}21×l
+where S′ = fθ
+�
+P ′, M(P ′)
+�
+(7)
+Due to the exponential search space of the substitutions,
+we use the similar continuous relaxation method as in the
+deletion approach. First, we relax the binary counterfactual
+embedding matrix P ′ to continuous space. We also relax
+the hard constraint in Eq.(7) and define the differentiable
+loss function as:
+L3 = max
+�
+0, TM(S, S′) − 0.5 + α
+�
++ λ3∥P − σ(P ′)∥1
+s.t. P ′ ∈ R21×l, where S′ = fθ
+�
+P ′, M(P ′)
+�
+(8)
+After optimization, we convert the learned continuous ma-
+trix σ(P ′) into binary by setting the maximum value of each
+column as 1 and others as 0. Then, the changed residues
+between P and P ′ are the radical substitution explanations.
+4.2.2. CONSERVATIVE SUBSTITUTION EXPLANATION
+From the conservative substitution perspective, we aim to
+find the maximal set of residue replacements which how-
+ever lead to the same folding, and then the learned substitu-
+ions are the most conservative substitutions for the protein.
+On the contrary to Eq.(7), we formulate an inverse optimiza-
+tion problem as:
+maximize ||P − P ′||0
+s.t. TM(S, S′) > 0.5, P ′ ∈ {0, 1}21×l
+where S′ = fθ
+�
+P ′, M(P ′)
+�
+(9)
+With the same relaxation process, the loss function is:
+L4 = max
+�
+0, 0.5 − TM(S, S′) + α
+�
+− λ4∥P − σ(P ′)∥1
+s.t. P ′ ∈ R21×l, where S′ = fθ
+�
+P ′, M(P ′)
+�
+(10)
+
+ExplainableFold: Understanding AlphaFold Prediction with Explainable AI
+After learning σ(P ′) and getting the binary matrix, again,
+the changed residues between P and P ′ are the conservative
+substitution explanations.
+4.3. Phased MSA Re-alignment
+It is impractical to re-compute MSAs in each training step.
+Therefore, we propose a phased MSA re-alignment strategy.
+When learning the explanations, we fix the generated MSAs
+and only learn the changes on the sequence embedding for
+t training steps (t = 100 by default), which is one pahse.
+Then, we re-align the MSAs and start another training phase.
+5. Experiments
+We first introduce the datasets and implementation details.
+Then, we introduce the evaluation results of the deletion
+approach and substitution approach, respectively.
+5.1. Datasets
+We test the ExplainableFold framework on the 14th Criti-
+cal Assessment of protein Structure Prediction (CASP-14)
+dataset1 (Moult et al.). CASP consecutively establishes
+protein data with detailed structural information as a stan-
+dard evaluation benchmark for protein structure prediction.
+Following Jumper et al. (2021), we remove all sequences
+for which fewer than 80 amino acids had the alpha carbon
+resolved and remove duplicated sequences. After filtering,
+55 protein sequences are selected.
+5.2. Implementation Details
+Though the ExplainableFold framework can be applied on
+any model that predicts protein 3D structures, we choose
+Alphafold2 (Jumper et al., 2021), the state-of-the-art model,
+as the base model in the experiments. More specifically, we
+use the OpenFold (Ahdritz et al., 2022) implementation and
+load the official pre-trained AlphaFold parameters2.
+When learning the explanations, the pre-trained parame-
+ters of AlphaFold are fixed, and only the perturbation vec-
+tor on the input (∆ for the deletion approach and P ′ for
+the substitution approach) will be optimized. However, it
+still requires computing the gradient through the entire Al-
+phafold network, as a result, the learning process requires
+extensive memory consumption. To solve the problem, we
+follow exactly the same training procedure as introduced in
+the original AlphaFold paper (Jumper et al., 2021). More
+specifically, we use the gradient checkpointing technique to
+reduce the memory usage (Chen et al., 2016). Meanwhile,
+if a protein has more than 384 residues, we cut it to differ-
+ent chunks for each consecutive 384 residues, and generate
+1https://predictioncenter.org/casp14/
+2https://github.com/deepmind/alphafold
+explanations for each of them.
+We employ the same training strategy for both deletion and
+substitution explanation methods: for each training phase
+between MSA re-alignments, we optimize the purturbation
+vector for 100 steps with Adam optimizer (Kingma & Ba,
+2014) and learning rate 0.1. After each training loop, we re-
+align the MSAs with the AlphaFold HHblits/JackHMMER
+pipeline. We repeat the training and alignment process for
+3 phases when generating explanations for each protein.
+All experiments are conducted on NVIDIA A5000 GPUs.
+The entire training process (including all 3 phases) for one
+protein takes approximately 5 hours. We set α = 0.1 and
+λ = 0.01 in Equations (4)(6)(8)(10). To realize an incre-
+mental substitution process, we initialize the counterfactual
+protein embedding matrix as a duplication of the original
+protein embedding matrix, i.e., we initialize ∆ with all 0’s
+and initialize σ(P ′) equal to the original P . Thus, the
+optimal explanations are gradually learned.
+5.3. Evaluation of the Deletion Approach
+Counterfactual explanations can be evaluated by their com-
+plexity, sufficiency and necessity (Glymour et al., 2016; Tan
+et al., 2022). First, according to the Occam’s Razor Princi-
+ple (Blumer et al., 1987), we hope an explanation can be as
+simple as possible so that it is cognitively easy to understand
+for humans. This can be evaluated by the complexity of the
+explanation, i.e., the percentage of residues that are included
+in the explanation:
+Complexity = |E|/l
+(11)
+where l is the length of the protein.
+Sufficiency and necessity measure how crucial the generated
+explanations are for the protein structure. We follow the def-
+inition in causal inference theory (Glymour et al., 2016) and
+existing explainable AI research (Tan et al., 2022) and mea-
+sure the explanations with two causal metrics: Probability
+of Necessity (PN) and Probability of Sufficiency (PS).
+PN measures the necessity of the explanation. A set of
+explanation residues is considered a necessary explanation
+if, by removing their effects from the protein sequence,
+the predicted structure of the protein will have a different
+folding (TM-score < 0.5). Suppose there are N proteins in
+the testing data, then PN is calculated as:
+PN =
+�N
+k=1 PNk
+N
+, PNk =
+�
+1, if TM(Sk, S∗
+k) ≤ 0.5
+0, else
+(12)
+Intuitively, PN measures the percentage of proteins whose
+explanation residues, if removed, will change the protein
+structure, and thus their explanation residues are necessary.
+PS measures the sufficiency of the explanation. A set of
+explanation residues is considered a sufficient explanation if,
+
+ExplainableFold: Understanding AlphaFold Prediction with Explainable AI
+Table 1. PN Evaluation. Deletion∗ is the necessity optimization.
+Ave Exp.
+Ave Comp.
+Ave TM-score
+PN
+Size (|E|) ↓
+(|E|/l) ↓
+TM(S, S∗) ↓
+score↑
+Random
+77.31
+0.30
+0.83
+0.07
+Evo (Masso et al., 2006)
+88.42
+0.33
+0.77
+0.16
+Deletion (necessity)∗
+74.54
+0.29
+0.48
+0.44
+Table 2. PS Evaluation. Deletion∗ is the sufficiency optimization.
+Ave Exp.
+Ave Comp.
+Ave TM-score
+PS
+Size (|E|) ↓
+(|E|/l) ↓
+TM(S, S∗) ↑
+score↑
+Random
+102.9
+0.40
+0.38
+0.31
+Evo (Masso et al., 2006)
+104.95
+0.42
+0.41
+0.40
+Deletion (sufficiency)∗
+95.20
+0.37
+0.61
+0.62
+by removing all of the other residues and only keeping the
+explanation residues, the protein still has the same folding.
+Similarty, PS is calculated as:
+PS =
+�N
+k=1 PSk
+N
+, PSk =
+�
+1, if TM(Sk, S∗
+k) > 0.5
+0, else
+(13)
+Intuitively, PS measures the percentage of proteins whose
+explanation residues alone can keep the protein structure
+unchanged, and thus their explanation resides are sufficient.
+5.3.1. BASELINES
+We compare the model performance with a common com-
+putational biology baseline (Masso et al., 2006), which
+analyzes a protein’s tolerance to the change on each residue
+by extracting the data from evolutionary database. More
+specifically, proteins are not tolerate to the mutations at evo-
+lutionary conserved positions. However, they are capable
+of withstanding certain mutations at other positions. When
+implementing the baseline, we refer to a protein’s MSAs
+and select the evolutionary conserved residues as the expla-
+nation. This is illustrated in Figure 2 using protein CASP14
+target T1030 as an example, where for each residue position,
+we count the number of MSAs that conserve the residue
+at this position, and show the top 30% and 40% conserved
+residues. We also randomly select residues as explanation
+and compute PN and PS scores as another baseline to mea-
+sure the general difficulty of the evaluation task, and more
+details are provided in the following subsection.
+5.3.2. RESULTS
+The results of PN and PS evaluation are reported in Table
+1 and Table 2, respectively. The explanation complexity
+of our method (29% for necessary explanation and 37%
+for sufficient explanation) are automatically decided by our
+optimization process. However, the baselines do not have
+the ability to decide the optimal explanation complexity. For
+fair comparison, we set the complexities of the baselines
+to be slightly larger than our method (30% for necessary
+explanation and 40% for sufficient explanation). Therefore,
+the baselines will have a small advantage over our method
+(a) Top 30% conserved residues (b) Top 40% conserved residues
+Figure 2. Evolutionary conserved residues are considered more
+important for the protein structures (the residues marked red).
+because they are allowed to use more residues to achieve
+the necessity or sufficiency goals.
+For PN evaluation, the results of the random baseline shows
+that protein structures tend to be robust to residue deletions.
+For example, when randomly removing the effects of 30%
+residues, only 7% of the proteins fold into different struc-
+tures, which indicates that finding necessary explanations
+is a challenging problem. The evolutionary baseline is able
+to select more necessary residues with a PN score of 0.16,
+which is 128.6% better than random selection. Compared to
+them, our method shows much better performance: with a
+smaller number of residues, the generated explanations are
+able to cause 44% of the proteins fold into different struc-
+tures, outperforming the evolutionary baseline by 175%.
+For PS evaluation, the evolutionary baseline is not notice-
+ably better than randomly selecting residues. The reason
+may be that despite the proteins’ less tolerance to the evo-
+lutionary conserved residues, there is no guarantee that the
+evolutionary conserved residues alone contain sufficient in-
+formation to preserve the protein structure. In comparison,
+our method does generate more sufficient explanations, out-
+performing the evolutionary baseline by 55% according to
+the PS score with less complex explanations. Meanwhile,
+our TM score is > 50%, indicating that the protein structure
+is indeed preserved under our sufficient explanation.
+Additionally, we show the learning curve of the optimization
+for CASP14 target protein T1030 in Figure 3. For necessary
+optimization, the algorithm gradually deletes the protein
+residues until the TM-score is smaller than 0.4 (i.e., 0.5−α,
+see Eq.(4)). Then, the explanation complexity slightly drops
+back while keeping the TM-score at the same level. For
+sufficient optimization, the L1-loss drastically increases at
+the beginning, which suggests that the algorithm is trying
+to delete as many residues as possible while keeping the
+original folding structure unchanged. However, after re-
+computing MSAs, the TM-score becomes too low. Thus,
+the algorithm increases the number of preserved residues
+to keep TM-score larger than 0.6 (i.e., 0.5 + α, see Eq.(6)).
+Note that the TM-scores change sharply when re-computing
+MSAs at the end of each training loop. The more frequently
+
+175
+150
+MSAs Num
+125
+100
+75
+50
+25
+0
+20
+40
+60
+80
+0
+100
+120
+Residue Position175
+150
+iSAs Num
+125
+100
+75
+50
+25
+20
+40
+60
+80
+0
+100
+120
+Residue PositionExplainableFold: Understanding AlphaFold Prediction with Explainable AI
+(a) Necessary Optimization
+(b) Sufficient Optimization
+Figure 3. Learning Curves of the Deletion Approach
+we re-align MSAs, the smoother the optimization will be.
+5.4. Evaluation of the Substitution Approach
+The substitution approach identifies the most radical or con-
+servative amino acid substitutions, which are of particular
+interest in biochemical research (Zhang, 2000). Previously,
+it was impractical to conduct wet-lab experiments to investi-
+gate the relative “safety” of replacing specific residues with
+alternative amino acids due to their prohibitive cost (Bordo
+& Argos, 1991). Alternatively, scientists infer the exchange-
+ability of two types of amino acids either through the use of
+heuristics based on their physical or chemical properties or
+through the analysis of evolutionary data, such as:
+• Epstein’s distance(Epstein, 1967): Predict the impact of
+switching two amino acids based on their size and polarity.
+• Miyata’s distance (Miyata et al., 1979): Predict the impact
+based on their volume and polarity.
+• Evolutionary indicator (Bordo & Argos, 1991): Detect
+“safe” substitutions based on evolutionary data.
+Note that these indicators are rather suggestions than ground-
+truth. They provide general trends that are better than ran-
+dom selection but cannot be expected to be precise in every
+scenario (Bordo & Argos, 1991). These methods are not
+perfectly consistent with each other, but are linearly related.
+Therefore, we utilize the amino acid substitution data gener-
+ated by our method to caculate the pair-wise exchangeability
+between the amino acids, and test the correlation between
+our exchangeability with the above three existing exchange-
+ability indicators. The details of the pair-wise substitution
+statistics and the calculation of pair-wise exchangeability
+are provided in the Appendix.
+In Table 3, we report the correlation of our generated pair-
+wise exchangeability with the three aforementioned indica-
+tors by a non-parametric method: Pearson’s correlation r.
+Besides, the correlation among the three biochemical meth-
+ods themselves range from 0.438 to 0.578. Additionally,
+the correlation is also visualized in Figure 4, where darker
+color indicates higher correlation. For Pearson’s correlation,
+a value greater than 0 indicates a positive association, where
+r > 0.1, r > 0.3, r > 0.5 represents small, medium, and
+Table 3. Correlation between our method and each of the biochem-
+ical indicators. Metrics with “*” are originally distance metrics,
+for which we take the inverse to reprenst the exchangeability. The
+results are significant at p < 0.001 under two-tailed test.
+Epstein∗
+Miyata∗
+Evolution
+Radical
+0.388
+0.602
+0.382
+Conservative
+0.494
+0.796
+0.405
+(a) Corr. w/ Epstein’s distance
+(b) Corr. w/ Miyata’s distance
+Figure 4. The correlation between the exchangeability provided by
+our conservative optimization method and (a) Epstein’s distance
+as well as (b) Miyata’s distance.
+large correlations, accordingly (Cohen et al., 2009). Both
+Table 3 and Figure 4 show that our method has clear posi-
+tive correlations with all of the three biochemical methods,
+indicating that ExplainableFold can provide informative
+exchangeability signals (Yampolsky & Stoltzfus, 2005). Be-
+sides, the results generated by ExplainableFold may further
+improve when larger protein datasets are available or applied
+on even better base models in the future.
+6. Conclusions and Future Work
+In this paper, we propose ExplainableFold—an Explain-
+able AI framework that helps to understand the deep learn-
+ing based protein structure prediction models such as Al-
+phaFold. Technically, we develop a counterfactual explana-
+tion framework and implement the framework based on two
+approaches: the residue deletion approach and the residue
+substitution approach. Intuitively, ExplainableFold aims to
+find simple explanations that are effective enough to keep
+or change the protein’s folding structure. Experiments are
+conducted on CASP-14 protein datasets and results show
+that our approach outperforms the results from traditional
+biochemical methods. We believe Explainable AI is funda-
+mentally important for AI-driven scientific research because
+science not only pursues the answers for the “what” ques-
+tions but also (or even more) for the “why” questions. In the
+future, we will further improve our framework by consider-
+ing more protein modification methods beyond deletion and
+substitution. We will also generalize our framework to other
+scientific problems due to the flexibility of our framework.
+
+1.0
+L1 Loss
+TM-score
+0.8
+0.6
+0.4
+0.2
+0.0
+0
+50
+100
+150
+200
+250
+300
+Training Steps1.0
+L1 Loss
+TM-score
+0.8
+0.6
+0.4
+0.2
+0
+50
+100
+150
+200
+250
+300
+Training StepsARN
+YV
+A
+R
+N -
+D
+0.8
+C-
+Q:
+E
+G
+0.6
+H -
+L-
+K-
+0.4
+M -
+F -
+S -
+0.2
+W-
+Y
+V
+0.0ARN
+D
+C
+H
+M
+STWYV
+A
+R
+N -
+0.8
+D
+0.7
+Q
+E
+0.6
+G
+H -
+0.5
+I-
+L-
+K-
+0.4
+M -
+F
+0.3
+P
+0.2
+T-
+W -
+0.1
+Y-
+V
+0.0ExplainableFold: Understanding AlphaFold Prediction with Explainable AI
+References
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+Dziubinska, M. M., Zhang, S., Ojewole, A., Guney, M. E.,
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+ExplainableFold: Understanding AlphaFold Prediction with Explainable AI
+Appendix
+A. Statistical Analysis of Amino Acid Substitutions
+Table 4 shows the total count of each amino acid in the
+testing proteins. In Table 5, we show how many times a
+specific type of substitution happens in the generated ex-
+planations learned by the conservative substitution method.
+For instance, the substitution of A → R happens 19 times.
+The exchangeability of X → Y can be easily calculated by
+|X → Y |/|X| (Bordo & Argos, 1991; Masso et al., 2006).
+The same statistics for radical substitution is provided in
+Table 6. For radical substitution, the higher the number
+in Table 6, the lower the exchangeability, and thus the ex-
+changeability of X → Y is calculated as the reciprocal
+|X|/|X → Y | (Bordo & Argos, 1991; Masso et al., 2006).
+Table 4. Total number of each amino acid in testing data
+A
+R
+N
+D
+C
+Q
+E
+G
+H
+I
+#
+782
+581
+827
+776
+175
+520
+848
+853
+309
+904
+L
+K
+M
+F
+P
+S
+T
+W
+Y
+V
+#
+1122
+911
+273
+600
+506
+916
+746
+149
+595
+780
+Table 5. Structural Conservative Statistics
+A
+R
+N
+D
+C
+Q
+E
+G
+H
+I
+L
+K
+M
+F
+P
+S
+T
+W
+Y
+V
+A
+0
+19
+25
+16
+50
+18
+21
+78
+23
+26
+22
+22
+19
+14
+39
+28
+15
+42
+35
+8
+R
+8
+0
+15
+23
+23
+11
+15
+37
+12
+12
+19
+18
+9
+15
+23
+18
+11
+49
+18
+9
+N
+15
+33
+0
+32
+51
+19
+18
+56
+16
+19
+11
+19
+50
+21
+33
+21
+16
+42
+22
+14
+D
+7
+29
+22
+0
+57
+21
+25
+42
+11
+19
+23
+19
+16
+21
+44
+16
+15
+32
+16
+11
+C
+2
+1
+1
+2
+0
+7
+1
+9
+8
+4
+5
+5
+2
+2
+2
+4
+0
+8
+2
+5
+Q
+5
+12
+18
+12
+33
+0
+14
+42
+15
+18
+7
+15
+21
+7
+33
+5
+7
+32
+21
+18
+E
+14
+14
+14
+50
+47
+30
+0
+62
+15
+35
+19
+19
+18
+29
+32
+16
+11
+79
+19
+11
+G
+18
+11
+19
+19
+62
+18
+18
+0
+23
+26
+29
+9
+18
+25
+29
+22
+12
+46
+15
+9
+H
+0
+15
+4
+15
+11
+14
+9
+28
+0
+5
+9
+7
+5
+12
+9
+9
+2
+21
+9
+7
+I
+9
+16
+16
+14
+46
+16
+25
+58
+33
+0
+49
+19
+56
+35
+29
+14
+14
+54
+18
+32
+L
+5
+28
+23
+50
+57
+30
+25
+70
+21
+53
+0
+19
+47
+40
+40
+21
+19
+51
+22
+33
+K
+2
+44
+22
+56
+78
+22
+28
+51
+22
+35
+18
+0
+29
+19
+47
+7
+11
+49
+21
+15
+M
+4
+5
+5
+5
+9
+8
+8
+26
+5
+12
+28
+5
+0
+7
+9
+5
+4
+16
+7
+8
+F
+8
+19
+16
+25
+35
+18
+9
+36
+14
+21
+19
+7
+21
+0
+21
+9
+5
+46
+21
+2
+P
+8
+11
+5
+12
+16
+9
+14
+25
+9
+28
+9
+14
+28
+16
+0
+4
+14
+30
+16
+2
+S
+21
+26
+22
+33
+49
+14
+28
+53
+23
+30
+26
+21
+29
+22
+36
+0
+40
+65
+36
+19
+T
+9
+19
+21
+35
+33
+22
+21
+40
+9
+33
+42
+22
+30
+28
+29
+22
+0
+47
+25
+19
+W
+0
+2
+4
+4
+7
+1
+1
+12
+4
+7
+5
+0
+5
+7
+7
+4
+0
+0
+7
+4
+Y
+7
+9
+16
+23
+25
+18
+15
+36
+11
+9
+5
+7
+29
+26
+15
+16
+8
+37
+0
+9
+V
+8
+12
+7
+29
+44
+25
+9
+54
+25
+49
+14
+14
+43
+19
+11
+15
+11
+40
+18
+0
+Table 6. Structural Radical Statistics
+A
+R
+N
+D
+C
+Q
+E
+G
+H
+I
+L
+K
+M
+F
+P
+S
+T
+W
+Y
+V
+A
+0
+28
+22
+16
+39
+5
+19
+22
+30
+28
+25
+5
+25
+22
+33
+14
+14
+64
+16
+14
+R
+8
+0
+11
+33
+19
+5
+28
+22
+16
+25
+8
+2
+11
+36
+25
+5
+5
+33
+11
+11
+N
+11
+16
+0
+8
+47
+11
+22
+16
+14
+19
+25
+22
+19
+25
+25
+5
+8
+25
+14
+19
+D
+11
+11
+11
+0
+25
+14
+22
+16
+19
+25
+11
+16
+44
+16
+16
+11
+0
+22
+14
+25
+C
+2
+0
+0
+0
+0
+2
+2
+11
+2
+0
+8
+8
+2
+2
+5
+8
+5
+2
+0
+5
+Q
+2
+19
+5
+11
+25
+0
+22
+16
+14
+14
+14
+16
+14
+16
+25
+5
+5
+19
+8
+8
+E
+5
+11
+5
+19
+58
+5
+0
+22
+14
+19
+14
+11
+28
+36
+19
+5
+8
+39
+25
+19
+G
+2
+25
+2
+16
+56
+11
+14
+0
+8
+33
+28
+22
+53
+28
+14
+14
+5
+44
+22
+8
+H
+2
+5
+0
+0
+16
+14
+8
+11
+0
+11
+8
+8
+0
+0
+14
+2
+0
+5
+5
+14
+I
+25
+28
+22
+36
+33
+5
+22
+44
+5
+0
+2
+8
+16
+8
+30
+11
+11
+47
+5
+11
+L
+22
+28
+25
+30
+64
+28
+28
+72
+19
+25
+0
+47
+33
+11
+56
+28
+22
+36
+28
+19
+K
+14
+2
+8
+25
+81
+22
+19
+30
+11
+14
+8
+0
+16
+30
+33
+5
+19
+64
+19
+19
+M
+2
+0
+16
+11
+5
+2
+2
+25
+8
+8
+0
+8
+0
+2
+19
+2
+0
+14
+5
+5
+F
+16
+11
+11
+22
+28
+8
+28
+19
+11
+16
+11
+19
+14
+0
+39
+19
+2
+11
+8
+5
+P
+8
+11
+2
+19
+22
+5
+16
+11
+8
+2
+14
+16
+36
+14
+0
+2
+5
+36
+22
+8
+S
+22
+8
+5
+14
+44
+16
+22
+30
+22
+28
+28
+28
+25
+22
+25
+0
+16
+58
+16
+14
+T
+11
+14
+16
+22
+56
+8
+19
+42
+14
+5
+19
+8
+33
+22
+19
+14
+0
+58
+16
+11
+W
+2
+0
+0
+2
+2
+8
+2
+14
+2
+2
+0
+0
+2
+2
+5
+5
+2
+0
+2
+8
+Y
+25
+14
+2
+5
+28
+5
+8
+25
+16
+11
+5
+19
+25
+8
+19
+8
+8
+14
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+k-planar Placement and Packing
+of ∆-regular Caterpillars
+Carla Binucci1, Emilio Di Giacomo1, Michael Kaufmann2,
+Giuseppe Liotta1, and Alessandra Tappini1
+1Dipartimento di Ingegneria, Universit`a degli Studi di Perugia, via
+G. Duranti 93, 06125, Perugia, Italy. {carla.binucci,
+emilio.digiacomo, giuseppe.liotta,
+alessandra.tappini}@unipg.it
+2Wilhelm-Schickard Institut f¨ur Informatik, Universit¨at T¨ubingen,
+Sand 13, 72076, T¨ubingen, Germany.
+[email protected]
+January 4, 2023
+Abstract
+This paper studies a packing problem in the so-called beyond-planar
+setting, that is when the host graph is “almost-planar” in some sense. Pre-
+cisely, we consider the case that the host graph is k-planar, i.e., it admits
+an embedding with at most k crossings per edge, and focus on families of
+∆-regular caterpillars, that are caterpillars whose non-leaf vertices have
+the same degree ∆. We study the dependency of k from the number h of
+caterpillars that are packed, both in the case that these caterpillars are all
+isomorphic to one another (in which case the packing is called placement)
+and when they are not. We give necessary and sufficient conditions for
+the placement of h ∆-regular caterpillars and sufficient conditions for the
+packing of a set of ∆1-, ∆2-, . . . , ∆h-regular caterpillars such that the
+degree ∆i and the degree ∆j of the non-leaf vertices can differ from one
+caterpillar to another, for 1 ≤ i, j ≤ h, i ̸= j.
+1
+Introduction
+Graph packing is a classical problem in graph theory.
+The original formu-
+lation requires to merge several smaller graphs into a larger graph, called the
+host graph, without creating multiple edges. More precisely, graphs G1, G2, . . . , Gh
+with Gi = (Vi, Ei) should be combined to a new graph G = (V, E) by injec-
+tive mappings ηi : Vi → V so that V = V1 ∪ V2 ∪ · · · ∪ Vh and the images of
+1
+arXiv:2301.01226v1  [math.CO]  3 Jan 2023
+
+the edge sets Ei do not intersect. It has been often assumed that |Vi| = n for
+all i = 1, 2, . . . h, and thus the mappings ηi are bijective. Many combinatorial
+problems can be regarded as packing problems. For example, the Hamiltonian
+cycle problem for a graph G can be stated as the problem of packing an n-vertex
+cycle with the complement of G.
+When no restriction is imposed on the host graph, we say that the host
+graph is Kn. Some classical results in this setting are those by Bollob´as and
+Eldridge [6], Teo and Yap [26], Sauer and Spencer [25], while related famous
+conjectures are by Erd˝os and S´os from 1963 [10] and by Gy´arf´as from 1978 [16].
+Within this line of research, Wang and Sauer [27], and Mah´eo et al. [22] char-
+acterized triples of trees that admit a packing into Kn. Haler and Wang [17]
+extended this result to four copies of a tree. Further notable work on graph
+packing into Kn is by Hedetniemi et al. [18], Wozniak and Wojda [28] and Aich-
+holzer et al. [2]. A packing problem with identical copies of a graph is also called
+a placement problem (see, e.g., [17, 27, 29]).
+A tighter relation to graph drawing was established when researchers did not
+consider Kn to be the host graph, but required that the host graph is planar.
+The main question here is how to pack two trees of size n into a planar graph
+of size n. After a long series of intermediate steps [11, 12, 13, 14, 24] where the
+class of trees that could be packed has been gradually generalized, Geyer et al.
+[15] showed that any two non-star trees can be embedded into a planar graph.
+Relaxing the planarity condition allows for packing of more (than two) trees,
+and restricting the number of crossings for each edge, i.e., in the so-called
+beyond-planar setting [9, 19, 21], still keeps the host graph sparse. The study
+of the packing problem in the beyond planarity setting was started by De Luca
+et al. [8], who consider how to pack caterpillars, paths, and cycles into 1-planar
+graphs (see, e.g., [21] for a survey and references on 1-planarity). While two
+trees can always be packed into a planar graph, it may not be possible to pack
+three trees into a 1-planar graph.
+In this work we further generalize the problem by allowing the host graph
+to be k-planar for any k ≥ 1, and we study the dependency of k on the number
+of caterpillars to be packed and on their vertex degree. We consider ∆-regular
+caterpillars, which are caterpillars whose non-leaf vertices all have the same
+degree. Our results can be briefly outlined as follows.
+• We consider the packing problem of h copies of the same ∆-regular cater-
+pillar into a k-planar graph. We characterize those families of h ∆-regular
+caterpillars which admit a placement into a k-planar graph and show that
+k ∈ O(∆h + h2).
+• We extend the study from the placement problem to the packing problem
+by considering a set of ∆1-, ∆2-, . . . , ∆h-regular caterpillars such that the
+degree ∆i and the degree ∆j of the non-leaf vertices can differ from one
+caterpillar to another, with 1 ≤ i, j ≤ h, i ̸= j. By extending the tech-
+niques of the bullet above, we give sufficient conditions for the existence
+of a k-planar packing of these caterpillars and show that k ∈ O(∆h2).
+2
+
+• Finally, we prove a general lower bound on k and show that this lower
+bound can be increased for small values of h and for caterpillars that are
+not ∆-regular.
+The rest of the paper is organized as follows. Preliminaries are in Section 2.
+The placement of h ∆-regular caterpillars into a k-planar graph is discussed
+in Section 3. Section 4 is devoted to k-planar h-packing, while Section 5 gives
+lower bounds on the value of k as a function of h. Concluding remarks and open
+problems can be found in Section 6.
+2
+Preliminaries
+We assume familiarity with basic graph drawing and graph theory terminology
+(see, e.g., [5, 20, 23]) and recall here only those concepts and notation that will
+be used in the paper.
+Given a graph G, we denote by degG(v) the degree of a vertex v in G. Let
+G1, G2, . . . , Gh be h graphs, all having n vertices, an h-packing of G1, G2, . . . , Gh
+is an n-vertex graph G that contains G1, G2, . . . , Gh as edge-disjoint spanning
+subgraphs.
+We also say that G1, G2, . . . , Gh can be packed into G and that
+G is the host graph of G1, G2, . . . , Gh. An h-packing of h graphs into a host
+graph G such that the h graphs are all isomorphic to a graph H, is called an
+h-placement of H into G. We also say that G1, G2, . . . , Gh can be placed into
+G. The following property establishes a necessary condition for the existence of
+an h-packing into any host graph.
+Property 1. A packing of h connected n-vertex graphs exists only if n ≥ 2h
+and degGi(v) ≤ n − h, for each i ∈ {1, 2, . . . , h} and for each vertex v.
+Proof. Each Gi has at least n−1 edges (because it is connected); thus, if n < 2h
+the h graphs have more edges in total than the number of edges of any graph
+with n vertices. But since graphs Gi must be edge-disjoint subgraphs of G, the
+number of edges of G must be at least the total number of edges of the graphs
+Gi.
+Since degGi(v) ≥ 1 for every i ∈ {1, 2, . . . , h} and for each v (because
+each Gi is connected) and since �h
+i=1 degGi(v) ≤ n − 1 (because G cannot
+have vertex-degree larger that n − 1), it holds that degGi(v) ≤ n − h, for each
+i ∈ {1, 2, . . . , h} and for each vertex v.
+A k-planar graph is a graph that admits a drawing in the plane such that each
+edge is crossed at most k times. If the host graph of an h-packing (h-placement)
+is k-planar, we will talk about a k-planar h-packing (k-planar h-placement).
+Sometimes, we shall simply say k-planar packing or k-planar placement, when
+the value of h is clear from the context or not relevant.
+A caterpillar is a tree such that removing all leaves we are left with a path,
+called spine. A caterpillar T is ∆-regular, for ∆ ≥ 2, if degT (v) = ∆ for every
+vertex v of the spine of T. The number of vertices of a ∆-regular caterpillar is
+n = σ(∆ − 1) + 2 for some positive integer σ, which is the number of vertices of
+the spine.
+3
+
+3
+h-placement of ∆-regular Caterpillars into k-
+planar Graphs
+Given h copies of a same ∆-regular caterpillar, we want to study under which
+conditions they admit a placement into a k-planar graph. We start by showing
+that the necessary condition stated in Property 1 is, in general, not sufficient to
+guarantee a placement even for ∆-regular caterpillars.
+Theorem 1. For every h ≥ 2, let ∆ be a positive integer such that
+h−1
+∆−1 is not
+an integer. A set of h ∆-regular caterpillars with n = 2h vertices does not admit
+a placement into any graph.
+Proof. Since each caterpillar has n − 1 edges and the number of caterpillars is
+h = n
+2 , the total number of edges is n(n−1)
+2
+and thus, if a placement exists, the
+host graph can only be Kn. We now prove that this is not possible. Denote by
+C1, C2, . . . , Ch the h caterpillars and suppose that a packing into Kn exists. Let
+v be a vertex of Kn and let v1, v2, . . . , vh be the h vertices that are mapped to
+v, with vi being a vertex of Ci. Each vertex vi has degree in Ci that is either ∆
+or 1 (because each Ci is ∆-regular). Denote by c the number of vertices among
+v1, v2, . . . , vh that have degree ∆; the degree of v in the packing is c∆ + (h − c)
+and since the degree of v in Kn is n − 1, it must be c∆ + (h − c) = n − 1, i.e.,
+c∆ + (h − c) = 2h − 1, which can be rewritten as c =
+h−1
+∆−1. But this is not
+possible because c is integer, while h−1
+∆−1 is not.
+In the rest of this section we shall establish necessary and sufficient condi-
+tions that characterize when a set of h isomorphic ∆-regular caterpillars admit
+a k-planar h-placement. Concerning the sufficiency, in Section 3.1 we describe
+a constructive argument that computes a set of so-called zig-zag drawings and
+study the properties of such drawings. In Section 3.2, we complete the charac-
+terization by also giving necessary conditions for an h-placement of ∆-regular
+caterpillars into a k-planar graph; in the same section, we give an upper bound
+on k as a function of h and ∆.
+We recall that a ∆-regular caterpillar has a number of vertices n that is equal
+to σ(∆ − 1) + 2 for some natural number σ, which is the number of vertices
+of the spine. While ∆-regular caterpillars are defined for any value of σ ≥ 1,
+when we want to pack a set of h ≥ 2 caterpillars, Property 1 requires that each
+caterpillar has at least two spine vertices, i.e., that σ ≥ 2 for each caterpillar.
+Otherwise, the unique spine vertex would have degree n − 1 and Property 1
+would not hold.
+3.1
+Zig-zag Drawings of ∆-regular caterpillars
+Let C be a ∆-regular caterpillar with n vertices; we construct a drawing Γ of
+C as shown in Fig. 1. The number of vertices of the spine of C is σ =
+n−2
+∆−1;
+consider a set of σ points on a circle γ and denote by u1, u2, . . . , uσ these points
+according to the circular clockwise order they appear along γ. Draw the spine
+4
+
+u1
+u2
+u3
+u4
+u5
+(a)
+v17
+v1
+v2
+v3
+v4
+v5
+v6
+v7
+v8
+v9
+v10
+v11
+v12
+v13
+v14
+v15
+v16
+upper part
+lower part
+hole short
+edge
+(b)
+v17
+v1
+v2
+v3
+v4
+v5
+v6
+v7
+v8
+v9
+v10
+v11
+v12
+v13
+v14
+v15
+v16
+(c)
+Figure 1:
+(a) A zig-zag drawing of a 4-regular caterpillar; (b) the upper and
+the lower part are highlighted; c) a 2-packing obtained by the drawing of (b)
+with a copy of it rotated by one step.
+of C by connecting, for i = 1, 2, . . . , ⌊ σ
+2 ⌋, the points ui and ui+1 to the point
+uσ−i+1; see Fig. 1(a).
+If σ is even and i =
+σ
+2 , the points ui+1 and uσ−i+1
+coincide and therefore the point u σ
+2 is connected only to u σ
+2 +1. Notice that
+all points ui have two incident edges, except u1 and u⌊ σ
+2 ⌋+1 which have only
+one. We add the leaves adjacent to each vertex ui ̸∈ {u1, u⌊ σ
+2 ⌋+1} by connecting
+uσ−i+1 to ∆−2 points between ui and ui+1; we then add the leaves adjacent to
+u1 by connecting it to ∆−1 points between uσ and u1; we finally add the leaves
+adjacent to u⌊ σ
+2 ⌋+1 by connecting it to ∆ − 1 points between u σ
+2 and u σ
+2 +1 if σ
+is even, or to ∆−1 points between u⌊ σ
+2 ⌋+1 and u⌊ σ
+2 ⌋+2 if σ is odd. The resulting
+drawing is called a zig-zag drawing of C.
+From now on, we assume that in a zig-zag drawing the points that represent
+vertices are equally spaced on the circle γ. Let χ be the convex hull of the points
+representing the vertices of C in Γ. A zig-zag drawing has exactly two sides of
+χ that coincide with two edges of C; we call these two edges short edges of Γ;
+each other side of χ is called a hole. Denote by v1, v2, . . . , vn the vertices of Γ
+according to the circular clockwise order they appear along χ with v1 ≡ u1; see
+Fig. 1(b). Notice that (v1, vn) is a short edge and vn is the degree-1 vertex of
+this edge.
+Consider a straight line s that intersects both short edges of Γ; line s inter-
+sects all the edges of the zig-zag drawing. Without loss of generality, assume
+that s is horizontal and denote by U the set of vertices that are above s and by
+L the set of vertices that are below s. The vertices in U form the upper part of
+Γ and those in L form the lower part of Γ. Without loss of generality assume
+that v1 is in the upper part (and therefore vn is in the lower part). It follows
+that each edge has the end-vertex with lower index in the upper part, and the
+end-vertex with higher index in the lower part. Hence the short edge different
+from (v1, vn), which we denote as (vr−1, vr), is such that vr−1 is in the upper
+part and vr is in the lower part. The first vertex of the upper part, i.e., vertex
+v1, is called starting point of Γ, while the first vertex of the lower part, i.e.,
+5
+
+vertex vr, is called ending point of Γ. We observe that r = n
+2 + 1 if the number
+of vertices of the spine σ =
+n−2
+∆−1 is even, while r = 1 + n−(∆−1)
+2
+if σ is odd.
+This can be written with a single formula as r = 1+ n−(∆−1)(σ
+mod 2)
+2
+. The two
+short edges separate two sets of consecutive holes, one completely contained in
+the upper part and one completely contained in the lower part; if σ is even,
+these two sets have the same number of holes equal to n−2
+2 ; if σ is odd, then
+one of the two sets has n−∆−1
+2
+holes, while the other has n+∆−3
+2
+. Note that the
+smaller set is in the upper part.
+Let ℓ be a positive integer and let Γ′ be the drawing obtained by re-mapping
+vertex vi to the point1 representing vi+ℓ in Γ. We say that Γ′ is the drawing
+obtained by rotating Γ by ℓ steps. Note that the starting point of Γ′ is vj with
+j = 1 + ℓ and the ending point is vr with r = 1 + ℓ + n−(∆−1)(σ
+mod 2)
+2
+=
+j + n−(∆−1)(σ
+mod 2)
+2
+.
+The drawing in Fig. 1(c) is the union of two zig-zag
+drawings Γ1 and Γ2, where Γ2 is obtained by rotating Γ1 by one step; the
+starting point of Γ1 is v1 while its ending point is v8; the starting point of Γ2 is
+v2, while its ending point is v9.
+Lemma 1. Let Γ1 be a zig-zag drawing of a ∆-regular caterpillar C with starting
+point j1 and ending point r1; let Γ2 be a zig-zag drawing of C with starting
+point j2. If 0 < j2 − j1 < n−(∆−1)(σ
+mod 2)
+2
+, where σ is the number of spine
+vertices of C, then Γ1 ∪ Γ2 has no multiple edges.
+Proof. We first observe that Γ2 is obtained by rotating Γ1 by ℓ steps, where
+ℓ = j2 − j1. Suppose that a multiple edge (vi, vg), with i < g exists in Γ1 ∪ Γ2.
+This implies that in the drawing Γ1 there must be an edge (vi′, vg′) that, when
+rotated by ℓ steps, coincides with (vi, vg). In other words, the two edges (vi, vg)
+and (vi′, vg′) must be such that: (i) i′ < i < r1 ≤ g < g′; (ii) g = i′ + ℓ; (iii) the
+number α of vertices encountered between vi and vg when going clockwise from
+vi to vg is the same as the number of vertices encountered when going clockwise
+from vg′ to vi′ (see Fig. 2). Denote by β the number of vertices encountered
+when going clockwise from vi′ to vi, and by ζ the number of vertices encountered
+when going clockwise from vg to vg′. We have 2α + β + ζ + 4 = n. If σ is even,
+then β = ζ (see Fig. 2(a)), which implies α + β + 2 = n
+2 . Notice that g = i′ + ℓ
+implies that ℓ = β + α + 2 (ℓ is equal to the number of vertices encountered
+clockwise between vi′ and vg plus one) and therefore (vi′, vg′) can coincide with
+(vi, vg) after a rotation of ℓ steps only if ℓ = n
+2 but, when σ is even, we have
+ℓ = j2 − j1 < n
+2 and therefore a multiple edge cannot exist.
+If σ is odd, then β − (∆ − 1) ≤ ζ ≤ β + (∆ − 1) (see Figs. 2(b) and 2(c)) and
+therefore 2α + 2β − (∆ − 1) + 4 ≤ 2α + β + ζ + 4 = n ≤ 2α + 2β + (∆ − 1) + 4,
+which can be rewritten as n−(∆−1)
+2
+≤ α + β + 2 ≤ n+(∆−1)
+2
+. It follows that,
+in order to have (vi′, vg′) and (vi, vg) coincident after a rotation of ℓ steps, the
+value of ℓ must be such that n−(∆−1)
+2
+≤ ℓ ≤ n+(∆−1)
+2
+; but, when σ is odd, we
+have ℓ = j2 − j1 < n−(∆−1)
+2
+and therefore a multiple edge cannot exist.
+1In a drawing in convex position the indices of the vertices are taken modulo n.
+6
+
+vj1
+vr1
+vi′
+vh′
+vi
+vh
+β
+ζ
+α
+α
+ℓ
+(a)
+vj1≡vi′
+vr1
+vh′
+vi
+vh
+β
+ζ
+α
+α
+ℓ
+(b)
+vj1
+vr1
+vi′
+vh′
+vi
+vh
+β
+ζ
+α
+α
+ℓ
+(c)
+Figure 2:
+Illustration for the proof of Lemma 1; (a) σ even; (b)-(c) σ odd.
+We conclude this section by computing the maximum number of crossings
+per edge in the union of two zig-zag drawings without overlapping edges. We
+state this lemma in general terms assuming that the two ∆-regular caterpillars
+can have different vertex degrees, as we are going to use the lemma to establish
+upper bounds on k both for k-planar h-placements and for k-planar h-packings
+(Section 4).
+Let Γ be a union of a set of zig-zag drawings.
+To ease the description
+that follows, we regard Γ as a sub-drawing of a straight-line drawing of Kn
+whose vertices coincide with those of Γ (and therefore are equally spaced along
+a circle). In particular, for each vertex vj, we denote by ej,0, ej,1, . . . , ej,n−2
+the edges incident to vj in Kn according to the circular counterclockwise order
+around vj starting from ej,0 = (vj, vj−1). Each of the zig-zag drawings that
+form Γ contains a subset of these edges and Γ is a valid packing if there is no
+edge that belongs to two different zig-zag drawings in the set whose union is Γ.
+We denote by Sn the (circular) sequence of slopes si = i · π
+n, for i =
+0, 1, . . . , n − 1; refer to Fig. 3. Notice that, without loss of generality, we can
+assume that the convex hull of Γ has a side with slope s0 and, as a consequence,
+every edge of Γ has a slope in the set Sn. Let vj be a vertex; if the slope of
+ej,0 is sij, then the slope of ej,p is sij+p (with indices taken modulo n); in other
+words, the edges incident to each vertex have slopes that form a sub-sequence of
+n − 1 consecutive elements of Sn; we denote such a sequence as ψ(ij), where ij
+indicates that the first element of ψ(ij) is sij. We say that vj uses the sequence
+ψ(ij). If we consider two different vertices vj and vj+p and vj uses the sequence
+ψ(ij), then vj+p uses the sequence ψ(ij − 2p) (with indices taken modulo n);
+in other words, the sequence used by a vertex shifts clockwise by two elements
+moving to the next vertex.
+Lemma 2. Let C1 be an n-vertex ∆1-regular caterpillar and let C2 be an n-
+vertex ∆2-regular caterpillar with ∆i ≤ n − 2 (for i = 1, 2). Let Γ1 be a zig-zag
+drawing of C1 with starting point vj1 and let Γ2 be a zig-zag drawing of C2 with
+starting point vj2 with 0 < j2 − j1 < n
+2 . If Γ1 ∪ Γ2 has no multiple edges, then
+any edge of Γ1 ∪ Γ2 is crossed at most 2(∆1 + ∆2) + 4(j2 − j1) times.
+7
+
+s0
+s1
+sn−1
+sn−2
+vj
+ej,0
+ej,n−2
+ej+1,0
+vj+1
+ψ(j)
+ψ(j + 1)
+Sn
+ej+1,n−2
+≡
+Figure 3: Illustration for the definition of slopes.
+Proof. We first observe that the edges of a zig-zag drawing of a ∆-regular cater-
+pillar are all drawn as segments whose slope belongs to a set of ∆ slopes. In
+particular, for every spine vertex v, the edges incident to v are drawn using all
+these ∆ slopes.
+Consider the starting vertex vj1 of Γ1; the edges incident to vj1 are drawn
+with the first ∆1 slopes of ψ(ij1). Analogously, the edges incident to the starting
+vertex vj2 of Γ2 are drawn with the first ∆2 slopes of ψ(ij2). The sequence ψ(ij2)
+is shifted clockwise by 2(j2−j1) units with respect to ψ(ij1). On the other hand,
+since j2−j1 < n
+2 , the first slope of ψ(ij2) is distinct from the first slope of ψ(ij1).
+Let e = (vi, vg) be an edge of Γ1. We now prove that the number of crossings
+along e is at most the one given in the statement. Let e1 = (vi, va) be the
+edge of Γ2 incident to vi that forms the smallest angle with e; analogously, let
+e2 = (vg, vb) be the edge of Γ2 incident to vg that forms the smallest angle with
+e. Notice that, in principle there are four possible clockwise orders of vi, va,
+vg, and vb (see cases (a)–(d) in Fig. 4 for an illustration). However the case (b)
+cannot happen. Namely, in case (b) the slopes used to draw the edges of Γ2
+would be shifted counterclockwise with respect to those used to represent the
+edges of Γ1; but, as observed above, the slopes used by Γ2 are shifted clockwise
+with respect to those used by Γ1.
+Let α1 be the angle between e and e1 and let α2 be the angle between e and
+e2. Let V1 be the set of vertices seen by the angle α1 including va and excluding
+vg; analogously let V2 be the set of vertices seen by the angle α2 including vb
+and excluding vi. In each of the three cases (a), (c), and (d), at least one of α1
+and α2 is such that e sweeps the angle moving clockwise. Let αl with l ∈ {1, 2}
+be the angle that satisfies this condition. In particular, for case (a) αl can be
+both α1 or α2, in case (c) αl is α2 and in case (d) αl is α1 (see Fig. 4). Every
+edge that crosses e has an end-vertex in V1 and one end-vertex in V2. To count
+the number of such edges (and therefore the number of crossings along e), we
+evaluate |Vl|. The value of |Vl| is at most the number of slopes of Sn that are
+encountered in counterclockwise order between the slope s ∈ Sn of el and the
+slope s′ ∈ Sn of e. In particular, in case (a) |Vl| is exactly this number, while in
+case (c) and (d) |Vl| is less than this number. The slope s′ is at most the last
+8
+
+vi
+vg
+va
+vb
+α1
+α2
+V1
+V2
+(a)
+vi
+vg
+va
+vb
+α1
+α2
+(b)
+vi
+vg
+va
+vb
+α1
+α2
+V2
+(c)
+vi
+vg
+va
+vb
+α1
+α2
+V1
+(d)
+Figure 4:
+Illustration for the proof of Lemma 2. The edges of Γ2 are dashed.
+9
+
+slope used by Γ1, which is sp with p = j1 + ∆1, while the slope s is at least the
+first slope used by Γ2, which is sq with q = j1 − 2(j2 − j1). Thus, the number of
+slopes between s′ (included) and s (excluded) is at most p−q = ∆1 +2(j2 −j1).
+Hence |Vl| ≤ ∆1 + 2(j2 − j1).
+We call a block a subset of consecutive vertices of Vl starting with a spine
+vertex and containing all the leaves that follow that spine vertex. The number
+of edges of Γ2 incident to the vertices of a block is 2(∆2 − 1) (since ∆2 edges
+are incident to the spine vertex and ∆2 − 2 is the number of leaves).
+The
+number of blocks in Vl is
+�
+|Vl|
+(∆2−1)
+�
+. It follows that the number of crossings χe
+along e is at most
+�
+|Vl|
+(∆2−1)
+�
+2(∆2 − 1) which is less than 2(|Vl| + ∆2). Since
+|Vl| ≤ ∆1 + 2(j2 − j1), we have χe ≤ 2(∆1 + ∆2) + 4(j2 − j1), which concludes
+the proof in the case when e belongs to Γ1. The case when the edge e belongs
+to Γ2 is analogous. In particular, when e belongs to Γ2, the cases (b), (c), and
+(d) apply, while case (a) does not happen.
+3.2
+Characterization
+We are now ready to characterize the ∆-regular caterpillars that admit an h-
+placement.
+Theorem 2. Let C be a ∆-regular caterpillar with n vertices. An h-placement
+of C exists if and only if: (i) ∆ ≤ n − h; and (ii) n ≥ 2h + (∆ − 1) · (σ mod 2),
+where σ is the number of spine vertices of C. Further, if an h-placement exists,
+there exists one that is k-planar for k ∈ O(∆h + h2).
+Proof. We first prove the sufficient condition. Let C1, C2, . . . , Ch be the h cater-
+pillars and assume that n ≥ 2h+(∆−1)(σ mod 2). We compute an h-placement
+of C1, C2, . . . , Ch starting from a zig-zag drawing Γ1 of C1 and obtaining the
+drawing Γi of Ci by rotating Γ1 by i − 1 steps, for i = 2, 3, . . . , h.
+Notice that, when the number of spine vertices σ of each Ci is even, h ≤ n
+2
+and therefore each Γi is rotated by less than n
+2 steps; when σ is odd h ≤ n−(∆−1)
+2
+and each Γi is rotated by less than n−(∆−1)
+2
+steps. In both cases, each pair of
+drawings Γi and Γj satisfies the conditions of Lemma 1 and therefore there
+are no multiple edges, that is, the union of all Γi is a valid h-placement of
+C1, C2, . . . , Ch.
+We now prove the necessary condition.
+If σ is even, then conditions (i)
+and (ii) are necessary by Property 1. Hence, consider the case when σ is odd.
+Condition (i) is necessary by Property 1. Assume, by contradiction, that (ii) is
+not necessary, i.e., there exists an h-placement of h caterpillars C1, C2, . . . , Ch
+such that n < 2h + (∆ − 1). Since C1, C2, . . . , Ch admit an h-placement, by
+Property 1 n must be at least 2h. Thus, it would be 2h ≤ n < 2h + (∆ − 1); in
+other words, n = 2h + α with 0 ≤ α ≤ ∆ − 2.
+Let G be the host graph of the h-placement and let v be the vertex of G to
+which the largest number of spine vertices of C1, C2, . . . , Ch is mapped. Let β
+be the number of spine vertices that are mapped to v. There are other h − β
+10
+
+leaf vertices that are mapped to v (because one vertex per caterpillar has to be
+mapped on each vertex of G). The degree of v in G is at most n − 1 and each
+of the spine vertices mapped to v has degree ∆. Hence, the β spine vertices
+mapped to v have degree β∆ in total. Vertex v can have at most other n−1−β∆
+edges and therefore it must be n − 1 − β∆ ≥ h − β, i.e., β ≤ n−1−h
+∆−1 . On the
+other hand, there are σh spine vertices in total and, since G has n vertices, there
+are at least ⌈ σh
+n ⌉ spine vertices mapped to v, i.e., β ≥ ⌈ σh
+n ⌉. Putting together
+the two conditions on β we obtain:
+�σh
+n
+�
+≤ β ≤ n − 1 − h
+∆ − 1
+.
+Since n = 2h + α, we have h = n−α
+2 ; replacing h in Section 3.2, we obtain:
+�σ
+2 − σα
+2n
+�
+≤ β ≤ n + α − 2
+2(∆ − 1) .
+Since n = σ(∆ − 1) + 2, we have:
+�σ
+2 −
+σα
+2(σ(∆ − 1) + 2)
+�
+≤ β ≤ σ(∆ − 1) + α
+2(∆ − 1)
+.
+(1)
+Eq. (1) implies that:
+�σ
+2 −
+α
+2(∆ − 1) + 4
+σ
+�
+≤ σ
+2 +
+α
+2(∆ − 1).
+(2)
+We now prove that Eq. (2) cannot be satisfied. Since σ is odd, it is σ = 2i+1
+for some i ∈ N, and thus:
+�
+i + 1
+2 − ζ
+�
+≤ k + 1
+2 + ζ′,
+(3)
+with ζ =
+α
+2(∆−1)+ 4
+σ and ζ′ =
+α
+2(∆−1). We have ζ < ζ′ and we prove that ζ′ < 1
+2:
+ζ′ =
+α
+2(∆ − 1) ≤
+∆ − 2
+2(∆ − 1) <
+∆ − 1
+2(∆ − 1) = 1
+2.
+The first term of Eq. (3) is i + 1 because 0 < 1
+2 − ζ < 1; the second term is less
+than i + 1 because 0 < 1
+2 + ζ′ < 1. It follows that Eq. (3) does not hold and
+therefore Eq. (2) does not hold.
+We now prove the bound on the number of crossings along an edge. We
+consider an edge of Γ1; the number of crossings along an edge of the drawing
+of another caterpillar is bounded by the same number. Let e be an edge of the
+drawing Γ1. By Lemma 2, the number of crossings χe along e due to the edges of
+another drawing Γl (with 2 ≤ l ≤ h) is at most 2(∆1+∆l)+4(jl−j1). Summing
+11
+
+u1
+u2
+u3
+u4
+u5
+(a)
+v17
+v1
+v2
+v3 v4
+v5
+v6
+v7
+v8
+v9
+v10
+v11
+v12
+v13
+v14
+v15
+v16
+(b)
+Figure 5:
+(a) An inner zig-zag drawing and (d) an outer zig-zag drawing of a
+4-regular caterpillar.
+over all drawings distinct from Γ1, we obtain χe ≤ �h
+l=2(2(∆1+∆l)+4(jl−j1)).
+Considering that ∆l = ∆ for every l and that jl − j1 = l − 1, we have
+χe ≤
+h
+�
+l=2
+(4∆ + 4(l − 1)) ≤ (4∆ − 2)h + 2h2 − 4∆.
+(4)
+We conclude by observing that the number of crossings given by Eq. (4) can
+be reduced, although not asymptotically. A zig-zag drawing can be embedded
+inside the circle (see Fig. 5(a)) or outside the circle (see Fig. 5(b)). Thus, the
+number given by Eq. (4) can be halved by embedding half of the caterpillars
+inside the circle and the other half outside the circle.
+4
+h-packing of ∆-regular Caterpillars in k-planar
+Graphs
+In this section we study h-packings of h ∆1-, ∆2-, . . . , ∆h-regular caterpillars
+such that the degree ∆i and the degree ∆j of the spine vertices can differ from
+one caterpillar to another, for 1 ≤ i, j ≤ h, i ̸= j.
+Lemma 3. Let C1 be an n-vertex ∆1-regular caterpillar and let C2 be an n-
+vertex ∆2-regular caterpillar such that ∆1 > ∆2 and ∆1 ≤ n − 2. Let Γ1 be
+a zig-zag drawing of C1 with starting point vj1 and ending point vr1, and let
+Γ2 be a zig-zag drawing of C2 with starting point vj2 and ending point vr2. If
+∆2
+2 ≤ j2 − j1 < n−(∆1−1)
+2
+, then Γ1 ∪ Γ2 has no multiple edges.
+12
+
+Proof. As described in the proof of Lemma 2, the edges of a zig-zag drawing of
+a ∆-regular caterpillar are all drawn as segments whose slope belongs to a set
+of ∆ slopes. In particular, for every spine vertex v, the edges incident to v are
+drawn using all these ∆ slopes. Based on this observation, we show that the ∆1
+slopes used to represent the edges of Γ1 are distinct from the ∆2 slopes used to
+represent the edges of Γ2. We use the same notation used in Theorem 2.
+Consider the staring vertex vj1 of Γ1; the edges incident to vj1 are drawn
+with the first ∆1 slopes of ψ(ij1). Analogously, the edges incident to the starting
+vertex vj2 of Γ2 are drawn with the first ∆2 slopes of ψ(ij2). Since j2 −j1 ≥ ∆2
+2 ,
+the sequence ψ(ij2) is shifted clockwise by ∆2 units with respect to ψ(ij1). On
+the other hand, since j2 − j1 ≤ n−(∆1−1)
+2
+, the sequence of the first ∆2 slopes of
+ψ(ij2) does not overlap with the first ∆1 slopes of ψ(ij1), which concludes the
+proof.
+Theorem 3. Let C1, C2, . . . , Ch be h caterpillars such that Ci is ∆i-regular,
+for 1 ≤ i ≤ h, and ∆h ≤ ∆h−1 ≤ · · · ≤ ∆1 ≤ n − h. If �h
+i=1 ∆i ≤ n − 1 and
+�h
+i=2
+� ∆i
+2
+�
+< n−(∆1−1)
+2
+, then there exists a k-planar packing with k ∈ O(∆1h2).
+Proof. We compute a zig-zag drawing of C1 with starting point vj1, with j1 = 1;
+for each Ci, with 2 ≤ i ≤ h, we compute a zig-zag drawing Γi with starting
+vertex vji where ji = ji−1 +
+� ∆i
+2
+�
+. Notice that, each vertex v of Γ1 ∪Γ2 ∪· · ·∪Γh
+has degree at most n−1; namely �h
+i=1 degCi(v) ≤ �h
+i=1 ∆i ≤ n−1. Moreover,
+given two caterpillars Ci and Ci′ with 1 ≤ i < i′ ≤ h, we have that: (i)
+ji′ − ji ≥ ji′ − ji′−1 =
+�
+∆i′
+2
+�
+; and (ii) ji′ − ji ≤ jh − j1 = �h
+i=2⌈ ∆i
+2 ⌉, which
+gives ji′ − ji < n−(∆1−1)
+2
+< n−(∆i−1)
+2
+. Putting together (i) and (ii), we obtain
+∆i′
+2
+< ji′ − ji < n−(∆i−1)
+2
+. Hence, Lemma 3 holds for every pair of caterpillars
+and the union of all the zig-zag drawings Γ1, Γ2, . . . , Γh is a valid packing of
+C1, C2, . . . , Ch.
+We now prove the bound on the number of crossings along an edge. We
+consider an edge of Γ1; the number of crossings along an edge of another drawing
+is bounded by the same number.
+Let e be an edge of the drawing Γ1.
+By
+Lemma 2, the number of crossings χe along e due to the edges of another
+drawing Γl (with 2 ≤ l ≤ h) is at most 2(∆1 + ∆l) + 4(jl − j1). Summing over
+all drawings distinct form Γ1, we obtain χe ≤ �h
+l=2(2(∆1 + ∆l) + 4(jl − j1)).
+Considering that jl ≥ jl−1 +
+� ∆i
+2
+�
+, we obtain that jl − j1 = �l
+i=2
+� ∆i
+2
+�
+. Since
+∆l ≤ ∆1 for every 2 ≤ l ≤ h, we have jl − j1 ≤ (l − 1)( ∆1
+2 + 1). Therefore, we
+obtain χe ≤ �h
+l=2(4∆1 + 4(l − 1)( ∆1
+2 + 1)) ≤ (∆1 + 2)h2 + 4∆1(h − 1).
+We now consider a special case of packing a set of h ∆1-, ∆2-, . . . , ∆h-
+regular caterpillars where, for each ∆i (1 ≤ i ≤ h), we have that ∆i − 1 is a
+multiple of ∆i+1 − 1. In this case, we show that the sufficient conditions of
+Theorem 3 can be relaxed. For example, consider the packing of a 17-regular
+caterpillar and two 9-regular caterpillars, each having 34 vertices. These three
+caterpillars do not satisfy the sufficient condition of Theorem 3. However, a k-
+planar packing of these caterpillars is possible, as proven in Theorem 4. We start
+13
+
+vj
+vi
+vr
+vg
+upper part
+lower part
+c = 0
+c = 1
+c = 2
+d = 0
+d = 1
+d = 2
+Figure 6: Illustration for Property 2; σ = 5. For each spine vertex, c and d are
+shown. Considering adjacent spine vertices, the sum of c and d is 2 or 3.
+with the following property, which immediately follows from the construction of
+a zig-zag drawing (see also Fig. 6 for an illustration).
+Property 2. Let Γ be a zig-zag drawing of a ∆-regular caterpillar with starting
+vertex vj and ending vertex vr. If vi is a spine vertex in the upper part of Γ,
+then i = j + c(∆ − 1) for some c ∈ N; if vg is a spine vertex in the lower part of
+Γ, then g = r + d(∆ − 1) for some d ∈ N. Moreover, if vi and vg are adjacent
+then either c + d =
+� σ
+2
+�
+− 1 or c + d =
+� σ
+2
+�
+, where σ is the number of spine
+vertices of Γ.
+Property 2 is extensively used in the proof of the following lemma.
+Lemma 4. Let C1 be an n-vertex ∆1-regular caterpillar and let C2 be an n-
+vertex ∆2-regular caterpillar such that ∆1−1 = q(∆2−1), for some q ∈ N+ and
+∆i ≤ n − 2 (for i = 1, 2). Let Γ1 be a zig-zag drawing of C1 with starting point
+vj1 and ending point vr1, and let Γ2 be a zig-zag drawing of C2 with starting
+point vj2 and ending point vr2. If 0 < j2 − j1 < n−(∆1−1)
+2
+, then Γ1 ∪ Γ2 has no
+multiple edges.
+Proof. Let (vi1, vg1) be an edge of Γ1 and (vi2, vg2) be an edge of Γ2. Assume
+that vi1 belongs to the upper part of Γ1 and vi2 belongs to the upper part of Γ2.
+Note that this implies that vg1 belongs to the lower part of Γ1 and vg2 belongs
+to the lower part of Γ2. We prove that (vi1, vg1) and (vi2, vg2) do not coincide.
+We first show that it does not happen that vi1 coincides with vi2 and vg1
+coincides with vg2. We then show that it does not happen that vi1 coincides
+with vg2 and vg1 coincides with vi2. In the rest of the proof we will express the
+four indices i1, i2, g1 and g2 in terms of the values j1, j2, r1 and r2, according to
+Property 2. Without loss of generality, we can assume that r2 ≤ n and j1 ≥ 1,
+i.e., that the vertices vr2, vn, v1, and vj1 appear in this clockwise order, with
+vr2 and vn possibly coincident and with v1 and vj1 possibly coincident. With
+these assumptions, we have j1 < j2 < r1 < r2 and vi1 can coincide with vg2
+14
+
+only if i1 = g2 − n, i.e., only if the value of g2 is greater than n and coincides
+with i1 modulo n. Thus, while assuming that vi1 coincides with vi2 implies that
+i1 = i2, assuming that vi1 coincides with vg2 implies that i1 = g2 − n.
+Case 1: It does not happen that vi1 coincides with vi2 and vg1 coincides
+with vg2.
+At least one vertex per edge is a spine vertex. We distinguish four sub-cases
+depending on which vertex is a spine vertex for each edge. Since all the cases
+are very similar, we give here only the first case and the others can be found in
+the appendix.
+Case 1.a: vi1 and vi2 are spine vertices. By Property 2 we have, for some
+c1, c2 ∈ N:
+i1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1).
+(5)
+and
+i2 = j2 + c2(∆2 − 1).
+(6)
+If vi1 coincides with vi2, we have i1 = i2; from Eq. (5) and Eq. (6) we obtain:
+j2 − j1 = (qc1 − c2)(∆2 − 1).
+(7)
+Concerning vg1 and vg2, we have:
+gm
+1 ≤ g1 ≤ gM
+1 ;
+gm
+2 ≤ g2 ≤ gM
+2 .
+with gm
+l
+= rl + dl(∆l − 1), gM
+l
+= rl + (dl + 1)(∆l − 1) for some dl ∈ N such that
+cl + dl =
+� σl
+2
+�
+− 1, where σl is the number of spine vertices of Cl, for l = 1, 2.
+We prove that gM
+1
+< gm
+2 , which implies g1 ̸= g2. To have gM
+1
+< gm
+2 it must be:
+r1 + (d1 + 1)(∆1 − 1) < r2 + d2(∆2 − 1)
+r1 + q(d1 + 1)(∆2 − 1) < r2 + d2(∆2 − 1)
+r2 − r1 > (qd1 + q − d2)(∆2 − 1).
+(8)
+Since rl = jl + n−(∆l−1)(σl
+mod 2)
+2
+, for l = 1, 2, Eq. (8) can be rewritten as:
+j2 − j1 >
+�
+(qd1 + q − d2) + (σ2
+mod 2)
+2
+− q(σ1
+mod 2)
+2
+�
+(∆2 − 1).
+(9)
+Combining Eq. (7) and Eq. (9) we obtain:
+qc1 − c2 > (qd1 + q − d2) + (σ2
+mod 2)
+2
+− q(σ1
+mod 2)
+2
+.
+Since cl + dl =
+� σl
+2
+�
+− 1, we have dl =
+σl+1(σl
+mod 2)
+2
+− cl − 1, for l = 1, 2;
+replacing d1 and d2 in the previous equation, we obtain:
+qc1 − c2 > qσ1
+2
++ q(σ1
+mod 2)
+2
+− qc1 − q + q − σ2
+2 − 1(σ2
+mod 2)
+2
++ c2+
++ 1 + σ2
+mod 2
+2
+− q(σ1
+mod 2)
+2
+15
+
+which, considering that σ2 =
+n−2
+∆2−1 = q(n−2)
+∆1−1 = qσ1, implies:
+qc1 − c2 > 1
+2
+(10)
+In summary, to have gM
+1
+< gm
+2 Eq. (10) must hold. On the other hand, from
+Eq. (7) and from the hypothesis that j2−j1 > 0 we obtain (qc1−c2)(∆2−1) > 0
+which, since (∆2 − 1) > 0, implies qc1 − c2 > 0 and, since qc1 − c2 is integer,
+can be rewritten as qc1 − c2 ≥ 1. This implies that Eq. (10) holds and therefore
+that gM
+1
+< gm
+2 and g1 ̸= g2.
+Case 2: It does not happen that vi1 coincides with vg2 and vg1 coincides
+with vi2.
+Also in this case we distinguish four sub-cases depending on which vertex is
+a spine vertex for each edge. As in Case 1, we give here only the first sub-case,
+while the others can be found in the appendix.
+Case 2.a:
+vi1 and vi2 are spine vertices.
+Since vg2 is a vertex in the
+lower part of Γ2, it must be g2 = r2 + d2(∆2 − 1) + α2, for some α2 such that
+0 ≤ α2 < ∆2 − 1. If vg2 coincides with vi1, as explained above, it must be
+i1 = g2 − n. Combining the expression of g2 with Eq. (5) we obtain:
+r2 − j1 = (qc1 − d2)(∆2 − 1) − α2 + n.
+(11)
+Concerning vg1, we have:
+gm
+1 ≤ g1 ≤ gM
+1 ;
+with gm
+1 = r1 + d1(∆1 − 1), gM
+1
+= r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such
+that c1 + d1 =
+� σ1
+2
+�
+− 1, where σ1 is the number of spine vertices of C1.
+We prove that i2 < gm
+1 , which implies i2 ̸= g1. To have i2 < gm
+1 it must be:
+j2 + c2(∆2 − 1) < r1 + d1(∆1 − 1)
+j2 + c2(∆2 − 1) < r1 + qd1(∆2 − 1)
+j2 − r1 < (qd1 − c2)(∆2 − 1).
+(12)
+Since rl = jl + n−(∆l−1)(σl
+mod 2)
+2
+, for l = 1, 2, Eq. (11) can be rewritten as:
+j2 − j1 = (qc1 − d2)(∆2 − 1) − α2 + n
+2 + (∆2 − 1)(σ2
+mod 2)
+2
+,
+(13)
+while Eq. (12) can be rewritten as:
+j2 − j1 <
+�
+(qd1 − c2) − q(σ1
+mod 2)
+2
+�
+(∆2 − 1) + n
+2 .
+(14)
+Combining Eq. (13) and Eq. (14) we obtain:
+qc1 − d2 < (qd1 − c2) − (σ2
+mod 2)
+2
+− q(σ1
+mod 2)
+2
++
+α2
+∆2 − 1.
+16
+
+Since cl + dl =
+� σl
+2
+�
+− 1, we have dl =
+σl+1(σl
+mod 2)
+2
+− cl − 1, for l = 1, 2;
+replacing d1 and d2 in the previous equation, we obtain:
+qc1 − σ2
+2 − σ2
+mod 2
+2
++ c2 + 1 < qσ1
+2
++ q(σ1
+mod 2)
+2
+− qc1 − q − c2−
+− σ2
+mod 2
+2
+− q(σ1
+mod 2)
+2
++
+α2
+∆2 − 1
+which, considering that σ2 =
+n−2
+∆2−1 = q(n−2)
+∆1−1 = qσ1, implies:
+qc1 − qσ1
+2
++ c2 < −q + 1
+2
++
+α2
+2(∆2 − 1)
+(15)
+In summary, to have iM
+2
+< gm
+1 Eq. (15) must hold. On the other hand, from
+Eq. (13) and from the hypothesis that j2 − j1 <
+n−(∆1−1)
+2
+=
+n−q(∆2−1)
+2
+we
+obtain:
+qc1 − d2 + 1
+2(σ2
+mod 2) < −q
+2 +
+α2
+∆2 − 1.
+Replacing again d2 with σ2+1(σ2
+mod 2)
+2
+− c2 − 1, we obtain:
+qc1 − qσ1
+2
++ c2 < −q + 2
+2
++
+α2
+∆2 − 1.
+(16)
+We have that − q
+2 − 1 +
+α2
+∆2−1 < − q
+2 − 1
+2 +
+α2
+2(∆2−1), since
+α2
+2(∆2−1) < 1
+2.
+In other words, Eq. (16) implies that Eq. (15) holds and therefore that
+i2 < gm
+1 and i2 ̸= g1.
+Theorem 4. Let C1, C2, . . . , Ch be h caterpillars such that Ci is ∆i-regular,
+∆i − 1 is a multiple of ∆i+1 − 1, with 1 ≤ i < h, and ∆i ≤ n − h (for i =
+1, 2, . . . , h). If n ≥ 2h + (∆1 − 1), then there exists a k-planar packing with
+k ∈ O(∆1h + h2).
+Proof. For each Ci, with 1 ≤ i ≤ h, we compute a zig-zag drawing Γi with
+starting vertex vi. Notice that, given two caterpillars Cj1 and Cj2 with 1 ≤
+j1 < j2 ≤ h, we have that ∆j1 − 1 is a multiple of ∆j2 − 1, and the zig-zag
+drawings Γj1 and Γj2 have starting vertices vj1 and vj2, respectively. Hence,
+0 < j2 − j1 < h and by hypothesis h ≤ n−(∆1−1)
+2
+. Hence, Lemma 4 holds for
+every pair of caterpillars and the union of all zig-zag drawings Γ1, Γ2, . . . , Γh is
+a valid packing of C1, C2, . . . , Ch.
+The proof of the bound on the number of crossings along an edge is the same
+as the one of Theorem 2, considering that ∆l ≤ ∆1 and that jl − j1 = l − 1 for
+every 2 ≤ l ≤ h.
+17
+
+5
+Lower bounds
+In this section we first give a general lower bound on the value of k for k-planar
+h-packings; we then increase this lower bound for some small values of h.
+Theorem 5. Every k-planar h-packing of h graphs with n vertices and m edges
+is such that k ≥
+h2m2
+14.6n2 .
+Proof. The number of edges of a k-planar graph with n vertices is at most
+3.81
+√
+k ·n, for k ≥ 2 [1]. Since the h graphs have h·m edges in total, a k-planar
+packing of these graphs can exist only if h ≤ 3.81
+√
+k n
+m, i.e., if k ≥
+h2m2
+14.6n2 .
+Since for a tree m = n − 1, we have the following.
+Corollary 1. Every k-planar h-packing of h trees is such that k ≥
+h2
+58.4.
+We now refine the lower bound above for small values of h in an h-placement
+of caterpillars. Specifically we show that for values of h equal to 3, 4, and 5 the
+corresponding lower bounds are 2, 3, and 5, respectively. Note that for all these
+cases the lower bound implied by Corollary 1 is 1.
+Theorem 6. For h = 3, 4 there exists a caterpillar C with at least h+7 vertices
+for which every k-planar h-placement of C is such that k ≥ h − 1. For h = 5
+there exists a caterpillar C with at least 24 vertices for which every k-planar
+5-placement of C is such that k ≥ h.
+Proof. Case h = 3, 4. Let n be an integer such that n ≥ h + 7, and let Cn,h be
+the n-vertex caterpillar shown in Fig. 7. Notice that the vertex of Cn,h denoted
+as v in Fig. 7 has degree n − h; we call it the center of Cn,h. Consider any
+h-placement of Cn,h into a graph G and denote as vi the vertex of G which the
+center of Ci is mapped to (i = 1, 2, . . . , h). The vertices v1, v2, . . . , vh must be
+distinct because, if two centers were mapped to the same vertex of G then this
+vertex would have degree larger than n − 1. Namely, if two centers are mapped
+to the same vertex, this vertex has degree 2n − 2h which is larger than n − 1
+if n > 2h − 1, i.e., if h + 7 > 2h − 1, which is true for h < 6. Since each vi
+(1 ≤ i ≤ h) has degree n − h in Ci and degree 1 in each of the h − 1 other
+caterpillars, its degree in G is n−1. Thus, G contains Kh,n−h. Thus, for h = 3,
+G contains K3,7 (n ≥ 10 in this case), which is not 1-planar [7]; for h = 4, G
+contains K4,7 (n ≥ 11 in this case), which is not 2-planar [4]. The case h = 5 is
+analogous with K5,19, which is not 4-planar [3].
+6
+Concluding Remarks and Open Problems
+This paper studied the placement and the packing of caterpillars into k-planar
+graphs. It proved necessary and sufficient conditions for the h-placement of ∆-
+regular caterpillars in a k-planar graph and sufficient conditions for the packing
+of a set of ∆1-, ∆2-, . . . , ∆h-regular caterpillars with k ∈ O(∆1h2) (∆1 is the
+18
+
+. . .
+. . .
+v
+h + 2
+h
+n − h − 2
+Cn,h
+Figure 7:
+A caterpillar as described in the proof of Theorem 6.
+maximum vertex degree in the set). The work in this paper contributes to the
+rich literature concerning the placement and the packing problem in planar and
+non-planar host graphs and it specifically relates with a recent re-visitation of
+these questions in the beyond-planar context.
+Many open problems naturally arise from the research in this paper. We
+conclude the paper by listing some of those that, in our opinion, are among the
+most interesting.
+• Extend the characterization of Theorem 2 to the placement of caterpillars
+that are not ∆-regular.
+• Theorems 4 and 3 give sufficient conditions for the k-planar packing of
+some families of caterpillars. It would be interesting to give a complete
+characterization of the packability of these families into k-planar graphs.
+• Theorem 6 improves the lower bound of Theorem 5 for caterpillars that
+are not ∆-regular. It would be interesting to find a similar result with
+∆-regular caterpillars.
+Finally, we point out that one could investigate what graphs can be packed/placed
+into a k-planar graph for a given value of k, instead of studying how k varies with
+the number h and the vertex degree of the caterpillars that are packed/placed.
+While the interested reader can refer to [3] for results with k = 1, the following
+theorem gives a preliminary result for k = 2 (see the appendix for a proof).
+Notice that Eq. (4) in the proof of Theorem 2 would give upper bounds in the
+range [86, 137] for the caterpillars considered by the following theorem.
+Theorem 7. A ∆-regular caterpillar with 4 ≤ ∆ ≤ 7 admits a 2-planar 3-
+placement.
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+21
+
+A
+Missing cases for the proof of Lemma 4
+Case 1.b: vg1 and vg2 are spine vertices. By Property 2 we have, for some
+d1, d2 ∈ N:
+g1 = r1 + d1(∆1 − 1) = r1 + qd1(∆2 − 1).
+(17)
+and
+g2 = r2 + d2(∆2 − 1).
+(18)
+If vg1 coincides with vg2, we have g1 = g2; from Eq. (17) and Eq. (18) we
+obtain:
+r2 − r1 = (qd1 − d2)(∆2 − 1).
+(19)
+Concerning vi1 and vi2, we have:
+im
+1 ≤ i1 ≤ iM
+1 ;
+im
+2 ≤ i2 ≤ iM
+2 .
+with im
+l = il + cl(∆l − 1), iM
+l
+= il + (cl + 1)(∆l − 1) for some cl ∈ N such that
+cl + dl =
+� σl
+2
+�
+− 1, where σl is the number of spine vertices of Cl, for l = 1, 2.
+We prove that iM
+1 < im
+2 , which implies i1 ̸= i2. To have iM
+1 < im
+2 it must be:
+j1 + (c1 + 1)(∆1 − 1) < j2 + c2(∆2 − 1)
+j1 + q(c1 + 1)(∆2 − 1) < j2 + c2(∆2 − 1)
+j2 − j1 > (qc1 + q − c2)(∆2 − 1).
+(20)
+Since rl = jl + n−(∆l−1)(σl
+mod 2)
+2
+, for l = 1, 2, Eq. (19) can be rewritten as:
+j2 − j1 =
+�
+(qd1 − d2) + (σ2
+mod 2)
+2
+− q(σ1
+mod 2)
+2
+�
+(∆2 − 1).
+(21)
+Combining Eq. (21) and Eq. (20) we obtain:
+c2 − qc1 − q > −1
+2.
+(22)
+In summary, to have iM
+1
+< im
+2 Eq. (22) must hold. On the other hand, from
+Eq. (21) and from the hypothesis that j2 − j1 > 0 we obtain c2 − qc1 − q > −1
+which, since c2 − qc1 − q is integer, can be rewritten as c2 − qc1 − q ≥ 0. This
+implies that Eq. (22) holds and therefore that iM
+1 < im
+2 and i1 ̸= i2.
+Case 1.c: vi1 and vg2 are spine vertices. By Property 2 we have, for some
+c1 ∈ N:
+i1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1).
+(23)
+We also have, for some c2 ∈ N and 0 ≤ α2 < ∆2 − 1:
+i2 = j2 + c2(∆2 − 1) + α2.
+(24)
+22
+
+If vi1 coincides with vi2, we have i1 = i2; from Eq. (23) and Eq. (24) we
+obtain:
+j2 − j1 = (qc1 − c2)(∆2 − 1) − α2.
+(25)
+Concerning vg1, we have:
+gm
+1 ≤ g1 ≤ gM
+1 ;
+with gm
+1 = r1 + d1(∆1 − 1), gM
+1
+= r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such
+that c1 + d1 =
+� σ1
+2
+�
+− 1, where σ1 is the number of spine vertices of C1.
+Since vg2 is a vertex in the lower part of Γ2, it must be g2 = r2 + d2(∆2 − 1).
+We prove that gM
+1
+< g2, which implies g1 ̸= g2. To have gM
+1
+< g2 it must be:
+r1 + (d1 + 1)(∆1 − 1) < r2 + d2(∆2 − 1)
+r1 + q(d1 + 1)(∆2 − 1) < r2 + d2(∆2 − 1)
+r2 − r1 > (qd1 + q − d2)(∆2 − 1).
+(26)
+Since rl = jl + n−(∆l−1)(σl
+mod 2)
+2
+, for l = 1, 2, Eq. (26) can be rewritten as:
+j2 − j1 >
+�
+(qd1 + q − d2) + (σ2
+mod 2)
+2
+− q(σ1
+mod 2)
+2
+�
+(∆2 − 1).
+(27)
+Combining Eq. (25) and Eq. (27) we obtain:
+qc1 − c2 −
+α2
+∆2 − 1 > (qd1 + q − d2) + (σ2
+mod 2)
+2
+− q(σ1
+mod 2)
+2
+.
+Since cl + dl =
+� σl
+2
+�
+− 1, we have dl =
+σl+1(σl
+mod 2)
+2
+− cl − 1, for l = 1, 2;
+replacing d1 and d2 in the previous equation, we obtain:
+qc1 − c2 −
+α2
+∆2 − 1 > qσ1
+2
++ q(σ1
+mod 2)
+2
+− qc1 − q + q − σ2
+2 − 1(σ2
+mod 2)
+2
++ c2+
++ 1 + σ2
+mod 2
+2
+− q(σ1
+mod 2)
+2
+which, considering that σ2 =
+n−2
+∆2−1 = q(n−2)
+∆1−1 = qσ1, implies:
+qc1 − c2 > 1
+2 +
+α2
+2(∆2 − 1)
+(28)
+In summary, to have gM
+1
+< g2 Eq. (28) must hold. On the other hand, from
+Eq. (25) and from the hypothesis that j2 −j1 > 0 we obtain (qc1 −c2)(∆2 −1)−
+α2 > 0 which, since (∆2 − 1) > 0, implies qc1 − c2 >
+α2
+∆2−1. Since 0 ≤
+α2
+∆2−1 < 1
+and qc1 − c2 is integer, we have qc1 − c2 ≥ 1. This implies that Eq. (28) holds
+and therefore that gM
+1
+< g2 and g1 ̸= g2.
+Case 1.d: vg1 and vi2 are spine vertices. By Property 2 we have, for some
+c2 ∈ N:
+i2 = j2 + c2(∆2 − 1).
+(29)
+23
+
+We also have, for some c1 ∈ N and 0 ≤ α1 < ∆2 − 1:
+i1 = j1 + c1(∆1 − 1) + α1 = j1 + qc1(∆2 − 1) + α1.
+(30)
+If vi1 coincides with vi2, we have i1 = i2; from Eq. (30) and Eq. (29) we
+obtain:
+j2 − j1 = (qc1 − c2)(∆2 − 1) + α1.
+(31)
+Concerning vg2, we have:
+gm
+2 ≤ g2 ≤ gM
+2 .
+with gm
+2 = r2 + d2(∆2 − 1), gM
+2
+= r2 + (d2 + 1)(∆2 − 1) for some d2 ∈ N such
+that c2 + d2 =
+� σ2
+2
+�
+− 1, where σ2 is the number of spine vertices of C2.
+Since vg1 is a vertex in the lower part of Γ1, it must be g1 = r1 + d1(∆1 − 1).
+We prove that g1 < gm
+2 , which implies g1 ̸= g2. To have g1 < gm
+2 it must be:
+r1 + d1(∆1 − 1) < r2 + d2(∆2 − 1)
+r1 + qd1(∆2 − 1) < r2 + d2(∆2 − 1)
+r2 − r1 > (qd1 − d2)(∆2 − 1).
+(32)
+Since rl = jl + n−(∆l−1)(σl
+mod 2)
+2
+, for l = 1, 2, Eq. (32) can be rewritten as:
+j2 − j1 >
+�
+(qd1 − d2) + (σ2
+mod 2)
+2
+− q(σ1
+mod 2)
+2
+�
+(∆2 − 1).
+(33)
+Combining Eq. (31) and Eq. (33) we obtain:
+qc1 − c2 +
+α1
+∆2 − 1 > (qd1 − d2) + (σ2
+mod 2)
+2
+− q(σ1
+mod 2)
+2
+.
+Since cl + dl =
+� σl
+2
+�
+− 1, we have dl =
+σl+1(σl
+mod 2)
+2
+− cl − 1, for l = 1, 2;
+replacing d1 and d2 in the previous equation, we obtain:
+qc1 − c2 +
+α1
+∆2 − 1 > qσ1
+2
++ q(σ1
+mod 2)
+2
+− qc1 − q − σ2
+2 − 1(σ2
+mod 2)
+2
++ c2+
++ 1 + σ2
+mod 2
+2
+− q(σ1
+mod 2)
+2
+which, considering that σ2 =
+n−2
+∆2−1 = q(n−2)
+∆1−1 = qσ1, implies:
+qc1 − c2 > 1 − q
+2
+−
+α1
+2(∆2 − 1)
+(34)
+In summary, to have g1 < gm
+2
+Eq. (34) must hold. On the other hand, from
+Eq. (31) and from the hypothesis that j2 −j1 > 0 we obtain (qc1 −c2)(∆2 −1)+
+α1 > 0 which, since (∆2−1) > 0, implies qc1−c2 > −
+α1
+∆2−1. Since 0 ≤
+α1
+∆2−1 < 1
+and qc1 − c2 is integer, we have qc1 − c2 > 0. Since q is a positive integer, this
+implies that Eq. (34) holds and therefore that g1 < gm
+2 and g1 ̸= g2.
+24
+
+Case 2.b: vg1 and vg2 are spine vertices. Since vg2 is a vertex in the lower
+part of Γ2, it must be g2 = r2+d2(∆2−1). If vg2 coincides with vi1, as explained
+above, it must be i1 = g2 − n. Combining the expression of g2 with Eq. (30) we
+obtain:
+r2 − j1 = (qc1 − d2)(∆2 − 1) + α1 + n.
+(35)
+Concerning vi2, we have:
+im
+2 ≤ i2 ≤ iM
+2 ;
+with iM
+2 = j2 + (c2 + 1)(∆2 − 1) for some c2 ∈ N such that c2 + d2 =
+� σ2
+2
+�
+− 1,
+where σ2 is the number of spine vertices of C2.
+We prove that iM
+2 < g1, which implies i2 ̸= g1. To have iM
+2 < g1 it must be:
+j2 + (c2 + 1)(∆2 − 1) < r1 + d1(∆1 − 1)
+j2 + (c2 + 1)(∆2 − 1) < r1 + qd1(∆2 − 1)
+j2 − r1 < (qd1 − c2 − 1)(∆2 − 1).
+(36)
+Since rl = jl + n−(∆l−1)(σl
+mod 2)
+2
+, for l = 1, 2, Eq. (35) can be rewritten as:
+j2 − j1 = (qc1 − d2)(∆2 − 1)/α1 + n
+2 + (∆2 − 1)(σ2
+mod 2)
+2
+,
+(37)
+while Eq. (36) can be rewritten as:
+j2 − j1 <
+�
+(qd1 − c2 − 1) − q(σ1
+mod 2)
+2
+�
+(∆2 − 1) + n
+2 .
+(38)
+Combining Eq. (37) and Eq. (38) we obtain:
+qc1 − d2 < (qd1 − c2 − 1) − (σ2
+mod 2)
+2
+− q(σ1
+mod 2)
+2
+−
+α1
+∆2 − 1.
+Since cl + dl =
+� σl
+2
+�
+− 1, we have dl =
+σl+1(σl
+mod 2)
+2
+− cl − 1, for l = 1, 2;
+replacing d1 and d2 in the previous equation, we obtain:
+qc1 − σ2
+2 − σ2
+mod 2
+2
++ c2 + 1 + σ2
+mod 2
+2
++
+α1
+∆2 − 1 < qσ1
+2
++ q(σ1
+mod 2)
+2
+−
+− qc1 − q − c2 − q(σ1
+mod 2)
+2
+− 1
+which, considering that σ2 =
+n−2
+∆2−1 = q(n−2)
+∆1−1 = qσ1, implies:
+qc1 − qσ1
+2
++ c2 + 1 < −q
+2 −
+α1
+2(∆2 − 1)
+(39)
+In summary, to have iM
+2
+< gm
+1 Eq. (39) must hold. On the other hand, from
+Eq. (37) and from the hypothesis that j2 − j1 <
+n−(∆1−1)
+2
+=
+n−q(∆2−1)
+2
+we
+obtain:
+qc1 − qσ1
+2
++ c2 + 1 < −q
+2 −
+α1
+∆2 − 1.
+(40)
+25
+
+We have that − q
+2 −
+α1
+∆2−1 < − q
+2 −
+α1
+2(∆2−1), since
+α1
+2(∆2−1) <
+1
+2. In other
+words, Eq. (40) implies that Eq. (39) holds and therefore that iM
+2
+< g1 and
+i2 ̸= g1.
+Case 2.c: vi1 and vg2 are spine vertices. Since vg2 is a vertex in the lower
+part of Γ2, it must be g2 = r2+d2(∆2−1). If vg2 coincides with vi1, as explained
+above, it must be i1 = g2 − n. Combining the expression of g2 with Eq. (23) we
+obtain:
+r2 − j1 = (qc1 − d2)(∆2 − 1) + n.
+(41)
+Concerning vg1, we have:
+gm
+1 ≤ g1 ≤ gM
+1 ;
+with gm
+1 = r1 + d1(∆1 − 1), gM
+1
+= r1 + (d1 + 1)(∆1 − 1) for some d1 ∈ N such
+that c1 + d1 =
+� σ1
+2
+�
+− 1, where σ1 is the number of spine vertices of C1.
+We prove that iM
+2 < gm
+1 , which implies i2 ̸= g1. To have iM
+2 < gm
+1 it must be:
+j2 + (c2 + 1)(∆2 − 1) < r1 + d1(∆1 − 1)
+j2 + (c2 + 1)(∆2 − 1) < r1 + qd1(∆2 − 1)
+j2 − r1 < (qd1 − c2 − 1)(∆2 − 1).
+(42)
+Since rl = jl + n−(∆l−1)(σl
+mod 2)
+2
+, for l = 1, 2, Eq. (41) can be rewritten as:
+j2 − j1 = (qc1 − d2)(∆2 − 1) + n
+2 + (∆2 − 1)(σ2
+mod 2)
+2
+,
+(43)
+while Eq. (42) can be rewritten as:
+j2 − j1 <
+�
+(qd1 − c2 − 1) − q(σ1
+mod 2)
+2
+�
+(∆2 − 1) + n
+2 .
+(44)
+Combining Eq. (43) and Eq. (44) we obtain:
+qc1 − d2 < (qd1 − c2 − 1) − (σ2
+mod 2)
+2
+− q(σ1
+mod 2)
+2
+.
+Since cl + dl =
+� σl
+2
+�
+− 1, we have dl =
+σl+1(σl
+mod 2)
+2
+− cl − 1, for l = 1, 2;
+replacing d1 and d2 in the previous equation, we obtain:
+qc1 − σ2
+2 − σ2
+mod 2
+2
++ c2 + 1 < qσ1
+2
++ q(σ1
+mod 2)
+2
+− qc1 − q − c2 − 1−
+− σ2
+mod 2
+2
+− q(σ1
+mod 2)
+2
+which, considering that σ2 =
+n−2
+∆2−1 = q(n−2)
+∆1−1 = qσ1, implies:
+qc1 − qσ1
+2
++ c2 + 1 < −q
+2
+(45)
+26
+
+In summary, to have iM
+2
+< gm
+1 Eq. (45) must hold. On the other hand, from
+Eq. (43) and from the hypothesis that j2 − j1 <
+n−(∆1−1)
+2
+=
+n−q(∆2−1)
+2
+we
+obtain:
+qc1 − d2 + 1
+2(σ2
+mod 2) < −q
+2.
+Replacing again d2 with σ2+1(σ2
+mod 2)
+2
+− c2 − 1, we obtain:
+qc1 − qσ1
+2
++ c2 + 1 < −q
+2.
+(46)
+Since Eq. (45) is equivalent to Eq. (46), we can conclude that Eq. (45) holds
+and therefore that iM
+2 < gm
+1 and i2 ̸= g1.
+Case 2.d: vg1 and vi2 are spine vertices. Since vg1 is a vertex in the lower
+part of Γ1, it must be g1 = r1 + d1(∆1 − 1). If vg1 coincides with vi2, combining
+the expression of g1 with Eq. (6) we obtain:
+j2 − r1 = (qd1 − c2)(∆2 − 1).
+(47)
+Concerning vi1 vg2, we have:
+im
+1 ≤ i1 ≤ iM
+1 ;
+and
+gm
+2 ≤ g2 ≤ gM
+2 .
+with im
+1 = j1 + c1(∆1 − 1) = j1 + qc1(∆2 − 1), gM
+2
+= r2 + (d2 + 1)(∆2 − 1) − n
+for some d2 ∈ N such that c2 + d2 =
+� σ2
+2
+�
+− 1, where σ2 is the number of spine
+vertices of C2.
+We prove that gM
+2
+< im
+1 , which implies g2 ̸= i1. To have gM
+2
+< im
+1 it must be:
+r2 + (d2 + 1)(∆2 − 1) − n < j1 + c1(∆1 − 1)
+r2 + (d2 + 1)(∆2 − 1) − n < j1 + qc1(∆2 − 1)
+r2 − j1 < (qc1 − d2 − 1)(∆2 − 1) + n(∆2 − 1).
+(48)
+Since rl = jl + n−(∆l−1)(σl
+mod 2)
+2
+, for l = 1, 2, Eq. (47) can be rewritten as:
+j2 − j1 = (qd1 − c2)(∆2 − 1) − q(∆2 − 1)(σ1
+mod 2)
+2
++ n
+2 ,
+(49)
+while Eq. (48) can be rewritten as:
+j2 − j1 < (qc1 − d2 − 1)(∆2 − 1) + (∆2 − 1)(σ2
+mod 2)
+2
++ n
+2 .
+(50)
+Combining Eq. (49) and Eq. (50) we obtain:
+qd1 − c2 − q(σ1
+mod 2)
+2
+< (qc1 − d2 − 1) + (σ2
+mod 2)
+2
+.
+27
+
+Since cl + dl =
+� σl
+2
+�
+− 1, we have dl =
+σl+1(σl
+mod 2)
+2
+− cl − 1, for l = 1, 2;
+replacing d1 and d2 in the previous equation, we obtain:
+qσ1
+2
++ q(σ1
+mod 2)
+2
+− qc1 − q − c2 − q(σ1
+mod 2)
+2
+< qc1 − σ2
+2 − σ2
+mod 2
+2
++
++ c2 + 1 − 1 + σ2
+mod 2
+2
+which, considering that σ2 =
+n−2
+∆2−1 = q(n−2)
+∆1−1 = qσ1, implies:
+2
+�
+qc1 − qσ1
+2
++ c2
+�
+> −q
+(51)
+In summary, to have gM
+2
+< im
+1 Eq. (51) must hold. On the other hand, from
+Eq. (49) and from the hypothesis that j2 − j1 <
+n−(∆1−1)
+2
+=
+n−q(∆2−1)
+2
+we
+obtain:
+qd1 − c2 − q
+2(σ1
+mod 2) < −q
+2.
+Replacing again d1 with σ1+1(σ1
+mod 2)
+2
+− c1 − 1, we obtain:
+2
+�
+qc1 − qσ1
+2
++ c2
+�
+> −q.
+(52)
+Since Eq. (51) is equivalent to Eq. (52), we can conclude that Eq. (51) holds
+and therefore that gM
+2
+< im
+1 and g2 ̸= i1.
+B
+Proof of Theorem 7
+Theorem 7. A ∆-regular caterpillar with 4 ≤ ∆ ≤ 7 admits a 2-planar 3-
+placement.
+Proof. Let C1, C2, and C3 be three copies (shown in red, blue and green, respec-
+tively, in Fig. 8) of a ∆-regular caterpillar C with 4 ≤ ∆ ≤ 7. We denote the
+vertices of caterpillar Cj for j = 1, 2, 3 as follows; the spine vertices are denoted
+as vj
+0, vj
+1, . . . , vj
+c−1 in the order they appear along the spine; the leaves adjacent
+to vertex vj
+i (for i = 1, 2, . . . , c−1) are denoted as uj
+i,l with l = 0, 1, . . . , d, where
+d = ∆ − 2 if i = 0 or i = c − 1 and d = ∆ − 3 if 0 < i < c − 1.
+Let p0, p1, . . . , pn−1 be n points on a circle in clockwise order (with indices
+taken modulo n). To construct the packing, we compute a drawing for each
+caterpillar such that the vertices are mapped to points p1, p2, . . . , pn and the
+union of the three drawings is a 2-planar drawing. We describe the construction
+for ∆ = 4, 5, 6 (see also Figs. 8(a) to 8(c)); the construction in the case ∆ = 7
+is slightly different and it is shown in Fig. 8(d).
+Caterpillar C1 is drawn outside the circle so that vertex v1
+0 is mapped to
+point p0, each vertex v1
+i , for i = 1, 2, . . . , c − 1 is mapped to pi(∆−1)+1, each
+leaf u1
+0,l is mapped to the point pl+1, and each leaf u1
+i,l is mapped to the point
+pi(∆−1)+2+l. In other words, each vertex of the spine is followed clockwise by
+28
+
+∆ = 4
+(a)
+∆ = 5
+(b)
+∆ = 6
+(c)
+∆ = 7
+(d)
+Figure 8:
+2-planar 3-placements of ∆-regular caterpillars.
+its leaves and the last of these leaves is followed by the next vertex of the spine.
+Caterpillar C2 is drawn inside the circle so that vertex v2
+i is mapped to the
+point immediately following clockwise the point hosting v1
+i and each leaf u2
+i,l is
+mapped to the point immediately following clockwise u1
+i,l. Clearly, the drawings
+of the first two caterpillars do not cross each other because they are on different
+sides of the circle; also, their union has no multiple edges. Concerning C3, the
+vertex v3
+i , for i = 0, 1, . . . , c−2 is mapped to the point that hosts u1
+i,d and u2
+i,d−1,
+i.e., the last leaf of v1
+i and the second last leaf of v2
+i ; the vertex v3
+c−1 is mapped
+to the point that hosts u1
+i,d−1 and u2
+i,d−2, i.e., the second last leaf of v1
+c−1 and
+the third last leaf of v2
+i . About this mapping, observe that if we draw the edges
+of the spine of C3 outside the circle, each edge of the spine of C3 intersects two
+consecutive edges of the spine of C1 and each edge of the spine of C1 intersects
+at most two consecutive edges of the spine of C3. To complete the drawing, we
+need to draw the leaves of C3. Consider two consecutive spine vertices v3
+i and
+v3
+i+1, with 0 ≤ i ≤ c − 2; between these two vertices there are ∆ − 2 points not
+yet used by C3, we connect the first two of these vertices in clockwise order to
+vi. Depending on the value of ∆, there remain 0, 1, or 2 points between vi and
+vi+1 not yet used by C3; we connect these points to vi+1. Notice that, there
+remain to map ∆ − 3 leaves adjacent to v3
+0 and 3 leaves adjacent to v3
+c−1. On
+the other hand, there are ∆ points not yet used by C3 that are between v3
+c−1
+and v3
+0 clockwise; we connect the three vertices following clockwise v3
+c−1 to v3
+c−1,
+and the remaining ones to v3
+0. All the edges of C3 that are incident to leaves
+are drawn inside the circle. This mapping of C3 does not create multiple edges
+and gives rise to at most two crossings along the edges of C2 and C3.
+29
+

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+Exploring bulk viscous unified scenarios with Gravitational Waves Standard Sirens
+Weiqiang Yang,1, ∗ Supriya Pan,2, 3, † Eleonora Di Valentino,4, ‡
+Celia Escamilla-Rivera,5, § and Andronikos Paliathanasis3, 6, 7, ¶
+1Department of Physics, Liaoning Normal University, Dalian, 116029, P. R. China
+2Department of Mathematics, Presidency University, 86/1 College Street, Kolkata 700073, India
+3Institute of Systems Science, Durban University of Technology,
+PO Box 1334, Durban 4000, Republic of South Africa
+4School of Mathematics and Statistics, University of Sheffield,
+Hounsfield Road, Sheffield S3 7RH, United Kingdom
+5Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,
+Circuito Exterior C.U., A.P. 70-543, M´exico D.F. 04510, M´exico
+6Instituto de Ciencias F´ısicas y Matem´aticas, Universidad Austral de Chile, Valdivia 5090000, Chile
+7Mathematical Physics and Computational Statistics Research Laboratory,
+Department of Environment, Ionian University, Zakinthos 29100, Greece
+We consider the unified bulk viscous scenarios and constrain them using the Cosmic Microwave
+Background observations from Planck 2018 and the Pantheon sample from Type Ia Supernovae.
+Then we generate the luminosity distance measurements from O(103) mock Gravitational Wave
+Standard Sirens (GWSS) events for the proposed Einstein Telescope. We then combine these mock
+luminosity distance measurements from the GWSS with the current cosmological probes in order to
+forecast how the mock GWSS data could be effective in constraining these bulk viscous scenarios.
+Our results show that a non-zero time dependent bulk viscosity in the universe sector is strongly
+preferred by the current cosmological probes and will possibly be confirmed at many standard
+deviations by the future GWSS measurements. We further mention that the addition of GWSS
+data can significantly reduce the uncertainties of the key cosmological parameters obtained from
+the usual cosmological probes employed in this work.
+I.
+INTRODUCTION
+Understanding the nature of dark matter and dark en-
+ergy has been a challenge for cosmologists. The standard
+cosmological model, namely, the so-called Λ-Cold Dark
+Matter (ΛCDM) model representing a mixture of two
+non-interacting fluids − a positive cosmological constant
+(Λ > 0) and a cold dark matter component, has un-
+doubtedly proved its unprecedented success by explain-
+ing a large span of astronomical data.
+However, this
+simplest cosmological scenario has some limitations. For
+example, the cosmological constant problem [1] and the
+coincidence problem [2] have already questioned the ex-
+isting assumptions in the ΛCDM model, e.g. constant
+energy density of the vacuum and the non-interacting na-
+ture between Λ and CDM. These limitations motivated
+the cosmologists to find alternative cosmological scenar-
+ios beyond ΛCDM by relaxing the above assumptions,
+and as a consequence, several new cosmological models
+were introduced, see [3–12] for a review of various dark
+energy and modified gravity models. Additionally, the
+appearance of cosmological tensions at many standard
+deviations between Planck [13] (assuming ΛCDM in the
+background) and other cosmological probes, such as dis-
+tance ladders [14–24] or weak lensing [25–29] and galaxy
+∗ [email protected]
+† [email protected]
+‡ e.divalentino@sheffield.ac.uk
+§ [email protected]
+¶ [email protected]
+cluster data [30–32] has further weakened the confidence
+in the ΛCDM cosmological model [33–37]. Thus, the list
+of cosmological models aiming to address the cosmolog-
+ical tensions is increasing in time, see the review arti-
+cles [38–44] and references therein. Given the fact that
+the origin of dark matter and dark energy is not clearly
+understood yet, thus, there is no reason to favor any par-
+ticular cosmological theory over others. As a result, var-
+ious ways have been proposed to interpret the dynamics
+of the dark sector in terms of dark matter and dark en-
+ergy.
+The simplest assumption is the consideration of
+independent evolution of these dark fluids. The general-
+ization of the above consideration is the assumption of a
+non-gravitational interaction between these dark sectors.
+On the other hand, a heuristic approach is to consider a
+unified dark fluid that can explain the dynamics of dark
+energy and dark matter at cosmological scales. The at-
+tempt to unify the dark sector of the Universe began long
+back ago. The most simplest unified dark sector mod-
+els can be constructed in the context of Einstein gravity
+with the introduction of a generalized equation of state
+p = F(ρ), where p and ρ are respectively the pressure and
+energy density of the unified dark sector and F is an an-
+alytic function of the energy density, ρ. The well known
+unified cosmological models, such as the Chaplygin gas
+model [45] and its successive generalizations, namely, the
+generalized Chaplygin gas, modified Chaplygin gas, see
+Refs. [46–57] and some other unified cosmological scenar-
+ios as well [58–60] belong to this classification. While it
+is essential to mention that a subset of the unified mod-
+els has been diagnosed with exponential blowup in the
+arXiv:2301.03969v1  [astro-ph.CO]  10 Jan 2023
+
+2
+matter power spectrum which is not consistent with the
+observations [61], however, this does not rule out the pos-
+sibility of unified models aiming to cover a wide region
+of the universe evolution because a new kind of unified
+fluid may avoid such unphysical activities. The unified
+cosmological models can also be developed by considering
+a relation like p = G(H) where G is an analytic function
+of H, the Hubble function of the Friedmann-Lemaˆıtre-
+Robertson-Walker (FLRW) line element.
+Apparently,
+theories with p = F(ρ) and p = G(H) seem identical,
+however, this is only true in spatially flat FLRW uni-
+verse. For a curved universe, the two approaches are not
+the same.
+In the present work we are interested to study a partic-
+ular class of unified models endowed with bulk viscosity.
+The cosmological fluids allowing bulk viscosity as an ex-
+tra ingredient can explain the accelerating expansion of
+the universe, and hence they are also enlisted as possi-
+ble alternatives to the standard ΛCDM cosmology in the
+literature [62, 63].
+Following an earlier work Ref. [64]
+where an evidence of non-zero bulk viscosity was pre-
+ferred by the current cosmological probes, in the present
+article, we use the simulated Gravitational Waves Stan-
+dard Sirens (GWSS) measurements from the Einstein
+Telescope [65]1 in order to quantify the improvements
+of the cosmological parameters, if any, from the future
+GWSS measurements. As the gravitational waves (GW)
+have opened a new window for astrophysics and cosmol-
+ogy, therefore, it will be interesting to investigate the
+contribution from the simulated GWSS data, once com-
+bined with the current cosmological probes. This moti-
+vated many investigators to use the mock GWSS data
+matching the expected sensitivity of the Einstein Tele-
+scope to constrain a class of cosmological models, see for
+instance, [66–77]. In particular, the combined analysis of
+simulated GWSS measurements from Einstein Telescope
+and the standard cosmological probes has proven to be
+very effective for a class of cosmological models, in the
+sense that the error bars in the key cosmological param-
+eters of these cosmological models are significantly re-
+duced thanks to the mock GWSS dataset [70, 71, 74, 78–
+81], however, in some specific f(R) theories of gravity, the
+generated mock GWSS from the Einstein Telescope may
+not be very much helpful to give stringent constraints on
+them during its first phase of running
+[82]. Thus, one
+may expect that the constraining power of the Einstein
+Telescope may depend on the underlying cosmological
+model. Aside from the future GWSS measurements from
+the Einstein Telescope, one can also use the simulated
+GWSS measurements from other GW observatories, such
+as, Laser Interferometer Space Antenna (LISA) [83–86]
+and DECi-heltz Interferometer Gravitational wave Ob-
+servatory (DECIGO) [87, 88], TianQin [89]. In this ar-
+ticle, we focus only on the simulated GWSS data from
+1 https://www.einsteintelescope.nl/en/
+Einstein Telescope to constrain the bulk viscous unified
+scenario.
+The paper has been organized as follows: in Sec. II we
+discuss the gravitational equations for the bulk viscous
+scenario. Sec. III describes the observational data that
+we have considered for the analysis in this work. Sec. IV
+presents the observational constraints on the bulk vis-
+cous models, and mainly we discuss how the inclusion
+of gravitational waves data from the Einstein Telescope
+improves the constraints. Finally, in Sec. V we present
+the conclusions.
+II.
+REVISITING THE BULK VISCOUS
+SCENARIOS: BACKGROUND AND
+PERTURBATIONS
+As usual, we consider the homogeneous and isotropic
+space
+time
+described
+by
+the
+Friedmann-Lemaˆıtre-
+Robertson-Walker (FLRW) line element
+ds2 = −dt2 + a2(t)
+�
+dr2
+1 − kr2 + r2 �
+dθ2 + sin2 θdφ2��
+,
+(1)
+where a(t) is the expansion scale factor and k denotes the
+spatial curvature of the universe. For k = 0, −1, +1, we
+have three different geometries of the universe, namely,
+spatially flat, open and closed, respectively. In this paper
+we restrict ourselves to the spatially flat scenario where
+we assume that (i) the gravitational sector is described
+by the Einstein’s gravity, (ii) the matter sector of the uni-
+verse consists of the relativistic radiation, non-relativistic
+baryons and a unified bulk viscous fluid which combines
+the effects of dark matter and dark energy, (iii) all the
+fluids are non-interacting with each other. Within this
+framework, we can write down the gravitational field
+equations as follows (in the units where 8πG = 1)
+H2 = 1
+3ρtot,
+(2)
+2 ˙H + 3H2 = − ptot,
+(3)
+where an overhead dot indicates the derivative with re-
+spect to the cosmic time t; H ≡ ˙a/a is the Hubble ex-
+pansion rate; (ρtot, ptot) = (ρr +ρb +ρu, pr +pb +pu) are
+the total energy density and total pressure of the cosmic
+components in which (ρr, pr), (ρb, pb), (ρu, pu) are the en-
+ergy density and pressure of radiation, baryons and the
+unified fluid, respectively. The conservation equation for
+each fluid follows the usual law ˙ρi + 3H(1 + wi)ρi = 0,
+where the subscript i refers to radiation (i = r), baryons
+(i = b) and the unified fluid (i = u) and wi are the stan-
+dard barotropic state parameters: wr = pr/ρr = 1/3,
+wb = pb/ρb = 0 and wu = pu/ρu = (γ − 1), where γ
+is a constant parameter.
+In general for different val-
+ues of γ, say for instance, γ = 0, we realize a cosmo-
+logical constant-like fluid endowed with the bulk viscos-
+ity and similarly γ = 1 results in a dust-like fluid en-
+dowed with the bulk viscosity.
+As the nature of the
+
+3
+fluid is not clearly understood and as the observational
+data play an effective role to understand this nature,
+thus, in order to be more transparent in this direction
+we consider γ lying in the interval [−3, 3] which includes
+both exotic (pu/ρu = (γ − 1) < −1/3) and non-exotic
+(pu/ρu = (γ − 1) > −1/3 ) fluids. As already mentioned,
+since the unified fluid has a bulk viscosity, therefore, it en-
+joys an effective pressure [90]: peff = pu−uν
+;νη(ρu), where
+uµ
+;µ is the expansion scalar of this fluid and η(ρu) > 0 is
+the coefficient of the bulk viscosity. Thus, in the FLRW
+background, the effective pressure of the bulk viscous
+fluid reduces to
+peff = pu − 3Hη(ρu).
+(4)
+Since there is no unique selection for the bulk viscous
+coefficient, η(ρu), therefore, we consider a well known
+choice for it in which the bulk viscous coefficient has a
+power law evolution of the form [90–92]:
+η(ρu) = αρm
+u ,
+(5)
+where α is a positive constant and m is any real number.
+Notice that for the case m = 0 we recover the scenario
+with a constant bulk viscous coefficient. Now, with the
+consideration of the bulk viscous coefficient in (5), the
+effective pressure of the unified fluid can be expressed as
+peff = (γ − 1)ρu −
+√
+3αρ1/2
+tot ρm
+u ,
+(6)
+and consequently, one can define the effective equation
+of state of the viscous dark fluid as
+weff = peff
+ρu
+= (γ − 1) −
+√
+3αρ1/2
+tot ρm−1
+u
+.
+(7)
+The adiabatic sound speed for the viscous fluid is given
+by
+c2
+a,eff = p′
+eff
+ρ′u
+= weff +
+w′
+eff
+3H(1 + weff).
+(8)
+where the prime denotes the derivative with respect to
+the conformal time τ and H is the conformal Hubble pa-
+rameter, H = aH. Note that depending on the nature of
+weff, c2
+a,eff could be negative, and hence ca,eff could be an
+imaginary quantity. This may invite instabilities in the
+perturbations. Thus, in order to avoid this possible un-
+physical situation, we consider the entropy perturbations
+(non-adiabatic perturbations) in the unified dark fluid
+following the analysis of generalized dark matter [93].
+Now we focus on the evolution of the unified bulk vis-
+cous fluid at the level of perturbations. In the entropy
+perturbation mode, the true pressure perturbation comes
+from the effective pressure given by
+δpeff = δpu − δη(∇σuσ) − η(δ∇σuσ)
+= δpu − 3Hδη − η
+a
+�
+θ + h′
+2
+�
+.
+(9)
+The effective sound speed of viscous dark fluid for the
+bulk viscous coefficient (5) can be defined as
+c2
+s,eff ≡
+�δpeff
+δρu
+�
+rf
+= c2
+s −
+√
+3αmρ1/2
+tot ρm−1
+u
+− αρm−1
+u
+aδu
+�
+θ + h′
+2
+�
+,
+(10)
+where ’|rf’ denotes the rest frame. Following the analysis
+in [93], the sound speed in the rest frame is assumed to
+be zero, i.e. c2
+s = 0.
+The density perturbation and the velocity perturba-
+tion can also be written as [93]
+δ′
+u = −(1 + weff)(θu + h′
+2 ) +
+w′
+eff
+1 + weff
+δu
+− 3H(c2
+s,eff − c2
+a,eff)
+�
+δu + 3H(1 + weff)θD
+k2
+�
+,
+(11)
+θ′
+u = −H(1 − 3c2
+s,eff)θu +
+c2
+s,eff
+1 + weff
+k2δu,
+(12)
+Thus, following the evolution at the background and per-
+turbation level prescribed above, one can now be able to
+understand the dynamics of the bulk viscous fluid. In
+this work we consider two different bulk viscous scenar-
+ios characterized as follows: the bulk viscous model 1
+(labeled as BVF1) where we consider γ = 1, and the
+bulk viscous model 2 (labeled as BVF2) where we keep
+γ as a free parameter. The common parameters in both
+BVF1 and BVF2 are α and m.
+III.
+STANDARD COSMOLOGICAL PROBES,
+SIMULATED GWSS DATA, AND
+METHODOLOGY
+In this section we describe the cosmological data
+sets employed to perform the statistical analyses of the
+present bulk viscous scenarios.
+• Cosmic Microwave Background (CMB): We
+use the CMB data from the Planck 2018 data re-
+lease.
+Precisely, we use the CMB temperature
+and polarization angular power spectra plikTT-
+TEEE+lowl+lowE [94, 95].
+• Pantheon sample from Type Ia Supernovae
+(SNIa) data: Type Ia Supernovae are the first
+astronomical data that probed the accelerating ex-
+pansion of the universe and hence indicated the
+existence of an exotic fluid with negative pressure
+(dark energy). Here we use the Pantheon compila-
+tion of the SNIa data comprising 1048 data points
+spanned in the redshift interval [0.01, 2.3] [96].
+• Gravitational
+Waves
+Standard
+Sirens
+(GWSS): We take the mock Gravitational Waves
+Standard
+Sirens
+(GWSS)
+data
+generated
+by
+matching
+the
+expected
+sensitivity
+of
+Einstein
+
+4
+Telescope in order to understand the constraining
+power of the future GWSS data from the Einstein
+Telescope.
+The Einstein Telescope is a proposed
+ground based third-generation (3G) GW detector.
+The telescope will take a triangular shape and
+its each arm length will be increased to 10 km,
+compared to 3 km arm length VIRGO and 4 km
+arm length LIGO [65, 97]. Thus, due to such in-
+creased arm length, the Einstein Telescope will be
+a potential GW detector by reducing all displace-
+ment noises [65, 97]. It is expected that after 10
+years of operation, Einstein Telescope will detect
+O(103) GWSS events. Although the detection of
+O(103) GWSS events is very optimistic while the
+number of detections could be low in reality [65].
+As argued in [65], the Einstein Telescope will likely
+to detect 20 − 50 events per year, i.e. 200 − 500
+events in 10 years. However, following the earlier
+works [66, 67, 70, 71, 74, 77, 79, 81, 82], in this
+article, we restrict ourselves to the detection of
+O(103) GWSS events by the Einstein Telescope
+to constrain the bulk viscous scenarios. For more
+features of the Einstein Telescope we refer the
+readers to [65].
+We originally generate the mock GWSS luminosity
+distance measurements matching the expected sen-
+sitivity of the Einstein Telescope after 10 years of
+full operation. Specifically we create 1000 triples
+(zi, dL(zi), σi) where zi is the redshift of a GW
+source, dL(zi) is the measured luminosity distance
+at redshift zi and σi is the uncertainty associated
+with the luminosity distance dL(zi). Let us briefly
+summarize the procedure of generating the mock
+GWSS dataset and we refer to Refs. [70, 71, 81]
+for more technical details.
+The initial step for
+generating the mock GWSS dataset is to identify
+the expected GW sources.
+We consider the GW
+events originating from two distinct binary systems,
+namely, (i) a combination of a Black Hole (BH) and
+a Neutron Star (NS) merger, identified as BHNS
+and (ii) the binary neutron star (BNS) merger.
+Following the mass distributions as described in
+Ref. [81], the ratio of the number of GW events for
+the BHNS merger versus BNS merger is taken to be
+0.03 as predicted for the Advanced LIGO-VIRGO
+network [98]. We then determine the merger rate
+R(z) of sources and from the merger rate of the
+sources, we determine the redshift distribution of
+the sources, P(z) given by [66, 67, 70, 79, 81, 99]
+P(z) ∝ 4πd2
+C(z)R(z)
+H(z)(1 + z) ,
+(13)
+where dC(z) ≡
+� z
+0 H−1(z′)dz′ is the co-moving dis-
+tance and for R(z) we take the following piece-
+wise linear function estimated in [100] (also see [66,
+67, 70, 79, 81, 101]): R(z) = 1 + 2z for z ≤ 1,
+R(z) = 3
+4(5 − z), for 1 < z < 5 and R(z) = 0 for
+z > 5. After having P(z), we sample 1000 values
+of redshifts from this distribution which represent
+the redshifts zi of our 1000 mock GWSS data.
+The next step is to choose a fiducial model because
+while going from the merger rate to the redshift dis-
+tributions, a fiducial cosmological model is needed
+since the expression for P(z) includes both the co-
+moving distance and expansion rate at redshift z,
+i.e. dL(z) and H(z) respectively. This H(z) cor-
+responds to the fiducial model.
+As in this arti-
+cle we are interested to investigate how the inclu-
+sion of GWSS data improves the constraints on the
+BVF models, therefore, we generate two different
+mock GWSS datasets choosing BVF1 and BVF2
+as the fiducial models.
+We take the fiducial val-
+ues of the cosmological parameters given by the
+best fit values of the same cosmological parame-
+ters of the BVF1 and BVF2 models obtained from
+the CMB+Pantheon data analysis. Now, for the
+chosen fiducial model(s), one can now estimate the
+luminosity distance at the redshift zi using the re-
+lation
+dL(zi) = (1 + zi)
+� zi
+0
+dz′
+H(z′) .
+(14)
+Thus, after having the luminosity distance dL(zi) of
+the GW source, our last job is now to determine the
+uncertainty σi associated with this luminosity dis-
+tance. The determination of the uncertainty σi di-
+rectly connects to the waveform of GW because the
+GW amplitude depends on the luminosity distance
+(also on the so-called chirp mass [66, 67, 79]) and
+hence one can extract the information about dL(zi)
+and σi. We refer to Refs. [66, 67, 70, 79, 81] for the
+technical details to calculate the uncertainties on
+the luminosity distance measurements. The lumi-
+nosity distance measurement dL(zi) has two kind of
+uncertainties, one is the instrumental uncertainty
+σinst
+i
+and the other one is the weak lensing uncer-
+tainty σlens
+i
+. The instrumental error can be derived
+to be σinst
+i
+(≃ 2dL(zi)/S where S is the combined
+signal-to-noise ratio of the Einstein Telescope) us-
+ing the Fisher matrix approach and assuming that
+the uncertainty on dL(zi) is not correlated with
+the uncertainties on the remaining GW parame-
+ters (see [66, 67, 70, 79, 81]) and the lensing error
+is σlens
+i
+≃ 0.05zidL(zi) [66]. Thus, the total uncer-
+tainty due to the instrumental and the weak lensing
+uncertainties on dL(zi) is σi =
+�
+(σinst
+i
+)2 + (σlens
+i
+)2.
+Finally, let us note that the combined signal-to-
+noise ratio of the GW detector is a very crucial
+quantity in this context since for the Einstein Tele-
+scope, the combined signal-to-noise ratio should be
+
+5
+Parameter Priors (BVF1) Priors (BVF2)
+Ωbh2
+[0.005, 0.1]
+[0.005, 0.1]
+τ
+[0.01, 0.8]
+[0.01, 0.8]
+ns
+[0.5, 1.5]
+[0.5, 1.5]
+ln(1010As)
+[2.4, 4]
+[2.4, 4]
+100θMC
+[0.5, 10]
+[0.5, 10]
+β
+[0, 1]
+[0, 1]
+m
+[−2, 0.5]
+[−2, 0.5]
+γ
+−
+[−3, 3]
+TABLE I. We show the flat priors on all the free parameters
+associated with the bulk viscous models.
+at least 8 for a GW detection [99]. Thus, in sum-
+mary, we generate 1000 GW sources up to redshift
+z = 2 with S > 8. For more technical details we
+refer the readers to Refs. [66, 67, 70, 79, 81, 99].
+To constrain the BVF scenarios we modify the pub-
+licly available CosmoMC package [102] which is an excel-
+lent cosmological code supporting the Planck 2018 like-
+lihood [95] and it has a convergence diagnostic follow-
+ing the Gelman-Rubin statistic R − 1 [103]. It is essen-
+tial to mention that for both BVF1 and BVF2 scenarios,
+we have used the dimensionless quantity β = αH0ρm−1
+tot,0
+where ρtot,0 is the present value of ρtot. We further men-
+tion here that β = 0 (equivalently, α = 0) implies no vis-
+cosity and hence the overall picture behaves like a unified
+cosmic fluid without bulk viscosity. Thus, in summary,
+the parameter space of BVF1 and BVF2 are as below:
+PBVF1 ≡ {Ωbh2, 100θMC, τ, ns, ln(1010As), β, m}
+PBVF2 = {Ωbh2, 100θMC, τ, ns, ln(1010As), β, m, γ}
+where the description of the free parameters are as fol-
+lows: Ωbh2 is the baryons density, 100θMC is the ratio
+of the sound horizon to the angular diameter distance;
+τ is the optical depth, ns is the scalar spectral index,
+As is the amplitude of the initial power spectrum. The
+flat priors on both cosmological scenarios are shown in
+Table I.
+IV.
+OBSERVATIONAL CONSTRAINTS:
+RESULTS AND ANALYSIS
+In this section we present the constraints on the
+bulk viscous scenarios considering CMB+Pantheon and
+CMB+Pantheon+GWSS. As we are interested to esti-
+mate the improvement of the cosmological parameters in
+presence of the GWSS measurements, and as the com-
+bined standard cosmological probes offer the most strin-
+gent constraints on the cosmological parameters, there-
+fore, the inclusion of GWSS with the combined standard
+cosmological probes is reasonable. As mentioned earlier,
+the key common free parameters of BVF1 and BVF2 are
+β and m, since β ̸= 0 indicates the preference for a non-
+zero bulk viscosity and m ̸= 0 indicates that the coeffi-
+cient of the bulk viscosity is not constant in the redshift
+range considered. In the following subsections we discuss
+the constraints on these two scenarios in detail.
+A.
+Constraints on the BVF1 scenario
+In
+Table
+II
+we
+have
+presented
+the
+constraints
+on
+the
+BVF1
+scenario
+for
+CMB+Pantheon
+and
+CMB+Pantheon+GWSS. The latter dataset is aimed to
+understand the improvement expected from GWSS on
+the constraints from CMB+Pantheon. In Fig. 1 we have
+compared these datasets graphically by showing the one
+dimensional marginalized distribution of some model pa-
+rameters and the two dimensional joint contours.
+As
+discussed, this scenario has two main key parameters,
+namely, β, quantifying the existence of bulk viscosity in
+the cosmic sector, and m, which tells us whether the bulk
+viscosity will have a dynamical nature (corresponding to
+m ̸= 0) or not.
+Since for CMB+Pantheon, we find an evidence for a
+non-zero bulk viscosity in the cosmic sector at many
+standard deviations, i.e. β = 0.430+0.017
+−0.016 at 68% CL,
+this is further strengthen for CMB+Pantheon+GWSS,
+where β = 0.4262+0.0079
+−0.0078 at 68% CL.2 One can clearly
+see that the inclusion of GWSS to CMB+Pantheon im-
+proves the error bars on β by a factor of at least 2.
+This reflects the constraining power of GWSS. On the
+other hand, focusing on the parameter m, which quanti-
+fies the time evolution of the bulk viscosity, we see that
+m remains non-zero at several standard deviations for
+CMB+Pantheon, where the 68% CL constraint on m
+is m = −0.557+0.068
+−0.059, and becomes m = −0.519+0.038
+−0.035
+for CMB+Pantheon+GWSS. From the constraints on
+m, one can clearly see that the uncertainty in m is re-
+duced by a factor of ∼ 1.7 − 1.8 when the GWSS data
+are included with the combined dataset CMB+Pantheon.
+Concerning the Hubble constant, we find that H0 as-
+sumes slightly higher values compared to the ΛCDM
+based Planck. Actually, we have H0 = 68.1+1.2
+−1.1 at 68%
+CL for CMB+Pantheon, while H0 = 68.30+0.46
+−0.45 at 68%
+CL for CMB+Pantheon+GWSS, again improving the
+uncertainty in H0 by a factor of 2.5. This shows that the
+effects of GWSS are clearly visible through these param-
+eters. In Fig. 1, one can compare the constraints on the
+model parameters obtained from CMB+Pantheon and
+CMB+Pantheon+GWSS.
+2 It is worthwhile to note here that the mean value of β is not
+significantly changed when the GWSS data are included, because
+we built the simulated data using the best fit obtained from
+CMB+Pantheon.
+
+6
+Parameters
+CMB+Pantheon
+CMB+Pantheon+GWSS
+Ωbh2
+0.02232+0.00015+0.00029
+−0.00015−0.00028
+0.02253+0.00014+0.00028
+−0.00014−0.00026
+100θMC
+1.02780+0.00058+0.0011
+−0.00055−0.0011
+1.02808+0.00037+0.00073
+−0.00038−0.00073
+τ
+0.0537+0.0074+0.016
+−0.0075−0.015
+0.0567+0.0079+0.016
+−0.0078−0.015
+ns
+0.9641+0.0043+0.0086
+−0.0043−0.0084
+0.9686+0.0041+0.0080
+−0.0040−0.0080
+ln(1010As)
+3.046+0.016+0.031
+−0.015−0.031
+3.048+0.016+0.033
+−0.016−0.033
+β
+0.430+0.017+0.033
+−0.016−0.034
+0.4262+0.0079+0.016
+−0.0078−0.015
+m
+−0.557+0.068+0.12
+−0.059−0.13
+−0.519+0.038+0.074
+−0.035−0.075
+H0
+68.1+1.2+2.2
+−1.1−2.3
+68.30+0.46+0.91
+−0.45−0.85
+TABLE II. We report the observational constraints on the BVF1 scenario at 68% and 95% CL for CMB+Pantheon and
+CMB+Pantheon+GWSS datasets.
+64
+66
+68
+70
+72
+H0
+−0.8
+−0.7
+−0.6
+−0.5
+−0.4
+m
+0.022
+0.0228
+Ωbh 2
+0.36
+0.39
+0.42
+0.45
+0.48
+β
+64
+66
+68
+70
+72
+H0
+0.8
+0.7
+0.6
+0.5
+0.4
+m
+0.0220
+0.0228
+Ωbh 2
+BVF1: CMB+Pantheon
+BVF1: CMB+Pantheon+GWSS
+FIG. 1.
+For the BVF1 scenario we show the 1-dimensional posterior distribution of some model parameters and the 2-
+dimensional joint contours of the model parameters at 68% and 95% CL for CMB+Pantheon and CMB+Pantheon+GWSS.
+Finally, through Fig. 2 we examine how the model af-
+fects the CMB TT power spectrum for different values
+of the model parameters, β and m with respect to the
+standard ΛCDM scenario. In the upper panel of Fig. 2
+we depict the evolution in the CMB TT power spectrum
+for different values of β while in the lower panel of Fig. 2
+we depict the evolution in the CMB TT power spectrum
+for different values of m. From both the graphs, we no-
+tice that as long as β or m increases, the model exhibits
+significant differences in the lower multipoles (ℓ ≤ 10).
+For ℓ ≥ 10, we observe that with the increasing values
+of β and m, the peaks of the CMB TT power spectrum
+increase significantly, particularly changing their mutual
+ratio.
+
+7
+Parameters
+CMB+Pantheon
+CMB+Pantheon+GWSS
+Ωbh2
+0.02241+0.00016+0.00032
+−0.00016−0.00032
+0.02238+0.00015+0.00030
+−0.00016−0.00031
+100θMC
+1.02907+0.00111+0.00180
+−0.00082−0.00198
+1.02921+0.00041+0.00080
+−0.00040−0.00079
+τ
+0.0516+0.0074+0.015
+−0.0072−0.015
+0.0521+0.0071+0.015
+−0.0078−0.014
+ns
+0.9575+0.0053+0.012
+−0.0066−0.012
+0.9583+0.0038+0.0075
+−0.0039−0.0077
+ln(1010As)
+3.038+0.016+0.032
+−0.017−0.032
+3.040+0.015+0.032
+−0.015−0.031
+β
+0.447+0.022+0.042
+−0.022−0.042
+0.425+0.018+0.032
+−0.016−0.034
+m
+−0.85+0.30+0.46
+−0.19−0.50
+−0.683+0.099+0.18
+−0.089−0.19
+γ
+0.9970+0.0015+0.0042
+−0.0024−0.0036
+0.99757+0.00049+0.0011
+−0.00058−0.0011
+H0
+65.2+1.7+4.4
+−2.6−3.9
+64.91+0.59+1.1
+−0.60−1.2
+TABLE III. We report the observational constraints on the BVF2 scenario at 68% and 95% CL for CMB+Pantheon and
+CMB+Pantheon+GWSS.
+101
+102
+103
+l
+0
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+l(l + 1)C TT
+l /(2π)[µK 2]
+ΛCDM
+BVF1 : β = 0.1
+BVF1 : β = 0.3
+BVF1 : β = 0.5
+101
+102
+103
+l
+0
+2000
+4000
+6000
+8000
+10000
+12000
+l(l + 1)C TT
+l /(2π)[µK 2]
+ΛCDM
+BVF1 : m = − 1.5
+BVF1 : m = − 0.5
+BVF1 : m = 0.5
+FIG. 2.
+The CMB CT T
+l
+power spectrum versus multipole
+moment l using the best fits values obtained for the BVF1
+model using the join data sets described, with three arbitrary
+β and m values.
+B.
+Constraints on the BVF2 scenario
+In Table III we present the constraints on the
+BVF2
+scenario
+for
+both
+CMB+Pantheon
+and
+CMB+Pantheon+GWSS.
+And
+in
+Fig.
+3,
+we
+com-
+pare the constraints from these datasets explicitly
+showing the one dimensional marginalized distribution
+of some model parameters and the two dimensional joint
+contours. As already discussed, this scenario has three
+main key parameters, namely, β, which quantifies the
+existence of bulk viscosity in the cosmic sector, m, which
+tells us whether the bulk viscosity enjoys a dynamical
+nature (corresponding to m ̸= 0) or not, and finally, the
+parameter γ, which indicates the fluid which endows
+the bulk viscosity.
+We note that γ = 1 refers to the
+dust fluid endowing the bulk viscosity in which we are
+interested in, for which we recover the previous scenario
+BVF1.
+For CMB+Pantheon, we find that β ̸= 0 at several
+standard deviations yielding β = 0.447 ± 0.022 at 68%
+CL which gives a clear indication of a non-zero bulk
+viscosity in the cosmic sector.
+When the GWSS are
+added to this combination, i.e. CMB+Pantheon+GWSS,
+the conclusion about β does not change significantly
+(β = 0.425+0.018
+−0.016 at 68% CL), indicating that for this
+scenario GWSS do not provide any additional constrain-
+ing power on β. Looking at the dynamical nature of the
+bulk viscosity, we see that for CMB+Pantheon, m re-
+mains nonzero at more than 2 standard deviations lead-
+ing to m = −0.85+0.30
+−0.19 at 68% CL. However, this evi-
+dence could be further strengthened by the inclusion of
+the GWSS data, that we forecast to be m = −0.683+0.099
+−0.089
+at 68% CL for CMB+Pantheon+GWSS, improving the
+error bars up to a factor of 3. Finally, focusing on the pa-
+rameter γ which directly connects with the nature of the
+cosmic fluid endowing the bulk viscosity, we can see that
+it is consistent with 1, which corresponds to a dust-like
+fluid, within 2 standard deviations for CMB+Pantheon
+(γ = 0.9970+0.0015
+−0.0024 at 68% CL). Also for this param-
+eter, the addition of the GWSS further improves the
+constraining power of a factor larger than 3 to 4, that
+we forecast to be γ = 0.99757+0.00049
+−0.00058 at 68% CL for
+CMB+Pantheon+GWSS. Therefore, with respect to the
+BVF1 case, where the inclusion of the forecasted GWSS
+was able to improve both β and m, in this BVF2 scenario,
+the improvement of the constraining power is displayed
+only on m and γ but does not affect anymore β signifi-
+
+8
+60
+64
+68
+72
+H0
+−1.6
+−1.2
+−0.8
+−0.4
+m
+0.993
+0.999
+1.005
+γ
+0.022
+0.0228
+Ωbh 2
+0.36 0.40 0.44 0.48 0.52
+β
+60
+64
+68
+72
+H0
+1.6
+1.2
+0.8
+0.4
+m
+0.993
+0.999
+1.005
+γ
+0.0220
+0.0228
+Ωbh 2
+BVF2: CMB+Pantheon
+BVF2: CMB+Pantheon+GWSS
+FIG. 3.
+For the BVF2 scenario we show the 1-dimensional posterior distributions of some model parameters and the 2-
+dimensional joint contours of the model parameters at 68% and 95% CL for CMB+Pantheon and CMB+Pantheon+GWSS.
+cantly.
+Furthermore, we find that for this scenario, the Hub-
+ble constant attains a very low value for CMB+Pantheon
+compared to Planck’s estimation within the ΛCDM
+paradigm. We also note that H0 is correlated to all three
+free parameters of this scenario, namely, β, m and γ.
+With β and γ, H0 is positively correlated while with m,
+it has a strong anti-correlation.
+For CMB+Pantheon,
+H0 = 65.2+1.7
+−2.6 km/s/Mpc at 68% CL and after the in-
+clusion of GWSS it becomes H0 = 64.91+0.59
+−0.60 km/s/Mpc
+at 68% CL, reducing the uncertainty in H0 by a factor
+of ∼ 4.
+Finally, in Fig. 4, we examine the CMB TT power
+spectrum for this bulk viscous scenario BVF2 consider-
+ing different values of the free parameter γ with respect
+to the standard ΛCDM scenario. As γ lies in the region
+[−3, 3] and the nature of the cosmic fluid characterized by
+its equation of state pu = (γ−1)ρu depends on the sign of
+γ, therefore, we have considered two separate plots, one
+where γ is non-negative (i.e. γ ≥ 0) and another plot
+where γ allows both positive and negative values includ-
+ing γ = 0. From both the panels of Fig. 4, we clearly see
+that any deviation from γ = 1 makes significant changes
+in the amplitude of the CMB TT power spectrum. In
+particular, we see that the peaks of the CMB TT spec-
+trum significantly increases and shift towards higher mul-
+tipoles for any value different from γ = 1 at small scales,
+as well as the Integrated Sachs Wolfe (ISW) plateau at
+large scales. As γ = 1 indicates a cosmological constant-
+like fluid endowed with the bulk viscosity, therefore, for
+γ = 1, we replicate an equivalent behaviour of the ΛCDM
+scenario.
+V.
+CONCLUSIONS
+Although the ΛCDM cosmological model is extremely
+successful in describing a large span of astronomical ob-
+servations, it cannot explain several theoretical and ob-
+servational issues.
+This motivated the scientific com-
+munity to construct a variety of cosmological proposals
+and testing them with the available astronomical data.
+Among these cosmological models, in this article, we fo-
+cus on the unified cosmological models allowing bulk vis-
+
+9
+101
+102
+103
+l
+0
+2000
+4000
+6000
+8000
+10000
+12000
+l(l + 1)C TT
+l /(2π)[µK 2]
+ΛCDM
+BVF2 : γ = − 1
+BVF2 : γ = 0
+BVF2 : γ = 1
+101
+102
+103
+l
+0
+2000
+4000
+6000
+8000
+10000
+12000
+l(l + 1)C TT
+l /(2π)[µK 2]
+ΛCDM
+BVF2 : γ = 0
+BVF2 : γ = 0.5
+BVF2 : γ = 1
+FIG. 4.
+The CMB CT T
+l
+power spectrum versus multipole
+moment l using the best fits values obtained for each BVF2
+models with three γ values, respectively using the join data
+sets described.
+cosity in the background. However, since these models
+do not recover the ΛCDM scenario as a special case, our
+only ability in distinguishing them, once the GWSS data
+will be available, will rely only on a Bayesian model com-
+parison for a better fit of the cosmological observations,
+as done in Ref. [64].
+The unified cosmological scenar-
+ios endowed with bulk viscosity are appealing from two
+different perspectives: the first one is the concept of a
+unified picture of dark matter and dark energy, and the
+second is the inclusion of bulk viscosity into that unified
+picture. Effectively, the unified bulk viscous scenario is
+a generalized cosmic picture combining two distinct cos-
+mological directions. According to a recent paper on the
+unified bulk viscous scenarios [64], current cosmological
+probes prefer a non-zero dynamical bulk viscosity in the
+dark sector at many standard deviations. So, in light of
+the current cosmological probes, unified bulk viscous cos-
+mological scenarios are attractive. In this line of thought,
+what about the future of such unified bulk viscous sce-
+narios?
+In this article we have focused on it with an
+answer.
+Following Ref. [64], in this work we have explored
+these scenarios with the GWSS aiming to understand
+how the future distance measurements from GWSS may
+improve the constraints on such scenarios. In order to
+proceed toward this confrontation, we have generated
+O(103) mock GWSS luminosity distance measurements
+matching the expected sensitivity of the Einstein Tele-
+scope and added these mock data to the current cosmo-
+logical probes, namely CMB from Planck 2018 release3
+and SNIa Pantheon sample. We find that the inclusion of
+GWSS luminosity distance measurements together with
+the current cosmological probes makes the possible fu-
+ture evidence for new physics stronger, by reducing the
+uncertainty in the parameters in a significant way. This
+is a potential behaviour of the GWSS luminosity distance
+measurements since this makes the parameter much de-
+terministic. Overall for both BVF1 and BVF2 scenar-
+ios, we find a very strong preference of a non-zero time
+dependent bulk viscous coefficient (alternatively, the vis-
+cous nature of the unified dark fluid) at many standard
+deviations.
+In conclusion, in the present paper we demonstrate
+that future GWSS distance measurements from the Ein-
+stein Telescope might be powerful to extract more infor-
+mation about the physics of the dark sector. Therefore,
+based on the present results, we feel that it might just
+be a matter of time before we convincingly detect the
+viscosity in the dark sector, if any.
+VI.
+ACKNOWLEDGMENTS
+The authors thank the referee for some important com-
+ments which helped us to improve the article consider-
+ably. WY was supported by the National Natural Science
+Foundation of China under Grants No. 12175096 and No.
+11705079, and Liaoning Revitalization Talents Program
+under Grant no. XLYC1907098. SP acknowledges the
+financial support from the Department of Science and
+Technology (DST), Govt.
+of India, under the Scheme
+“Fund for Improvement of S&T Infrastructure (FIST)”
+[File No.
+SR/FST/MS-I/2019/41].
+EDV is supported
+by a Royal Society Dorothy Hodgkin Research Fellow-
+ship. CE-R is supported by the Royal Astronomical So-
+ciety as FRAS 10147 and by PAPIIT UNAM Project
+TA100122. This article/publication is based upon work
+from COST Action CA21136 Addressing observational
+tensions in cosmology with systematics and fundamen-
+tal physics (CosmoVerse) supported by COST (European
+Cooperation in Science and Technology). AP was sup-
+ported in part by the National Research Foundation of
+South Africa (Grant Number 131604). Also AP thanks
+the support of Vicerrector´ıa de Investigaci´on y Desarrollo
+Tecnol´ogico (Vridt) at Universidad Cat´olica del Norte
+3 We mention that in the earlier work [64], CMB data from Planck
+2015 were used to constrain the bulk viscous scenarios.
+
+10
+through N´ucleo de Investigaci´on Geometr´ıa Diferencial
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+SUBMISSION TO IEEE TRANSACTION ON MULTIMEDIA
+1
+Multi-Stage Spatio-Temporal Aggregation
+Transformer for Video Person
+Re-identification
+Ziyi Tang, Ruimao Zhang, Member, IEEE, Zhanglin Peng, Jinrui Chen, Liang Lin, Senior
+Member, IEEE
+Abstract—In recent years, the Transformer architec-
+ture has shown its superiority in the video-based person
+re-identification task. Inspired by video representation
+learning, these methods mainly focus on designing mod-
+ules to extract informative spatial and temporal fea-
+tures. However, they are still limited in extracting local
+attributes and global identity information, which are
+critical for the person re-identification task. In this paper,
+we propose a novel Multi-Stage Spatial-Temporal Aggre-
+gation Transformer (MSTAT) with two novel designed
+proxy embedding modules to address the above issue.
+Specifically, MSTAT consists of three stages to encode
+the attribute-associated, the identity-associated, and the
+attribute-identity-associated information from the video
+clips, respectively, achieving the holistic perception of
+the input person. We combine the outputs of all the
+stages for the final identification. In practice, to save the
+computational cost, the Spatial-Temporal Aggregation
+(STA) modules are first adopted in each stage to conduct
+the self-attention operations along the spatial and tem-
+poral dimensions separately. We further introduce the
+Attribute-Aware and Identity-Aware Proxy embedding
+modules (AAP and IAP) to extract the informative
+and discriminative feature representations at different
+stages. All of them are realized by employing newly
+designed self-attention operations with specific meanings.
+Moreover, temporal patch shuffling is also introduced to
+further improve the robustness of the model. Extensive
+experimental results demonstrate the effectiveness of
+the proposed modules in extracting the informative and
+discriminative information from the videos, and illustrate
+the MSTAT can achieve state-of-the-art accuracies on
+various standard benchmarks.
+Index
+Terms—Video-based
+Person
+Re-ID,
+Trans-
+former, Spatial Temporal Modeling, Deep Representation
+Learning
+I. INTRODUCTION
+P
+ERSON Re-identification (re-ID) [6], [26], [28],
+which aims at matching pedestrians across dif-
+ferent camera views at different times, is a critical
+Ziyi Tang, Ruimao Zhang, and Jinrui Chen are with The Chi-
+nese University of Hong Kong (Shenzhen), and Ziyi Tang is
+also with Sun Yat-sen University (e-mail: [email protected],
+[email protected], and [email protected] ).
+Zhanglin Peng is with the Department of Computer Science,
+The University of Hong Kong, Hong Kong, China (e-mail:
+[email protected] ).
+Liang Lin is with the School of Computer Science and Engineer-
+ing, Sun Yat-sen University (e-mail: [email protected]).
+This paper was done when Ziyi Tang was working as a Research
+Assistant at The Chinese University of Hong Kong (Shenzhen).
+The Corresponding Author is Ruimao Zhang
+Fig. 1: Comparison between different Transformer-
+based frameworks for video re-ID. (a) shows the
+framework where the Transformer fuse post-CNN fea-
+tures of the entire video. (b) is Trigeminal Trans-
+former [51], including three separate streams for tem-
+poral, spatial, and spatio-temporal feature extraction.
+(c) displays a multi-stage spatio-temporal aggregation
+Transformer, which consists of three stages, all with a
+spatio-temporal view but different meanings.
+task of visual surveillance. In the earlier stage, the
+studies have mainly focused on image-based person
+re-ID [26], [28], [46], which mine the discrimina-
+tive information in the spatial domain. With the de-
+velopment of the monitoring sensors, multi-modality
+information has been introduced to re-ID task [33],
+[71], [72]. Numerous methods have been proposed to
+break down barriers between modalities regarding their
+image styles [86], structural features [81], [84], [97],
+or network parameters [33], [82].
+On the other hand, some studies have exploited
+multi-frame data and proposed various schemes [40],
+[62], [100] to extract informative temporal represen-
+tations to pursue video-based person re-ID. In such a
+setting, each time a non-labeled query tracklet clip is
+given, its discriminative feature representation needs to
+be extracted to retrieve the clips of the corresponding
+person in the non-labeled gallery. In practice, how to
+simultaneously extract such discriminative information
+arXiv:2301.00531v1  [cs.CV]  2 Jan 2023
+
+Cross-viewTransfomer
+Spatio-temporalTransfomer
+Temporal
+Spatial
+Spatio-
+temporal
+Transfomer
+Transfomer
+Transformer
+CNN
+CNN
+(a)Post-fusion Transformer
+(b)TrigeminalTransformer
+i Concatenation
+个
+企
+Space-related module/
+Attribute-Aware
+Identity-Aware
+Identity-Aware
+featurerepresentation
+Proxy Embedding
+ProxyEmbedding
+ProxyEmbedding
+Module
+Module
+Module
+Time-related module/
+featurerepresentation
+AttributeSpatio
+Space-time-relatedmodule/
+Spatio-Temporal
+Spatio-Temporal
+Temporal
+featurerepresentation
+Aggregation
+Aggregation
+Aggregation
+Attribute-relatedmodule
+Attribute-Associated
+Identity-Associated
+Attribute-ldentity-
+/featurerepresentation
+Stage
+Stage
+AssociatedStage
+Identity-related module
+(c) MSTATTANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION
+2
+from spatial and temporal dimensions is the key to
+improving the accuracy of video-based re-ID.
+To address such an issue, traditional methods [20]
+usually employ hierarchically convolutional architec-
+tures to update local patterns progressively. Further-
+more, some attempts [14], [15], [48], [73], [94] adopt
+attention-based modules to dynamically infer discrim-
+inative information from videos. For instance, Wu et
+al. [72] embed body part prior knowledge inside the
+network architecture via dense and non-local region-
+based attention. Although recent years have witnessed
+the success of convolution-based methods [12], [13],
+[20], [38], [43], [74], [94], [104], they have encoun-
+tered a bottleneck of accuracy improvement, as con-
+volution layers suffer from their intrinsic limitations
+of spatial-temporal dependency modeling and infor-
+mation aggregation [96].
+Recently, the Transformer architecture [24], [32],
+[54], [89] has attracted much attention in the com-
+puter vision area because of its excellent context
+modeling ability. The core idea of such a model is
+to construct interrelationships between local contents
+via global attention operation. In the literature, some
+hybrid network architectures [19], [34], [51] have
+been proposed to tackle long-range context modeling
+in video-based re-ID. A widely used paradigm is
+to leverage Transformer as the post-processing unit,
+coupled with a convolutional neural network (CNN)
+as the basic feature extractor. For example, as sum-
+marized in Fig. 1 (a), He et al. [35] and Zhang
+et al. [95] adopt a monolithic Transformer to fuse
+frame-level CNN feature. As shown in Fig. 1 (b),
+Liu et al. [51] take a step further and put forward
+multi-stream Transformer architecture in which each
+stream emphasizes a particular dimension of the video
+features. In a hybrid architecture, however, the 2D
+CNN bottom encoder restricts the long-range spatio-
+temporal interactions among local contents, which
+hinders the discovery of contextual cues. Later, to
+address this problem, some pure Transformer-based
+approaches are introduced to video-based re-ID. Nev-
+ertheless, the existing Transformer-based frameworks
+are mainly motivated by those in video understanding
+and concentrate on designing the architecture to learn
+spatial-temporal representations efficiently. Most of
+them are still limited in extracting informative and
+human-relevant discriminative information from the
+video clips, which are critical for large-scale matching
+tasks [39], [92], [98], [104].
+To address the above issues, we propose a novel
+Multi-stage
+Spatial-Temporal
+Aggregation
+Trans-
+former framework, named MSTAT, which consists
+of three stages to respectively encode the attribute-
+associated, the identity-associated, and the attribute-
+identity-associated
+information
+from
+video
+clips.
+Firstly, to save the computational cost, the Spatial-
+Temporal Aggregation (STA) modules [4], [7] are
+firstly adopted in each stage as their building blocks
+to conduct the self-attention operations along the
+spatial and temporal dimensions separately. Further,
+as shown in Fig. 1, we introduce the plug-and-play
+Attribute-Aware Proxy and Identity-Aware Proxy
+(AAP and IAP) embedding modules into different
+stages, for the purpose of reserving informative at-
+tribute features and aggregating discriminative identity
+features respectively. They are both implemented by
+self-attention operations but with different learnable
+proxy embedding schemes. For the AAP embedding
+module, AAPs play the role of attribute queries to
+reserve a diversity of implicit attributes of a person.
+Arguably, the combination of these attribute repre-
+sentations is informative and provides discriminative
+power, complementary to the identity-only prediction.
+In contrast, the IAP embedding module maintains a
+group of IAPs as key-value pairs. With explicit con-
+straints, they learn to successively match and aggregate
+the discriminative identity-aware features embedded
+in patch tokens. During similarity measurement, the
+output feature representations of the three stages are
+concatenated to form a holistic view of the input
+person.
+In practice, a Transformer-specific data augmenta-
+tion scheme, Temporal Patch Shuffling, is also intro-
+duced, which randomly rearranges the patches tem-
+porally. With such a scheme, the enriched training
+data effectively improve the ability to learn invariant
+appearance features, leading to the robustness of the
+model. Extensive experiments on three public bench-
+marks demonstrate our proposed framework is superior
+to the state-of-the-art on different metrics. Concretely,
+we achieve the best performance of 91.8% rank-1
+accuracy on MARS, which is the largest video re-ID
+dataset at present.
+In summary, our contributions are three-fold. (1) We
+introduce a Multi-stage Spatial-Temporal Aggregation
+Transformer framework (MSTAT) for video-based per-
+son re-ID. Compared to existing Transformer-based
+frameworks, MSTAT better learns informative attribute
+features and discriminative identity features. (2) For
+different stages, we devise two different proxy embed-
+ding modules, named Attribute-Aware and Identity-
+Aware Proxy embedding modules, to extract infor-
+mative attribute features and aggregate discriminative
+identity features from the entire video, respectively.
+(3) A simple yet effective data augmentation scheme,
+referred to as Temporal Patch Shuffling, is proposed
+to consolidate the network’s invariance to appearance
+shifts and enrich training data.
+II. RELATED WORKS
+A. Image-Based Person Re-ID
+Image-based person re-ID mainly focuses on person
+representation learning. Early works focus primarily
+on carefully designed handcraft features [6], [26], [28],
+[44], [46], [103]. Recently, The flourishing deep learn-
+ing has become the mainstream method for learning
+
+3
+IEEE TRANSACTIONS ON MULTIMEDIA
+representation in person ReID [43], [65], [67], [74],
+[77], [88]. For the last few years, CNN has been a
+widely-used feature extractor [1], [17], [41], [43]–[45],
+[65], [76], [94]. OSNet [104] fuses multi-scale features
+in an attention-style sub-network to obtain informative
+omni-scale features. Some works
+[18], [87], [98]
+focus on extracting and aligning semantic information
+to address misalignment caused by pose/viewpoint
+variations, imperfect person detection, etc. To avoid
+the misleading by noisy labels, Ye et al. [83] presents
+a self-label refining strategy, deeply integrating anno-
+tation optimization and network training. So far, some
+works [19], [34] also explore Image-based person re-
+ID based on Vision Transformer [24]. For example,
+TransReID [34] adopts Transformer as the backbone
+and extracts discriminative features from randomly
+sampled patch groups.
+B. Video-Based Person ReID
+Compared to image-based person re-ID, video-
+based person re-ID usually performs better because it
+provides temporal information and mitigates occlusion
+by taking advantage of multi-frame information. For
+capturing more robust and discriminative representa-
+tion from frame sequences, traditional video-based re-
+ID methods usually focus on two areas: 1) encoding
+of temporal information; 2) aggregation of temporal
+information.
+To encode additional temporal information, early
+methods [40], [62], [100] directly use temporal infor-
+mation as additional features. Some works [1], [49],
+[55], [73] use recurrent models, e.g., RNNs [56] and
+LSTM [37], to process the temporal information. Some
+other works [1], [12], [13], [53], [55], [60], [105]
+go further by introducing the attention mechanism to
+apply dynamic temporal feature fusion. Another class
+of works [21] introduces optical flow that captures
+temporal motion. What is more, some works [2],
+[42], [63], [75], [91], [102] directly implement spatio-
+temporal pooling to video sequences and generate a
+global representation via CNNs. Recently, 3D CNNs
+[29], [45] learn to encode video features in a joint
+spatio-temporal manner. M3D [41] endows 2D CNN
+with multi-scale temporal feature extraction ability via
+multi-scale 3D convolutional kernels.
+For the sake of aggregation that aims to generate
+discriminative features from full video features, a
+class of approaches [55], [93], [105] applies average
+pooling on the time dimension to aggregate spatio-
+temporal feature maps. Recently, some attention-based
+methods [2], [15], [72], [80] attained significant per-
+formance improvement by dynamically highlighting
+different video frames/regions so as to filter more dis-
+criminative features from these critical frames/regions.
+For instance, Liu et al. [51] introduce cross-attention
+to aggregate multi-view video features by pair-wise
+interaction between these views. Apart from the explo-
+ration of more effective architectural design, a branch
+of works study the effect of pedestrian attributes [10],
+[61], [101], such as shoes, bag, and down color, or
+the gait [11], [57], i.e. walking style of pedestrians, as
+a more comprehensive form of pedestrian description.
+Chang et al. [11] closely integrate two coherent tasks:
+gait recognition and video-based re-ID by using a
+hybrid framework including a set-based gait recogni-
+tion branch. Some works [61], [101] embed attribute
+predictors into the network supported by annotations
+obtained from a network pretrained on an attribute
+dataset. Chai et al. [10] separate attributes into ID-
+relevant and ID-irrelevant ones and propose a novel
+pose-invariant and motion-invariant triplet loss to mine
+the hardest samples considering the distance of pose
+and motion states.
+Although the above methods have made significant
+progress in performance, Transformer [66], which is
+deemed a more powerful architecture to process se-
+quence data, may raise the performance ceiling of
+video-based re-ID. To illustrate this, Transformer can
+readily adapt to video data with the support of the
+global attention mechanism to capture spatio-temporal
+dependencies and temporal positional encoding to or-
+der spatio-temporal positions. In addition, the class
+token is off-the-shelf for Transformer-based models
+to aggregate spatio-temporal information. However,
+Transformer suffers from multiple drawbacks [24],
+[70], [89], [90], and few works have been released so
+far on video-based person re-ID based on Transformer.
+In this work, we attempt to explore the potential of
+intractable Transformer in video-based person re-ID.
+C. Vision Transformer
+Recently, Transformer has shown its ability as an
+alternative to CNN. Inspired by the great success of
+Transformer in natural language processing, recent
+researchers [24], [54], [54], [70] have extended Trans-
+former to CV tasks and obtained promising results.
+Bertasius et al. [7] explores different video self-
+attention schemes considering their cost-performance
+trade-off, resulting in a conclusion that the di-
+vided space-time self-attention is optimal. Similarly,
+ViViT [4] factorizes self-attention to compute self-
+attention spatially and then temporally. Inspired by
+these works, we divide video self-attention into spa-
+tial attention followed by temporal attention, and we
+further propose a attribute-aware variant for video-
+based re-ID. Furthermore, little research has been
+done on Transformer for Video-based person re-ID.
+Trigeminal Transformers (TMT) [51] puts the input
+patch token sequence through a spatial, a temporal, and
+a spatio-temporal minor Transformer, respectively, and
+a cross-view interaction module fuses their outputs.
+Differently, MSTAT has three stages, all extracting
+spatio-temporal features but with different meanings:
+(1) attribute features, (2) identity features, (3) attribute-
+identity features.
+
+TANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION
+4
+c
+Tokenization
+Class
+Token
+Patch
+Tokens
+Attribute-aware Prxoy 
+Embedding Module
+…
+…
+L CE + L Tri
+N3
+Stage III
+2
+Spatio-Temporal 
+Aggregation
+N2
+Stage II
+Class Token Re-init
+L CE + L Tri
+L CE + L Tri
+c
+Inference
+Element-wise 
+Addition
+Spatial Positional 
+Encoding
+Concatenation
+Attribute
+Representation 
+c
+M
+X
+E
+×
+3
+M
+×
+×
+×
+A-Spatio-Temporal 
+Aggregation
+Spatio-Temporal 
+Aggregation
+N
+×
+Stage I
+1
+Identity-aware Prxoy 
+Embedding Module
+Identity-aware Prxoy 
+Embedding Module
+Fig. 2: The overall architecture of our proposed MSTAT which consists of three stages, all based on the
+Transformer architecture. Stage I updates the spatio-temporal patch token sequence of the input video
+and aggregates them into a group of attribute-associated representations. Subsequently, Stage II aggregates
+discriminative identity-associated features and Stage III attribute-identity-associated features, relying upon
+their stage-specific class tokens. Here, we omit the input and output of each module except the attribute-
+aware proxy embedding module in Stage I. At inference time, all these feature representations are combined
+through concatenation to infer the pedestrian’s identity jointly.
+III. METHOD
+In Sec. III-A, we first overview the proposed
+MSTAT framework. Then, Spatio-Temporal Aggrega-
+tion (STA), the normal spatial-temporal feature extrac-
+tor in MSTAT, is formulated in section Sec. III-B.
+Along with it, we introduce the proposed Attribute-
+Aware Proxy (AAP) and Identity-Aware Proxy (IAP)
+embedding modules in Sec. III-C. Finally, Tem-
+poral Patch Shuffling (TPS), a newly introduced
+Transformer-specific data augmentation scheme, is
+presented in section III-E.
+A. Overview
+This section briefly summarizes the workflow of
+MSTAT. The overall MSTAT framework is shown in
+Fig. 2. Given a video tracklet V P RT ˆ3ˆHˆW with
+T frames and the resolution of each frame is H ˆ W,
+the goal of MSTAT is to learn a mapping from a video
+tracklet V to a d-dimension representation space in
+which each identity is discriminative from the others.
+Specifically, as shown on the left of Fig. 2, MSTAT
+first linearly projects non-overlapping image patches
+of size 3 ˆ P ˆ P into d-dimensional patch tokens,
+where d “ 3P 2 denotes the embedded dimension of
+tokens. Thus, a patch token sequence X P RT ˆNˆd is
+obtained, where the number of patch tokens in each
+frame is denoted by N “ HˆW
+P 2 . Meanwhile, spatial
+positional encoding E P RNˆd is added to X in a
+element-wise manner for reserving spatial structure
+in each frame. Notably, we do not insert temporal
+positional encoding into X, since the temporal order
+is usually not conducive to video-based re-ID, which
+is also demonstrated in [92]. Finally, a class token
+c P Rd is associated with X to aggregate global identity
+representation.
+Next, we feed the token sequence X into Stage I of
+MSTAT. It takes X and c as input, and employs a stack
+of eight Spatio-Temporal Aggregation (STA) blocks
+for inter-frame and intra-frame correlation modeling.
+The output tokens are then fed into an Attribute-Aware
+Proxy (AAP) embedding module to mine rich visual
+attributes, a composite group of semantic cues that im-
+ply identity information, e.g., garments, handbags
+and so on. The Stage II includes a series of STA
+blocks (three in our experiments), followed by an
+Identity-Aware Proxy (IAP) embedding module which
+is able to screen out discriminative identity-associated
+information by inspecting the entire sequence in par-
+allel. In the Stage III, we first introduce a novel class
+token to directly aggregate higher-level features. In
+addition, a stack of Attribute-STA (A-STA) blocks is
+used to fuse attributes from different frames. At last,
+an IAP embedding module is adopted to generate a
+discriminative representation for the person. In the
+training phase, the attribute representations extracted
+from Stage I and the class tokens of Stage II and
+Stage III are supervised separately by a group of
+losses. During the testing, the attribute representations
+and the class tokens from the last two stages are
+concatenated for similarity measurement.
+B. Spatio-temporal Aggregation
+To begin with, we make a quick review of the vanilla
+Transformer self-attention mechanism first proposed in
+[66]. In practice, visual Transformer embeds an image
+into a sequence of patch tokens, and self-attention
+operation first linearly projects these tokens to the
+corresponding query Q, key K and value V respec-
+tively. Then, the scaled product of Q and K generates
+an attention map A, indicating estimated relationships
+
+s:/iblog.csdn.net/qqn34182315
+IEEE TRANSACTIONS ON MULTIMEDIA
+between token representations in Q and K. Then, V
+performs a re-weighting by multiplying the attention
+map A, to obtain the output of Transformer self-
+Attention. In this way, patch tokens are reconstructed
+by leveraging interaction with each other. Formally,
+self-Attention operation SAp.q can be formulated as
+follows:
+Q, K, V “ ˆSWq, ˆSWk, ˆSWv
+A “ SoftmaxpQKTq{
+?
+d
+SApˆSq “ AV
+(1)
+where ˆS P R ˆ
+Nˆd denotes an 2-dimensional input
+token sequence, and Wq, Wk, Wv P Rdˆd1 denote
+three learnable parameter matrices of size d ˆ d1. In
+the multi-head setting, we let d1 “ d{n, where n
+indicates the number of attention heads. The function
+Softmaxp¨q denotes the softmax operation for each
+row. And the scaling operation in Eqn.(1) eliminates
+the influence from the scale of embedded dimension
+d1.
+In our Spatio-Temporal Aggregation block (STA),
+self-attention operation along time axis and along
+space axis (i.e. temporal attention and spatial attention)
+are separately denoted as SAtp¨q and SAsp¨q. Let
+S P R ˆT ˆ ˆ
+Nˆd denote an input spatio-temporal token
+sequence. Formally, SAtp¨q and SAsp¨q can be written
+as:
+SAtpSq “ SApConcatpS:,0, ..., S:,n, ..., S:,N´1qq
+SAspSq “ SApConcatpS0,:, ..., St,:, ..., ST´1,:qq
+(2)
+where T indicates the total number of frames in
+video clip, N is the total spatial position index, and
+Concatp¨q denotes the concatenation operation in the
+split dimension, e.g., the spatial position dimension in
+Eqn.(2).
+Given SAtp.q and SAsp.q, the STA block consecu-
+tively integrates these two self-attention modules to ex-
+tract spatial-temporal features. As illustrated in Fig. 3,
+STA further extracts discriminative information from
+patch tokens to the class token through spatial attention
+SAsp¨q, which can be realized by concatenating the
+copies of class token to the token sequence of each
+frame before SAsp.q, and taking the average of class
+token copies after SAsp.q to further apply the later
+temporal aggregation. In this way, the general form
+of STA can be presented as:
+S1 “ S ` α ˆ SAtpLNpSqq
+STApS, cq “ ConcatpS1, cq
+`β ˆ SAspLNpConcatpS1, cqqq
+(3)
+where LNp¨q denotes Layer Normalization [5]. The
+hyper-parameter α and β are learnable scalar residual
+weights to balance temporal attention and spatial at-
+tention. Compared with the space-time joint attention
+in [7] and [4], which jointly processes all patches of a
+video, STA is more computation-efficient by reducing
+Fig. 3: The detailed comparison between (a) Spatio-
+Temporal Aggregation block (STA) and (b) Attri-
+bution Spatio-Temporal Aggregation block (A-STA).
+Two additional Attribute-Aware Proxy (AAP) embed-
+ding modules are placed into the latter, before and
+after the temporal attention module. The class token
+broadcasting operation duplicates the class token for
+each frame to attend spatial attention within a specific
+frame. Oppositely, class token averaging calculates the
+average of all class token copies. Note that the Pre-
+Norm [79] layers before temporal attention and spatial
+attention are omitted.
+Fig. 4: The detailed module design of the Attribute-
+Aware
+Proxy
+(AAP)
+embedding
+module.
+The
+Attribute-Aware
+Proxy
+Embedding
+denotes
+a
+learnable matrix that is used as the query of the
+attention operation. For simplicity, this figure only
+shows the single-head version of the AAP embedding
+module and the scaling operation before the softmax
+operation is omitted.
+complexity from OpT 2N 2q to OpT 2 ` N 2q. Actually,
+it avoids operating on a long sequence, whose length
+always leads to quadratic growth of computational
+complexity [31], [68].
+C. Attribute-Aware Proxy Embedding Module
+Local patch tokens usually contain rich attributive
+information, e.g., glasses, umbrellas, logos,
+and so on. Even if a single attribute is not discrimina-
+tive enough to recover one’s identity, the combinations
+
+00000
+00000
+个个个个个
+个个个个个
+ClassToken
+Class Token
+Averaging
+-
+Averaging
+-
+-
+-
+-
+SpatialAttention
+Spatial Attention
+-
+-
+ClassToken
+-
+ClassToken
+-
+Broadcasting
+Broadcasting
+-
+-
+-
+-
+-
+-
+AAP embeddingmodule
+-
+-
+-
+TemporalAttention
+-
+TemporalAttention
+-
+-
+-
+-
+-
+AAP embeddingmodule
+-
+个个个个个
+个不不个个
+00000
+00000
+(a) STA
+(b) A-STAAttribute
+Representation
+Linear
+Attribute-Aware
+i Proxy Embedding
+: Module
+Softmax
+Q
+K
+V
+Attribute-Aware
+Linear
+Linear
+Proxy Embeddings
+Class
+0000000
+Patch
+Token
+TokenTANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION
+6
+of a pedestrian’s rich attributes should be discrimina-
+tive as each attribute eliminates a certain degree of
+uncertainty. Rather than directly aggregating into a
+“coarse” class token, we introduce the Attribute-Aware
+Proxy (AAP) embedding module to directly extract
+attribute features from a single-frame or multi-frame
+patch token sequence. Practically, AAP embeddings
+are formed by a learnable matrix with anisotropic
+initialization for the richness of learned attributes. It
+can be considered as the “attribute bank” to serve as
+the query of the attention operation to match with
+the feature representations of the input patch tokens.
+Specifically, AAP embeddings interact with the keys
+of the patch token sequence. Finally, the resulting
+attention map is used to re-weight the value, generating
+the attribute representations of the specific video clip
+with the same dimension of AAPs. Formally, an AAP
+embedding module can be written as follows,
+Q, K, V “ PQ, SWk, SWv,
+AAPpSq “ SoftmaxpQKTq
+?
+d
+V
+(4)
+here we use the multi-head version of AAP embedding
+module in practice, which has the same multi-head
+setting as SAp¨q in Eqn.(1). Note that the spatio-
+temporal input S here can also be ˆS P R ˆ
+Nˆd for
+spatial-only use. Compared with SAp¨q, the newly
+proposed AAP module consider the query Q in Eqn.(1)
+as the a set of learnable parameters PQ P RNaˆd1,
+where Na ! N is a hyper-parameter that indicates the
+number of AAPs. By controlling Na, the AAP module
+could have a manually defined capacity, which leads
+to flexibility for various real applications.
+As shown in Fig. 2, both Stage I and Stage
+III employ the proposed AAP embedding modules.
+Specifically, in Stage I, the proposed AAP module is
+firstly used to generate attribute representations from a
+multi-frame sequence of patch tokens S P R ˆT ˆ ˆ
+Nˆd for
+similarity measurement. Although we do not have any
+attribute-level annotations, we hope the AAP module
+can automatically learn a rich set of implicit attributes
+from the entire training dataset, while these resultant
+attribute representations could also present discrimina-
+tive power complementary to ID-only representations.
+To achieve this goal, the ID-level supervision signal is
+first imposed on the combination of learned attribute
+representations to constrain its discriminative power.
+In addition, we initialize the AAPs with anisotropic
+distributions to capture diverse implicit attribute rep-
+resentations. In practice, we surprisingly find that
+such anisotropy can maintain after the model training,
+which means such optimized AAP could respond to a
+set of differentiated attributes. Moreover, the number
+of AAPs can be relatively large compared with the
+class token to cover rich attribute information. In
+this sense, both the richness and diversity of learned
+implicit attributes can be guaranteed.
+In Stage III, we further insert two intra-frame
+AAP embedding modules before and after the tem-
+poral attention of each STA to conduct attribute-aware
+temporal interaction. Such a modified STA block is
+named A-STA, which is illustrated in Fig. 3. In A-STA,
+semantic-related attributes in different frames experi-
+ence inter-frame interaction to model their temporal
+relations. In the end, after temporal attention, we set
+Na equal to N for the second AAP embedding module
+so that it has N tokens as output to keep the input-
+output consistency.
+D. Identity-Aware Proxy Embedding Module
+Extracting discriminative identity representation is
+also crucial for video-based re-ID. To this end, the
+Identity-Aware Proxy (IAP) embedding module is pro-
+posed for effective and efficient discriminative repre-
+sentation generation. In previous works, joint space-
+time attention has shown promising results [4], [7], as
+it accelerates information aggregation by applying self-
+attention over spatial and temporal dimensions jointly.
+However, the quadratic computational overheads limit
+its applicability. The IAP embedding module is pro-
+posed to address such an issue, which performs joint
+space-time attention with high efficiency while main-
+taining the discrimination of the identity feature rep-
+resentation.
+The IAP module contains a set of identity proto-
+types, which are presented as two learnable matrics.
+In practice, we exploit them to replace the keys
+tpi
+KuM
+i“1 P PK and values tpi
+V uM
+i“1 P PV of the
+attention operation. Both PK, PV
+P RMˆd1, where
+M P N` denotes the number of identity prototypes
+and determine the capacity of the IAP module (usually
+M ! N). As shown in Fig. 5, an attention map
+A P RMˆN is first calculated to present the affinity
+between prototype-patch pairs. Thus each element in A
+reflects how close a patch token is to a specific identity
+prototype. Then this attention map is sparsified by suc-
+cessively applying an L1 normalization and softmax
+normalization along M and N, respectively. At last,
+the class token c, i.e. the first row of V, is updated
+by the multiplication of V and A. Such an operation
+aggregates the most discriminative identity features
+from the entire patch token sequence. Formally, given
+the spatio-temporal token sequence S, the output of
+the IAP module can be calculated as follows:
+Q, K, V “ SWq, PK, PV
+A “ SoftmaxpL1NormpQKTqq
+?
+d
+IAPpSq “ AV
+(5)
+where K and V are not conditioned on input S
+but are learnable parameters. Here we insert an L1
+normalization layer before the softmax operation in
+Eqn. (5), resulting in double normalization [30], [31].
+Such a scheme performs patch token re-coding to
+reduce the noise of patch representations, leading to
+
+7
+IEEE TRANSACTIONS ON MULTIMEDIA
+Fig. 5: The detailed module design of the Identity-
+Aware Proxy (IAP) embedding module. The IAP em-
+bedding denotes the learnable matrix used to calculate
+the key or value of the attention operation. Here
+we only show the single-head version of the IAP
+embedding module and omit the scaling operation.
+In such a scheme, The output token sequence can
+be considered as reconstruction by a group of IAPs,
+which tend to reserve the most discriminative identity
+features.
+robust identification results. Specifically, the learnable
+matrix PK matches the input tokens through the double
+normalization operation to generate the affinity map
+A. Then these input tokens are thereupon re-coded
+through a projection of PV along A. Since the numbers
+of learnable vectors in PK and PV are much smaller
+than the number of input tokens, the above operation
+has been able to represent each token in a more
+compact space (i.e. linear combination of the vectors
+in PV ), effectively suppressing irrelevant information
+for re-ID. Moreover, IAPp¨q has OpNq computational
+complexity since the number of identity prototypes M
+is fixed and is usually much less than the total number
+of patch tokens of a specific video tracklet (e.g., 64
+in our experiments). So, the proposed IAP embedding
+module allows all spatio-temporal patch tokens to be
+processed in parallel for effective and efficient feature
+extraction.
+E. Temporal Patch Shuffling
+To improve the robustness of the model, we propose
+a novel data augmentation scheme termed Temporal
+Patch Shuffling (TPS). Suppose we have one patch
+sequences Ri “ tri1, ..., rit, ..., riT u from the same
+video clip, where the sub-index i denotes specific
+spatial locations. As shown in Fig. 6, the proposed
+TPS randomly permutes the patch tokens in Ri and
+refill the shuffled sequence ˆ
+Ri to form the new video
+clip for training. As illustrated in Fig. 6, we could
+simultaneously select multiple spatial regions in one
+video clip for shuffling. While in the inference phase,
+the original video clip is directly fed into the model
+for identification. TPS brings firm appearance shifts
+and motion changes from which the network learns
+to extract generalizable and invariant visual clues. In
+addition, the scale of available training data can be
+f1
+f2
+f3
+f4
+f5
+r21
+r22
+r23
+r24
+r25
+r11
+r12
+r13
+r14
+r15
+r15
+r14
+r11
+r12
+r13
+r24
+r23
+r25
+r22
+r21
+X
+x’
+shuffling
+f1
+f2
+f3
+f4
+f5
+Fig. 6: Visualization of Temporal Patch Shuffling
+(TPS). ft represents tth frame, rit the patch in spatial
+position i and tth frame. TPS is a built-in data aug-
+mentation scheme that randomizes the order of a patch
+sequence sampled from spatial position i. As a result,
+for example, the patch in the red box is transferred
+from the 5th frame to the 1st frame.
+greatly extended based on such a scheme, which helps
+to prevent the network from overfitting.
+In our experiments, we treat TPS as a plug-and-play
+operation and implement it at the stem of the network
+to promote the entire network for the best performance.
+The following section will conduct ablation studies to
+explore where to insert TPS and to what extent TPS
+should be for optimal training results.
+IV. EXPERIMENT
+A. Datasets and evaluation protocols
+In this paper, we evaluate our proposed MSTAT
+on three widely-used video-based person re-ID bench-
+marks:
+iLIDS-VID
+[69],
+DukeMTMC-VideoReID
+(DukeV) [59], and MARS [102].
+1) iLIDS-VID [69] is comprised of 600 video track-
+lets of 300 persons captured from two cameras. In
+these video tracklets, frame numbers range from 23
+and 192. The test set shares 150 identities with the
+training set.
+2) DukeMTMC-VideoReID [59] is a large-scale
+video-based benchmark which contains 4, 832 videos
+sharing 1, 404 identities. In the following sections, we
+use the abbreviation “DukeV” for the DukeMTMC-
+VideoReID dataset. The video sequences in the DukeV
+dataset are commonly longer than videos in other
+datasets, which contain 168 frames on average.
+3) MARS [102] is one of the largest video re-
+ID benchmarks which collects 1, 261 identities exist-
+ing in around 20, 000 video tracklets captured by 6
+cameras. Frames within a video tracklet are relatively
+more misaligned since they are obtained by a DPM
+detector [27] and a GMMCP tracker [22] rather than
+hand drawing. Furthermore, around 3, 200 distractor
+tracklets are mixed into the dataset to simulate real-
+world scenarios.
+For evaluation on MARS and DukeV datasets, we
+use two metrics: the Cumulative Matching Character-
+istic (CMC) curves [8] and mean Average Precision
+(mAP) following previous works [16], [51], [94], [99].
+However, in the gallery set of iLIDS-VID, there is
+merely one correct match for each query. For this
+benchmark, only cumulative accuracy is reported.
+
+00000000
+Identity-Aware
+: Proxy Embedding
+i Module
+Softmax
+L1Norm
+K
+V
+Identity-Aware
+Identity-Aware
+Linear
+Proxy Embeddings
+Proxy Embeddings
+Class
+Patch
+Token
+TokensXXTANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION
+8
+Method
+Source
+Backbone
+MARS
+Duke-V
+iLIDS-VID
+Rank-1
+Rank-5
+mAP
+Rank-1
+Rank-5
+mAP
+Rank-1
+Rank-5
+SCAN [94]
+TIP19
+Pure-CNN
+87.2
+95.2
+77.2
+-
+-
+-
+88.0
+96.7
+VRSTC [39]
+CVPR19
+Pure-CNN
+-
+89.8
+85.1
+96.9
+-
+96.2
+86.6
+-
+M3D [39]
+AAAI19
+Pure-CNN
+-
+-
+-
+96.9
+-
+96.2
+74.0
+94.3
+MG-RAFA [99]
+CVPR20
+Pure-CNN
+88.8
+97.0
+85.9
+-
+-
+-
+88.6
+98.0
+AFA [16]
+ECCV20
+Pure-CNN
+90.2
+96.6
+82.9
+-
+-
+-
+88.5
+96.8
+AP3D [29]
+ECCV20
+Pure-CNN
+90.7
+-
+85.6
+97.2
+-
+96.1
+88.7
+-
+TCLNet [16]
+ECCV20
+Pure-CNN
+89.8
+-
+85.1
+96.9
+-
+96.2
+86.6
+-
+A3D [15]
+TIP20
+Pure-CNN
+86.3
+95.5
+80.4
+-
+-
+-
+86.7
+98.6
+GRL [52]
+CVPR21
+Pure-CNN
+90.4
+96.7
+84.8
+95.0
+98.7
+93.8
+90.4
+98.3
+STRF [3]
+ICCV21
+Pure-CNN
+90.3
+-
+86.1
+97.4
+-
+96.4
+89.3
+-
+Fang et al. [25]
+WACV21
+Pure-CNN
+87.9
+97.2
+83.2
+-
+-
+-
+88.6
+98.6
+TMT [51]
+Arxiv21
+CNN-Transformer
+91.2
+97.3
+85.8
+-
+-
+-
+91.3
+98.6
+Liu et al. [47]
+CVPR21*
+CNN-Transformer
+91.3
+-
+86.5
+96.7
+-
+96.2
+-
+-
+STT [95]
+Arxiv21
+CNN-Transformer
+88.7
+-
+86.3
+97.6
+-
+97.4
+87.5
+95.0
+ASANet [10]
+TCSVT22
+Pure-CNN
+91.1
+97.0
+86.0
+97.6
+99.9
+97.1
+-
+-
+MSTAT(ours)
+-
+Pure-Transformer
+91.8
+97.4
+85.3
+97.4
+99.3
+96.4
+93.3
+99.3
+TABLE I: Result comparison with state-of-the-art video-based person re-ID methods on MARS, DukeMTMC-
+VideoReID, and iLIDS-VID. * denotes the workshop of the conference.
+B. Implementation details
+Our proposed MSTAT framework is built based on
+Pytorch toolbox [58]. In our experiments, it is running
+on a single NVIDIA A100 GPU (40G memory). We
+resize each video frame to 224 ˆ 112 for the above
+benchmarks. Typical data augmentation schemes are
+involved in training, including horizontal flipping, ran-
+dom cropping, and random erasing. For all stages,
+STA modules are pretrained on an action recognition
+dataset, K600 [9], while other aforementioned modules
+are randomly initialized.
+In the training phase, if not specified, we sample
+L “ 8 frames each time for a video tracklet and
+set the batch size as 24. In each mini-batch, we
+randomly sample two video tracklets from different
+cameras for each person. We supervise the network
+by cross-entropy loss with label smoothing [64] asso-
+ciated with widely used BatchHard triplet loss [36].
+Specifically, we impose supervision signals separately
+on the concatenated attribute representation from the
+AAP embedding module in Stage I, the output class
+tokens from Stage II, and Stage III. The learning
+rate is initially set to 1e-3, which would be multiplied
+by 0.75 after every 25 epochs. The entire network is
+updated by an SGD optimizer in which the weight
+decay and Nesterov momentum are set to 5 ˆ 10´5
+and 0.9, respectively.
+In the test phase, following [29], [95], we randomly
+sample 32 frames as a sequence from each original
+tracklet in either query or gallery. For each sequence,
+The attribute representation from Stage I, the output
+class tokens from stage Stage II and Stage III are
+concatenated as the overall representation. Following
+the widely-used protocol, we compute the cosine sim-
+ilarity between each query-gallery pair using their
+overall representations. Then, the CMC curves and the
+mAP can be calculated based on the predicted ranking
+list and the ground truth identity of each query. Note
+that we do not use any re-ranking technique.
+C. Compared with the state of the arts
+In Table I, we make a comparison on three bench-
+mark datasets between our method and video-based
+person re-ID methods from 2019 to 2021, including
+M3D [39], GRL [52], STRF [3], Fang et al. [25],
+TMT [51], Liu et al. [47], ASANet [10]. According to
+their backbones, these re-ID methods can be roughly
+divided into the following types: Pure-CNN, CNN-
+Transformer Hybrid, and Pure-Transformer methods.
+In real-world applications, rank-1 accuracy [8] re-
+flects what extent a method can find the most confident
+positive sample [85], and relatively high rank-1 accu-
+racy can save time in confirmation. As the first method
+based on Pure-Transformer for video-based re-ID so
+far, we achieve state-of-the-art results in rank-1 ac-
+curacy on three benchmarks. Our approach especially
+attains rank-1 accuracy of 91.8% and rank-5 accuracy
+of 97.4% on the largest-scale benchmark, MARS. It
+is noteworthy that our MSTAT outperforms the best
+pure CNN-based methods using ID annotations only
+by a margin of 1.1% and a CNN-Transformer hybrid
+method, TMT, by 0.6% in MARS rank-1 accuracy.
+Compared to our proposed method, TCLNet [16]
+explicitly captures complementary features over dif-
+ferent frames, and GRL [52] devises a guiding mech-
+anism for reciprocating feature learning. However, the
+designed modules in these methods commonly take
+as input the deep spatial feature maps extracted by
+a CNN backbone (e.g. ResNet50) that may overlook
+attribute-associated or identity-associated information
+without explicit modeling. Similar to ours, TMT [51]
+and M3D [3] process video tracklets in multiple
+views to extract and fuse multi-view features. No-
+tably, in all stages of MSTAT, intermediate features
+are spatio-temporal and can be iteratively updated to
+capture spatio-temporal cues with different emphases.
+ASANet [10] exploits explicit ID-relevant attributes
+(e.g., gender, clothes, and hair) and ID-irrelevant at-
+tributes (e.g., pose and motion) on a multi-branch net-
+
+9
+IEEE TRANSACTIONS ON MULTIMEDIA
+Method
+Test Protocol
+Rank-1
+Rank-5
+mAP
+MSTAT
+Stage I
+89.2
+96.7
+82.4
+Stage II
+89.2
+96.5
+83.0
+Stage III
+89.8
+96.5
+83.0
+Stage I & II
+91.2
+97.3
+85.0
+Stage I & III
+90.5
+97.2
+83.9
+Stage II & III
+90.6
+96.9
+84.6
+Stage I, II, & III
+91.8
+97.4
+85.3
+TABLE II: Ablation study on three stages of MSTAT
+on MARS. Test Protocol means the final feature rep-
+resentation used for similarity measurement. The net-
+work architecture and training hyper-parameter setting
+remain the same for each experiment.
+work. Despite the performance growth, the demand for
+attribute annotations may limit its applications in large-
+scale scenarios. In comparison with existing methods,
+our method aggregates spatio-temporal information in
+a unified manner and explicitly capitalizes on implicit
+attribute information to improve recognizability un-
+der challenging scenarios. Conclusively, our method
+achieves the state-of-the-art performance of 91.8% and
+93.3% rank-1 accuracy, respectively, on MARS and
+iLIDS-VID.
+D. Effectiveness of Multi-Stage Framework Architec-
+ture
+To evaluate the effectiveness of the three stages
+in our proposed MSTAT, we carry out a series of
+ablation experiments whose results are displayed in
+Table II. After the three stages are jointly trained, we
+first separately evaluate each stage using its output
+feature representation. Then, we concatenate two or
+more stages to evaluate whether each is effective.
+For three single stages, each has rank-1 accuracy
+ranging from 89.2 to 89.8. However, their combina-
+tions result in a significant increase of over 0.8%.
+Remarkably, while Stage I and Stage II secure only
+89.2 rank-1 accuracy, their integration attains up to
+91.2%, surpassing them by a 2% margin. One can
+attribute such a result to their emphases: one stage on
+attribute-associated features and the other on identity-
+associated features. Eventually, when all three stages
+are used, MSTAT reaches a 91.8% rank-1 accuracy,
+higher than all two-stage combinations. Overall, these
+experiments demonstrate that the three stages have dif-
+ferent preferences toward features and can complement
+each other by simple concatenation.
+E. Effectiveness of Key Components
+To demonstrate the effectiveness of our proposed
+MSTAT, we conduct a range of ablative experiments
+on the largest public benchmark MARS.
+1) Effectiveness of Attribute-Aware Proxy Embed-
+ding Module: As shown in Fig. 7, we evaluate MSTAT
+with different AAP numbers (i.e. Na in Sec. III-C) in
+the AAP embedding module in the last layer of Stage
+Fig. 7: Ablation study on the attribute-aware proxy
+(AAP) embedding module for attribute extraction in
+MARS. ”Base” is the network without attribute ex-
+traction using AAP in training and testing. AAP-
+k indicates the network where the AAP embedding
+module in Stage I has k AAPs.
+Fig. 8: Ablation study on A-STA. ”Base” is the net-
+work that consists of STA only. A-STA-k represents
+the network in which Stage III is equipped with A-
+STA layers each of k AAPs.
+I. The figure reveals that 24 proxies are optimal for
+attributive information extraction as it attains the best
+performance in terms of rank-1 and rank-5 accuracy.
+In contrast to the baseline, MSTAT has seen over 2%
+growth in rank-1 accuracy and around 1% in rank-5
+accuracy. However, a redundant or insufficient number
+of AAPs may cause a minor performance drop since
+they may pay attention to noisy or useless attributes.
+In summary, the AAP embedding module for clue
+extraction gives a boost to the performance in rank-
+1 and rank-5 accuracy, with negligible computational
+overhead.
+Attribute-Aware Proxy (AAP) embedding modules
+are also used for A-STA, a variant of STA for attribute-
+aware temporal feature fusion in Stage III. As shown
+in Fig. 8, we conduct a series of experiments to explore
+whether A-STA is effective and how many AAPs for
+A-STA are appropriate (also corresponding to Na in
+III-C)). The experiment results reveal that the baseline
+model fails to reach 90% rank-1 accuracy or 97% rank-
+5 accuracy. As the number of AAPs increases, these
+two metrics grow to 91.8% and 97.4%.
+Therefore, we can attribute the performance soar
+to A-STA, allowing for attribute-aware temporal in-
+teraction. A-STA offers a different viewpoint from
+that of Stage II on videos. Moreover, due to the
+redundancy of temporal information in many video re-
+ID scenarios discussed in [16], A-STA with too many
+AAPs incurs meaningless attributes. This can be why
+the performance descends once A-STA has too many
+AAPs.
+
+Rank-1 accuracy
+Rank-5 accuracy
+9160
+0.93
+0.974
+0.92
+0.918
+0.972
+0.908
+0.971
+0.91
+0.907
+0.906
+0L60
+0.969
+0.90
+0.968
+0.968
+0.892
+0.967
+0.B9
+0.966
+Bese
+AAP-BAAP-16 AAP-24 AAP-32
+Bease
+AAP-BAAP-16 AAP-24 AAP-32Rank-1 accuracy
+Rank-5 accuracy
+0.976
+0.93
+0.974
+0.92
+0.918
+0.910
+0.912
+0.972
+0.971
+0.91
+0.906
+0.970
+0.970
+0.90
+0.968
+0.968
+0.892
+0.B9
+0.566
+0.966
+BaBBASTA-16 ASTA-32 ASTA4B ASTA-64
+Baiga
+ASTA-16 ASTA-32 ASTA-4B ASTA-64TANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION
+10
+Fig. 9: Study on the effect of training video sequence
+length on MARS.
+In conclusion, our proposed AAP embedding mod-
+ule can be used for: (1) the extraction of informative
+attributes as plugged into any Transformer layer and
+(2) attribute-aware temporal interaction when a tempo-
+ral attention module is sandwiched between two. Both
+of the two functionalities cause a significant increase
+in performance, demonstrating their effectiveness.
+2) Effectiveness of Identity-Aware Proxy Embedding
+Module: In Table III, MSTAT that discards IAP em-
+bedding modules leads to only 88.2% rank-1 accuracy
+and 96.4% rank-5 accuracy. However, it boosts rank-1
+performance by 2.8% or 2.2% by taking the place of
+STA in Stage II or Stage III. Finally, IAP embedding
+modules in the last layers in both Stage II and
+Stage III further improve 0.8% rank-1 accuracy and
+0.4% rank-5 accuracy. The IAP embedding module’s
+ablation results demonstrate its ability to generate
+discriminative representations efficiently. Intuitively,
+we place the IAP embedding module only in the last
+few depths because it may discard non-discriminative
+features that should be preserved in shallow layers.
+3) Effectiveness of Temporal Patch Shuffling: To
+evaluate the effectiveness of Temporal Patch Shuffling
+(TPS), we assign different probabilities to implement
+TPS for each training video sample. Note that in the
+following experiments, the number of spatial positions
+to shuffle is set to 5 if we implement TPS on this sam-
+ple. As shown in Table IV, 20% probability provides
+the best result over others, which leads to a growth of
+0.3% in rank-1 accuracy. However, the 60% or 80%
+probability results in a 0.1% or 0.2% rank-1 accuracy
+drop mainly due to heavy noise. In summary, a proper
+level of TPS would be an effective data augmentation
+method for the Transformer for video-based person
+re-ID. Further, rather than reserving temporal motion
+(an ordered sequence of patches), TPS stimulates re-
+identification accuracy by learning temporal coherence
+from shuffled patch tokens.
+F. Effect of video sequence length
+To investigate how temporal noise influences the
+training of MSTAT, we conduct experiments on videos
+with varied lengths. In Fig. 9, experiments provide
+length-varying video tracklets for training, while all
+experiments are implemented under the identical eval-
+uation setting with a fixed video length of 32. All
+Method
+Position
+Rank-1
+Rank-5
+w/o IAP embedding module
+-
+88.2
+96.4
+w/ IAP embedding module
+Stage II
+91.0
+97.0
+Stage III
+90.4
+97.0
+Stage II&III
+91.8
+97.4
+TABLE III: Ablation study on the IAP embedding
+module. Stage II and Stage III in this table means that
+an IAP embedding module is appended to the last layer
+of Stage II and Stage III respectively. This table shows
+that the IAP embedding module brings improvements
+to every single stage. When it is placed on both two
+stages, MSTAT shows the best performance.
+Methods
+Prob.
+Rank-1
+Rank-5
+mAP
+MSTAT w/o TPS
+0%
+91.5
+97.5
+85.2
+MSTAT w/ TPS
+20%
+91.8
+97.4
+85.3
+40%
+91.7
+97.3
+85.2
+60%
+91.4
+97.5
+85.1
+80%
+91.3
+97.1
+85.1
+TABLE IV: Ablation study on Temporal Patch Shuf-
+fling. The table shows that the proper level of shuffling
+can bring slight improvement. However, it may de-
+grade the learning while the shuffling degree becomes
+increasingly overwhelming.
+experiments shut down until the loss stops decreasing
+for ten epochs.
+On the one hand, rank-1 accuracy shows an upward
+trend as temporal noise gradually decreases, reach-
+ing a peak at 8. On the other hand, temporal noise
+shows no apparent correlation with rank-5 accuracy
+and mAP. These results show that our model gains
+up to 0.6% rank-1 accuracy through learning better
+temporal features from data. However, rank-5 accuracy
+and mAP benefit little from noise reduction, from
+which we can speculate that in most cases in video
+re-ID, learning temporal features is less important than
+learning appearance features as they only account for
+0.6% of rank-1 and 0.2% of rank-5 accuracy. Similar
+results can be found in [51].
+G. Comparison among metric learning methods
+Metric learning aims to regularize the sample distri-
+bution on feature space. Usually, metric learning losses
+constrain the compactness of intra-class distribution
+and sparsity of the overall distribution. To explore
+which strategy cooperates with our framework better,
+we compare a range of classic metric learning loss
+functions on iLIDS-VID, as shown in Table V. Note
+that these losses are scaled to the same magnitude
+to ensure fairness. Significantly, OIM loss [78] and
+BatchHard triplet loss [36], widely used in re-ID,
+outperform Arcface [23] and SphereFace [50] losses
+by a large margin since the latter two loss functions
+suffer from untimely overfitting in our experiments.
+
+Rank-l accuracy
+Rank-5 accuracy
+0.920
+0.980
+0.918
+0.918
+0.916
+0.976
+0.975
+0.975
+0.914 -
+0.914
+160
+0.974
+0.913
+0.973
+0.912
+0.912
+0.972
+0.91
+6
+0.970
+4
+8
+4
+6
+1
+training video sequence length
+training video sequence length11
+IEEE TRANSACTIONS ON MULTIMEDIA
+Metric learning loss
+Rank-1
+Rank-5
+w/o Metric learning
+66.0
+90.0
+Arcface [23]
+73.3
+90.7
+SphereFace [50]
+66.7
+89.3
+OIM [78]
+89.3
+98.3
+BatchHard* [36]
+93.3
+99.3
+TABLE V: Comparison among metric learning loss
+functions
+on
+iLIDS-VID,
+where
+*
+denotes
+the
+method used in our implementation. For Arcface and
+SphereFace, we test three margins and report the best
+result: (1) by default, (2) 20% larger than the default,
+(3) 20% smaller than the default.
+Fig. 10: Visualization of the similarity matrix of
+attribute-aware proxies trained on MARS. The maxi-
+mal similarity between all pairs is around 0.2, demon-
+strating that AAPs learn to capture diverse attributes.
+H. Visualization
+To better understand how the proposed framework
+works, we conduct visualization on the AAP embed-
+ding module. In Fig. 10, we show the diversity of
+implicit attributes by the similarity matrix of 24 AAPs.
+This figure implies that AAPs are anisotropic, covering
+different attribute features that appear in the given
+training dataset.
+Specifically, as shown in Fig. 11, we randomly
+select two pedestrians’ tracklets. Attention map visu-
+alization is adopted as a sign of each AAP’s concen-
+tration. In practice, we process the raw attention maps
+first by several average filters and then by thresholding
+to deliver smooth visual effects instead of grid-like
+maps. In these heap maps, the brighter color denotes
+the higher attention value. Despite the absence of
+attribute-level supervision, Fig.11 shows that some
+AAPs learn to pay attention to a local region with
+special meanings as an identity cue. For example, the
+AAP in white color in video clips (a) automatically
+learns to cover the logo in the T-shirt, while the one
+in (b) captures the head of the woman.
+Moreover, we display the t-SNE visualization result
+on iLIDS-VID in Fig. 12. It only contains the first 1/3
+of the IDs in the test set for a better visual effect.
+We also provide the corresponding quantitative evalu-
+Methods
+IntraÓ
+IntraÓ
+InterÒ
+Rank-1Ò
+Baseline [7]
+0.4572
+0.4495
+0.4704
+0.873
+MSTAT w/o attr.
+0.4517
+0.4469
+0.4644
+0.913
+MSTAT (ours)
+0.4410
+0.4389
+0.5012
+0.933
+TABLE VI: Quantitative evaluation on iLIDS-VID.
+”Intra” denotes the averaged normalized intra-class
+distance, and ”Inter” is the minimum inter-class dis-
+tance. Here, * means that the metric is computed on
+samples with the correct rank-1 match.
+ 
+ 
+(a)
+(b)
+Fig. 11: Visualization of attribute-aware proxies for
+two different pedestrians on MARS. Attention heat
+maps of four consecutive frames from the AAP em-
+bedding module on Stage I are displayed.
+ation results in Table VI measured by the normalized
+averaged intra-class distance and the minimum inter-
+class distance (0-2) on the entire test set. As a result,
+MSTAT drops the average intra-class distance from
+0.4572 of the baseline to 0.4410 and enlarges the
+minimum inter-class distance from 0.4704 to 0.5012.
+Further, to eliminate the influence of accuracy, we
+measure the intra-class distance between correctly
+matched samples, from which we witness a similar
+result. These results explain why MSTAT’s t-SNE
+visualization seems sparser.
+V. CONCLUSION
+This paper proposes a novel framework for video-
+based person re-ID, referred to as Spatial-Temporal
+Aggregation Transformer (MSTAT). To tackle simulta-
+neous extraction for local attributes and global identity
+information, MSTAT adopts a multi-stage architecture
+to extract (1) attribute-associated, (2) the identity-
+associated, and (3) the attribute-identity-associated in-
+formation from video clips, with all layers inherited
+from the vanilla Transformer. Further, for reserving
+informative attribute features and aggregating discrim-
+inative identity features, we introduce two proxy em-
+bedding modules (Attribute-Aware Proxy embedding
+module and Identity-Aware Proxy embedding module)
+into different stages. In addition, a patch-based data
+augmentation scheme, Temporal Patch Shuffling, is
+proposed to force the network to learn invariance
+to appearance shifts while enriching training data.
+Massive experiments show that MSTAT can extract
+attribute-aware features consistent across frames while
+reserving discriminative global identity information on
+
+0
+1.0
+2
+3
+4-
+0.8
+5
+6
+7
+8-
+- 0.6
+9-
+10
+11
+12
+13
+0.4
+14
+15
+16
+17
+ 0.2
+18-
+19
+20
+21
+22
+-
+0.0
+23
+456780TANG et al.: MULTI-STAGE SPATIO-TEMPORAL AGGREGATION TRANSFORMER FOR VIDEO PERSON RE-IDENTIFICATION
+12
+Fig. 12: T-SNE Visualization of the iLIDS-VID test
+set. The numbers on the plots indicate person IDs.
+MSTAT shows an increase in intra-class compactness
+and the minimum inter-class distance over the entire
+test set compared to the baseline.
+different stages to attain high performance. Finally,
+MSTAT outperforms most existing state-of-the-arts on
+three public video-based re-ID benchmarks.
+Future work may focus on mining the hard instances
+or local informative attribute locations to conduct con-
+trastive learning to promote the model’s accuracy fur-
+ther. Moreover, leveraging more unlabeled and multi-
+modal data to improve the model’s effectiveness is also
+a potential research direction.
+ACKNOWLEDGMENT
+The work is supported in part by the Young Sci-
+entists Fund of the National Natural Science Foun-
+dation of China under grant No. 62106154, by Na-
+tional Key R&D Program of China under Grant No.
+2021ZD0111600, by Natural Science Foundation of
+Guangdong Province, China (General Program) un-
+der grant No.2022A1515011524, by Guangdong Ba-
+sic and Applied Basic Research Foundation under
+Grant No. 2017A030312006, by CCF-Tencent Open
+Fund, by Shenzhen Science and Technology Program
+ZDSYS20211021111415025, and by the Guangdong
+Provincial Key Laboratory of Big Data Computing,
+The Chinese Univeristy of Hong Kong (Shenzhen).
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+
+15
+IEEE TRANSACTIONS ON MULTIMEDIA
+Ziyi Tang is now pursuing his Ph.D.
+degree at Sun Yat-Sen University. Before
+that, he was a research assistant at The
+Chinese University of Hong Kong, Shen-
+zhen (CUHK-SZ), China. He received the
+B.E. degree from South China Agriculture
+University (SCAU), Guangzhou, China in
+2019 and M.S. degree from The Univer-
+sity of Southampton, Southampton, U.K.
+in 2020. He has won top places in data
+science competitions hosted by Kaggle and
+Huawei respectively. His research interests include Computer Vision,
+Vision-Language Joint Modeling, and Casual Inference.
+Ruimao Zhang is currently a Research
+Assistant Professor in the School of Data
+Science, The Chinese University of Hong
+Kong, Shenzhen (CUHK-SZ), China. He
+is also a Research Scientist at Shenzhen
+Research Institute of Big Data. He received
+the B.E. and Ph.D. degrees from Sun Yat-
+sen University, Guangzhou, China in 2011
+and 2016, respectively. From 2017 to 2019,
+he was a Post-doctoral Research Fellow in
+the Multimedia Lab, The Chinese Univer-
+sity of Hong Kong (CUHK), Hong Kong. After that, he joined at
+SenseTime Research as a Senior Researcher until 2021. His research
+interests include computer vision, deep learning and related multi-
+media applications. He has published about 40 peer-reviewed articles
+in top-tier conferences and journals such as TPAMI, IJCV, ICML,
+ICLR, CVPR, and ICCV. He has won a number of competitions and
+awards such as Gold medal in 2017 Youtube 8M Video Classification
+Challenge, the first place in 2020 AIM Challenge on Learned Image
+Signal Processing Pipeline. He was rated as Outstanding Reviewer
+of NeurIPS in 2021. He is a member of IEEE.
+Zhanglin Peng is now pursuing her Ph.D.
+degree with the Department of Computer
+Science, The University of Hong Kong,
+Hong Kong, China. She received her B.E.
+and M.S. degrees from Sun Yat-Sen Uni-
+versity, Guangzhou, China in 2013 and
+2016, respectively. From 2016 to 2020, she
+was a researcher at SenseTime Research.
+Her research interests are computer vision
+and machine learning.
+Jinrui Chen is currently pursuing the B.A.
+degree in Financial Engineering conferred
+jointly by the School of Data Science,
+the School of Science and Engineering,
+and the School of Management and Eco-
+nomics, The Chinese University of Hong
+Kong, Shenzhen (CUHK-SZ), China. His
+research interests include deep learning
+and financial technology.
+Liang Lin (M’09, SM’15) is a Full Pro-
+fessor of computer science at Sun Yat-
+sen University. He served as the Exec-
+utive Director and Distinguished Scien-
+tist of SenseTime Group from 2016 to
+2018, leading the R&D teams for cutting-
+edge technology transferring. He has au-
+thored or co-authored more than 200 pa-
+pers in leading academic journals and con-
+ferences, and his papers have been cited by
+more than 22,000 times. He is an associate
+editor of IEEE Trans. Multimedia and IEEE Trans. Neural Networks
+and Learning Systems, and served as Area Chairs for numerous
+conferences such as CVPR, ICCV, SIGKDD and AAAI. He is the
+recipient of numerous awards and honors including Wu Wen-Jun
+Artificial Intelligence Award, the First Prize of China Society of
+Image and Graphics, ICCV Best Paper Nomination in 2019, Annual
+Best Paper Award by Pattern Recognition (Elsevier) in 2018, Best
+Paper Dimond Award in IEEE ICME 2017, Google Faculty Award
+in 2012. His supervised PhD students received ACM China Doctoral
+Dissertation Award, CCF Best Doctoral Dissertation and CAAI Best
+Doctoral Dissertation. He is a Fellow of IET and IAPR.
+

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+Many-body resonances in the avalanche instability of many-body localization
+Hyunsoo Ha,1 Alan Morningstar,1, 2 and David A. Huse1
+1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
+2Department of Physics, Stanford University, Stanford, California 94305, USA
+(Dated: January 13, 2023)
+Many-body localized (MBL) systems fail to reach thermal equilibrium under their own dynamics,
+even though they are interacting, nonintegrable, and in an extensively excited state. One instability
+towards thermalization of MBL systems is the so-called “avalanche”, where a locally thermalizing
+rare region is able to spread thermalization through the full system. The spreading of the avalanche
+may be modeled and numerically studied in finite one-dimensional MBL systems by weakly coupling
+an infinite-temperature bath to one end of the system. We find that the avalanche spreads primarily
+via strong many-body resonances between rare near-resonant eigenstates of the closed system. Thus
+we find and explore a detailed connection between many-body resonances and avalanches in MBL
+systems.
+Introduction— Many-body localized (MBL) systems
+are a class of isolated many-body quantum systems that
+fail to thermalize due to their own unitary dynamics,
+even though they are interacting, nonintegrable and ex-
+tensively excited [1–7]. This happens for one-dimensional
+systems with short-range interactions in the presence
+of strong enough quenched randomness, which yields a
+thermal-to-MBL phase transition of the dynamics.
+In
+the MBL phase, there are an extensive number of emer-
+gent localized conserved operators [8–11].
+One instability of the MBL phase which is believed to
+play a central role in the asymptotic, long-time, infinite-
+system MBL phase transition is the avalanche [12–14].
+Rare locally thermalizing regions necessarily exist, how-
+ever sparse they may be, due to the randomness. Starting
+from such a local thermalizing region, this “thermal bub-
+ble” spreads through the adjacent typical MBL regions
+until the relaxation rate of the adjacent spins becomes
+smaller than the many-body level spacing of the ther-
+mal bubble, in which case the spreading of this avalanche
+halts. If the strength of the randomness is insufficient,
+the relaxation rate remains larger than the level spacing
+and the avalanche does not stop: the full system then
+slowly thermalizes and is no longer in the MBL phase (al-
+though it is likely in a prethermal MBL regime [15, 16]).
+The avalanche has been numerically simulated in
+small-sized
+systems
+[15,
+17–21]
+and
+experimentally
+probed [22]. Recent work shows that the instability of
+MBL to avalanches occurs at much stronger randomness
+than had been previously thought [15, 20]. This leaves a
+large intermediate prethermal-MBL regime in the phase
+diagram between the onset of MBL-like behavior in small
+samples (or correspondingly short times) and the asymp-
+totic MBL phase transition.
+Clear numerical evidence
+has been obtained for many-body resonances being an
+important part of the physics in the near-thermal part
+of this regime [15, 16, 23–29], while no such evidence for
+the expected thermalizing rare regions has been found
+yet. In the part of this intermediate prethermal MBL
+regime that is farther from the thermal regime, it re-
+Bath (T=
+)
+0
+After Long time
+(a)
+(c)
+(b)
+n
+n
+n
+Slowest Mode (
+)
+near-resonance
+dominant decay
+B
+A
+B
+A
+1
+2
+. . .
+. . .
+. . .
+. . .
+. . .
+. . .
+. . .
+L
+Figure 1.
+Schematic illustrations showing (a) the avalanche
+model, (b) the long time dynamics governed by the slowest
+mode, and (c) the dominant decay processes involving four
+eigenstates with a near-resonance. (a) We connect the bath
+in the weak-coupling limit with the one-dimensional MBL sys-
+tem. Specifically, we analyze the decay of the slowest mode
+(ˆτS), which is localized near the end of the system farthest
+from the bath; ˆτS is a “localized integral of motion” in the
+MBL phase. (b) Thermalization at the latest times is gov-
+erned by ˆτS. (c) Schematic decay of ˆτS. A large fraction of
+the probability current in the decay of ˆτS passes through four
+eigenstates associated with a rare near-resonance.
+mains unclear what is the primary mechanism that leads
+to thermalization for samples larger than those that can
+be diagonalized.
+In this work, we explore how an avalanche spreads
+through typical MBL regions for systems that are near
+the avalanche instability. We do not simulate the rare
+region that initiates the avalanche. Instead, we assume
+a large avalanche is spreading and we model that as an
+infinite-temperature bath (see Fig. 1) weakly coupled to
+one end of our MBL spin chain [15, 20]. We find that
+particular many-body near-resonances of the closed sys-
+tem play a key role in facilitating the spreading of the
+avalanche.
+These many-body near-resonances are the
+arXiv:2301.04658v1  [cond-mat.stat-mech]  11 Jan 2023
+
+2
+dominant process by which the bath at one end of the
+chain thermalizes the spins at the other end of the chain
+and thus propagates the avalanche.
+Model— Our model consists of a chain of L spin-
+1/2 degrees of freedom (or qubits).
+The dynamics of
+the closed system is given by the random-circuit Floquet
+MBL model studied in Ref. [15], which has unitary Flo-
+quet operator ˆUF . The disorder strength in this model
+is given by the parameter α, with the MBL regime be-
+ing at large α, while the thermal regime is at small α
+(see the Appendix for the full description of this model).
+To investigate avalanche spreading, we weakly connect
+an infinite-temperature Markovian bath to spin L at the
+right end of the system [15, 20]. The quantum state of
+this open system is the density matrix ˆρ(t).
+In our open-system Floquet model, the bath is repre-
+sented by the super-operator Sbath that acts once each
+time period:
+Sbath[ˆρ] =
+ˆρ
+1 + 3γ +
+γ
+1 + 3γ
+3
+�
+j=1
+ˆEj ˆρ ˆE†
+j ,
+(1)
+where ( ˆE1, ˆE2, ˆE3) = ( ˆXL, ˆYL, ˆZL) are the jump opera-
+tors acting on the last spin at site L (connected to the
+bath). We will take the weak coupling limit γ → 0. The
+open-system Floquet super-operator Speriod that takes
+our system through one full time-period is
+Speriod[ˆρ(t)] = Sbath[ ˆUF ˆρ(t) ˆU †
+F ] = ˆρ(t + 1) .
+(2)
+The time evolution of the system’s state is given by
+ˆρ(t) = ˆI/2L + p1e−r1tˆτ1 +
+�
+k≥2
+pke−rktˆτk,
+(3)
+where e−rk is the kth largest eigenvalue of Speriod with
+eigenoperator ˆτk. Note that the largest (0th) eigenvalue
+is 1 and is nondegenerate when γ > 0, with eigenoper-
+ator proportional to the identity, which is the long-time
+steady state of this system. The mode with the slowest
+relaxation for γ > 0 is ˆτS := ˆτ1, which relaxes with rate
+rS := Re(r1). The relaxation rate rS is proportional to
+γ, and we work to first order in γ [20].
+Relaxation of the slowest mode— In the weak-
+coupling limit, one can obtain ˆτS as a superposition of
+the diagonal terms |n⟩ ⟨n|, where |n⟩ are the eigenstates
+of the closed system such that ˆUF |n⟩ = eiθn |n⟩. When
+γ = 0, then |m⟩ ⟨n| are eigenoperators of Speriod with
+eigenvalues ei(θm−θn), so all diagonal terms |n⟩ ⟨n| are
+degenerate, with rk = 0. Therefore, in the γ ≪ 1 limit,
+one can obtain ˆτS in degenerate perturbation theory by
+diagonalizing the (super)operator S[ˆρ] := 1
+3
+�3
+j=1 ˆEj ˆρ ˆE†
+j
+in this degenerate subspace [20], where the matrix ele-
+ments are
+Smn = ⟨m| S[|n⟩ ⟨n|] |m⟩ = 1
+3
+3
+�
+j=1
+| ⟨m| ˆEj |n⟩ |2.
+(4)
+0.5
+Probability Density (a.u.)
+4
+L=
+MBL
+12
+Thermal
+Figure 2.
+Probability distribution over samples of the ratio
+R = Dµν/(2LΓ) (see text). In the thermal regime (dark red to
+yellow), the ratio exponentially decays with increasing L. In
+comparison, R barely drifts with L for the MBL case (blue).
+We observe that R never exceeds the value 0.5 (gray dashed
+line), which is explained with the minimal model in the main
+text. In this figure, we used α = 30 and α = 1 for MBL and
+thermal regimes, respectively.
+Note that in the degenerate subspace, S is a symmet-
+ric stochastic matrix with real eigenvalues.
+In partic-
+ular, ˆτS = �
+n cn |n⟩ ⟨n| where ⃗c is the eigenvector of
+S with the smallest spectral gap Γ from the steady
+state (�
+m Snmcm = (1 − Γ)cn). The relaxation rate is
+rS = 3γΓ. We normalize ˆτS so Tr{ˆτ 2
+S} = �
+n |cn|2 = 2L.
+Naively, the slowest mode is the local integral of motion
+(LIOM) that is farthest from the bath. More precisely,
+as we show in the Appendix, the slowest mode, ˆτS, is
+a traceless superposition of projectors on to the eigen-
+states of the closed system. Among such operators it is
+the one with the smallest weight of Pauli strings with
+non-identity at site L (connected to the bath). It is a
+LIOM that is indeed localized far from the bath, but it
+is different in detail from the ℓ-bits and LIOMs discussed
+in Refs. [8–10, 30].
+As illustrated in Fig. 1(b), the latest time dynamics
+are determined by ˆτS, with ˆρ(t) ≃ ˆI/2L + pSe−rStˆτS for
+any initial conditions that contain ˆτS. This assumes a
+nonzero gap between (r1/γ) and (r2/γ), which is indeed
+the case for all samples examined. We can view the slow
+relaxation of ˆτS in terms of probability currents that flow
+between the eigenstates of the isolated system, leading to
+the final ˆρ = ˆI/2L equilibrium where all eigenstates have
+equal weight. Specifically, we may quantify the contribu-
+tion Dmn of the pair of eigenstates m, n to the relaxation
+of ˆτS as
+Dmn := Smn(cm − cn)2 ≥ 0 .
+(5)
+One can show (see Appendix) that the relaxation rate of
+ˆτS is given by the sum of the contributions from all pairs
+
+3
+of eigenstates of the closed system:
+�
+m<n
+Dmn = Γ Tr{ˆτ 2
+S} = 2LΓ.
+(6)
+We find the pair of eigenstates |µ⟩, |ν⟩ of the closed sys-
+tem that gives the strongest contribution to the relax-
+ation of ˆτS by finding the pair with the largest Dmn. We
+quantify the fraction of the full relaxation that is due
+to this pair by the ratio R ≡ Dµν/(2LΓ). The distribu-
+tion of R over disorder realizations and for different sys-
+tem sizes is shown in Fig. 2, both for the thermal regime
+and for the MBL regime near the avalanche instability.
+In the thermal regime, R decreases exponentially with
+increasing L, indicating that many pairs of eigenstates
+are contributing similar amounts to the relaxation of the
+slowest mode, which is as should be expected for this
+thermal regime.
+In the MBL regime, on the other hand, we find that
+this pair of eigenstates contributes an order-one fraction
+of the relaxation of ˆτS, and this fraction does not decrease
+substantially with increasing system length L. This is
+consistent with previous works [15, 16, 31, 32] that found
+extremely broad distributions for the matrix elements of
+local operators among eigenstates of the closed system
+within the MBL regime. On looking at the relaxation
+more thoroughly, we find that in this MBL regime the
+strongest contribution to the relaxation actually involves
+a set of 4 eigenstates of the closed system, at least two
+of which are involved in a many-body near-resonance, as
+we will now describe in more detail.
+Near-resonant eigenstate set— Deep in the MBL
+regime, at inverse interaction α near the estimated
+avalanche instability, we typically find that the strongest
+contributions to the relaxation of the slow mode ˆτS come
+from a set of 4 eigenstates of the closed system, which
+include a near-resonant pair of eigenstates |A⟩, |B⟩ (we
+explain what “near-resonant” means below). The other
+two eigenstates involved are obtained by “flipping” the
+ℓ-bit ˆτL adjacent to the bath:
+|A′⟩ = ˆτ x
+L |A⟩ ,
+|B′⟩ = ˆτ x
+L |B⟩ .
+(7)
+In some cases, |A′⟩ and |B′⟩ are also near-resonant, as we
+discuss in the Appendix with details. The polarization
+cA = ⟨A|ˆτS |A⟩ of the slow mode differs substantially
+between the “A” eigenstates (|A⟩ and |A′⟩) and the “B”
+eigenstates, while it is essentially unchanged by flipping
+ˆτL, so cA ∼= cA′ and cB ∼= cB′. Thus it is the transitions
+between {A, A′} eigenstates and {B, B′} eigenstates that
+relax ˆτS. If we “de-mix” [15] the near-resonance to make
+a more localized pair of orthonormal states |a⟩, |b⟩, we
+obtain:
+|A⟩ ∼= |a⟩ − ϵ |b⟩ ,
+|B⟩ ∼= |b⟩ + ϵ |a⟩ ,
+(8)
+where |a⟩ and |b⟩ differ by spin flips at a nonzero fraction
+of all sites, including the sites that are most polarized in
+(a)
+(b)
+Z
+A
+B
+B
+A
+XY
+A
+B
+B
+A
+XY case
+Z case
+(2/3)
+XY
+(2/3)
+(4/3)
+Figure 3.
+Two distinct cases for the set of 4 important
+eigenstates of ˆUF that are most important for the avalanche
+to propagate. The near-resonant pair {A, B} are coupled with
+ˆ
+XL and ˆYL jump operators to B′ and A′, respectively, i.e.,
+they have anomalously large matrix elements. In what we call
+the “Z case” (b), the near-resonant pair involves a spin flip
+next to the bath (site L), and the bath can additionally couple
+the resonant pair with the ˆZL jump operator with matrix
+element SAB twice as large as SA′B and SAB′. The two states
+with the largest Dµν are colored red. In the XY case (a), this
+pair is not the near-resonant pair, and {µ, ν} = {A, B′}; while
+they are the same in the Z case, so {µ, ν} = {A, B}. In some
+samples that are of the XY case, eigenstates |A′⟩ and |B′⟩
+are also near-resonant (not shown).
+ˆτS, so this is a long-range many-body resonance, in that
+sense, and it includes a “flip” of ˆτS. The spin that is
+flipped between |a⟩ and |b⟩ that is closest to the bath is
+at site x. As we show in the Appendix, for the largest
+L that we can access, the most probable location of x is
+near x/L = 0.8.
+In the regime near the estimated avalanche instability,
+the “mixing” ϵ in the near-resonance is typically exponen-
+tially small in L, although it is exponentially larger than
+the mixing between other typical pairs of eigenstates that
+differ over a similar distance range. This is why we call
+this a “near-resonance”: the mixing is relatively strong,
+partly due to the two eigenstates being near degeneracy
+in the spectrum of UF , but it is not fully resonant, since
+ϵ ≪ 1.
+In many samples, |A′⟩ and |B′⟩ are not near-resonant
+and are well-approximated by |A′⟩ ∼= ˆXL |a⟩ and |B′⟩ ∼=
+ˆXL |b⟩. As a consequence of the near-resonance between
+|A⟩ and |B⟩, i.e., the relatively large ϵ compared to other
+pairs of states, there are anomalously large matrix ele-
+ments | ⟨A| ˆXL |B′⟩ |2 ∼= | ⟨B| ˆYL |A′⟩ |2 ∼= ϵ2 of the two
+jump operators ˆE1 = ˆXL and ˆE2 = ˆYL. These drive two
+particularly large contributions, DAB′ and DBA′ (and
+their transposes), to the total relaxation rate of ˆτS be-
+cause SAB′ ∼= SBA′ ∼= 2ϵ2/3.
+In addition, when the near-resonance involves flipping
+the polarization of site L next to the bath (so x = L),
+there is another anomalously large matrix element. This
+time it is the matrix element | ⟨A| ˆZL |B⟩ |2 ∼= 4ϵ2 of the
+jump operator ˆE3 = ˆZL between the two near-resonant
+
+4
+eigenstates themselves. This results in SAB ∼= 4ϵ2/3 and
+another large contribution DAB ∼= 2DAB′ ∼= 2DBA′ (and
+its transpose) to the total relaxation rate of ˆτS.
+Thus there are two cases for Dµν, the largest contribu-
+tion to Eq. 6, corresponding to whether or not the reso-
+nance involves flipping the spin nearest to the bath. The
+bath can drive relaxation of the slowest mode through
+a resonance indirectly (with X and Y jump operators),
+i.e., the pairs of eigenstates involved are not the near-
+resonant pair, and sometimes directly (with Z jump op-
+erator). We depict the two cases in Fig. 3(a,b) and call
+them the “XY ” and “Z” cases. Our claim is that the
+relaxation of the slowest mode, and thus the spread of
+the avalanche, proceeds via these dominant processes in-
+volving 4 eigenstates that include a particularly strong
+near-resonance, as explained in this section. This struc-
+ture implies that the largest contribution Dµν comes from
+either the Z jump operator or the X and Y jump op-
+erators of the bath, depending on which of the above
+cases is relevant.
+In the Z case, which is x = L, the
+largest contribution (A-B pair) is accompanied by two
+XY contributions of half the magnitude (A-B′ and B-A′
+pairs). In the XY case, which is x < L, there are two
+roughly equal largest contributions (A-B′ and B-A′). In
+both cases, the largest contribution Dµν doesn’t exceed
+half of the total relaxation. The phenomenology of the
+near-resonant eigenstate set explained in this section is
+corroborated by numerical observations in the next sec-
+tion. An extension of this simple description is discussed
+in the Appendix.
+Numerical Observations— We numerically deter-
+mine the eigenstates of ˆUF , the slowest mode of S, and
+the contributions Dmn to its relaxation rate associated
+with probability currents between eigenstates. We exam-
+ine the pairs of eigenstates that contribute the most (|µ⟩
+and |ν⟩), identify the 4 important eigenstates discussed
+earlier, and quantify how they are related to each other
+to verify that the near-resonant set of eigenstates does
+indeed have that structure.
+Recall that in our model, the polarization has a pre-
+ferred spin direction: Z. Deep in the MBL regime, the
+local ℓ-bit operators typically are very close to the lo-
+cal single-spin Pauli operators.
+Therefore, we identify
+|γ′⟩ ≡ τ x
+L |γ⟩ for γ ∈ {µ, ν} simply by finding the eigen-
+state with largest |⟨γ| ˆXL |n⟩| among all eigenstates |n⟩.
+To determine which jump operator from the bath dom-
+inantly couples |µ⟩ and |ν⟩ we define a tool
+Zjump(m, n) :=
+| ⟨m| ˆZL |n⟩ |2
+�3
+j=1 | ⟨m| ˆEj |n⟩ |2 .
+(9)
+Note that ˆE3 = ˆZL.
+We find that Zjump(µ, ν) is ex-
+tremely close to 0 or 1 in any one sample, as shown in
+the inset of Fig. 4(a,b). This separation is captured by
+the minimal model since Zjump(µ, ν) ∼= 0 (or 1) is asso-
+ciated to the XY (or Z) case where the pair {µ, ν} isn’t
+(a)
+(b)
+(c)
+(d)
+0
+0.5
+1
+0
+0.5
+1
+0
+1
+0
+1
+XY and Z
+AB’ and A’B
+AB
+only Z
+XY case
+system size (L)
+system size (L)
+Z case
+jump
+jump
+Figure 4.
+Consistency with the near-resonant eigenstate
+set. All data here are for α = 30. (a) The near-resonance
+between {A, B} causes the AB′ and A′B matrix elements to
+be roughly equal, as described in the minimal model. The
+distribution of the ratio of SAB′ and SA′B is indeed peaked
+near 1. The inset shows Zjump of AB′ and A′B are peaked at
+zero, implying that these pairs are coupled with the ˆ
+XL and
+ˆYL jump operators. (b) When the near-resonant pair involves
+the spin flip of the last site (Z case), the bath can directly
+couple |A⟩ and |B⟩. The distribution of the shown ratio of
+matrix elements demonstrates that SAB ≃ 2SAB′ ≃ 2SA′B.
+The inset shows Zjump of AB is peaked at 1, implying A and
+B are coupled with ˆZL.
+The bottom two panels show the
+demixing angles for (c) the XY case and (d) the Z case. The
+points correspond to AB (red dot), A′B (gray ×), AB′ (gray
+dot), and A′B′ (blue dot). The near-resonant pair identified
+as in the main text (AB) has the largest demixing angles for
+all cases. The calculations in (a) and (b) are done with L = 11
+and 104 disorder realizations.
+(or is) the near-resonant pair {A, B}. We therefore iden-
+tify the near-resonant pair for each sample based on the
+minimal model as follows. In the case that |µ⟩ and |ν⟩—
+the pair with the largest Dmn—are “connected” with Z
+(Zjump ∼= 1), these states are identified as |A⟩ and |B⟩.
+Otherwise, if they are connected with X and Y instead
+(Zjump ∼= 0), we associate {|A⟩ , |B⟩)} to {|µ′⟩ , |ν⟩} or
+{|µ⟩ , |ν′⟩}, whichever pair has the smaller quasienergy
+splitting.
+We first checked that cγ ∼= cγ′ indeed holds for γ ∈
+{A, B}. This relation is satisfied because Sγ′γ is of or-
+der one, so the slow mode ˆτS contains negligible pop-
+ulation difference between γ and γ′. Also, we checked
+that |cµ − cν| is of order one, which should be true as
+
+5
+we picked the pair with the largest Dµν = Sµν(cµ − cν)2.
+Therefore, the matrix elements Smn (Eq. 4) are a good
+proxy for Dmn (Eq. 5) within this set of important eigen-
+states, so we compare the matrix elements going forward.
+We compare these matrix elements to confirm the re-
+sults are consistent with the minimal model described
+in Fig. 3(a,b). As we show in Fig. 4(a), SAB′ ∼= SBA′
+holds (and their transposes), and XY jump operators
+couple them (inset). Furthermore, in the case that spin
+L is flipped between |A⟩ and |B⟩, we additionally observe
+SAB ∼= 2SAB′ ∼= 2SBA′ as presented in Fig. 4(b). In this
+case |A⟩ and |B⟩ are coupled with ˆZL (inset).
+We note that the near-resonant eigenstate set of the
+previous section explains why R < 1/2. When the near-
+resonance doesn’t involve a spin flip next to the bath,
+SAB′ = SA′B, so DAB′ = DA′B and no single pair con-
+tributes more than 1/2 of the total.
+If the resonance
+flips the last spin, DAB = 2DAB′ = 2DA′B, so again the
+maximum contribution can’t be greater than half of the
+total.
+We finally tested our picture using the “demixing” pro-
+cedure introduced in Ref. [15]. In that procedure, one
+calculates the most localized basis of a subspace spanned
+by two eigenstates, and thus also the corresponding ba-
+sis rotation. The basis rotation corresponds to a location
+on a Bloch sphere; the polar angle, called the “demixing
+angle”, is a measure of the resonance strength between
+the two eigenstates (ϵ in Eq. 8). As shown in Fig. 4(c,d),
+the demixing angle between the eigenstates we identified
+as |A⟩ and |B⟩ is by far the largest among other pairs in
+our eigenstate set, consistent with the idea that that pair
+is indeed a near-resonance that enables the bath to relax
+the slowest mode, i.e., the avalanche to propagate.
+Conclusion— Assuming the avalanche has proceeded
+for a sufficiently large distance, we model the putative
+thermal bubble as an infinite Markovian bath with infi-
+nite temperature. In this model, we discovered the exis-
+tence of dominant processes in the avalanche, involving
+only a few pairs of eigenstates of the closed system, in-
+cluding a strong near-resonance. The avalanche proceeds
+through these rare eigenstate pairs, leveraging many-
+body near-resonances to relax the spins some distance
+away along the chain. The inner structure of the domi-
+nant set of eigenstates is dictated by whether or not the
+associated resonance involves flipping a spin at the site
+next to the bath. This sets what jump operators effec-
+tively use the resonance present in the closed system to
+spread the avalanche. We presented a minimal model in-
+volving two near-resonant eigenstates and two additional
+auxiliary states to explain how this works in detail, and
+verified our picture with numerical observations.
+Our
+work advances the understanding of the avalanche in-
+stability of many-body localization and provides a de-
+tailed connection to rare many-body resonances present
+in MBL systems.
+Acknowledgements— We thank Sarang Gopalakr-
+ishnan, Vedika Khemani and Jacob Lin for discussions.
+The work at Princeton was supported in part by NSF
+QLCI grant OMA-2120757. A.M. was supported in part
+by the Stanford Q-FARM Bloch Postdoctoral Fellowship
+in Quantum Science and Engineering and the Gordon
+and Betty Moore Foundation’s EPiQS Initiative through
+Grant GBMF8686. Simulations presented in this work
+were performed on computational resources managed and
+supported by Princeton Research Computing.
+Appendix
+Random Floquet Model
+1
+2
+L
+L-1
+...
+Figure 5.
+Random Floquet Model introduced in Ref. [15].
+Details of the model are described in the Appendix.
+We use the Floquet circuit model of Morningstar, et
+al. [15] and depicted in Fig. 5.
+The Floquet unitary
+operator UF is obtained by applying a layer of one-site
+gates Ud followed by L−1 two-site gates—one per bond—
+denoted by Uu. The onsite gates are first sampled from
+the 2 × 2 circular unitary ensemble (CUE) and then di-
+agonalized, so the localizing disorder of the model is in
+the Z direction. The two-site gates ui act on sites i and
+i + 1 and can be written as
+ui = exp
+� i
+αMi
+�
+(10)
+where Mi are sampled from the 4 × 4 Guassian unitary
+ensemble (GUE). 1/α controls the interaction strength so
+that the effect of the disorder gets stronger as α increases.
+The order of applying the ui’s is determined randomly for
+each sample.
+Slowest Mode
+As explained in the main text, one can obtain the slow-
+est mode ˆτS in the weak-coupling limit by diagonalizing
+
+6
+(a)
+(b)
+(c)
+Disorder
+Strength
+site ( )
+site ( )
+site ( )
+Pauli string weight (
+)
+Figure 6.
+Pauli String weight wn for (a) τ1, (b) ¯τ1, and (c) the slowest mode τS, averaged (after taking the logarithm) for 103
+samples with system size L = 10.
+the superoperator S[ˆρ] :=
+1
+3
+�
+µ ˆEµˆρ ˆE†
+µ, in the degen-
+erate subspace of Floquet eigenstates |n⟩ ⟨n|, where the
+matrix elements are
+Snm = ⟨m| S[|n⟩ ⟨n|] |m⟩ = 1
+3
+�
+µ
+| ⟨m| ˆEµ |n⟩ |2
+(11)
+and ˆEµ = ( ˆXL, ˆYL, ˆZL) are the jump operators at the
+spin next to the bath. In particular, ˆτS = �
+n cn |n⟩ ⟨n|
+where ⃗cn is the eigenvector of Snm with the smallest spec-
+tral gap Γ. As the slowest mode is constructed from the
+degenerate subspace of |n⟩ ⟨n|, it is a conserved quantity
+(a LIOM) of the purely unitary dynamics given by UF .
+We derive useful features of the slowest mode ˆτS below.
+Lemma 1. For any Pauli-string operator P ′ acting
+on the (L − 1) spins away from the bath, and ˆEµ =
+( ˆXL, ˆYL, ˆZL) acting on the last spin next to the bath, the
+following holds:
+S[P ′ ⊗ IL] = P ′ ⊗ IL,
+S[P ′ ⊗ ˆEµ] = −1
+3P ′ ⊗ ˆEµ.
+(12)
+Proof. The super-operator S only acts on site L, such
+that for an arbitrary matrix M ∈ M(2,2), S[P ′ ⊗ M] =
+P ′ ⊗
+�
+1
+3
+�
+µ ˆEµM ˆE†
+µ
+�
+. Eq. 12 follows by substituting IL
+and ˆEµ in M.
+Theorem 2. The slowest mode is the conserved quantity
+of the closed system (which is not an identity) with the
+smallest weight from the Pauli strings with non-identity
+at site L (connected to the bath).
+Proof. Consider a conserved quantity written as ˆτ =
+�
+n cn |n⟩ ⟨n| where cn ∈ R. The normalization condi-
+tion is �
+n c2
+n = 2L as we defined in the main text. We
+can write the slowest mode in the Pauli-string basis like
+ˆτS = �
+P ¯cP P with P being the Pauli-strings. The coef-
+ficients are ¯cP = Tr(ˆτSP)/2L. In the Pauli-string basis,
+the normalization condition is rewritten as �
+P ¯c2
+P = 1.
+The total weight of the Pauli-strings with non-identity at
+site L is
+wL =
+�
+P ′
+¯c2
+P ′⊗ ˆ
+XL +
+�
+P ′
+¯c2
+P ′⊗ ˆYL +
+�
+P ′
+¯c2
+P ′⊗ ˆ
+ZL
+=
+�
+µ
+�
+P ′
+¯c2
+P ′⊗ ˆ
+Eµ
+(13)
+where P ′ is a Pauli-string defined on L-1 sites and ˆEµ =
+( ˆXL, ˆYL, ˆZL) are the Pauli operators from site L.
+Next, we define a functional R(⃗c) := �
+n,m cnSnmcm.
+Below, we show how the functional R is related to wL.
+Tr (ˆτS[ˆτ])) = Tr
+��
+l
+cl |l⟩ ⟨l|
+�
+n
+cnS[|n⟩ ⟨n|]
+�
+= Tr
+��
+l
+cl |l⟩ ⟨l|
+�
+n,m
+cnSnm |m⟩ ⟨m|
+�
+=
+�
+n,m
+cnSnmcm = R
+(14)
+We can rewrite the same equation under the Pauli-string
+basis, ˆτ = �
+P ¯cP P. Here, we use Lemma 1 introduced
+earlier.
+Tr (ˆτS[ˆτ])) =
+�
+P ′
+¯c2
+P ′⊗I − 1
+3
+�
+µ
+�
+P ′
+¯c2
+P ′⊗ ˆ
+Eµ
+= 1 − 4
+3
+�
+µ
+�
+P ′
+¯c2
+P ′⊗ ˆ
+Eµ
+= 1 − 4
+3wL
+(15)
+From the above two equations, we derive the relation
+R = 1 − (4/3)wL.
+
+7
+In the final step, let’s define f := �
+n c2
+n. Using the
+Lagrange multiplier method, functional R is local ex-
+tremum under the constraint f = 2L when ∇R−λ∇f = 0
+holds, which is equivalent to the eigenvalue problem:
+∀n,
+�
+m Snmcm = λcn. In this condition, we obtain
+R = λ, and the weight of the Pauli-strings with non-
+identity at site L is wL = (3/4)(1−λ). Therefore, finding
+ˆτ which minimizes wL is equivalent to finding the eigen-
+vector of the super-operator S with the second largest
+eigenvalue (largest eigenvalue is one which corresponds
+simply to an identity matrix).
+Hence, the conserved
+quantity with the smallest wL except the identity is the
+slowest mode ˆτS.
+Theorem 3. The slowest mode is traceless.
+Proof. From the definition of the slowest mode, it is
+an eigenmatrix of superoperator S with an eigenvalue
+1 − Γ.
+Therefore, Tr(ˆτS)
+=
+Tr(S[ˆτS])/(1 − Γ)
+=
+Tr( 1
+3
+�
+µ ˆEµˆτS ˆE†
+µ)/(1 − Γ) = Tr(ˆτS)/(1 − Γ).
+As 0 <
+Γ < 1, it should be traceless Tr(ˆτS) = 0.
+We numerically compared the slowest mode ˆτS with
+the LIOM (τ1) and ℓ-bit (¯τ1) introduced in Refs. [8–10].
+The explicit forms of τ1 and ¯τ1 are
+τ1 ≡
+�
+n
+⟨n| ˆZ1 |n⟩ |n⟩ ⟨n|
+(16)
+¯τ1 ≡
+�
+n
+sgn(⟨n| ˆZ1 |n⟩) |n⟩ ⟨n| ,
+(17)
+where |n⟩s are the eigenfunctions of the closed system.
+Note that τ1 and ¯τ1 are the conserved quantities that
+are localized near site 1, farthest from the bath. In par-
+ticular, we calculate the Pauli string weight wi(ˆτ) for
+ˆτ ∈ {τ1, ¯τ1, ˆτS}, which is the total weight of the Pauli
+string bases with the last non-identity matrix placed at
+i-th site. These Pauli bases are products of identity ma-
+trices for sites farther than i.
+We compare the Pauli string weight in Fig. 6. For suf-
+ficiently large disorder strength α, all three operators are
+localized far from the bath. It is clear that the slowest
+mode is different from τ1 and ¯τ1. Compared to the other
+two quantities, the slowest mode has the least weight at
+the last site, as we proved in Theorem 2.
+The differ-
+ence manifests when we compare the three operators for
+a single sample. While τ1 and ¯τ1 always have maximum
+weight on the first site, the slowest mode often has max-
+imum weight other than the first site but is still far away
+from the bath. What constrains the slowest mode is not
+the first site to have the maximum weight but to have
+the least weight on the last site next to the bath, which
+was indeed true for every sample.
+Derivation of Equation 6
+As explained in the main text, the slowest mode is
+ˆτS = �
+n cn |n⟩ ⟨n| where ⃗cn is the eigenvector of a sym-
+metric Markovian matrix Snm with the smallest spec-
+tral gap Γ (�
+m Snmcm = (1 − Γ)cn).
+The operator
+ˆτS is normalized such that ∥ˆτS∥2
+1 = �
+n c2
+n = 2L.
+In
+our model, the relaxation rate of the slowest mode is
+rS = 3γΓ. We assume a gap between the smallest and
+second-smallest spectral gaps.
+With this assumption,
+the density matrix at the latest time is asymptotically
+ˆρ(t) ≃ I/2L + ϵe−rStˆτS. Therefore, ˆρ(t) thermalizes and
+converges to ˆρ(∞) = I/2L asymptotically at the latest
+time with the rate:
+d
+dt (∥ˆρ(t) − ˆρ(∞)∥1)2
+≃ d
+dt
+�
+∥ϵe−rStˆτS∥1
+�2
+= d
+dt
+�
+|ϵ|2 e−2rSt ∥ˆτS∥2
+1
+�
+= d
+dt
+�
+|ϵ|2 e−2rSt 2L�
+= −2rS |ϵ|2 e−2rSt 2L
+= −3γ |ϵ|2 e−2rSt 2L+1 Γ
+(18)
+Also, one can view ˆτS as specifying a structure of
+probability currents that flow between eigenstates of the
+isolated system during the late-time approach to equi-
+librium.
+Specifically, if we define a current jnm :=
+Snm(cn − cm), then
+d
+dt ˆρmm(t) = 3ϵγe−rSt �
+n jnm holds
+at late times, which implies that jnm obtained from ˆτS
+determines the structure of the late time thermalization.
+One can estimate how ˆρ(t) converges to ˆρ(∞) with the
+inner structure of ˆτS:
+d
+dt (∥ˆρ(t) − ˆρ(∞)∥1)2
+≃ d
+dt
+��
+m
+|ˆρmm(t) − 1/2L|2
+�
+= 6γ|ϵ|2 �
+m
+e−rStcn
+�
+e−rSt �
+n
+jnm
+�
+= 3γ|ϵ|2e−2rSt �
+n,m
+(cn − cm)jmn
+= −3γ|ϵ|2e−2rSt �
+n,m
+(cn − cm)2Snm
+= −3γ|ϵ|2e−2rSt �
+n,m
+Dnm
+(19)
+Here, we used the fact that at the latest time, only the
+diagonal terms in ˆρ(t) remain (under the weak coupling
+limit γ ≪ 1, the slowest mode consists of only the diago-
+nal terms). Dnm quantifies the contribution of a pair |n⟩
+and |m⟩ to the relaxation of the slowest mode.
+
+8
+Comparing the two equations above, one can derive
+the following:
+�
+n,m
+Dnm =
+�
+n,m
+(cn − cm)2Snm = 2L+1Γ.
+(20)
+This can also be directly derived by:
+�
+n,m
+Dnm =
+�
+n,m
+(cn − cm)2Snm
+=
+�
+n,m
+(c2
+n + c2
+m)Snm − 2
+�
+n
+cn
+��
+m
+Snmcm
+�
+= 2L+1 − 2L+1(1 − Γ)
+= 2L+1Γ.
+(21)
+From the above derivations, the largest contribution
+Dµν is related with Γ by
+Dµν = R
+�
+n<m
+Dnm = 2LRΓ.
+(22)
+Therefore, one can naively derive the relation between
+the thermalization rate and the matrix element Sµν be-
+tween the most important pair of states with an effective
+O(1) value of R as follows.
+rS = 3γΓ = 3γ(cµ − cν)2
+R
+2−LSµν
+(23)
+As the observed ratio R and cµ −cν are both O(1) values
+for MBL regimes, we estimate a relation rS ∼ 2−LSµν.
+Detuning of near-resonance by flipping ˆτL
+In this section, we examine to what extent there is
+also a near-resonance between eigenstates |A′⟩ and |B′⟩.
+These two eigenstates differ from the near-resonant states
+|A⟩ and |B⟩ by the flipping of ℓ-bit ˆτL.
+This change
+may strongly detune the near-resonance, which is what
+happens for samples where the near-resonance involves
+flipping spins that are close to or include spin L. But
+in other samples, the near-resonance only involves spin
+flips that are rather far from spin L, so flipping ˆτL has a
+rather small effect on the near-resonance. We denote the
+location of the spin-flip that is closest to L as x.
+The minimal wavefunction model introduced in the
+main text (Eq. 8) neglected the possibility of a near-
+resonance between (A′, B′). Revising our minimal model
+by including and “de-mixing” the possible near-resonance
+between (A′, B′), one can write:
+|A⟩ = |a⟩ − ϵ |b⟩
+|B⟩ = |b⟩ + ϵ |a⟩
+|A′⟩ = |a′⟩ − ϵ′ |b′⟩
+|B′⟩ = |b′⟩ + ϵ′ |a′⟩ .
+(24)
+The quasi-energy splittings of these two near-resonances
+are ∆ = |θA − θB| and ∆′ = |θA′ − θB′|; we have labelled
+them so that ∆ < ∆′. We can partially quantify how
+much flipping ˆτL alters the near-resonance by comparing
+ϵ′ and ∆′ to ϵ and ∆.
+The distributions of ϵ′/ϵ and ∆′/∆ for different system
+sizes are shown in Fig. 7(a,b). Both distributions show
+long tails to |ϵ′| ≪ |ϵ| and ∆′ ≫ ∆. This shows that in a
+substantial fraction of the samples the near-resonance is
+strongly detuned by flipping ˆτL. Fig. 7(d,e) shows that
+this mostly happens due to the near-resonance extending
+to near site L, so x/L is close to or equal to one.
+There are also many samples that have |ϵ|, |ϵ′|, and
+|ϵ − ϵ′| all of the same order, as indicated by the peaks
+near ϵ′/ϵ ∼= ∆′/∆ ∼= 1 in Fig. 7(a,b). These are the cases
+where flipping ˆτL does alter the near-resonance, but not
+by a particularly large or small amount, which mostly
+happens for smaller x/L [see Fig. 7(d,e)].
+The cases where flipping ˆτL has very little effect on
+the near resonance should have ϵ′/ϵ and ∆′/∆ very close
+to one, so would make a very sharp peak in those distri-
+butions, which is not seen in Fig. 7(a,b). This suggests
+that only a small fraction of samples are in this category.
+For such samples SAB′ is small due to a near cancella-
+tion between the contribution from |a⟩ coupling to |a′⟩
+and that from |b⟩ coupling to |b′⟩. This near-cancellation
+will not happen if we replace |A⟩ with |a⟩ and look in-
+stead at SaB′. Thus in Fig. 7(c) we show the distribu-
+tion of SAB′/SaB′ (∼ |(ϵ − ϵ′)/ϵ′|2). The much weaker
+tail in this distribution to very small SAB′/SaB′ are the
+samples where the near-resonance is only very weakly af-
+fected by flipping ˆτL, while the much stronger tail to very
+large SAB′/SaB′ are those more common samples where
+this flip strongly detunes the near-resonance. Fig. 7(f)
+shows the median SAB′/SaB′, with the expected trend of
+stronger detuning (larger SAB′/SaB′) as x/L increases
+and the near-resonance thus extends closer to the flipped
+ˆτL.
+Range of Near-Resonance
+The previous sections described how the bath uses
+near-resonances to flip the spins at the other end of an
+MBL chain and thermalize the system. Therefore, it is
+perhaps natural to expect the near-resonance involved to
+be an end-to-end resonance such that the last spin-flip
+between |A⟩ and |B⟩ should not be far away from the
+bath. Strikingly, however, the distance from the bath to
+the last spin-flip appears to increase in proportion to the
+system length L. In particular, we compare the spin po-
+larization on every site for the near-resonant eigenstate
+pair (A, B) and find the location x ≤ L where the last
+spin flip occurs. These two eigenstates are directly cou-
+pled with the ˆZx operator (| ⟨A| ˆZx |B⟩ |2 ∼= 4ϵ2), and in
+the special case when x = L, the jump operator from the
+bath can directly couple the near-resonant pair (which
+we defined as the Z case).
+
+9
+Distribution (a.u.)
+Distribution (a.u.)
+Distribution (a.u.)
+(a)
+(b)
+(d)
+(e)
+(c)
+(f)
+4
+L=
+13
+median (
+)
+median (
+)
+median (
+)
+Figure 7. Comparing (A, B) and (A′, B′). All data here are for α = 30 and L ∈ [4, 13]. Distributions of (a) ϵ′/ϵ, (b) ∆′/∆,
+and (c) SAB′/SaB′ show there are many samples satisfying |ϵ′| ≪ |ϵ| or |ϵ| ∼ |ϵ′| ∼ |ϵ − ϵ′|, but cases with |ϵ − ϵ′| ≪ |ϵ| are
+less common (see text of the Appendix). The median of (d) ϵ′/ϵ, (e) ∆′/∆, and (f) SAB′/SaB′ vs. the location of the last
+spin-flip (x/L) of the near-resonance are illustrated. The detuning due to flipping ˆτL gets more significant as the last spin-flip
+gets closer to the bath.
+L= 4
+Normalized Last Spin-Flip Position (x/L)
+13
+Distribution (a.u.)
+Figure 8.
+Distribution of the last spin flip location.
+We
+normalized the location with system size (x/L).
+The normalized position of the last spin flip (x/L) is
+illustrated in Fig. 8 with different colors for different sys-
+tem sizes (L ∈ [4, 13]). It is clear that as the system size
+gets larger, the curve peaks when it is smaller than 1 (the
+peak is near x/L ≈ 0.8 < 1), indicating that an extensive
+number of sites are in between the bath and the last spin
+flip.
+As we explained in the previous section, the min-
+imal model including the possible near-resonance be-
+tween (A′, B′) predicts the matrix element proportional
+to |ϵ−ϵ′|2. The value of x that yields the largest matrix el-
+ement is determined by two competing factors: First, as x
+becomes smaller, the resonance’s range becomes shorter
+and fewer degrees of freedom are involved.
+Therefore
+the resonance becomes stronger, with a typically larger
+ϵ. However, decreasing x by too much also causes the
+resonance to decouple from the bath: flipping the ℓ-bit
+beside the bath may then produce a near copy of the res-
+onance and result in small |ϵ − ϵ′| even though |ϵ| and
+|ϵ′| are large, which will result in that resonance not be-
+ing useful to the bath in relaxing the slowest mode. In
+the other direction, as x becomes larger the resonance
+becomes longer-ranged and thus weaker (|ϵ| and |ϵ|′ be-
+come smaller), but it also couples more strongly to the
+bath so that |ϵ − ϵ′| ∼= |ϵ| ≫ |ϵ′|.
+
+10
+Therefore, in most samples for large L the bath uses
+near-resonances with x/L < 1. The range of the optimal
+near-resonance is determined by a balance between the
+two considerations mentioned above.
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+

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+Nonlinear Optimization Filters for Stochastic Time-Varying Convex
+Optimization
+Andrea Simonetto a
+Paolo Massioni b
+February 1, 2023
+Abstract
+We look at a stochastic time-varying optimization problem and we formulate online algorithms to
+find and track its optimizers in expectation. The algorithms are derived from the intuition that standard
+prediction and correction steps can be seen as a nonlinear dynamical system and a measurement equation,
+respectively, yielding the notion of nonlinear filter design. The optimization algorithms are then based
+on an extended Kalman filter in the unconstrained case, and on a bilinear matrix inequality condition
+in the constrained case. Some special cases and variations are discussed, notably the case of parametric
+filters, yielding certificates based on LPV analysis and, if one wishes, matrix sum-of-squares relaxations.
+Supporting numerical results are presented from real data sets in ride-hailing scenarios. The results are
+encouraging, especially when predictions are accurate, a case which is often encountered in practice when
+historical data is abundant.
+1
+Introduction
+We look at time-varying optimization problems of the form
+min
+xPRn fpx; yptqq ` gpxq,
+t ě 0,
+(1)
+where f : RnˆRd Ñ R is a smooth strongly convex function in x (for all y), parametrized over a time-varying
+data stream yptq P Rd (t represents the continuous time), and g is a closed convex and proper function (such
+as the indicator function, or an ℓ1 regularization).
+Problem (1) appears naturally in a number of application scenarios, where optimal decisions have to be
+taken online and they change as new data arrive. Examples stem from video processing [1] to robot control [2],
+and to large-scale optimal management of smart infrastructures [3].
+Solving Problem (1) means to find and track the optimizer trajectory x‹ptq, as yptq changes and it is
+revealed. In order to accomplish this, in the standard time-varying literature, one can sample Problem (1)
+at discrete time instant tk, k “ 0, 1, . . ., with h :“ tk ´ tk´1 being the sampling time, and solve the sequence
+of time-invariant problems
+x‹ptkq “ arg min
+xPRn fpx; yptkqq ` gpxq,
+k P N,
+(2)
+as they are revealed in time. Then one can set up online algorithms that find approximate x‹ptkq’s, say
+xk’s, that eventually converge to1 the solution trajectory, within an error bound. The reader is referred to
+the surveys [4, 5] for an ample treatment of the methods in both discrete and continuous time. One of the
+important aspects to keep in mind is that online algorithms are sought that are computationally frugal, so
+that one can approximate the solution of x‹ptkq within the sampling time h, and the key performance metric
+is how good the algorithms are with respect to an algorithm that has infinite computational time.
+A tacit assumption in the above methods is that one wants to converge to the solution trajectory generated
+by the evolution of the data stream yptq. However, this may be not the best course of action, since the data
+aUMA, ENSTA Paris, Institut Polytechnique de Paris, 91120 Palaiseau, France; [email protected].
+bUniv Lyon, INSA Lyon, Universit´e Claude Bernard Lyon 1, Ecole Centrale de Lyon, CNRS, Amp`ere, UMR5505, 69621
+Villeurbanne, France; [email protected]
+1Somebody could rather say: “track”.
+1
+arXiv:2301.13646v1  [math.OC]  31 Jan 2023
+
+are often noisy and convergence to a noisy optimizer may be not advisable. A better angle is to ask whether
+we can set up algorithms to converge to a filtered version of the trajectory.
+In this context, we re-interpret Problem (1) as a stochastic problem, where we would like to find and
+track the filtered solution trajectory as
+ˆx‹ptq “ arg min
+xPRn EyPYptqrfpx; yqs ` gpxq,
+(3)
+where the expectation EyPYptqr¨s is with respect to the random variable y, which is drawn from a time-varying
+probability distribution Yptq.
+Were the probability distribution time-invariant, Formulation (3) would be common in stochastic opti-
+mization. With our setting, we are considering instead a gradual distribution shift of an unknown distribution,
+which renders the formulation less common and more challenging. Recent papers have started to look into
+this formulation [6–8]; especially the third one is the closest to our goal, however the authors “just” adapt
+the standard prediction-correction algorithms to the stochastic setting by properly tuning the prediction and
+the correction step sizes. Also they do not consider a non-smooth component as our function gpxq.
+In this paper, we look from a different angle and ask whether we can use a perturbed version of the
+optimality condition as a suitable dynamical model to do filtering. This will have the advantage to combine
+prediction and correction in novel and more performant ways. To fix the ideas on this novel angle, consider
+the unconstrained problem:
+x‹ptq “ arg min
+xPRn fpx; yptqq,
+(4)
+with a strongly convex, doubly differentiable, and smooth function f in x. By perturbing the optimality
+condition ∇xfpx‹ptq; yptqq “ 0n, we can derive the ordinary differential equation that describes how the
+optimizer evolves as [9]:
+d
+dtx‹ptq
+“
+´r∇xxfpx‹ptq; yptqqs´1∇yxfpx‹ptq; yptqq d
+dtyptq
+“:
+Fpx‹ptq, yptq, 9yptqq.
+(5)
+This is a nonlinear dynamical system. To derive a filter, we need to couple this model with a measurement
+equation, which tells us how far from the optimizer trajectory we are. We can thus use,
+zptq “ ∇xfpx‹ptq; yptqq,
+(6)
+as a measurement equation (i.e., it will be different from zero at any point but on the optimizer trajectory,
+where zptq “ 0n).
+Armed with (5) and (6), we could design a dynamical filter to reconstruct x‹ptq based on a noisy data
+stream yptq.
+1.1
+Contributions
+Starting from the continuous-time intuition from (5) and (6), we develop discrete-time filters for unconstrained
+and constrained time-varying problems. In particular,
+‚ We derive an extended Kalman filter for unconstrained and differentiable convex problems in the discrete-
+time setting starting from an algorithmic viewpoint;
+‚ We generalize the filtering procedure for constrained and non-differentiable problems leveraging the non-
+linear dynamical systems and nonlinear measurement equations coming from forward-backward algorithms
+and fixed-point residuals. We are then able to derive the optimal “Kalman-style” gain via both a worst-case
+approach and via dissipativity theory. We also present a possibly less conservative linear parameter-varying
+(LPV) methodology.
+‚ We showcase the benefit of the approach with respect to state-of-the-art prediction-correction methods
+(which our filters generalize), in numerical simulations stemming from a ride-hailing example with multiple
+companies in New York City.
+2
+
+1.2
+Related work
+Time-varying and stochastic optimization are vibrant research fields, and we do not plan to give an exhaustive
+account here. The reader is referred to [4,5,10] for the first and [11–14] for a sub-sample of the second, and
+the many references therein. Jointly Stochastic and time-varying optimization is a less studied area, and as
+we have mentioned [6–8] scratch the surface in this direction. The celebrated [15] paper is also somewhat in
+the general area of interest, though their approach does not include prediction, it uses restart and is more
+directed at finding optimal regret rates for non-stationary objectives rather than noise.
+We will build on techniques from dissipativity theory for analyzing and designing optimization algorithms.
+This is a recent and fertile area brought to fame by L. Lessard and collaborators’ seminal paper [16], and
+now gaining momentum [17–22]. Our novel insight in this direction is to use the techniques to determine
+Kalman-style gains for optimization algorithms with errors. Finally, we use LPV techniques from [23–25],
+especially in the context of matrix sum-of-squares relaxations.
+Organization. The remaining of the paper is organized as follows. Section 2 discusses formulation and
+main assumptions. We focus on the unconstrained case in Section 3, and on the general case in Section 4.
+Section 5 describes our gain design. We then conclude with some numerical simulations in Section 6. All
+proofs are given in the Appendix.
+Notation. Notation is wherever possible standard. For a differentiable function f, we define a step of
+the gradient method starting from a point xk as xk`1 “ rI ´ α∇xfp‚qsxk ” xk ´ α∇xfpxkq, where α ą 0 is
+the step size and I is the identity operator. Then, s steps are indicated as,
+xk`1 “ xs
+k “ rI ´ α∇xfp‚qs˝sxk.
+(7)
+We let also x0
+k “ xk when needed.
+We further indicate with proxαgpxq the proximal operator,
+proxαgpxq “ arg min
+vPRn
+!
+gpvq ` 1
+2α}v ´ x}2)
+.
+(8)
+Finally, spaces are indicated as R, N, probability distributions are calligraphic, i.e., Y, matrices and vectors
+are boldfaced, e.g., x P Rn, A P Rnˆm, operators are in sans-serif, e.g., I, J, constants are in standard roman.
+2
+Problem formulation and assumptions
+Let us now consider the sequence of problems,
+ˆx‹ptkq “ arg min
+xPRn EyPYptkqrfpx; yqs ` gpxq,
+k P N,
+(9)
+with a strongly convex, doubly differentiable and smooth cost function f uniformly in x and a generic convex
+function g. Let us also introduce the shorthand notation ˆx‹
+k “ ˆx‹ptkq, yk “ yptkq, and Yk “ Yptkq. Notice
+that, as done in stochastic optimization, the data point yk “ yptkq is supposed to be a random vector drawn
+from the distribution Yk “ Yptkq.
+First, let us derive a discrete-time dynamical system on how the optimizers evolve in time. Quite naturally,
+one could be attempted at discretizing (5), but the presence of the inverse of the Hessian and the derivative of
+the data stream makes it quite cumbersome, especially if one has then to linearize it for an extended Kalman
+filter.
+Instead we use a Bayesian approach, assuming that we have a (noisy) prior on how the data stream
+evolves, and we start from what we can be computed algorithmically. Let us denote with Jk`1pxq : Rn Ñ Rn
+an approximation at time tk of EyPYk`1r∇xfpx; yqs.
+Noisy prior on how the data evolves and how the
+gradient evolves can come from linear filters, or more sophisticated neural network models, or kernel models
+(see also [26] for what they call predictable sequences). For now think about one of the simplest model:
+Jk`1pxq “ 2∇xfpx; ykq ´ ∇xfpx; yk´1q (obtained via an extrapolation technique [27]).
+Then, we use an algorithmic view. At time k, we would like to solve
+ˆx‹
+k`1 “ arg min
+xPRn EyPYk`1rfpx; yqs ` gpxq,
+(10)
+3
+
+yet it is not possible with data up to time tk. So, we introduce the noisy dynamical system
+ˆx‹
+k`1 “ rproxαgpI ´ αJk`1p‚qqs˝P ˆx‹
+k ` qk “: Φk,gpˆx‹
+kq ` qk,
+(11)
+where rproxαgpI ´ αJk`1p‚qqs˝P means the application of the proximal gradient method of step size α ą 0
+for P times, and qk is the process error. The error comes both from a modelling error (truncating after P
+iterations), and from the noisy predicted gradient Jk`1pxq. We will see how to characterize the error later,
+but for now it is useful to keep in mind that qk is not-zero mean in general. If the gradient were exact and
+P Ñ 8, then Equation (11) would solve (10) with no noise (qk “ 0).
+The “pseudo”-dynamical system (11) will be our computationally affordable nonlinear dynamical model.
+In par with (11), we introduce a measurement equation,
+zk`1
+“
+´ˆx‹
+k ` rproxβgpI ´ β∇xfp‚; yk`1qqs˝C ˆx‹
+k ` rk
+“:
+Ψgpˆx‹
+k, yk`1q ` rk,
+(12)
+where C is the number proximal gradient steps, β ą 0 is the correction step size, and rk is a noise term
+coming from the noisy character of yk`1 and it is in general not zero-mean.
+Again, zk`1 “ 0n on the
+optimizer trajectory.
+The right-hand side of (12) represents the fixed-point residual of our C-steps proximal gradient method,
+that we use to compute the measurement.
+2.1
+Properties, requirements of the gradient approximators
+Before going further, it is useful to understand a bit better the properties of the gradient approximations
+Jk`1pxq and ∇xfpx; yk`1q. Both are attempting at approximating EyPYk`1r∇xfpx; yqs, but there are a few
+differences.
+The first is based on data available up to time tk and it is in general a biased estimator. This is not a
+problem per se, since even in deterministic prediction-correction methods, the predicted gradient is in general
+a biased prediction.
+Example 1 (Recurring stochastic example) Consider the case in which ∇xfpx; yq is linear in y (e.g.,
+for linear models and Least-Squares estimators, for example when fpx; yq :“ 1
+2}x ´ Ay}2 for a given matrix
+A), and more generally the case in which:
+fpx; yq “ f 1pxq ` yTAx,
+with f 1pxq strongly convex and smooth. In this case, let }∇yxfpx; yq} “ }A} ď C0.
+Further, suppose yptq is generated by a nominal (doubly differentiable in t) trajectory to which we add
+a Gaussian zero-mean noise at each sampling time tk.
+And in particular, set yptkq “ ¯yptkq ` ek, with
+ek „ Np0, Σkq for a given time-varying covariance matrix Σk. We also know that Er}ek}s ď
+a
+trpΣkq, and
+we set
+a
+trpΣkq ď Σ.
+Choose the extrapolation prediction: Jk`1pxq “ 2∇xfpx; ykq ´ ∇xfpx; yk´1q “ ∇xfpx; 2yk ´ yk´1q.
+Consider a nominal trajectory ¯yptq for which we assume the bounds:
+maxt}∇t ¯yptq} , }∇tt ¯yptq}u ď C,
+@x, t.
+(13)
+Then, in the Appendix we show that:
+Er}Jk`1pˆx‹
+k`1q ´ EyPYk`1r∇xfpˆx‹
+k`1; yqs}s
+“
+EwPYk,zPYk´1r}∇xfpˆx‹
+k`1; 2w ´ zq `
+´EyPYk`1r∇xfpˆx‹
+k`1; yqs}s
+ď
+C0Ch2 ` 3C0Σ,
+EyPYk`1r}∇xfpx; yq ´ EyPYk`1r∇xfpx; yqs}s
+ď
+C0Σ.
+■
+The second estimator, meaning using ∇xfpx; yk`1q instead of EyPYk`1r∇xfpx; yqs, is evaluated with data
+coming at time tk`1 and it is unbiased [11].
+For the two approximations we require the following.
+4
+
+Assumption 1 Let the cost function fpx; yq be µ-strongly convex and L-smooth in x uniformly in y (i.e.,
+for all y). The chosen gradient predictor Jk`1pxq is then µ-strongly monotone and L-Lipschitz in x for all
+k’s.
+Assumption 2 The noise processes and gradient prediction errors are bounded as follows:
+paq
+Er}Jk`1pˆx‹
+k`1q ´ EyPYk`1r∇xfpˆx‹
+k`1; yqs}s ď τ,
+pbq
+EyPYk`1r}∇xfpx; yq ´ EyPYk`1r∇xfpx; yqs}s ď σ,
+for finite scalars τ and σ, for all k P N, and (b) for all x P Rn.
+Assumption 1 is often required for time-varying optimization [5].
+The assumption on the predictor
+is also reasonable, for instance is verified in Example 1, and with Taylor-based and extrapolation-based
+predictions [27].
+For Assumption 2: Property paq is in par with some deterministic and stochastic assumptions appeared
+in past years. For example, it can be seen in parallel with the quality of the hint or predictable sequences
+in [26,28]. In Example 1, Property paq is verified with τ “ C0Ch2 ` 3C0Σ. Stochastic versions of Property
+paq have appeared, e.g., in [8], with example-based constructions for determining a suitable Jk`1. Property
+pbq is also commonly asked in stochastic frameworks [6].
+Usually, one asks that EyPYk`1r}∇xfpx; yq ´
+EyPYk`1r∇xfpx; yqs}2s ď σ2, but due to the convexity of p¨q2 and Jensen’s inequality, Property pbq is implied
+by the squared one (in fact pEr} ¨ }sq2 ď Er} ¨ }2s). For us Property pbq can be time-varying. In Example 1,
+Property pbq is verified with σ “ C0Σ. Finally, Property pbq can be tightened to be valid only on the algorithm
+iterates pxkqkPN, and interpret it as gradient noise with a small theoretical effort [7].
+3
+An extended Kalman filter
+We are now ready to derive a filter to track the filtered optimizer trajectory ˆx‹ptkq. Since the linear system
+and the measurement equations are nonlinear, we will use an extended Kalman filter. For it, we require
+that both ∇xfpx; yq and Jk`1pxq are differentiable with respect to x, and we will require knowledge of the
+covariance of the noise processes qk, rk at all time instances. We will also assume that g ” 0, to be able to
+differentiate, which puts us in an unconstrained differentiable problem setting, where we use gradient (the
+proximal operator is the identity in this case).
+Recall the notation xs
+k “ rI ´ α∇xfp‚, yk`1qs˝sxk defined in (7), indicating the effect of s steps of the
+gradient method, or an approximate gradient if fp‚, yk`1q is substituted with Jk`1p‚q. Define the derivative
+quantities (we drop the mention to g, since it does not exist in this case),
+Fk
+“
+∇xΦkpxkq “
+P
+ź
+p“1
+´
+In ´ α∇xJk`1pxP ´p
+k
+q
+¯
+(14)
+Hk`1
+“
+∇xΨpxk`1|k, yk`1q “ In ´
+C
+ź
+c“1
+´
+In ´ β∇xxfpxC´c
+k`1|k; yk`1q
+¯
+.
+(15)
+We also let Rk P Rnˆn and Qk P Rnˆn be the covariance matrices of the noise processes rk and qk,
+respectively.
+With this in place, the extended Kalman filter (TV-EKF) represented in Algorithm 1 is able to filter and
+track the optimizer trajectory ˆx‹ptq. We notice that we have presented the filter with correction first, to
+highlight the standard workflow within a sampling period. We notice also that the filter can be extended to
+include a filtering process for the data stream yptq, if a dynamical model for it is available.
+Algorithmically, the presented TV-EKF requires several computations. In the correction step, it comprises
+the computation of the Hessian of f at various points for determining Hk and taking a matrix inverse for
+determining Kk. The update for xk requires also C gradient steps. In the prediction pass, the filter includes
+a process to determine any prediction Jk`1, computing its derivatives, and P gradient steps. Considering
+a n-dimensional state, and letting g, h, j, dj be the computational effort to determine the gradient, Hessian,
+predicted gradient, and its derivatives, then the overall computational complexity of TV-EKF is OpCph `
+gq ` Ppdj ` gq ` n3q.
+We will study the empirical performance of TV-EKF in Section 6, but we close here with some remarks.
+5
+
+Algorithm 1 An extended Kalman filter (TV-EKF)
+Input: Initialize: x1|0 “ 0, P1|0 “ In. Number of prediction and correction steps P, C, sampling time h,
+step sizes α, β, covariance matrices Rk, Qk for all k, as well as prediction strategy J.
+Output: A sequence pxkqkPN
+1: for k P N, k ě 1 do
+2:
+Receive yk
+3:
+Compute Hk as in Eq. (15).
+4:
+Correction step:
+Kk
+“
+Pk|k´1HkpHkPk|k´1HT
+k ` Rkq´1
+xk
+“
+xk|k´1 ` KkpΨpxk|k´1, ykqq
+Pk
+“
+pI ´ KkHkqPk|k´1
+5:
+Compute Fk as in Eq. (14).
+6:
+Prediction step:
+xk`1|k “ Φkpxkq,
+Pk`1|k “ FkPkFT
+k ` Qk
+7: end for
+Remark 1 (Prediction-Correction methods) We can see how xk is updated as
+xk “ Φkpxk´1q ` KkpΨpΦkpxk´1q; ykqq “ pIn ´ Kkq rI ´ αJkp‚qs˝P xk´1
+looooooooooomooooooooooon
+prediction
+`
+` Kk rI ´ β∇xfp‚; ykqs˝C ˝ rI ´ αJkp‚qs˝P xk´1
+looooooooooooooooooooooooooomooooooooooooooooooooooooooon
+correcting the prediction
+,
+(16)
+and if we let Kk ” In, then we obtain back the standard prediction-correction methods.
+Remark 2 The TV-EKF algorithms that is presented here could be extended to equality-constrained opti-
+mization problems, once formulated as saddle-points, but we leave this for future endeavors.
+The TV-EKF algorithm that we have presented offers several advantages, and above all the ease of
+implementation. However, from an optimization perspective, it is lacking in good convergence guarantees2,
+and from a noise perspective, we do not have a good intuition or recipe on how to set the covariances Rk, Qk
+in meaningful ways.
+This, combined with the fact that TV-EKF will not work for non-smooth costs, pushes us to look beyond
+to a more general setting. However, before moving on, we give some intuition on why the TV-EKF does
+perform well empirically in the numerical settings that are presented in Section 6.
+Proposition 1 (Equivalence to a damped Newton’s step) When the noise on the measurement is neg-
+ligible, i.e., Rk « 0, and we take only one step of correction C “ 1, then TV-EKF is a damped Newton’s
+method, with update
+xk “ xk|k´1 ´ βr∇xxfpxk|k´1; ykqs´1∇xfpxk|k´1; ykq.
+■
+Proposition 1 implies that TV-EKF includes second-order information and could be thought of as a
+stochastic quasi-Newton method. In this sense, if the noise covariances are well estimated, or small, then
+TV-EKF is expected to do better than standard prediction-correction methods that only use first-order
+information. This is not surprising, but the connection Kalman-Newton in optimization is interesting.
+2Besides the fact that the prediction and correction steps represent contractive operators for α ă 2µ{L2, β ă 2{L.
+6
+
+4
+The general case
+The standard extended Kalman filter can be easily derived when the cost is differentiable. We generalize
+now our filter design to the case in which one has also the term gpxq in the cost, modeling constraints and
+non-smooth regularizations.
+Reconsider the dynamical system (11) in par with the measurement equation (12). Under Assumption 1,
+from standard operator theory, we know that if α and β are chosen small enough, and in particular3 α ă
+2µ{L2, β ă 2{L, both the prediction and the correction represent contraction operators, which converge to
+their respectives unique fixed points. In particular, their contraction factors are [29]:
+ρp “
+a
+1 ´ 2αµ ` α2L2,
+ρc “ maxt|1 ´ βµ|, |1 ´ βL|u.
+(17)
+for prediction and correction operators, respectively.
+4.1
+A static gain filter
+With our dynamical model (11) and measurement equation (12), we are now ready to build our filter. Here,
+since the model and the measurements are non-differentiable equations, we will focus on static gain filters,
+that can be computed off-line, before running the time-varying algorithm. We let Ψ1
+gpx, yq “ Ψgpx, yq ` x.
+Therefore, we start by considering the update equation
+xk`1 “ Φk,gpxkq´KpΦk,gpxkq´Ψ1
+gpΦk,gpxkq, yk`1qq “ pIn´KqΦk,gpxkq`KΨ1
+gpΦk,gpxkq, yk`1q,
+k P N
+(18)
+consisting of running a prediction Φk,gpxkq and then correcting it via the correction Ψ1
+gpΦk,gpxkq, yk`1q. As
+mentioned in Remark 1, by setting K ” In, we obtain back the standard prediction-correction methods.
+Algorithm 2 makes the update explicit along with all the involved computations, for a generic choice of
+J and K. As we can see the computational complexity here does not involve matrix inversions, but it adds
+proximal mapping computations. If the proximal step is easy to perform compared to the other computations,
+then the complexity is OpCph ` gq ` Ppdj ` gqq, which is better than TV-EKF, as expected.
+Algorithm 2 A static contractive filter (TV-CONTRACT)
+Input: Initialize: x1|0 “ 0. Number of prediction and correction steps P, C, sampling time h, step sizes
+α, β, prediction strategy J, as well as a filter gain K.
+Output: A sequence pxkqkPN
+1: for k P N, k ě 1 do
+2:
+Receive yk
+3:
+Correction step: xk “ pIn ´ Kqxk|k´1 ` KΨ1
+gpxk|k´1, ykq
+4:
+Compute Jk`1pxkq
+5:
+Prediction step: Compute xk`1|k “ Φk,gpxkq
+6: end for
+As mentioned before, in this paper, we are interested in designing K in such a way to reduce the tracking
+error of the sequence txkukPN, and in particular: which K would deliver the smallest lim supkÑ8 Er}xk´ˆx‹
+k}s?
+4.2
+Scalar, worst-case convergence results
+We start by analyzing the easier case of determining the best scalar gain χ P r0, 1s, for the update,
+xk`1 “ p1 ´ χqΦk,gpxkq ` χΨ1
+gpΦk,gpxkq, yk`1q,
+k P N.
+(19)
+While this is restrictive in practice, it will give us some intuition on the general problem. Also, by restricting
+χ in r0, 1s, we are considering all convex combinations of prediction and correction phases. The latter is
+important if g represents a feasible set and we want the sequence txkukPN to be feasible for every k. We have
+the following Theorem.
+3This is due to the fact that Jk is µ-strongly monotone and L-Lipschitz, but not a gradient per se. Sharper conditions can
+be derived if Jk would be a gradient, like in the correction case, for which we can choose α ă 2{L.
+7
+
+Theorem 1 Let Assumptions 1-2 hold. Assume furthermore that the optimizer trajectory is bounded as,
+}ˆx‹
+k`1 ´ ˆx‹
+k} ď ∆ ă 8,
+@k P N.
+(20)
+Choose α ă 2µ{L2, β ă 2{L. Consider Algorithm 2 with the selection of K “ χIn, χ P r0, 1s, leading to the
+update (19), and its sequence txkukPN. Define functions ζ and ξ as:
+ζℓ,ρ “ t1 if ℓ “ 0;
+ρℓ otherwise u,
+ξℓ,ρ “ t0 if ℓ “ 0;
+1 ` ρℓ otherwise u.
+Recall the contraction parameters (17) and choose the number of prediction and correction steps P and C
+such that ζC,ρcζP,ρp ă 1. Then, by calling ϱχ “ p1 ´ χqζP,ρp ` χζC,ρcζP,ρp, the asymptotic error is upper
+bounded as,
+lim sup
+kÑ8
+Er}xk ´ ˆx‹
+k}s “
+1
+1 ´ ϱχ
+”
+p1 ´ χq
+“
+ζP,ρp∆ ` ξP,ρpτµ
+‰
+` χ
+“
+ζC,ρcrζP,ρp∆ ` ξP,ρpτµs ` σc
+‰ ı
+,
+(21)
+with τµ “ τ{µ, and σc “ βσ{p1 ´ ρcq
+Finally, under the setting of Example 1, ∆ “ C0h{µ, τµ “ pC0Ch2 ` 3C0Σq{µ and σ “ C0Σ.
+■
+Theorem 1 captures the asymptotic tracking error of the proposed TV-CONTRACT algorithm, when
+K “ χIn. The requirement (20) is standard, assuring that the trajectory is regular enough to be tracked,
+see [5]. The condition ζC,ρcζP,ρp ă 1 is verified, whenever P ` C ě 1, since ρp, ρc P r0, 1q with the choice of
+α, β.
+For χ “ 1, Theorem 1 extends [27, Proposition 5.1] to stochastic settings and [8, Theorem 2.7] to f ` g
+settings with multiple prediction and correction steps.
+The question we have for filter design is how to tune χ to lower the asymptotical error, given all
+the rest fixed?
+4.3
+Tuning χ
+The filter design problem can be now formulated as,
+min
+χPr0,1s (21)
+(22)
+Problem (22) is a linear-fractional programming, that can be solved by transforming it into a linear program,
+once all the coefficients are fixed. We do not give the details of this, since easily found in standard books [30,
+Ch. 4.3.2]. Nonetheless we report an interesting fact on the nature of the solution.
+Proposition 2 The solution of the tuning problem (22) is either χ‹ “ 1, χ‹ “ 0, or in a special case, any
+χ P r0, 1s.
+■
+What Proposition 2 says is that from a worst-case perspective (i.e., from an asymptotic tracking error)
+we are better off to either just do predictions, or just do prediction-correction (and in a very special case, we
+can take any choice). The choice is made a priori, from the size of the prediction or correction errors (see the
+proof for exact conditions).
+This is hardly satisfactory.
+We see next how to extend the above to a generic matrix gain K, via
+dissipativity theory, which has a richer behavior.
+5
+Dissipativity theory and filter design
+We move now to the general case of designing a matrix K in an optimal fashion. We will be using recent
+tools from dissipativity theory applied to optimization algorithm design, and we were particularly inspired
+by [18,19].
+8
+
+�
+��� 0 B
+I 0
+�
+���
+ΦΦg
+ΨΨ′
+g
+xx
+ww
+uu
+�
+��� 0 B
+I 0
+�
+���
+T 1
+T 2
+Qqq
+Rrr
+abstraction
+xx
+ww
+uu
+Figure 1: The algorithmic choice (26) is an abstraction of Algorithm 2. The matrix block indicates the
+algorithmic update.
+5.1
+Filter design
+To start our filter design, we need to recast Algorithm 2 as a block diagram, where the optimization algorith-
+mic updates are interpreted as nonlinear blocks and modeled as quadratic constraints. Consider then noisy
+algorithmic update,
+$
+&
+%
+wk`1 “ ¯wk`1 ` Qqk`1,
+¯wk`1 “ T 1
+k`1pxkq
+uk`1 “ ¯uk`1 ` Rrk`1,
+¯uk`1 “ T 2
+k`1p ¯wk`1q
+xk`1 “ pI ´ Kqwk`1 ` Kuk`1
+(23)
+where qk`1 and rk`1 are noise terms, whose expected norm is bounded, as we will see shortly, and Q, R are
+tuning matrices that can model the relative amount of error or correlations.
+The algorithmic choice (26) is an abstraction of Algorithm 2, as we can see in Figure 1, where T 1
+k`1 and
+T 2
+k`1 are the ideal operators representing the ideal prediction and correction steps, respectively, and both
+with fixed point ˆx‹
+k`1 “ ¯w‹
+k`1 “ ¯u‹
+k`1. The fact that, technically, one should consider ¯uk`1 “ T 2
+k`1pwk`1q,
+and the noise terms qk`1 and rk`1 are in fact correlated, since rk`1 should depend on a noisy prediction,
+can be ignored here, since we will only look at worst-case performance guarantees, from bounded errors to
+bounded output.
+Under Assumptions 1 and 2, with the same notation of Theorem 1 and from the proof of [27, Proposition
+5.1] with Er}τk}s ď τµ, we can bound,
+}wk`1 ´ ¯w‹
+k`1} “ }xk`1|k ´ ˆx‹
+k`1} ď ζP,ρp}xk ´ ˆx‹
+k`1} ` ξP,ρpτk.
+(24)
+On the other hand,
+}wk`1 ´ ¯w‹
+k`1} “ } ¯wk`1 ´ ¯w‹
+k`1 ` Qqk`1} ď ζP,ρp}xk ´ ˆx‹
+k`1} ` }Qqk`1},
+(25)
+so we can look at bounded errors as }Qqk`1} ď ξP,ρpτk and Er}Qqk`1}s ď ξP,ρpτµ.
+Similarly for the error term Rrk, we can look only for bounded errors as Er}Rrk}s ď ζC,ρcξP,ρpτµ ` σc.
+In particular, we consider Er}qk}s ď 1{
+?
+2, Er}rk}s ď 1{
+?
+2 and model the matrices Q, R to have their
+largest singular value at
+?
+2pξP,ρpτµq and
+?
+2pζC,ρcξP,ρpτµ ` σcq, respectively.
+For the sake of ease of notation, in (26), we define matrices B, Be, and the nominal input and error signal
+¯z, e as,
+B “ rpIn ´ Kq, Ks,
+Be “ rpIn ´ KqQ, KRs,
+(26)
+¯z “ r ¯wT, ¯uTsT,
+e “ rqT, rTsT,
+xk`1 “ B¯zk`1 ` Beek`1,
+(27)
+and Er}ek}s ď 1. We also introduce point-wise-in-time quadratic constraints for T 1 and T 2 as,
+„
+xk ´ ˆx‹
+k`1
+¯wk`1 ´ ¯w‹
+k`1
+ȷT„
+ω2
+1In
+0
+0
+´In
+ȷ„
+xk ´ ˆx‹
+k`1
+¯wk`1 ´ ¯w‹
+k`1
+ȷ
+ě 0,
+(28)
+„
+xk ´ ˆx‹
+k`1
+¯uk`1 ´ ¯u‹
+k`1
+ȷT„
+ω2
+1ω2
+2In
+0
+0
+´In
+ȷ„
+xk ´ ˆx‹
+k`1
+¯uk`1 ´ ¯u‹
+k`1
+ȷ
+ě 0,
+(29)
+with ω1 “ ζP,ρp and ω2 “ ζC,ρcζP,ρp, which are due to the contracting properties of T 1 and T 2, respectively.
+With this in place, convergence and asymptotic tracking error performance can be formulated as follows.
+9
+
+Theorem 2 Consider Algorithm 2 and its abstraction (27), to find and track the filtered optimizer trajectory
+ˆx‹ptq of the time-varying stochastic optimization problem minxPRn Erfpx; yptqqs ` gpxq. Assume that the
+optimizer trajectory varies in a bounded way as }ˆx‹ptk`1q ´ ˆx‹ptkq} ď ∆, for all k P N and ∆ ă 8. Let
+Assumptions 1-2 hold as well.
+Introduce matrix W P Rnˆn and X P Rnˆn, X ą 0, scalars λ1 ě 0, λ2 ě 0, in addition to supporting
+scalars γ1, γ2, and consider the tuning matrix K.
+For any fixed scalar ρ P p0, 1q, by solving the convex
+problem4
+minimize
+X ą 0, λ1 ě 0, λ2 ě 0,
+W, γ2
+1, γ2
+2
+γ2
+1ρ2∆2 ` γ2
+2
+(30a)
+subject to
+ρ2X ľ pλ1ω2
+1 ` λ2ω2
+1ω2
+2qIn,
+(30b)
+X ľ In,
+X ĺ γ2
+1In
+(30c)
+»
+————–
+´λ1In
+0
+´λ2In
+02nˆ2n
+X ´ WT
+WT
+´γ2
+2I2n
+QpX ´ WTq
+RWT
+‹
+´X
+fi
+ffiffiffiffifl
+ĺ 0,
+(30d)
+then Algorithm 2 with K “ WX´1 generates a sequence txkukPN that converges as,
+AE :“ lim sup
+kÑ8
+Er}xk ´ ˆx‹
+k}s ď
+1
+1 ´ ρ pγ1ρ∆ ` γ2q .
+(31)
+Furthermore, for any fixed ρ, solving Problem (30) diminishes the asymptotical error AE.
+■
+Theorem 2 describes how to best tune the matrix K to minimize the asymptotic tracking error.
+In
+particular, doing a grid search on ρ P p0, 1q, one can identify the best K that minimizes the worst-case
+asymptotical error bound. Two remarks are in order.
+First the convex problem (30) grows linearly in the dimension of the problem n. However, if the matrices
+R and Q have some particular structure (i.e., diagonal, block diagonal), we can reduce the problem size
+considerably. In the case of Q “ qIn, R “ rIn, then choosing X “ pIn, W “ wIn, the problem becomes
+independent of the problem size n, as it happens in other examples [22].
+Second, the matrices R and Q are gateways through which one can include variable correlations and
+dependency, which is not present in a standard prediction-correction algorithm, nor in the scalar worst-case
+convergence tuning χ.
+5.2
+LPV filter design
+The presented filter design, captured by the conditions in Theorem 2, could suffer from conservatism, since
+the matrices Q and R are selected as worst cases. For example, we have modeled Q at having its largest
+singular value at
+?
+2pξP,ρpτµq. In practice, one could wish to model these matrices as parameter-varying, and
+directly dependent on how the data changes in time.
+We propose here an LPV filter design that accomplishes this task and reduce conservatism of the design
+process. We focus on a simplified setting to convey the basic ideas, and the reader is left to generalize the
+approach.
+Consider the change in time of the data ∇typtq, and let θ P r0, 1s be a normalized parameter that capture
+this change.
+For example, we let θ “ p}∇typtq}8q{pmaxt }∇typtq}8q.
+We then consider Q as an affine
+parameter-varying matrix: Qpθq “ Q0 ` θQ1 . We consider here a static R for simplicity, thereby assuming
+that the change in the data only affects the prediction accuracy, which is reasonable to assume (and easy to
+lift if needed).
+In parallel, we will be looking for an affine parameter-varying Lyapunov matrix Xpθq “ X0 ` θX1, and a
+parameter-varying filter gain matrix Kpθq. The following theorem is then in place.
+4For readability, we indicate in blue the decision variables.
+10
+
+Theorem 3 Consider Algorithm 2 and its abstraction (27), to find and track the filtered optimizer trajectory
+ˆx‹ptq of the time-varying stochastic optimization problem minxPRn Erfpx; yptqqs ` gpxq. Assume that the
+optimizer trajectory varies in a bounded way as }ˆx‹ptk`1q ´ ˆx‹ptkq} ď ∆, for all k P N and ∆ ă 8. Let
+Assumptions 1-2 hold as well.
+Introduce matrix function Wpθq : r0, 1s Ñ Rnˆn “ W0 ` θW1 and Xpθq : r0, 1s Ñ Rnˆn “ X0 `
+θX1, Xpθq ą 0, scalars λ1 ě 0, λ2 ě 0, in addition to supporting scalars γ1, γ2, and consider the tuning
+matrix function Kpθq. Assume that Qpθq “ Q0 ` θQ1 and that the value of θ at subsequent time step is
+upper bounded as |θs`1 ´ θs| ď ν. For any fixed scalar ρ P p0, 1q, by solving the problem
+minimize
+X0, X1, λ1 ě 0, λ2 ě 0,
+W0, W1, γ2
+1, γ2
+2
+γ2
+1ρ2∆2 ` γ2
+2
+(32a)
+subject to
+ρ2rX0 ` X1s ľ pλ1ω2
+1 ` λ2ω2
+1ω2
+2qIn,
+(32b)
+rX0 ` X1s ľ In,
+X0 ĺ γ2
+1In
+(32c)
+X1 ĺ 0
+(32d)
+»
+————–
+´λ1In
+0
+´λ2In
+02nˆ2n
+Ypθq ´ WpθqT
+WT
+´γ2
+2I2n
+QpθqpYpθq ´ WpθqTq
+RWpθqT
+‹
+´Ypθq
+fi
+ffiffiffiffifl
+ĺ 0,
+Ypθq “ X0 ´ νX1 ` θX1
+,
+/
+/
+/
+/
+/
+/
+/
+/
+.
+/
+/
+/
+/
+/
+/
+/
+/
+-
+@θ P r0, 1s(32e)
+then Algorithm 2 with Kpθq “ WpθqYpθq´1 generates a sequence txkukPN that converges as,
+AE :“ lim sup
+kÑ8
+Er}xk ´ ˆx‹
+k}s ď
+1
+1 ´ ρ pγ1ρ∆ ` γ2q .
+(33)
+Furthermore, for any fixed ρ, solving Problem (30) diminishes the asymptotical error AE.
+■
+Theorem 3 describes parametric conditions for Algortihm 2 to converge to the optimizer trajectory in
+expectation and within an error ball. We remark here the extra constraint X1 ĺ 0, which requires some
+explanation.
+As one can see in the proof of Theorem 3, the key step in proving convergence of the algorithm is to
+ensure that a Lyapunov function decreases at subsequent times. The Lyapunov function that we consider is
+Lkpθkq “ pxk ´ ˆx‹
+kqJXpθkqpxk ´ ˆx‹
+kq, and therefore we have to deal with matrices Xpθq at subsequent times.
+By imposing X1 ĺ 0, we can write however,
+Xpθs`1q “ Xpθsq ` pθs`1 ´ θsqX1 ĺ Xpθsq ´ |θs`1 ´ θs|X1 “ Ypθsq,
+from which we can derive Theorem 3 and this renders the derivation of a solvable program easier.
+The constraint X1 ĺ 0 induces conservatism in the design, but in all our numerical simulations we found
+it to be redundant (meaning that the optimal X‹
+1 was ĺ 0 with or without the constraint), and therefore not
+conservative in our application. We remark that a more correct approach would be to consider both extremes
+(`ν and ´ν) as done in [25] and remove X1 ĺ 0, but this would lead to an ambiguous definition of Kpθq
+and an harder problem to solve.
+Another possible approach is to remove X1 ĺ 0 and to consider only slowly changing parameters, for
+which ν ! 1. Since in our simulations ν « 0.4, we have preferred to focus on the former approach. Finally,
+note that considering a X1 ĺ 0 is not totally unreasonable, since we can assume that increasing θ, the
+convergence performance would be negatively affected.
+For fixed ρ, the problem to be solved is infinite dimensional (yet convex once θ is fixed). We recall that
+the constraints in (32e) are quadratic in θ, due to the product with Qpθq.
+A possible way to solve problem (32) is to discretize the domain with a uniform grid Θ :“ t0, θ1, . . . , θq, 1u
+and impose (32e) for all the points of the grid.
+Another possible way is to introduce another variable
+η “ θ2 P r0, 1s, and render affine (32e).
+A more sophisticated way to proceed is to introduce a more
+conservative yet convex condition in θ, hinging on the concept of matrix sum of squares.
+11
+
+Definition 1 Let Ppθq be a symmetric matrix of polynomials of degree up to 2d P N in the variable θ P R.
+A matrix is sum of squares (MSOS) if there exists a finite number l of symmetric matrices of polynomials
+Πipθq such that
+Ppθq “
+lÿ
+i“1
+ΠipθqTΠipθq.
+The decomposition implies that Ppθq ľ 0 for all θ. The constraint “Ppθq is MSOS” is convex.
+We are now ready for the following result.
+Theorem 4 Consider the matrix multiplicator Λ P R5nˆ5n, Λ ľ 0. In Theorem 3, condition (32e) can be
+substituted with the more conservative, yet convex and finite dimensional condition:
+´
+»
+————–
+´λ1In
+0
+´λ2In
+02nˆ2n
+Ypθq ´ WpθqT
+WT
+´γ2
+2I2n
+QpθqpYpθq ´ WpθqTq
+RWpθqT
+‹
+´Ypθq
+fi
+ffiffiffiffifl
+´ Λθp1 ´ θq
+is MSOS,
+(34a)
+Λ ľ 0,
+Ypθq “ X0 ´ νX1 ` θX1
+(34b)
+where Λ is now a new decision variable in the optimization problem.
+■
+Theorems 3 and 4 describe an LPV design strategy for our filter gain design. This is a particular choice
+due to the affine parameter-varying Qpθq and static R.
+More complex choices can be made (relatively)
+straightforwardly following the same pattern of the presented theorems. We now look at some numerical
+simulations.
+6
+Numerical simulations
+We focus now on showcasing the performance of the proposed algorithms on a real dataset and problem
+stemming from ride-hailing. We obtain trips data from the New York City dataset5 for the yellow taxi cab
+in the month of November 2019, totalling over 6.8 millions trips. We group the trips in 5 minutes intervals
+and divide the trip requests among n “ 5 different ride-hailing companies (such as taxi cab, Uber, Lyft, etc.).
+The trip requests are divided randomly among the companies in a way that different companies do not have
+the same number of requests.
+In the modern context of mobility as a service with ride-hailing orchestration [31], it makes sense for the
+city to provide software platforms to decide caps on the number of vehicles that each company can put on the
+streets depending on trading-off satisfying the demand and limiting traffic. A natural optimization problem
+that a city can formulate is
+ˆx‹ptq “ arg
+min
+xPrx,xsn
+n
+ÿ
+i“1
+´
+E
+”1
+2}xi ´ ciyiptq}2ı
+` logp1 ` κ exppxiqq ` ς
+2
+n
+ÿ
+j“1
+}xi ´ xj}2¯
+,
+(35)
+where xi for each i represents the upper limit on vehicles on the roads for company i. The constraint rx, xsn
+represents box constraints on the number of allowed vehicles. The term ci ą 0 multiplies the trip requests for
+company i at time t, yiptq to be able to match most of the requests as possible. The logistic term with κ ą 0
+is a regularization to make the cost non-quadratic but still convex and favor a smaller number of vehicles on
+the roads. Finally, the coupling term }xi ´xj}2 is set to have a similar regulation among different companies.
+For the sake of the simulations, we take ci “ 1, κ “ 0.02, ς “ 0.1, and we sample Problem (35) at every 5
+minutes. We simulate also the ground truth considering a smoothed version of the data.
+5Open data from the NYC Taxi and Limousine Commission Data Hub.
+12
+
+6.1
+Unconstrained case
+We start by considering the unconstrained case, where x P R5, and we look at different noise regimes and
+algorithms.
+As for the algorithms, we consider our TV-EKF (Algorithm 1), a standard prediction-correction [27], and
+the stochastic prediction-correction version of [8]. For the latter, with their optimal choice of window size
+and weights for two-point evaluation, we can see that it is equivalent to the AGT algorithm of [32], i.e., exact
+prediction with Hessian inversion, but with a finite difference evaluation of ∇txf.
+For the three algorithms, we consider different choices of prediction steps P and correction steps C. For
+the stochastic prediction-correction version of [8], prediction is exact, so P is not a free parameter. We
+simulate three noise regimes:
+‚ The case of very good prediction Q{R « 0. For this, J generates as predictive signal a random signal
+around the ground truth with variance 10.
+We use for the correction the true data stream.
+This case
+represents a realistic scenario of having a very good predictor (based on accurate historical data and, e.g.,
+on a periodic Kernel method). This could be typical in ride-hailing systems.
+‚ The case of poor prediction R{Q « 0. For this, J generates as predictive signal a random signal around
+the ground truth with variance 200. We use for the correction a convex combination of the true data stream
+and the ground truth (weighting the true data 0.05). This case represents a potential scenario of situation in
+which the system transitions (e.g., a lock-down happens) and we have poor prediction. This is in general less
+typical, since prediction can be built online on current data, based, e.g., on extrapolation, but still interesting
+to analyze.
+‚ The case of J generated by the true data based on an extrapolation-based prediction [27], and correction
+also based on true data. We label this case Q « R. The error between the true data and the ground truth
+has variance « 50, and the prediction « 200. We design Qk and Rk accordingly, also taking into account
+that the data streams are correlated, and therefore Qk, Rk are full.
+Table 1 displays the results obtained in these settings. As we can see, Algorithm 1 performs the best
+by a significant margin, when prediction is accurate (Q{R « 0).
+When prediction is poor (R{Q « 0),
+then Algorithm 1 behaves close to a damped Newton’s method, which significantly outperforms a standard
+prediction-correction algorithm, but it is in par with its stochastic version (since the latter still uses the very
+accurate data stream to built its exact prediction).
+Finally, for the setting of Q « R, then all the three methods perform very similarly, which is almost
+expected since taking prediction, or prediction and then correction incur the same “error”, so any combination
+could achieve similar results. In this case, Algorithm 1 chooses a Kk « In.
+The results in Table 1 support Algorithm 1 as an algorithm that can automatically tune prediction and
+correction; based on this tuning it can be significantly better than the competitors; and in the worst case it
+performs as state-of-the-art methods. We finally remark that the good prediction case is considered to be
+typical in this application scenario, and that the method from [8] could also be updated by designing an EKF
+for it, in the same way we did for the standard prediction-correction.
+6.2
+Constrained case
+We analyze now the constrained case, for which we set x “ 100 and x “ 1000. These constraints are not
+overly restrictive, but the point here is to see how our BMI-based method performs with respect to a standard
+prediction-correction method in different scenarios. Here, the stochastic variant [8] cannot be applied, but
+one could use the methods in [5] with exact prediction and finite-difference computations for ∇txf. We do
+not look into that, since typically these methods are more computationally demanding.
+The settings we investigate are similar to the unconstrained case, with the difference that we use our
+second algorithm TV-CONTRACT with a static K generated via Problem (30), and uniform grid-search on
+ρ. We also consider two cases for Q « R, the first has Q « 200In and R « 50In, the second Q « 67In and
+R « 50In. In both cases, as before, Q, R are full.
+In Table 2, we displays the results obtained in these settings. As we can see, Algorithm 2 has the best
+advantage when prediction is accurate and K can be chosen different from In. In general, the performance
+of Algorithm 2 is comparable with the competition, unless there is a clear advantage to choose a different
+from In gain. In this latter case, the performance gain can be significant. In Table 2, we have also added
+the selected best K, which is full whenever we use the « sign, with diagonal elements close to the indicated
+values.
+13
+
+Table 1: Performance of the considered algorithm in an unconstrained setting. For each row, the first line
+represents the average error }xk ´ ˆx‹
+k}, the second line the 25% percentile, and the third line the 75%
+percentile. In bold, the smallest error for the selected case and parameter choice. With ˚ we indicate a
+ą 10% error reduction with respect to the closest competitor.
+Regime
+Algorithm
+Extrap. P-C, [27], pP, Cq
+Stoch. P-C [8], C
+TV-EKF: Algortihm 1, pP, Cq
+p1, 1q
+p5, 1q
+p1, 5q
+p5, 5q
+1
+5
+p1, 1q
+p5, 1q
+p1, 5q
+p5, 5q
+Q
+R « 0
+80.0
+47.2
+134.0
+118.9
+160.1
+151.1
+70.1˚
+10.8˚
+61.2˚
+14.0˚
+26.3
+17.0
+49.7
+44.7
+59.9
+56.0
+20.2
+3.8
+17.7
+4.6
+111.3
+64.7
+183.9
+163.5
+219.1
+206.6
+107.1
+14.9
+90.8
+19.3
+R
+Q « 0
+90.5
+195.7
+19.4
+48.6
+11.2
+8.0
+7.5
+7.6
+7.5
+7.5
+66.3
+148.7
+12.7
+32.2
+4.1
+2.9
+2.8
+2.9
+2.8
+2.8
+109.3
+236.4
+23.8
+60.6
+14.3
+10.6
+10.4
+10.4
+10.5
+10.5
+Q « R
+135.0
+162.4
+144.7
+149.5
+160.1
+151.1
+141.7
+152.4
+149.5
+150.2
+47.8
+60.9
+53.6
+55.5
+59.9
+56.0
+52.0
+56.4
+56.4
+56.4
+185.7
+225.9
+198.0
+204.3
+219.1
+206.6
+193.4
+210.6
+205.0
+206.0
+Su. 25
+Mo. 26
+Tu. 27
+We. 28
+Th. 28
+Fr. 29
+Sa. 30
+0
+200
+400
+600
+800
+1000
+Allowed Vehicles
+Solutions for a week in November 2019
+Trip count
+P-C method
+TV-Contract
+True solution
+Figure 2: Display of selected trajectories in Thanksgiving week of 2019.
+The setting is Q « Rp2q and
+P “ C “ 5, and the trajectories are for one of the five companies.
+The results in Table 2 support Algorithm 2 as an algorithm that can automatically tune prediction and
+correction; based on this tuning it can be better than the competitors; and in the worst case it performs
+in par with state-of-the-art methods. We finally remark that the good prediction case is considered to be
+typical in this application scenario.
+For display purposes, Figure 2 illustrates the different trajectories for one of the five companies during
+Thanksgiving week of the selected month of November 2019, for the case Q « Rp2q and P “ C “ 5.
+6.3
+Variable K case
+We finish the simulation assessment showing the tracking results obtained solving Problem (32) for an affine
+parametric-varying Q. In particular, we let Q0 “ Q{5 and Q1 “ 4Q{5, where Q is numerically defined as
+before, and run Algorithm 2 on all the four cases that we have looked at in Table 2. We solve Problem (32)
+by uniform gridding with 4 points. As mentioned, in our case ν « 0.4.
+In Table 3, we report the results for the Q « R cases, since we do not observe any substantial difference
+for the other cases of Table 2. We indicate in bold if we have a gain w.r.t. a static gain, and with a dagger,
+if we have also a gain w.r.t. the state of the art. We also report how the maximal element of the diagonal of
+K changes in time in a selected week.
+As we can see, the results are very similar to the static results, but the gains can be important in some
+cases. As we see, the filter gain does not change over a wide range. However, it appears that even these
+small changes are enough to reduce the asymptotical error in selected scenarios, and behaving in par with
+the static approach in the others. As one can infer, parametric-varying gain design does depend on the
+modelling choices for Qpθq and Xpθq, and one could expect possibly more performant results in the case of
+more complex dependencies on θ. We leave this analysis for future endeavors.
+14
+
+Table 2: Performance of the considered algorithm in a constrained setting.
+For each row, the first line
+represents the average error }xk ´ ˆx‹
+k}, the second line the 25% percentile, and the third line the 75%
+percentile. In bold, the smallest error for the selected case and parameter choice. With ˚ we indicate a
+ą 10% error reduction with respect to the closest competitor.
+Regime
+Algorithm
+Kp5,5q
+Extrapolation P-C, [27], pP, Cq
+TV-CONTRACT: Algorithm 2, pP, Cq
+p1, 1q
+p5, 1q
+p1, 5q
+p5, 5q
+p1, 1q
+p5, 1q
+p1, 5q
+p5, 5q
+Q
+R « 0
+68.7
+58.2
+84.3
+79.7
+71.1
+51.3˚
+73.0˚
+54.6˚
+25.3
+20.3
+33.5
+32.6
+32.8
+9.0
+28.9
+15.5
+K « 0.24In
+102.4
+93.1
+122.1
+117.1
+103.1
+94.4
+106.3
+92.4
+R
+Q « 0
+88.6
+143.2
+54.5
+66.9
+88.6
+143.2
+54.5
+66.9
+58.1
+80.3
+16.4
+34.1
+58.1
+80.3
+16.4
+34.1
+K “ In
+119.6
+200.2
+95.5
+99.1
+119.6
+200.2
+95.5
+99.1
+Q « R
+85.4
+93.3
+87.7
+89.2
+85.3
+94.2
+87.7
+89.2
+p1q
+32.5
+34.4
+34.2
+34.9
+32.3
+34.4
+34.2
+34.9
+K « In
+118.8
+135.6
+126.3
+128.8
+118.7
+137.7
+126.3
+128.8
+Q « R
+68.3
+58.9
+84.1
+79.3
+68.3
+56.8
+84.1
+68.7˚
+p2q
+25.3
+23.0
+33.3
+32.2
+27.1
+22.1
+33.3
+28.1
+K « 0.86In
+101.5
+92.2
+122.0
+116.7
+101.3
+91.6
+122.0
+103.1
+Table 3: Performance of the considered algorithm in a constrained setting.
+For each row, the first line
+represents the average error }xk ´ ˆx‹
+k}, the second line the 25% percentile, and the third line the 75%
+percentile. We indicate in bold if we have a gain w.r.t. a static gain of Table 2, and with a dagger, if we have
+also a gain w.r.t. the state of the art of Table 2. Finally, with ˚ we indicate a ą 10% error reduction with
+respect to the closest competitor.
+Regime
+Algorithm
+TV-CONTRACT-LPV: Algorithm 2, pP, Cq
+p1, 1q
+p5, 1q
+p1, 5q
+p5, 5q
+Q « R
+85.5
+93.9
+87.3:
+89.8
+Su. 25 Mo. 26Tu. 27We. 28Th. 28 Fr. 29 Sa. 30
+0.94
+0.96
+0.98
+1.00
+Max ( Diag (K) )
+K (1,5)
+K (5,5)
+p1q
+32.0
+34.3
+33.7
+35.1
+118.6
+136.7
+125.4
+129.2
+Q « R
+70.9
+56.9
+74.4:˚
+65.3:
+Su. 25 Mo. 26Tu. 27We. 28Th. 28 Fr. 29 Sa. 30
+0.82
+0.86
+0.90
+Max ( Diag (K) )
+K (1,5)
+K (5,5)
+p2q
+32.7
+22.5
+29.3
+26.6
+102.7
+94.2
+108.9
+99.5
+15
+
+7
+Conclusions
+We have discussed several methods to generalize time-varying optimization algorithms to the case of noisy
+data streams. The methods are rooted in the intuition that prediction and correction can be seen as a nonlin-
+ear dynamical system and a nonlinear measurement equation, respectively. This leads to extended Kalman
+filter formulations as well as contractive filters based on bilinear matrix inequalities (BMI’s). Numerical
+results are promising, even when using possibly conservative BMI conditions.
+A
+Proofs
+A.1
+An additional example
+Example 2 (Deterministic example) Consider a deterministic method with an extrapolation predictor [27],
+meaning: Jk`1pxq “ 2∇xfpx; ykq ´ ∇xfpx; yk´1q, where now yptq is deterministic. Assume fpx; yptqq is
+strongly convex and smooth, uniformly in y, assume that the Hessian of fpx; yptqq does not depend on y,
+and assume the following bounds on data and mixed derivatives:
+maxt}∇typtq} , }∇ttyptq} , }∇yxfpx; yptqq} , }∇yyxfpx; yptqq}u ď C,
+@x, t.
+(36)
+Then we have that
+}Jk`1px‹
+k`1q ´ ∇xfpx‹
+k`1; yk`1q} ď pC2 ` C3qh2.
+■
+Proof:
+From [27, Lemma 4.5], we know that there exists a τ P rtk´1, tk`1s such that
+››Jk`1px‹
+k`1q ´ ∇xfpx‹
+k`1; yk`1q
+›› ď
+››∇tt∇xfpx‹
+k`1; ypτqq
+›› h2.
+(37)
+Let the i-th component of ∇xfpx; yptqq be Dipx; yptqq.
+By using the higher-derivatives Fa`a di Bruno’s chain rule:
+r∇tt∇xfpx; yptqqsi “ B2
+Bt2 Dipx; yptqq “
+ÿ
+j
+ˆ B
+Byj
+Dipx; yptqq B2yjptq
+Bt2
+˙
+`
+ÿ
+j,ℓ
+ˆ
+B2
+ByjByℓ
+Dipx; yptqq Byjptq
+Bt
+Byℓptq
+Bt
+˙
+,
+(38)
+from which the thesis follows.
+♦
+A.2
+Derivations for Example 1
+We choose to write x‹ “ x‹
+k`1 as a short-hand notation, in this proof only.
+By linearity of ∇xfpx‹; yptqq with respect to the parameter yptq, and the linearity of the expectation, we
+can write
+EwPYk,zPYk´1r}∇xfpx‹; 2w ´ zq ´ EyPYk`1r∇xfpx‹; yqs}s “ EwPYk,zPYk´1r}∇xfpx‹; 2w ´ z ´ EyPYk`1rysq}s
+“ EwPYk,zPYk´1r}∇xfpx‹; 2w ´ z ´ ¯yk`1q}s
+“ Ee1PN p0,Σkq,e2PN p0,Σk´1qr}∇xfpx‹; 2¯yk ´ ¯yk´1 ´ ¯yk`1q ` ∇xfpx‹; 2e1 ´ e2q}s.
+We now use the Triangle inequality, the result (38), and the mean value theorem for the nominal trajectory,
+to upper bound the last inequality as
+Ee1PN p0,Σkq,e2PN p0,Σk´1qr}∇xfpx‹; 2¯yk ´ ¯yk´1 ´ ¯yk`1q} ` }∇xfpx‹; 2e1 ´ e2q}s
+“ }∇xfpx‹; 2¯yk ´ ¯yk´1 ´ ¯yk`1q} ` Ee1PN p0,Σkq,e2PN p0,Σk´1qr}∇xfpx‹; 2e1 ´ e2q}s
+ď C0Ch2 ` C0 Ee1PN p0,Σkq,e2PN p0,Σk´1qr}2e1 ´ e2}s ď C0Ch2 ` 3C0Σ,
+from which the first claim is proven.
+16
+
+For the second,
+EyPYk`1r}∇xfpx; yq ´ EyPYk`1r∇xfpx; yqs}s “ EyPYk`1r}∇xfpx; yq ´ ∇xfpx; ¯yk`1qs}s
+“ EePN p0,Σk`1qr}∇xfpx; eqs}s
+ď C0 EePN p0,Σk`1qr}e}s ď C0Σ,
+as claimed.
+■
+A.3
+Proof of Proposition 1
+Consider C “ 1 in Algorithm 1, as well as a negligible Rk. Then, we can simplify the Kalman gain as:
+Kk
+“
+Pk|k´1HkrHkPk|k´1HT
+ks´1 “ H´1
+k .
+Therefore, the state update reads
+xk
+“
+xk|k´1 ` KkpΨpxk|k´1, ykqq “ xk|k´1 ` H´1
+k p´xk|k´1 ` xk|k´1 ´ β∇xfpxk|k´1; ykqq
+“
+xk|k´1 ´ βr∇xxfpx; ykqs´1∇xfpxk|k´1; ykq,
+from which the thesis is proven.
+■
+A.4
+Supporting results for Theorem 1
+Lemma 1 Let Assumptions 1-2 hold. Choose α ă 2µ{L2. Let xf
+k`1 be the fixed point of the prediction
+“pseudo”-dynamical model: xf
+k`1 “ Φk,gpxf
+k`1q. Then the distance between xf
+k`1 and the optimizer trajectory
+is bounded in expectation as,
+Er}xf
+k`1 ´ ˆx‹
+k`1}s ď 1
+µEr}Jk`1pˆx‹
+k`1q ´ EyPYk`1r∇xfpˆx‹
+k`1; yqs}s ď τ
+µ “: τµ.
+Furthermore, let the setting of Example 1 hold. Then,
+Er}xf
+k`1 ´ ˆx‹
+k`1}s ď C0Ch2{µ ` 3C0Σ{µ.
+■
+Proof:
+Choosing α ă 2µ{L2 and under Assumption 1, we know that the prediction is a contractive operator
+and its fixed point exists and it is unique. By implicit function theorems, see for instance [33, Theorem 2F.9]
+and [27, Theorem B.1 and Lemma 4.1], being careful to J being strongly monotone and not generally the
+gradient of a strongly convex function, then,
+}xf
+k`1 ´ ˆx‹
+k`1} ď 1
+µ }Jk`1pˆx‹
+k`1q ´ EyPYk`1r∇xfpˆx‹
+k`1; yqs}
+loooooooooooooooooooooooooomoooooooooooooooooooooooooon
+p˛q
+.
+(39)
+Passing in expectations, and by using Assumption 2, the first thesis follows.
+As for the second statement, it follows from the derivations of Example 1.
+♦
+Lemma 2 Let Assumptions 1-2 hold. Choose α ă 2µ{L2, β ă 2{L. Consider the prediction update xk`1|k “
+Φk,gpxkq with P prediction steps, and the correction update x1
+k “ Ψ1
+gpxk`1|k, yk`1q with C correction steps.
+Let the contraction factors ρp, ρc be defined as in (17). Then, the following error bounds are in place.
+Er}xk`1|k ´ ˆx‹
+k`1}s
+ď
+ρP
+p Er}xk ´ xf
+k`1}s ` τµ
+(40)
+Er}x1
+k ´ ˆx‹
+k`1}s
+ď
+ρC
+c Er}xk`1|k ´ ˆx‹
+k`1}s ` σc,
+(41)
+where σc “
+β σ
+1´ρc .
+Furthermore, under the setting of Example 1, τµ “ pC0Ch2 ` 3C0Σq{µ and σ “ C0Σ.
+■
+17
+
+Proof:
+Choosing α ă 2µ{L2, β ă 2{L and under Assumption 1, we know that the prediction and correction
+are contractive operators and their fixed points are unique.
+For the prediction part, by using Equation (39), we obtain,
+}xk`1|k ´ ˆx‹
+k`1}
+“
+}xk`1|k ˘ xf
+k`1 ´ ˆx‹
+k`1}
+ď
+}rproxαgpI ´ αJk`1p‚qqs˝P xk ´ xf
+k`1} ` 1
+µp˛q ď ρP
+p }xk ´ xf
+k`1} ` 1
+µp˛q,
+(42)
+and passing in expectation with Assumption 2 the claim is proven.
+For the second claim, we can write
+}x1
+k ´ ˆx‹
+k`1} ď }rproxβgpI ´ β∇xfp‚; yk`1q ˘ βEyPYk`1r∇xfp‚; yqsqs˝Cxk`1|k ´ ˆx‹
+k`1}.
+(43)
+Call ϵk`1pxq :“ ∇xfpx; yk`1q ´ EyPYk`1r∇xfpx; yqs. Then each proximal gradient step will incur in an
+additive }ϵk`1pxq} error, where x will be different at each step:
+}x1
+k ´ ˆx‹
+k`1} ď ρc}xC´1
+k`1|k ´ ˆx‹
+k`1} ` β}ϵk`1pxC´1
+k`1|kq} ď ρC
+c }xk`1|k ´ ˆx‹
+k`1} ` β
+C
+ÿ
+c“1
+ρC´c
+c
+}ϵk`1pxC´c
+k`1|kq}
+looooooooooooooomooooooooooooooon
+p˛˛q
+. (44)
+Passing in expectation, with Assumption 2 and the sum of geometric series, the second claim is also proven.
+♦
+A.5
+Proof of Theorem 1
+The proof follows the one of [27, Proposition 5.1], combining Lemma 1 and Lemma 2. Start by considering
+χ “ 1, so a classical prediction-correction method. We can use [27, Proposition 5.1], with Erτks “ τµ, and
+the correction with an additional error term to say,
+}xk`1 ´ ˆx‹
+k`1} ď ζC,ρc
+´
+ζP,ρp}xk ´ ˆx‹
+k} ` ζP,ρp∆ ` ξP,ρpτk
+¯
+` p˛˛q “: E1.
+(45)
+Looking at prediction only, χ “ 0, we obtain instead,
+}xk`1 ´ ˆx‹
+k`1} ď
+´
+ζP,ρp}xk ´ ˆx‹
+k} ` ζP,ρp∆ ` ξP,ρpτk
+¯
+“: E2.
+(46)
+For a generic χ P r0, 1s, we can combine the errors as
+}xk`1 ´ ˆx‹
+k`1} ď p1 ´ χqE2 ` χE1.
+(47)
+Then, we can recursively compute the error via geometric series summation. By passing through expectations,
+the claim follows.
+For Example 1, with ∇yxf bounded, by implicit function theorems [5], we have that ∆ “ C0h{µ, from
+which the thesis.
+■
+A.6
+Proof of Proposition 2
+For Problem (22), we look at the minimum of the curve,
+min
+χPr0,1s
+aχ ` b
+cχ ` d “: Fpχq.
+(48)
+For our problem cχ ` d “ 1 ´ ζP,ρp ` χpζP,ρp ´ ζP,ρpζC,ρcq ą 0, b “ ζP,ρp∆ ` ξP,ρpτµ ą 0, while a “
+pζC,ρc ´ 1qpζP,ρp∆ ` ξP,ρpτµq ` ζC,ρcσc can be positive, negative, or zero. Since function Fpχq is a linear-
+fractional function in one dimension, for χ ě 0, function Fpχq is monotone. In particular, for a{c ă pąqb{d
+the function is decreasing (increasing), leading to the optimal choices of χ‹ “ 1p0q. The condition means,
+pζC,ρc ´ 1qpζP,ρp∆ ` ξP,ρpτµq ` ζC,ρcσc
+ζP,ρp ´ ζP,ρpζC,ρc
+ă pąqζP,ρp∆ ` ξP,ρpτµ
+1 ´ ζP,ρp
+.
+For the special case a{c “ b{d, Fpχq ” 1 and any χ is optimal.
+■
+18
+
+A.7
+Proof of Theorem 2
+To impose convergence and performance, we look at the following matrix condition, featuring semidefinite
+matrix X, the scalar ρ, λ1, λ2, γ2, and matrix K which is implicit in B, Be:
+p‚qT
+„
+X
+0
+0
+´X
+ȷ „
+0
+B
+Be
+ρIn
+012
+012
+ȷ
+` λ1p‚qT
+„
+ω2
+1In
+0
+0
+´In
+ȷ „
+In
+0
+0
+012
+0
+In
+0
+012
+ȷ
+`
+` λ2p‚qT
+„
+ω2
+1ω2
+2In
+0
+0
+´In
+ȷ „
+In
+0
+0
+012
+0
+0
+In
+012
+ȷ
+`
+»
+–
+0
+022
+´γ2
+2In
+fi
+fl ĺ 0,
+(49)
+where p‚qT means that what is post-multiplied is also pre-multiplied transposed and 0 “ 0nˆn, 0ij “ 0inˆjn.
+Condition (49) combines the system, the quadratic constraints (i.e., the contractivity) via an S-procedure,
+and the performance criterion.
+We now develop the multiplications, we let } ¨ }2
+X :“ p¨qTXp¨q, and pre and post multiply with the vector
+rpxk ´ ˆx‹
+k`1qT, p ¯wk`1 ´ ˆx‹
+k`1qT, p¯uk`1 ´ ˆx‹
+k`1qT, eT
+k`1sT and we obtain,
+´ ρ2}xk ´ ˆx‹
+k`1}2
+X ` }xk`1 ´ ˆx‹
+k`1}2
+X ď ´ λ1
+“
+ω2
+1}xk ´ ˆx‹
+k`1}2 ´ } ¯wk`1 ´ ˆx‹
+k`1}2‰
+loooooooooooooooooooooooooomoooooooooooooooooooooooooon
+ě0
+`
+´ λ2
+“
+ω2
+1ω2
+2}xk ´ ˆx‹
+k`1}2 ´ }¯uk`1 ´ ˆx‹
+k`1}2‰
+looooooooooooooooooooooooooomooooooooooooooooooooooooooon
+ě0
+`γ2}ek`1}2 ď γ2}ek`1}2.
+(50)
+Define the error Ei :“ xi ´ ˆx‹
+i and the drift δk “ ˆx‹
+k`1 ´ ˆx‹
+k, then
+}Ek`1}2
+X ď ρ2}Ek ´ δk}2
+X ` γ2}ek`1}2.
+(51)
+Taking the square root of both sides, since ě 0
+}Ek`1}X ď
+b
+ρ2}Ek ´ δk}2
+X ` γ2}ek`1}2 ď ρ}Ek}X ` ρ}δk}X ` γ}ek`1}.
+(52)
+Let }δk} ď ∆, also note that Er}ek`1}s ď 1 since Er}qk`1}s ď 1{
+?
+2, Er}rk`1}s ď 1{
+?
+2. Since we impose X ľ
+In without loss of generality (since the problem remains unchanged for any scalar scaling), then }Ek`1}X ě
+}Ek`1} and }Ek}X ď }X1{2}}Ek} “ γ1}Ek}. Here the equality sign is due to the fact that we minimize over
+γ1.
+Similarly, ρ}δk}X ď ργ1}δk}. Then, iterating on k, and taking the expectations, we obtain,
+Er}Ek}s ď γ1ρkEr}E0}s `
+1
+1 ´ ρ pγ1ρ∆ ` γ2q ,
+(53)
+lim sup
+kÑ8
+Er}Ek}s ď
+1
+1 ´ ρ pγ1ρ∆ ` γ2q .
+(54)
+As such, for any fixed ρ, minimizing γ1ρ∆ ` γ2 minimizes the asymptotic tracking error. Furthermore,
+since }x}1 ď ?n}x}2 for x P Rn, we know that
+a
+γ2
+1ρ2∆2 ` γ2
+2 ě pγ1ρ∆ ` γ2q{
+?
+2. So our cost majorizes the
+asymptotical error γ1ρ∆`γ2 and therefore by minimizing our cost, we diminish the latter (notice, we do not
+minimize the latter, in general, since we have a constrained problem).
+To finish the proof, we need to transform (49) into (30d). We develop the matrix multiplications, and we
+observe that the resulting matrix is block diagonal. The first block is ρ2X ľ pλ1ω2
+1 `λ2ω2
+1ω2
+2qIn. The second
+block is
+p‚qTX rB
+Bes ` λ1
+»
+–
+´In
+0
+012
+0
+012
+022
+fi
+fl ` λ2
+»
+–
+0
+0
+012
+´In
+012
+022
+fi
+fl `
+»
+–
+0
+0
+012
+0
+012
+´γ2I2n
+fi
+fl ĺ 0.
+(55)
+Expand X into XX´1X and introduce the variable W “ KX. Then, taking the Schur’s complement, we
+obtain (30d), from which the thesis.
+■
+19
+
+A.8
+Proof of Theorem 3
+The proof follows the proof of Theorem 2. We focus here on the different parts.
+Starting from Eq. (49), we adapt the first term to:
+p‚qT
+„
+Xpθs`1q
+0
+0
+´Xpθsq
+ȷ „
+0
+Bpθsq
+Bepθsq
+ρIn
+012
+012
+ȷ
+.
+(56)
+The bottom diagonal leads to conditions
+ρ2rX0 ` θsX1s ľ pλ1ω2
+1 ` λ2ω2
+1ω2
+2qIn,
+(57)
+rX0 ` θsX1s ľ In,
+rX0 ` θsX1s ĺ γ2
+1In
+(58)
+which needs to be valid for the extreme points θs “ 0, 1, since affine in θs. However, given the constraint
+X1 ĺ 0, the above simplify into (32b) and (32c), respectively.
+Bu using again the constraint X1 ĺ 0, the upper diagonal can be upper bounded as,
+p‚qTXpθs`1qrBpθsq
+Bepθsqs “ p‚qTrX0 ` θs`1X1srBpθsq
+Bepθsqs ĺ
+p‚qTrX0 ´ νX1 ` θsX1srBpθsq
+Bepθsqs “ p‚qTYpθsqrBpθsq
+Bepθsqs,
+(59)
+so imposing a ĺ condition on the latter, would imply a condition on the former. In particular, adapting (55),
+the condition
+p‚qTYpθsqrBpθsq
+Bepθsqs`λ1
+»
+–
+´In
+0
+012
+0
+012
+022
+fi
+fl`λ2
+»
+–
+0
+0
+012
+´In
+012
+022
+fi
+fl`
+»
+–
+0
+0
+012
+0
+012
+´γ2I2n
+fi
+fl ĺ 0 (60)
+would imply a similar condition on Xpθs`1q, thus the upper diagonal of (56), and for proof of Theorem 2,
+convergence of the algorithm as indicated in Theorem 3.
+Condition (60) leads to condition (32e), by Schur complement and dropping the now-redundant subscript
+s.
+We remark here the importance of the constraint X1 ĺ 0, without which we would need two upper bounds
+in (59), one for ´ν and one for `ν, which would render the substitution Wpθq “ KpθqYpθq ambiguous, and
+determining Kpθq harder.
+■
+A.9
+Proof of Theorem 4
+The condition θ P r0, 1s is equivalent to θp1 ´ θq ě 0. Then we apply the generalized S-procedure as in [25].
+■
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+22
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+Efficient On-device Training via Gradient Filtering
+Yuedong Yang
+Guihong Li
+Radu Marculescu
+The University of Texas at Austin
+{albertyoung, lgh, radum}@utexas.edu
+Abstract
+Despite its importance for federated learning, continu-
+ous learning and many other applications, on-device train-
+ing remains an open problem for EdgeAI. The problem
+stems from the large number of operations (e.g., floating
+point multiplications and additions) and memory consump-
+tion required during training by the back-propagation al-
+gorithm. Consequently, in this paper, we propose a new
+gradient filtering approach which enables on-device DNN
+model training.
+More precisely, our approach creates a
+special structure with fewer unique elements in the gradi-
+ent map, thus significantly reducing the computational com-
+plexity and memory consumption of back propagation dur-
+ing training. Extensive experiments on image classification
+and semantic segmentation with multiple DNN models (e.g.,
+MobileNet, DeepLabV3, UPerNet) and devices (e.g., Rasp-
+berry Pi and Jetson Nano) demonstrate the effectiveness
+and wide applicability of our approach. For example, com-
+pared to SOTA, we achieve up to 19× speedup and 77.1%
+memory savings on ImageNet classification with only 0.1%
+accuracy loss. Finally, our method is easy to implement and
+deploy; over 20× speedup and 90% energy savings have
+been observed compared to highly optimized baselines in
+MKLDNN and CUDNN on NVIDIA Jetson Nano. Conse-
+quently, our approach opens up a new direction of research
+with a huge potential for on-device training.
+1. Introduction
+Existing approaches for on-device training are neither
+efficient nor practical enough to satisfy the resource con-
+straints of edge devices (Figure 1). This is because these
+methods do not properly address a fundamental problem in
+on-device training, namely the computational and memory
+complexity of the back-propagation (BP) algorithm. More
+precisely, although the architecture modification [6] and
+layer freezing [19, 21] can help skipping the BP for some
+layers, for other layers, the complexity remains high. Gra-
+dient quantization [4, 7] can reduce the cost of arithmetic
+operations but cannot reduce the number of operations (e.g.,
+multiplications); thus, the speedup in training remains lim-
+ited.
+Moreover, gradient quantization is not supported
+by existing deep-learning frameworks (e.g., CUDNN [9],
+MKLDNN [1], PyTorch [26] and Tensorflow [2]). To en-
+able on-device training, there are two important questions
+must be addressed:
+• How can we reduce the computational complexity of
+back propagation through the convolution layers?
+• How can we reduce the data required by the gradient
+computation during back propagation?
+In this paper, we propose gradient filtering, a new research
+direction, to address both questions. By addressing the first
+question, we reduce the computational complexity of train-
+ing; by addressing the second question, we reduce the mem-
+ory consumption.
+In general, the gradient propagation through a convolu-
+tion layer involves multiplying the gradient of the output
+variable with a Jacobian matrix constructed with data from
+either the input feature map or the convolution kernel. We
+aim at simplifying this process with the new gradient filter-
+ing approach proposed in Section 3. Intuitively, if the gradi-
+ent map w.r.t. the output has the same value for all entries,
+then the computation-intensive matrix multiplication can be
+greatly simplified, and the data required to construct the Ja-
+cobian matrix can be significantly reduced. Thus, our gra-
+dient filtering can approximate the gradient w.r.t. the output
+by creating a new gradient map with a special (i.e., spatial)
+structure and fewer unique elements. By doing so, the gra-
+dient propagation through the convolution layers reduces to
+cheaper operations, while the data required (hence memory)
+for the forward propagation also lessens. Through this fil-
+tering process, we trade off the gradient precision against
+the computation complexity during BP. We note that gradi-
+ent filtering does not necessarily lead to a worse precision,
+i.e., models sometimes perform better with filtered gradi-
+ents when compared against models trained with vanilla BP.
+In summary, our contributions are as follows:
+• We propose gradient filtering, which reduces the com-
+putation and memory required for BP by more than
+1
+arXiv:2301.00330v1  [cs.CV]  1 Jan 2023
+
+two orders of magnitude compared to the exact gradi-
+ent calculation.
+• We provide a rigorous error analysis which shows that
+the errors introduced by the gradient filtering have only
+a limited influence on model accuracy.
+• Our experiments with multiple DNN models and com-
+puter vision tasks show that we can train a neural net-
+work with significantly less computation and memory
+costs, with only a marginal accuracy loss compared to
+baseline methods. Side-by-side comparisons against
+other training acceleration techniques also suggest the
+effectiveness of our method.
+• Our method is easy to deploy with highly optimized
+deep learning frameworks (e.g., MKLDNN [1] and
+CUDNN [9]).
+Evaluations on resource-constrained
+edge (Raspberry Pi and Jetson Nano) and high-
+performance devices (CPU/GPU) show that our
+method is highly suitable for real life deployment.
+The paper is organized as follows. Section 2 reviews rel-
+evant work. Section 3 presents our method in detail. Sec-
+tion 4 discusses error analysis, computation and memory
+consumption. Experimental results are presented in Section
+5. Finally, Section 6 summarizes our main contributions.
+2. Related Work
+Architecture Modification: Authors of [6] propose to at-
+tach small branches to the original neural network. Dur-
+ing training, the attached branches and biases in the orig-
+inal model are updated. Though memory consumption is
+reduced, updating these branches still needs gradient prop-
+agation through the entire network; moreover, a large com-
+putational overhead for inference is introduced.
+Layer Freezing: Authors of [19, 21] propose to only train
+parts of the model. [19] makes layer selection based on layer
+importance metrics, while [21] uses evolutionary search.
+However, the layers selected by all these methods are typ-
+ically computationally heavy layers (e.g., the last few lay-
+ers in ResNet [15]) which consume most of the resources.
+Thus, the speedup achieved by these approaches is limited.
+Gradient Quantization: [3,5] quantize gradient after back-
+propagation, which means these methods cannot accelerate
+the training on a single device. Work in [4, 7, 16, 18, 29,
+30, 34] accelerates training by reducing the cost for every
+arithmetic operation. However, these methods do not re-
+duce the number of operations, which is typically huge for
+SOTA CNNs, so their achievable speedup is limited. Also,
+all these methods are not supported by the popular deep
+learning frameworks [1,2,9,26].
+In contrast to the prior work, our method opens up a new
+research direction. More precisely, we reduce the number of
+Arch. Modification
+Example: [6]
+Drawbacks:
+Large overhead;
+Limited to specific model
+Layer/Channel Freezing
+Example: [19, 21]
+Drawbacks:
+High search cost;
+Limited to simple models.
+Gradient Quantization
+Example: [4, 7, 16]
+Drawbacks:
+Not supported by 
+existing DL frameworks
+Gradient Filtering [Ours]
+Advantages:
+Very fast and accurate;
+Well supported by 
+existing DL frameworks
+Efficiency
+Applicability
+Orthogonal Research Directions for On-device Training
+Figure 1. Matrix of orthogonal directions for on-device training.
+“Arch” is short for “architecture”. Our approach opens up a new
+direction of research for on-device training for EdgeAI.
+computations and memory consumption required for train-
+ing a single layer via gradient filtering. Thus, our method
+can be combined with any of the methods mentioned above.
+For example, in Section G in the Supplementary, we illus-
+trate how our method can work together with the gradient
+quantization methods to enable a higher speedup.
+3. Proposed Method
+In this section, we introduce our gradient filtering ap-
+proach to accelerate BP. To this end, we target the most
+computation and memory heavy operation, i.e., convolution
+(Figure 2(a)). Table 1 lists some symbols we use.
+Cx
+Number of channels of x
+Wx, Hx
+Width and height of x
+θ
+Convolution kernel
+θ′
+Rotated θ, i.e., θ′ = rot180(θ)
+r
+Patch size (r × r )
+gx, gy, gθ
+Gradients w.r.t. x, y, θ
+˜gy
+Approximated gradient gy
+˜x, ˜θ′
+Sum of x and θ′ over
+spatial dimensions (height and width)
+x[n, ci, h, w]
+Element for feature map x
+at batch n, channel ci, pixel (h, w)
+θ[co, ci, u, v]
+Element for convolution kernel θ
+at output channel co, input channel ci,
+position (u, v)
+Table 1. Table of symbols we use.
+2
+
+Loss
+Loss
+Gradient Filter
+1.0 -1.0 -0.2 0.0
+0.0 4.0 2.0 3.0
+0.2 0.4 2.0 -1.0
+0.5 4.1 -1.4 -4.0
+4.0
+4.8
+5.2
+-4.4
+0.1 0.1 -0.2 0.5
+0.0 0.2 0.1 0.4
+0.5 0.2 0.2 -0.9
+0.4 0.1 -0.1 0.4
+0.1
+0.2
+0.3
+-0.1
+3.36
+������������
+�������������
+������������������������
+�������������������������
+�������������������������
+0.1
+0.3
+0.4
+0.1
+0.2
+0.1
+-0.4
+-0.2
+-0.5
+������������𝜃
+0.1
+�������������𝜃
+0.01
+0.02
+0.03
+-0.01
+�������������������������
+Ⓕ
+⊙
+������������
+������������������������
+������������𝜃
+⊛
+������������
+������������������������
+Memory
+Ⓕ
+������������
+�������������������������
+Memory
+��������������
+Ⓕ
+������������
+�������������������������
+������������𝜃
+�������������𝜃 ⊙
+������������
+������������
+�������������������������
+������������������������
+⊛
+������������
+������������
+������������������������
+⊛
+Vanilla Conv.
+Our Conv.
+Ⓐ
+(a)
+(b)
+Ⓐ
+Spatial Sum
+=
+������������. ������������ × ������������. ������������ + ������������. ������������ × ������������. ������������ +
+������������. ������������ × ������������. ������������ + −������������. ������������ × (−������������. ������������)
+Patch size ������������ × ������������ = 2 × 2
+Forward Propagation
+Backward Propagation
+Average Filter
+Ⓐ
+������������ Spatial Sum
+⊛Convolution
+ⒻFrobenius Inner Product
+⊙Element-wise Product
+������������
+������������
+������������
+������������
+Average Value
+Other Layers
+Other Layers
+Height
+Width
+Figure 2. (a) Computation procedures for vanilla training method (upper) and our method (lower). (b) Example of gradient propagation
+with gradient filtering. Numbers in this example are chosen randomly for illustration purposes. In this case, the patch size selected for the
+gradient filter is 2 × 2. Thus, the 4 × 4 gradient map gy is approximated by ˜gy, which has four 2 × 2 patches with one unique value for
+each patch. Also, input feature map x and mirrored convolution kernel θ′ are spatial summed to ˜x and ˜θ′. Since ˜x has fewer unique values
+than x, memory consumption is reduced. Finally, with ˜gy, ˜x and ˜θ, we compute the gradient w.r.t. kernel and input feature map with much
+fewer operations than the standard back propagation method.
+3.1. Problem Setup
+The computations for both forward and backward paths
+are shown in Figure 2(a). For the standard (vanilla) ap-
+proach (upper Figure 2(a)), starting with input x, the for-
+ward propagation convolves the input feature map x with
+kernel θ and returns output y, which is further processed
+by the other layers in the neural network (dotted arrow) un-
+til the loss value l is calculated. As shown in Figure 2(a),
+the BP of the convolution layer starts with the gradient map
+w.r.t. output y (gy). The gradient w.r.t. input (gx) is calcu-
+lated by convolving gy with the rotated convolution kernel
+θ′, i.e., gx = gy ⊛ rot180(θ) = gy ⊛ θ′. The gradient w.r.t.
+convolution kernel, namely gθ, is calculated with the Frobe-
+nius inner product [17] between x and gy, i.e., gθ = gy
+F x.
+The lower half of Figure 2(a) shows our method, where
+several changes are made: We introduce the gradient filter
+“ A ” after gy to generate the approximate gradient for BP.
+Also, instead of using the accurate x and θ′ values for gra-
+dient computation, we sum over spatial dimensions (height
+and width dimensions), i.e., ˜x and ˜θ′, respectively. Finally,
+the convolution layer now multiplies the approximate gra-
+dient ˜gy with spatial kernel ˜θ′ instead of convolving with it
+to calculate ˜gx. Figure 2(b) shows an example of gradient
+propagation with our gradient filter.
+3.2. Preliminary Analysis
+Consider the vanilla BP for convolution in Figure 2(a).
+Equation (1) shows the number of computations (#FLOPs)
+required to calculate gx given gy:
+#FLOPs = 2CxCy · WyHy · WθHθ
+(1)
+The computation requirements in Equation (1) belong to
+three categories: number of channels, number of unique el-
+ements per channel in the gradient map, and kernel size. Our
+method focuses on the last two categories.
+i. Unique elements: (WyHy) represents the number of
+unique elements per channel in the gradient w.r.t. output
+variable y (gy). Given the high-resolution images we use,
+this term is huge, so if we manage to reduce the number
+of unique elements in the spatial dimensions (height and
+width), the computations required are greatly reduced too.
+ii.
+Kernel size: (WθHθ) represents the number of
+unique elements in the convolution kernel. If the gradient gy
+has some special structure, for example gy = 1Hy×Wy · v
+(i.e., every element in gy has the same value v), then the
+convolution can be simplified to (� θ′)v1Hy×Wy (with
+boundary elements ignored). With such a special structure,
+only one multiplication and (WθHθ − 1) additions are re-
+quired. Moreover, � θ′ is independent of data so the result
+can be shared across multiple images until θ gets updated.
+3.3. Gradient Filtering
+To reduce the number of unique elements and create the
+special structure in the gradient map, we apply the gradi-
+ent filter after the gradient w.r.t. output (gy) is provided.
+During the backward propagation, the gradient filter A ap-
+proximates the gradient gy by spatially cutting the gradient
+map into r×r-pixel patches and then replacing all elements
+in each patch with their average value (Figure 2(b)):
+˜gy[n, co, h, w] = 1
+r2
+⌈h/r⌉r
+�
+i=⌊h/r⌋r
+⌈w/r⌉r
+�
+j=⌊w/r⌋r
+gy[n, co, i, j]
+(2)
+3
+
+For instance in Figure 2(b), we replace the 16 distinct values
+in the gradient map gy with 4 average values in ˜gy. So given
+a gradient map gy with N images per batch, C channels,
+and H × W pixels per channel, the gradient filter returns
+a structured approximation of the gradient map containing
+only N × C × ⌈ H
+r ⌉ × ⌈ W
+r ⌉ blocks, with one unique value
+per patch. We use this matrix of unique values to represent
+the approximate gradient map ˜gy, as shown in Figure 2(b).
+3.4. Back Propagation with Gradient Filtering
+We describe now the computation procedure used after
+applying the gradient filter. Detailed derivations are pro-
+vided in Supplementary Section A.
+Gradient w.r.t. input: The gradient w.r.t. input is cal-
+culated by convolving θ′ with gy (Figure 2(a)). With the
+approximate gradient ˜gy, this convolution simplifies to:
+˜gx[n, ci, h, w] =
+�
+co
+˜gy[n, co, h, w] ⊙ ˜θ′[co, ci]
+(3)
+where ˜θ′[co, ci] = �
+u,v θ′[co, ci, u, v] is the spatial sum of
+convolution kernel θ, as shown in Figure 2(b).
+Gradient w.r.t. kernel: The gradient w.r.t. the kernel is
+calculated by taking the Frobenius inner product between x
+and gy, i.e., gθ[co, ci, u, v] = x F gy, namely:
+gθ[co, ci, u, v] =
+�
+n,i,j
+x[n, ci, i+u, j +v]gy[n, co, i, j] (4)
+With the approximate gradient ˜gy, the operation can be sim-
+plified to:
+˜gθ[co, ci, u, v] =
+�
+n,i,j
+˜x[n, ci, i, j]˜gy[n, co, i, j]
+(5)
+with ˜x[n, ci, i, j] = �⌈i/r⌉r
+h=⌊i/r⌋r
+�⌈j/r⌉r
+w=⌊j/r⌋r x[n, ci, h, w].
+As shown in Figure 2(b), ˜x[n, ci, i, j] is the spatial sum of
+x elements in the same patch containing pixel (i, j).
+4. Analyses of Proposed Approach
+In this section, we analyze our method from three per-
+spectives: gradient filtering approximation error, computa-
+tion reduction, and memory cost reduction.
+4.1. Error Analysis of Gradient Filtering
+We prove that the approximation error introduced by our
+gradient filtering is bounded during the gradient propaga-
+tion. Without losing generality, we consider that all vari-
+ables have only one channel, i.e., Cx0 = Cx1 = 1.
+Proposition 1: For any input-output channel pair (co, ci)
+in the convolution kernel θ, assuming the DC component
+has the largest energy value compared to all components in
+the spectrum1, then the signal-to-noise-ratio (SNR) of ˜gx is
+greater than SNR of ˜gy.
+Proof:
+We use Gx, Gy and Θ to denote the gradients
+gx, gy and the convolution kernel θ in the frequency domain;
+Gx[u, v] is the spectrum value at frequency (u, v) and δ is
+the 2D discrete Dirichlet function. To simplify the discus-
+sion, we consider only one patch of size r × r.
+The gradient returned by the gradient filtering can be
+written as:
+˜gy = 1
+r2 1r×r ⊛ gy
+(6)
+where ⊛ denotes convolution.
+By applying the discrete
+Fourier transformation, Equation (6) can be rewritten in the
+frequency domain as:
+˜Gy[u, v] = 1
+r2 δ[u, v]Gy[u, v]
+(7)
+˜gy is the approximation of gy (i.e., the ground truth for ˜gy
+is gy), and the SNR of ˜gy equals to:
+SNR˜gy =
+�
+(u,v) G2
+y[u, v]
+�
+(u,v)(Gy[u, v] − 1
+r2 δ[u, v]Gy[u, v])2
+= (1 − 2r2 − 1
+r4
+G2
+y[0, 0]
+�
+(u,v) G2y[u, v])−1
+(8)
+For the convolution layer, the gradient w.r.t. the approxi-
+mate variable ˜x in the frequency domain is2:
+˜Gx[u, v] = Θ[−u, −v] ˜Gy[u, v]
+= 1
+r2 Θ[−u, −v]δ[u, v]Gy[u, v]
+(9)
+and its ground truth is:
+Gx[u, v] = Θ[−u, −v]Gy[u, v]
+(10)
+Similar to Equation (8), the SNR of g˜x is:
+SNR˜gx = (1 − 2r2 − 1
+r4
+(Θ[0, 0]Gy[0, 0])2
+�
+(u,v) (Θ[u, v]Gy[u, v])2 )−1
+(11)
+Equation (11) can be rewritten as:
+r4(1 − SNR−1
+˜gx )
+2r2 − 1
+=
+(Θ[0, 0]Gy[0, 0])2
+�
+(u,v)(Θ[−u, −v]Gy[u, v])2
+=
+G2
+y[0, 0]
+�
+(u,v)( Θ[−u,−v]
+Θ[0,0] Gy[u, v])2
+(12)
+Furthermore, the main assumption (i.e., the DC component
+dominates the frequency spectrum of Θ) can be written as:
+Θ2[0, 0]/max(u,v)̸=(0,0)Θ2[u, v] ≥ 1
+(13)
+1As a reminder, the energy of a signal is the sum of energy of the DC
+component and the energy of its AC components.
+2Because gy is convolved with the rotated kernel θ′, in the frequency
+domain, we use Θ[−u, −v] instead of Θ[u, v].
+4
+
+1 × 1
+10 × 10
+20 × 20
+30 × 30
+Patch Size r × r
+1M
+10M
+100M
+1G
+#FLOPs
+Baseline
+Reduced
+Unique
+Elements
+Reduced
+Unique
+Elements
++Structured
+Gradient
+Actual
+Minimum
+Achievable Computation
+Figure 3. Computation analysis for a specific convolution layer3.
+Minimum achievable computation is given in Equation (16). By
+reducing the number of unique elements, computations required
+by our approach drop to about 1/r2 compared with the standard
+BP method. By combining it with structured gradient map, com-
+putations required by our approach drop further, getting very close
+to the theoretical limit.
+that is, ∀(u, v), Θ2[−u,−v]
+Θ2[0,0]
+≤ 1; thus, by combining Equa-
+tion (12) and Equation (13), we have:
+G2
+y[0, 0]
+�
+(u,v)( Θ[−u,−v]
+Θ[0,0] Gy[u, v])2 ≥
+G2
+y[0, 0]
+�
+(u,v)(Gy[u, v])2
+⇔
+r4(1 − SNR−1
+˜gx )
+2r2 − 1
+≥
+r4(1 − SNR−1
+˜gy )
+2r2 − 1
+(14)
+which means that: SNR˜gx ≥ SNR˜gy. This completes our
+proof for error analysis. ■
+In conclusion, as the gradient propagates through the net-
+work, the noise introduced by our gradient filter becomes
+weaker compared to the real gradient signal. This property
+ensures that the error in gradient has only a limited influ-
+ence on the quality of BP. We validate Proposition 1 later in
+the experimental section.
+4.2. Computation and Overhead Analysis
+In this section, we analyse the computation required to
+compute gx, the gradient w.r.t. input x. Figure 3 compares
+the computation required to propagate the gradient through
+this convolution layer under different patch sizes r × r. A
+patch size 1 × 1 means the vanilla BP algorithm which we
+use as the baseline. As discussed in the preliminary analysis
+section (Section 3.2), two terms contribute to the computa-
+tion savings: fewer unique elements in the gradient map and
+the structured gradient map.
+Fewer unique elements: In vanilla BP, there are HyWy
+unique elements in the gradient map. After applying gradi-
+ent filtering with a patch size r × r, the number of unique
+3The layer is from U-Net [27]. The size of the input is assumed to be
+120 × 160 pixels with 192 channels; the output has the same resolution,
+but with only 64 channels. The kernel size of the convolution layer is 3×3.
+Analysis for ResNet is included in the supplementary material.
+elements reduces to only ⌈ Hy
+r ⌉⌈ Wy
+r ⌉. This reduction con-
+tributes the most to the savings in computation (orange line
+in Figure 3).
+Structured Gradient Map: By creating the structured gra-
+dient map, the convolution over the gradient map ˜gy is sim-
+plified to the element-wise multiplication and channel-wise
+addition. Computation is thus reduced to (HθWθ)−1 of its
+original value. For instance, the example convolution layer
+in Figure 3 uses a 3 × 3 convolution kernel so around 89%
+computations are removed. The blue line in Figure 3 shows
+the #FLOPs after combining both methods. Greater reduc-
+tion is expected when applying our method with larger con-
+volution kernels. For instance, FastDepth [31] uses 5 × 5
+convolution kernel so as much as 96% reduction in compu-
+tation can be achieved, in principle.
+Minimum Achievable Computation: With the two reduc-
+tions mentioned above, the computation required to propa-
+gate the gradient through the convolution layer is:
+#FLOPs(r) = ⌈Hy
+r ⌉⌈Wy
+r ⌉Cx(2Cy −1)+o(HyWy) (15)
+where o(HyWy) is a constant term which is independent of
+r and negligible compared to HyWy. When the patch is as
+large as the feature map, our method reaches the minimum
+achievable computation (blue dashed line in Figure 3):
+minr #FLOPs(r) = 2CxCy − Cx + o(HyWy)
+(16)
+In this case, each channel of the gradient map is represented
+with a single value, so the computation is controlled by the
+number of input and output channels.
+Overhead: The overhead of our approach comes from ap-
+proximating the feature map x, gradient gy, and kernel θ.
+As the lower part of Figure 2(a) shows, the approximation
+for x is considered as part of forward propagation, while
+the other two as back propagation. Indeed, with the patch
+size r, the ratio of forward propagation overhead is about
+1/(2CoWθHθ), while the ratio of backward propagation
+overhead is about (r2 − 1)/(2Cx).
+Given the large number of channels and spatial dimen-
+sions in typical neural networks, both overhead values take
+less than 1% computation in the U-Net example above.
+4.3. Memory Analysis
+As Figure 2(a) shows, the standard back propagation for
+a convolution layer relies on the input feature map x, which
+needs to be stored in memory during forward propagation.
+Since every convolution layer requiring gradient for its ker-
+nel needs to save a copy of feature map x, the memory
+consumption for storing x is huge. With our method, we
+simplify the feature map x to approximated ˜x, which has
+only ⌈ Hx
+r ⌉⌈ Wx
+r ⌉ unique elements for every channel. Thus,
+by saving only these unique values, our method achieves
+around (1 − 1
+r2 ) memory savings, overall.
+5
+
+MobileNetV2 [28]
+#Layers
+Accuracy
+FLOPs
+Mem
+ResNet-18 [15]
+#Layers
+Accuracy
+FLOPs
+Mem
+No Finetuning
+0
+4.2
+0
+0
+No Finetuning
+0
+4.7
+0
+0
+Vanilla
+BP
+All
+75.1
+1.13G
+24.33MB
+Vanilla
+BP
+All
+73.1
+5.42G
+8.33MB
+2
+63.1
+113.68M
+245.00KB
+2
+70.4
+489.20M
+196.00KB
+4
+62.2
+160.00M
+459.38KB
+4
+72.3
+1.14G
+490.00KB
+TinyTL [6]
+N/A
+60.2
+663.51M
+683.00KB
+TinyTL [6]
+N/A
+69.2
+3.88G
+1.76MB
+Ours
+2
+63.1
+39.27M
+80.00KB
+Ours
+2
+68.6
+28.32M
+64.00KB
+4
+63.4
+53.96M
+150.00KB
+4
+68.5
+61.53M
+112.00KB
+MCUNet [20]
+#Layers
+Accuracy
+FLOPs
+Mem
+ResNet-34 [15]
+#Layers
+Accuracy
+FLOPs
+Mem
+No Finetune
+0
+4.1
+0
+0
+No Finetune
+0
+0
+0
+Vanilla
+BP
+All
+68.5
+231.67M
+9.17MB
+Vanilla
+BP
+All
+70.8
+11.17G
+13.11MB
+2
+62.1
+18.80M
+220.50KB
+2
+69.6
+489.20M
+196.00KB
+4
+64.9
+33.71M
+312.38KB
+4
+72.3
+1.21G
+392.00KB
+TinyTL [6]
+N/A
+53.1
+148.01M
+571.5KB
+TinyTL [6]
+N/A
+72.9
+8.03G
+2.95MB
+Ours
+2
+61.8
+6.34M
+72.00KB
+Ours
+2
+68.6
+28.32M
+64.00KB
+4
+64.4
+11.01M
+102.00KB
+4
+70.6
+64.07M
+128.00KB
+Table 2. Experimental results for ImageNet classification with four neural networks (MobileNet-V2, ResNet18/34, MCUNet). “#Layers”
+is short for “the number of active convolutional layers”. For example, #Layers equals to 2 means that only the last two convolutional layers
+are trained. For memory consumption, we only consider the memory for input feature x. Strategy “No Finetuning” shows the accuracy on
+new datasets without finetuning the pretrained model. Since TinyTL [6] changes the architecture, “#Layers” is not applicable (N/A).
+PSPNet [33]
+#Layers
+GFLOPs
+mIoU
+mAcc
+PSPNet-M [33]
+#Layers
+GFLOPs
+mIoU
+mAcc
+FCN [22]
+#Layers
+GFLOPs
+mIoU
+mAcc
+Calibration
+0
+0
+12.86
+19.74
+Calibration
+0
+0
+14.20
+20.46
+Calibration
+0
+0
+10.95
+15.69
+Vanilla
+BP
+All
+166.5
+55.01
+68.02
+Vanilla
+BP
+All
+42.4
+48.48
+61.48
+Vanilla
+BP
+All
+170.3
+45.22
+58.80
+5
+15.0
+39.54
+51.86
+5
+12.22
+36.35
+47.09
+5
+59.5
+27.41
+37.90
+10
+110.6
+53.15
+67.10
+10
+22.46
+46.01
+58.70
+10
+100.9
+43.87
+57.58
+Ours
+5
+0.14
+39.34
+51.86
+Ours
+5
+0.11
+36.14
+46.86
+Ours
+5
+0.58
+27.42
+37.88
+10
+0.79
+50.88
+64.73
+10
+0.76
+44.90
+57.50
+10
+0.96
+36.30
+48.82
+DLV3 [8]
+#Layers
+GFLOPs
+mIoU
+mAcc
+DLV3-M [8]
+#Layers
+GFLOPs
+mIoU
+mAcc
+UPerNet [32]
+#Layers
+GFLOPs
+mIoU
+mAcc
+Calibration
+0
+0
+13.95
+20.62
+Calibration
+0
+0
+21.96
+36.15
+Calibration
+0
+0
+14.71
+21.82
+Vanilla
+BP
+All
+151.2
+58.32
+71.72
+Vanilla
+BP
+All
+54.4
+55.66
+68.95
+Vanilla
+BP
+All
+541.0
+64.88
+77.13
+5
+18.0
+40.85
+53.16
+5
+14.8
+38.21
+49.35
+5
+503.9
+47.93
+61.67
+10
+102.0
+54.65
+68.64
+10
+33.1
+47.95
+61.49
+10
+507.6
+48.83
+63.02
+Ours
+5
+0.31
+33.09
+44.33
+Ours
+5
+0.26
+35.47
+46.35
+Ours
+5
+1.97
+47.04
+60.44
+10
+2.96
+47.11
+60.28
+10
+1.40
+45.53
+58.99
+10
+2.22
+48.00
+62.07
+Table 3. Experimental results for semantic segmentation task on augmented Pascal VOC12 dataset [8]. Model name with postfix “M”
+means the model uses MobileNetV2 as backbone, otherwise ResNet18 is used. “#Layers” is short for “the number of active convolutional
+layers” that are trained. All models are pretrained on Cityscapes dataset [11]. Strategy “Calibration” shows the accuracy when only the
+classifier and normalization statistics are updated to adapt different numbers of classes between augmented Pascal VOC12 and Cityscapes.
+5. Experiments
+In this section, we first present our experimental results
+on ImageNet classification [12] and semantic segmentation.
+Then, we study the impact of different hyper-parameter se-
+lections. Furthermore, we present the evaluation result run-
+ning our method on real hardware. Lastly, we empirically
+validate the assumption in Section 4.1.
+5.1. Experimental Setup
+Classification: Following [25], we split every dataset into
+two highly non-i.i.d. partitions with the same size. Then,
+we pretrain our models on the first partition with a vanilla
+training strategy, and finetune the model on the other par-
+tition with different configurations for the training strat-
+egy (i.e., with/without gradient filtering, hyper-parameters,
+number of convolution layers to be trained). More details
+(e.g., hyper-parameters) are in the Supplementary.
+Segmentation: Models are pretrained on Cityscapes [11]
+by MMSegmentation [10]. Then, we calibrate and finetune
+these models with different training strategies on the aug-
+mented Pascal-VOC12 dataset following [8], which is the
+combination of Pascal-VOC12 [13] and SBD [14]. More
+details are included in the supplementary material.
+On-device Performance Evaluation:
+For CPU per-
+formance evaluation, we implement our method with
+MKLDNN [1] (a.k.a. OneDNN) v2.6.0 and compare it with
+the convolution BP method provided by MKLDNN. We test
+on three CPUs, namely Intel 11900KF, Quad-core Cortex-
+A72 (Jetson Nano) and Quad-core Cortex-A53 (Raspberry
+Pi 3b). For GPU performance evaluation, we implement
+our method on CUDNN v8.2 [9] and compare with the BP
+method provided by CUDNN. We test on two GPUs, RTX
+3090Ti and the edge GPU on Jetson Nano.
+Since both
+6
+
+MKLDNN and CUDNN only support float32 BP, we test
+float32 BP only. Additionally, for the experiments on Jet-
+son Nano, we record the energy consumption for CPU and
+GPU with the embedded power meter. More details (e.g.,
+frequency) are included in the supplementary material.
+5.2. ImageNet Classification
+Table 2 shows our evaluation results on the ImageNet
+classification task. As shown, our method significantly re-
+duces the FLOPs and memory required for BP, with very
+little accuracy loss. For example, for ResNet34, our method
+achieves 18.9× speedup with 1.7% accuracy loss when
+training four layers; for MobileNetV2, we get a 1.2% bet-
+ter accuracy with 3.0× speedup and 3.1× memory savings.
+These results illustrate the effectiveness of our method. On
+most networks, TinyTL has a lower accuracy while consum-
+ing more resources compared to the baselines methods.
+5.3. Semantic Segmentation
+Table 3 shows our evaluation results on the augmented
+Pascal-VOC12 dataset.
+On a wide range of networks,
+our method constantly achieves significant speedup with
+marginal accuracy loss. For the large network UPerNet, our
+method achieves 229× speedup with only 1% mIoU loss.
+For the small network PSPNet, our method speedups train-
+ing by 140× with only 2.27% mIoU loss. This shows the
+effectiveness of our method on a dense prediction task.
+5.4. Hyper-Parameter Selection
+Figure 4 shows our experimental results for ResNets un-
+der different hyper-parameter selection, i.e. number of con-
+volution layers and patch size of gradient filter r × r. Of
+note, the y-axis (MFLOPs) in Figure 4 is log scale. More
+results are included in Supplementary Section F. We high-
+light three qualitative findings in Figure 4:
+a. For a similar accuracy, our method greatly reduces
+the number of operations (1 to 2 orders of magni-
+tude), while for a similar number of computations, our
+method achieves a higher accuracy (2% to 5% better).
+This finding proves the effectiveness of our method.
+b. Given the number of convolution layers to be trained,
+the more accurate method returns a better accuracy.
+Baseline (i.e., standard BP) uses the most accurate gra-
+dient, Ours-R4 (BP with gradient filter with patch size
+4 × 4) uses the least accurate gradient; thus, Baseline
+> Ours-R2 > Ours-R4.
+This finding is intuitive since the more accurate method
+should introduce smaller noise to the BP, e.g., the gradi-
+ent filtering with patch size 2 × 2 (Ours-R2) introduces less
+noise than with patch size 4 × 4 (Ours-R4). In Figure 5, we
+92
+93
+94
+Accuracy [%]
+ResNet18-CIFAR10
+101
+102
+103
+#MFLOPs
+14.1x OPs
+2.1x OPs
+2.2% Acc
+68.3x
+OPs
+Baseline
+Ours-R2
+Ours-R4
+78
+80
+82
+Accuracy [%]
+ResNet34-CIFAR100
+101
+102
+103
+18.8x OPs
+1% Acc
+7.6x OPs
+2.0x OPs
+5.1% Acc
+63.6x OPs
+Baseline
+Ours-R2
+Ours-R4
+Figure 4. Computation (#MFLOPs, log scale) and model accuracy
+[%] under different hyper-parameter selection. “Baseline” means
+vanilla BP; “Ours-R2/4” uses gradient filtering with patch size 2×
+2/4 × 4 during BP.
+evaluate the relationship between accuracy and noise level
+introduced by gradient filtering. With a higher SNR (i.e., a
+lower noise level), a better accuracy is achieved.
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+0.225
+SNR [db]
+71.6
+71.8
+Top 1 Accuracy [%]
+Figure 5. Relationship between accuracy and noise level intro-
+duced by the gradient filtering. As shown, accuracy increases as
+the SNR increases, i.e., noise level decreases.
+c. Given the number of computations, the less accurate
+method returns the better accuracy by training more
+layers, i.e., Ours-R4 > Ours-R2 > baseline.
+This finding suggests that for neural network training with
+relatively low computational resources, training more layers
+with less accurate gradients is preferable than training fewer
+layers with more accurate gradients.
+5.5. On-device Performance Evaluation
+Figure 6 and Table 4 show our evaluation results on real
+devices. More results are included in the Supplementary
+Section H. As Figure 6 shows, on CPU, most convolution
+layers achieve speedups over 20× with less than 50% mem-
+ory consumption for gradient filtering with patch sizes 2×2;
+for gradient filtering with patch size 4 × 4, the speedups are
+much higher, namely over 60×. On GPU, the speedup is
+a little bit lower, but still over 10× and 25×, respectively.
+Furthermore, as Table 4 shows, our method saves over 95%
+7
+
+0
+1
+2
+3
+4
+5
+6
+0×
+20×
+40×
+60×
+80×
+100×
+Speedup (×times)
+114×
+CPU Speedup
+Jetson-R2
+Jetson-R4
+11900KF-R2
+11900KF-R4
+RPi3-R2
+RPi3-R4
+0
+1
+2
+3
+4
+5
+6
+0×
+10×
+20×
+30×
+40×
+50×
+60×
+GPU Speedup
+Jetson-R2
+Jetson-R4
+RTX3090Ti-R2
+RTX3090Ti-R4
+0
+1
+2
+3
+4
+5
+6
+Test Case - Baseline: MKLDNN
+0
+20
+40
+60
+80
+100
+Percentage [%]
+Baseline: MKLDNN
+50% Memory Cost
+Normalized CPU Memory Cost
+Jetson-R2
+Jetson-R4
+11900KF-R2
+11900KF-R4
+RPi3-R2
+RPi3-R4
+0
+1
+2
+3
+4
+5
+6
+Test Case - Baseline: CUDNN
+0
+20
+40
+60
+80
+100
+Baseline: CUDNN
+50% Memory Cost
+Normalized GPU Memory Cost
+Jetson-R2
+Jetson-R4
+RTX3090Ti-R2
+RTX3090Ti-R4
+Figure 6. Speedup and normalized memory consumption results on multiple CPUs and GPUs under different test cases (i.e. different
+input sizes, numbers of channels, etc.) Detailed configuration of these test cases are included in the supplementary material. “R2”, “R4”
+mean using gradient filtering with 2 × 2 and 4 × 4 patch sizes, respectively. Our method achieves significant speedup with low memory
+consumption compared to all baseline methods. For example, on Jetson CPU with patch size 4 × 4 (“Jetson-R4” in left top figure), our
+method achieves 114× speedup with only 33% memory consumption for most test cases.
+Device
+Patch Size
+Normalized Energy Cost [STD]
+Edge
+CPU
+2 × 2
+4.13% [0.61%]
+4 × 4
+1.15% [0.18%]
+Edge
+GPU
+2 × 2
+3.80% [0.73%]
+4 × 4
+1.22% [1.10%]
+Table 4. Normalized energy consumption for BP with gradient
+filtering for different patch sizes. Results are normalized w.r.t. the
+energy cost of standard BP methods. For instance, for edge CPU
+with a 4 × 4 patch, only 1.15% of energy in standard BP is used.
+Standard deviations are shown within brackets.
+energy for both CPU and GPU scenarios, which largely re-
+solves one of the most important constraints on edge de-
+vices. All these experiments on real devices show that our
+method is practical for the real deployment of both high-
+performance and IoT applications.
+Model
+Ratio
+Model
+Ratio
+(Wide)ResNet18-152
+1.462
+VGG(bn)11-19
+1.497
+DenseNet121-201
+2.278
+EfficientNet b0-b7
+1.240
+Table 5. Evaluation of energy ratio defined in Equation (13) on
+models published on Torchvision. The ratio greater than 1 empiri-
+cally verifies our assumption.
+5.6. Main Assumption Verification
+We now empirically verify the assumption that the DC
+component dominates the frequency spectrum of the convo-
+lution kernel (Section 4.1). To this end, we collect the en-
+ergy ratio shown in Equation (13) from trained models pub-
+lished in Torchvision [24]. As Table 5 shows, for the con-
+volution kernels in all these networks, we get a ratio greater
+than one, which means that the energy of DC components
+is larger than energy of all AC components. Thus, our as-
+sumption in Section 4.1 empirically holds true in practice.
+6. Conclusions
+In this paper, we have addressed the on-device model
+training for resource-constrained edge devices. To this end,
+a new gradient filtering method has been proposed to sys-
+tematically reduce the computation and memory consump-
+tion for the back-propagation algorithm, which is the key
+bottleneck for efficient model training.
+In Section 3, a new gradient filtering approach has been
+proposed to reduce the computation required for propagat-
+ing gradients through the convolutional layers. The gradient
+filtering creates an approximate gradient feature map with
+fewer unique elements and a special structure; this reduces
+the computation by more than two orders of magnitude.
+Furthermore, we proved that the error introduced during
+back-propagation by our gradient filter is bounded so the
+influence of gradient approximation is limited.
+Extensive experiments in Section 5 have demonstrated
+the efficiency and wide applicability of our method. Indeed,
+models can be finetuned with orders of magnitudes fewer
+computations, while having only a marginal accuracy loss
+compared to popular baseline methods.
+8
+
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+In this supplementary material, we present:
+• A: Detailed derivation for gradient filtering described
+in Section 3.
+• B: Detailed proof for Proposition 1 in Section 4.1.
+• C: Visualized computation analysis for ResNet18.
+• D: Detailed experimental setup for Section 5.1.
+• E: More experimental results for Semantic Segmenta-
+tion in Section 5.3.
+• F: More experimental results for hyper-parameter ex-
+ploration on CIFAR datasets in Section 5.4.
+• G: Experimental results for combining gradient filter-
+ing (our method) with existing INT8 gradient quanti-
+zation approaches [4,7].
+• H: More experimental results for on-device perfor-
+mance evaluation in Section 5.5.
+A. Gradient Filtering Derivation
+In this section, we present the complete derivations for
+Equation (3) and Equation (5) in Section 3, namely the back
+propagation with gradient filtering. For convenience, Ta-
+ble 6 (reproduced from Table 1 in paper) lists commonly
+used symbols.
+A.1. Gradient Filtering
+We have:
+˜gy[n, co, h, w] = 1
+r2
+⌈i/r⌉r
+�
+h=⌊i/r⌋r
+⌈j/r⌉r
+�
+w=⌊j/r⌋r
+gy[n, co, i, j] (17)
+Cx
+Number of channels of x
+Wx, Hx
+Width and height of x
+θ
+Convolution kernel
+θ′
+Rotated θ, i.e., θ′ = rot180(θ)
+r
+Patch size (r × r)
+gx, gy, gθ
+Gradients w.r.t. x, y, θ
+˜gy
+Approximated gradient gy
+˜x, ˜θ′
+Sum of x and θ′ over
+spatial dimensions (height and width)
+x[n, ci, h, w]
+Element for feature map x
+at batch n, channel ci, pixel (h, w)
+θ[co, ci, u, v]
+Element for convolution kernel θ
+at output channel co, input channel ci,
+position (u, v)
+Table 6. Table of symbols we use.
+Thus, for any entry in the approximated gradient ˜gy, the
+value equals to the average of all neighboring elements
+within the same r × r patch, as shown in the Figure 2 in the
+main manuscript. For the approximated gradient ˜gy with
+batch size n, channel c, resolution (Hy, Wy), there will be
+(n × c × ⌈ Hy
+r ⌉ × ⌈ Wy
+r ⌉) unique numbers in ˜gy. To simplify
+the following derivations, we rewrite the approximated gra-
+dient ˜gy as follows:
+˜gp
+y[n, co, hp, wp, i, j] = ˜gy[n, co, hp∗r+i, wp∗r+j] (18)
+where (hp, wp) is the position of the patch and (i, j) is the
+offset within the patch. Since every element in the same
+patch has the exact same value, we denote this unique value
+with ˜gu
+y , i.e.,
+˜gu
+y [n, co, hp, wp] = ˜gp
+y[n, co, hp, wp, i, j], ∀0 ≤ i, j < r
+(19)
+A.2. Approximation for Rotated Convolution Ker-
+nel θ′
+˜θ′[co, ci] =
+�
+u,v
+θ′[co, ci, u, v]
+=
+�
+u,v
+rot180(θ)[co, ci, u, v]
+=
+�
+u,v
+θ[co, ci, u, v]
+(20)
+A.3. Approximation for Input Feature x
+˜x[n, ci, h, w] =
+⌈i/r⌉r
+�
+h=⌊i/r⌋r
+⌈j/r⌉r
+�
+w=⌊j/r⌋r
+x[n, ci, i, j]
+(21)
+Thus for every entry in approximated feature map ˜x, the
+value equal to the sum of all neighboring elements within
+the same r × r patch. Following the definition of the gra-
+dient filter in Section A.1, we use the following symbols to
+simplify the derivation:
+˜xp[n, ci, hp, wp, i, j] = ˜x[n, ci, hp ∗r +i, wp ∗r +j] (22)
+and
+˜xu[n, ci, hp, wp] = ˜xp[n, ci, hp, wp, i, j], ∀0 ≤ i, j < r
+(23)
+A.4. Boundary Elements
+As mentioned in Section 3, given the structure created
+by the gradient filters, the gradient propagation in a con-
+volution layer can be simplified to weights summation and
+multiplication with few unique gradient values. This is true
+11
+
+for all elements far away from the patch boundary because
+for these elements, the rotated kernel θ′ only covers the ele-
+ments from the same patch, which have the same value, thus
+the computation can be saved. However, for the elements
+close to the boundary, this is not true, since when convolv-
+ing with boundary gradient elements, the kernel may cover
+multiple patches with multiple unique values instead of just
+one. To eliminate the extra computation introduced by the
+boundary elements, we pad each patch sufficiently such that
+every element is far away from boundary:
+˜gp
+y[n, ci, hp, wp, i, j] = ˜gu
+y [n, ci, hp, wp], ∀i, j ∈ Z
+(24)
+For example, with the patch size 4 × 4, the element at the
+spatial position (3, 3) is on the boundary, so when we calcu-
+late ˜gx[n, ci, 3, 3] by convolving the rotated kernel θ′ with
+the approximated gradient ˜gy:
+˜gx[n, ci, 3, 3] =
+�
+i,j
+θ′[co, ci, i, j]˜gy[n, co, 3+i, 3+j] (25)
+values of ˜gy are from multiple patches and have differ-
+ent values (e.g., ˜gy[n, co, 3, 3] is from patch (0, 0) while
+˜gy[n, co, 4, 4] is from patch (1, 1); they have different val-
+ues).
+In our method, we simplify the Equation (25) by
+rewriting it in the following way:
+˜gx[n, ci, 3, 3]
+≈
+1
+�
+i,j=−1
+θ′[co, ci, i, j]˜gp
+y[n, co, ⌊3
+4⌋, ⌊3
+4⌋, 3 + i, 3 + j]
+(26)
+=
+1
+�
+i,j=−1
+θ′[co, ci, i, j]˜gu
+y [n, co, ⌊3
+4⌋, ⌊3
+4⌋]
+(27)
+=
+1
+�
+i,j=−1
+θ′[co, ci, i, j]˜gu
+y [n, co, 0, 0]
+(28)
+where Equation (26) is derived from Equation (25) by con-
+sidering that patch (0, 0) is sufficiently padded so that for
+elements with all offsets (3 + i, 3 + j), they have the same
+value, which is the unique value gu
+y [n, co, 0, 0].
+For approximated input feature map ˜x, we apply the
+same approximation for the boundary elements.
+A.5. Gradient w.r.t. Input (Equation (3) in the Pa-
+per)
+˜gx[n, ci, h, w]
+(29)
+=
+�
+co,u,v
+θ[co, ci, −u, −v]˜gy[n, co, h + u, w + v]
+(30)
+≈
+�
+co,u,v
+θ[co, ci, −u, −v]·
+˜gp
+y[n, co, ⌊h
+r ⌋, ⌊w
+r ⌋, (h mod r) + u, (w mod r) + v]
+(31)
+=
+�
+co,u,v
+θ[co, ci, −u, −v]˜gu
+y [n, co, ⌊h
+r ⌋, ⌊w
+r ⌋]
+(32)
+=
+�
+co
+˜gu
+y [n, co, ⌊h
+r ⌋, ⌊w
+r ⌋]
+�
+u,v
+θ[co, ci, −u, −v]
+(33)
+=
+�
+co
+˜gu
+y [n, co, ⌊h
+r ⌋, ⌊w
+r ⌋]˜θ′[co, ci]
+(34)
+By expanding ˜gu
+y to ˜gy, we have:
+˜gx[n, ci, h, w] =
+�
+co
+˜gy[n, co, h, w] ⊙ ˜θ′[co, ci]
+(35)
+which is the Equation (3) in Section 3 in the paper.
+From Equation (30) to Equation (32), we consider that
+the patch in the approximated gradient ˜gy is padded suffi-
+ciently so they have the same value for all possible offsets
+((h mod r) + u, (w mod r) + v). If there is only one in-
+put channel and output channel for the convolutional layer
+as the Figure 2 in the paper shows, then Equation (34)
+become an element-wise multiplication, which is Equa-
+tion (35) (also the Equation (3) in the paper).
+A.6. Gradient w.r.t. Convolution Kernel (Equation
+(5) in the Paper)
+˜gθ[co, ci, u, v]
+(36)
+=
+�
+n,h,w
+x[n, ci, h + u, w + v]˜gy[n, co, h, w]
+(37)
+≈
+�
+n,h,w
+˜xp[n, ci, ⌊h
+r ⌋, ⌊w
+r ⌋, (h mod r) + u, (w mod r) + v]·
+˜gu
+y [n, co, ⌊h
+r ⌋, ⌊w
+r ⌋]
+(38)
+=
+�
+n,h,w
+˜xu[n, ci, ⌊h
+r ⌋, ⌊w
+r ⌋]˜gu
+y [n, co, ⌊h
+r ⌋, ⌊w
+r ⌋]
+(39)
+=
+�
+n,h,w
+˜xu[n, ci, ⌊h
+r ⌋, ⌊w
+r ⌋]˜gu
+y [n, co, ⌊h
+r ⌋, ⌊w
+r ⌋]
+(40)
+12
+
+By expanding ˜xu and ˜gu
+y to ˜x and ˜gy, respectively, we have:
+˜gθ[co, ci, u, v] =
+�
+n,i,j
+˜x[n, ci, i, j]˜gy[n, co, i, j]
+(41)
+which is precisely Equation (5) in Section 3.
+From Equation (37) to Equation (39), we consider that
+the patch in the approximated input feature map ˜x is padded
+sufficiently thus they have the same value for all possible
+offsets ((h mod r) + u, (w mod r) + v). For every given
+input/output channel pair (co, ci), Equation (40) represents
+the Frobenius inner product between ˜xu and ˜gu
+y .
+B. Detailed Proof for Proposition 1
+In this section, we provide more details to the proof in
+Section 4.1. We use Gx, Gy and Θ to denote the gradients
+gx, gy and the convolution kernel θ in the frequency domain,
+respectively. Gx[u, v] is the spectrum value at frequency
+(u, v) and δ is the 2D discrete Dirichlet function. Without
+losing generality and to simplify the proof, we consider the
+batch size is 1, the number of input/output channels is 1,
+namely Cx = Cy = 1, and there is only one patch in ˜gy.
+The gradient returned by the gradient filtering can be
+written as:
+˜gy = 1
+r2 1r×r ⊛ gy
+(42)
+where ⊛ denotes convolution.
+By applying the discrete
+Fourier transformation, Equation (42) can be rewritten in
+the frequency domain as:
+˜Gy[u, v] = 1
+r2 δ[u, v]Gy[u, v]
+(43)
+˜gy is the approximation for gy(so the ground truth for ˜gy is
+gy), and the SNR of ˜gy equals to:
+SNR˜gy = (
+�
+(u,v)(Gy[u, v], − ˜Gy[u, v])2
+�
+(u,v) G2y[u, v]
+)−1
+= (
+�
+(u,v)(Gy[u, v] − 1
+r2 δ[u, v]Gy[u, v])2
+�
+(u,v) G2y[u, v]
+)−1
+(44)
+where the numerator can be written as:
+�
+(u,v)
+(Gy[u, v] − 1
+r2 δ[u, v]Gy[u, v])2
+=
+�
+(u,v)̸=(0,0)
+(Gy[u, v] − 1
+r2 δ[u, v]Gy[u, v])2
++ (Gy[0, 0] − 1
+r2 δ[0, 0]Gy[0, 0])2
+(45)
+Because δ[u, v] =
+�
+1
+(u, v) = (0, 0)
+0
+(u, v) ̸= (0, 0), Equation (45) can
+be written as:
+�
+(u,v)̸=(0,0)
+G2
+y[u, v] + (r2 − 1)2
+r4
+G2
+y[0, 0]
+=
+�
+(u,v)̸=(0,0)
+G2
+y[u, v] + G2
+y[0, 0] − G2
+y[0, 0]
++ (r2 − 1)2
+r4
+G2
+y[0, 0]
+=
+�
+(u,v)
+G2
+y[u, v] − 2r2 − 1
+r4
+G2
+y[0, 0]
+(46)
+By substituting the numerator in Equation (44) with Equa-
+tion (46), we have:
+SNR˜gy = (
+�
+(u,v) G2
+y[u, v] − 2r2−1
+r4
+G2
+y[0, 0]
+�
+(u,v) G2y[u, v]
+)−1
+= (1 − 2r2 − 1
+r4
+G2
+y[0, 0]
+�
+(u,v) G2y[u, v])−1
+= (1 − 2r2 − 1
+r4
+Energy of DC Component in Gy
+Total Energy4in Gy
+)−1
+(47)
+For the convolution layer, the gradient w.r.t. approximated
+variable ˜x in the frequency domain is:
+˜Gx[u, v] = Θ[−u, −v] ˜Gy[u, v]
+= 1
+r2 Θ[−u, −v]δ[u, v]Gy[u, v]
+(48)
+and its ground truth is:
+Gx[u, v] = Θ[−u, −v]Gy[u, v]
+(49)
+Similar to Equation (47), the SNR of g˜x is:
+SNR˜gx = (1 − 2r2 − 1
+r4
+Θ2[0, 0]G2
+y[0, 0]
+�
+(u,v) Θ2[u, v]G2y[u, v])−1
+= (1 − 2r2 − 1
+r4
+G2
+x[0, 0]
+�
+(u,v) G2x[u, v])−1
+= (1 − 2r2 − 1
+r4
+Energy of DC Component in Gx
+Total Energy5in Gx
+)−1
+(50)
+Equation (50) can be rewritten as:
+r4(1 − SNR−1
+˜gx )
+2r2 − 1
+=
+(Θ[0, 0]Gy[0, 0])2
+�
+(u,v)(Θ[−u, −v]Gy[u, v])2
+=
+G2
+y[0, 0]
+�
+(u,v)( Θ[−u,−v]
+Θ[0,0] Gy[u, v])2
+(51)
+4As reminder, the total energy of a signal is the sum of energy in DC
+component and energy in AC components.
+13
+
+Besides, the proposition’s assumption (the DC component
+dominates the frequency spectrum of Θ) can be written as:
+Θ2[0, 0]
+max(u,v)̸=(0,0)Θ2[u, v] ≥ 1
+(52)
+which is:
+∀(u, v), Θ2[−u, −v]
+Θ2[0, 0]
+≤ 1
+(53)
+thus, by combining Equation (51) and Equation (53), we
+have:
+r4(1 − SNR−1
+˜gx )
+2r2 − 1
+=
+G2
+y[0, 0]
+�
+(u,v)( Θ[−u,−v]
+Θ[0,0] Gy[u, v])2
+≥
+G2
+y[0, 0]
+�
+(u,v)(Gy[u, v])2
+=
+r4(1 − SNR−1
+˜gy )
+2r2 − 1
+(54)
+which means that: SNR˜gx ≥ SNR˜gy. This completes our
+proof for error analysis.■
+In conclusion, as the gradient propagates, the noise in-
+troduced by the gradient filter becomes weaker and weaker
+compared to the real gradient signal. This property ensures
+that the error in gradient has only a limited influence on the
+quality of BP.
+This proof can be extended to the more general case
+where batch size and the number of channels are greater
+than 1 by introducing more dimensions (i.e., batch dimen-
+sion, channel dimension) into all equations listed above.
+C. Computation Analysis for ResNet18
+In this section, we provide two more examples for com-
+putation analysis in Section 4.2. Figure 7 shows the com-
+putation required by the convolution layers from ResNet18
+with different patch sizes for gradient filtering. With re-
+duced unique elements, our approach reduces the num-
+ber of computations to 1/r2 of standard BP method; with
+structured gradient, our approach further reduces the num-
+ber of computations to about 1/(r2HθWθ) of standard BP
+method.
+D. Detailed Experimental Setup
+In this supplementary section, we extend the experimen-
+tal setup in Section 5.1.
+D.1. ImageNet Classification
+D.1.1
+Environment
+ImageNet related experiments are conducted on IBM Power
+System AC922, which is equipped with a 40-core IBM
+Power 9 CPU, 256 GB DRAM and 4 NVIDIA Tesla V100
+16GB GPUs. We use PyTorch 1.9.0 compiled with CUDA
+10.1 as the deep learning framework.
+1 × 1
+3 × 3
+5 × 5
+7 × 7
+Patch Size r × r
+1M
+10M
+100M
+FLOPs
+Baseline
+Reduced
+Unique
+Elements
++Structured
+Gradient
+Actual
+Minimum
+Achievable Computation
+(a) Last convolutional layer in block 4 of ResNet18 with 512 input/output
+channels; the resolution of input feature map is 7 × 7.
+1 × 1
+4 × 4
+8 × 8
+12 × 12
+Patch Size r × r
+100K
+1M
+10M
+100M
+FLOPs
+Baseline
+Reduced
+Unique
+Elements
++Structured
+Gradient
+Actual
+Minimum
+Achievable Computation
+(b) Last convolutional layer in block 3 of ResNet18 with 256 input/output
+channels; the resolution of input feature map is 14 × 14.
+1 × 1
+10 × 10
+20 × 20
+Patch Size r × r
+100K
+1M
+10M
+100M
+FLOPs
+Baseline
+Reduced
+Unique
+Elements
++Structured
+Gradient
+Actual
+Minimum
+Achievable Computation
+(c) Last convolutional layer in block 2 of ResNet18 with 128 input/output
+channels; the resolution of input feature map is 28 × 28.
+Figure 7. Computation analysis for three convolution layers in
+of ResNet18 model. Since convolutional layers in every block
+of ResNet18 is similar, we use the last convolutional layer as the
+representative of all convolutional layers in the block. Minimum
+achievable computation is presented in Equation (16) in the pa-
+per. By reducing the number of unique elements, computations
+required by our approach drop to about 1/r2 compared with the
+standard BP method. By combining it (“+” in the figure) with
+structured gradient map, computations required by our approach
+drop further.
+D.1.2
+Dataset Split
+We split the dataset into two non-i.i.d. partitions following
+the FedAvg method [25]. The label distribution is shown
+in Figure 8.
+Among all 1000 classes for the ImageNet,
+pretrain and finetune partitions overlap on only 99 classes,
+which suggests that our method can efficiently adapt the
+14
+
+Model
+Accuracy
+Model
+Accuracy
+ResNet-18
+73.5%
+MobileNet-V2
+74.3%
+ResNet-34
+76.4%
+MCUNet
+71.4%
+Table 7. Model pretraining accuracy on ImageNet.
+DNN model to data collected from new environments. For
+each partition, we randomly select 80% data as training data
+and 20% as validation data.
+0
+200
+400
+600
+800
+1000
+Class Index
+0
+200
+400
+600
+800
+1000
+Image Count
+ImageNet Data Split
+Pretrain
+Finetune
+Figure 8. Label distribution for pretraining and finetuning datasets.
+Pretraining and finetuning partitions are split from ImageNet
+dataset.
+D.1.3
+Pretraining
+We pretrain ResNet 18, ResNet 34, MobileNet-V2 and
+MCUNet with the same configuration. We use SGD opti-
+mizer. The learning rate of the optimizer starts at 0.05 and
+decays according to cosine annealing method [23] during
+training. Additionally, weight decay is set to 1 × 10−4 and
+momentum is set to 0.9. We set batch size to 64. We ran-
+domly resize, randomly flip and normalize the image for
+data augmentation. We use cross entropy as loss function.
+Models are trained for 200 epochs and the model with the
+highest validation accuracy is kept for finetuning. Table 7
+shows the pretrain accuracy.
+D.1.4
+Finetuning
+We adopt the hyper-parameter (e.g., momentum, weight de-
+cay, etc.) from pretraining. Several changes are made: mod-
+els are finetuned for 90 epochs instead of 200; we apply
+L2 gradient clipping with threshold 2.0; linear learning rate
+warm-up for 4 epochs is introduced at the beginning of fine-
+tuning, i.e., for the first 4 epochs, the learning rate grows
+linearly up to 0.05, then the learning rate decays accord-
+ing to cosine annealing method in the following epochs. Of
+note, to ensure a fair comparison, we use the same hyper-
+parameters for all experiments, regardless of model type
+and training strategy.
+D.2. CIFAR Classification
+D.2.1
+Environment
+CIFAR related experiments are conducted on a GPU work-
+station with a 64-core AMD Ryzen Threadripper PRO
+3995WX CPU, 512 GB DRAM and 4 NVIDIA RTX A6000
+GPUs. We use PyTorch 1.12.0 compiled with CUDA 11.6
+as the deep learning framework.
+D.2.2
+Dataset Split
+We split the dataset into two non-i.i.d. partitions following
+FedAvg method. The label distribution is shown in Figure
+9. For CIFAR10, pretrain and finetune partitions overlap
+on 2 classes out of 10 classes in total. For CIFAR100, pre-
+train and finetune partitions overlap on 6 classes out of 100
+classes.
+0
+2
+4
+6
+8
+Class Index
+0
+1000
+2000
+3000
+4000
+Image Count
+CIFAR10 Data Split
+Pretrain
+Finetune
+0
+20
+40
+60
+80
+100
+Class Index
+0
+100
+200
+300
+400
+Image Count
+CIFAR100 Data Split
+Pretrain
+Finetune
+Figure 9. Label distribution for pretraining and finetuning datasets
+on CIFAR10 and CIFAR100. Pretraining and finetuning partitions
+are split from CIFAR10/100, respectively.
+D.2.3
+Pretraining
+We pretrain ResNet18 and ResNet34 with the same config-
+uration. We use the ADAM optimizer with a learning rate
+of 3 × 10−4 and weight decay 1 × 10−4 with no learning
+rate scheduling method. We use cross entropy as loss func-
+tion. We set batch size to 128, and normalize the data be-
+fore feeding it to the model. Models are trained for 30 and
+50 epochs for CIFAR10 and CIFAR100, respectively. Then,
+the model with the highest accuracy is kept for finetuning.
+Table 8 shows the pretrain accuracy.
+D.2.4
+Finetuning
+We adopt the training configuration from PSQ [7] with
+some changes. We use cross entropy loss with SGD opti-
+15
+
+ResNet18
+ResNet34
+CIFAR10
+95.1%
+97.6%
+CIFAR100
+75.5%
+83.5%
+Table 8. Model pretraining accuracy on CIFAR10/100.
+mizer for training. The learning rate of the optimizer starts
+at 0.05 and decays according to cosine annealing method
+during training. Momentum is set to 0 and weight decay
+is set to 1 × 10−4. We apply L2 gradient clipping with
+a threshold 2.0. Batch normalization layers are fused with
+convolution layers before training, which is a common tech-
+nique for inference acceleration.
+D.3. Semantic Segmentation
+D.3.1
+Environment
+ImageNet related experiments are conducted on IBM Power
+System AC922, which is equipped with a 40-core IBM
+Power 9 CPU, 256 GB DRAM and 4 NVIDIA Tesla V100
+16GB GPUs. We use PyTorch 1.9.0 compiled with CUDA
+10.1 as the deep learning framework. We implement our
+method based on MMSegmentation 0.27.0.
+D.3.2
+Pretraining
+We use models pretrained by MMSegmentation. Consider-
+ing that the numbers of classes, image statistics, and model
+hyper-parameters may be different when applying on dif-
+ferent datasets, we calibrate the model before finetuning.
+We use SGD optimizer. The learning rate of the optimizer
+starts at 0.01 and decays exponentially during training. Ad-
+ditionally, weight decay is set to 5 × 10−4 and momentum
+is set to 0.9. We set batch size to 8. We randomly crop, flip
+and photo-metric distort and normalize the image for data
+augmentation. We use cross entropy as loss function. For
+DeepLabV3, FCN, PSPNet and UPerNet, we calibrate the
+classifier (i.e., the last layer) and statistics in batch normal-
+ization layers for 1000 steps on the finetuning dataset. For
+DeepLabV3-MobileNetV2 and PSPNet-MobileNetV2, be-
+cause the number of channels for convolutional layers in
+the decoder are different for models applied on different
+datasets, we calibrate the decoder and statistics in batch nor-
+malization layers for 5000 steps on the finetuning dataset.
+D.3.3
+Finetuning
+We finetune all models with the same configuration. We use
+the SGD optimizer. The learning rate of the optimizer starts
+at 0.01 and decays according to cosine anneling method dur-
+ing training. Additionally, weight decay is set to 5 × 10−4
+and momentum is set to 0.9. We set batch size to 8. We
+randomly crop, flip and photo-metric distort and normalize
+the image for data augmentation. We use cross entropy as
+loss function. Models are finetuned for 20000 steps. Exper-
+iments are repeated three times with random seed 233, 234
+and 235.
+D.4. On-device Performance Evaluation
+D.4.1
+NVIDIA Jetson Nano
+We use NVIDIA Jetson Nano with quad-core Cortex-A57,
+4 GB DRAM, 128-core Maxwell edge GPU for perfor-
+mance evaluation on both edge CPU and edge GPU. We
+use the aarch64-OS Ubuntu 18.04.6 provided by NVIDIA.
+During evaluation, the frequencies for CPU and GPU are
+1.5 GHz and 921 MHz, respectively. Our code and library
+MKLDNN (a.k.a. OneDNN) are compiled on Jetson Nano
+with GCC 7.5.0, while libraries CUDA and CUDNN are
+compiled by NVIDIA. For CPU evaluations, our code and
+baseline are implemented with MKLDNN v2.6. For GPU
+evaluations, our code and baseline are implemented with
+CUDA 10.2 and CUDNN 8.2.1.
+Before the evaluation for every test case, we warm up
+the device by running the test once. Then we repeat the test
+10 times and report the average value for latency, energy
+consumption, etc.
+Energy consumption is obtained by reading the embed-
+ded power meter in Jetson Nano every 20 ms.
+D.4.2
+Raspberry Pi 3b
+We use Raspberry Pi 3b with quad-core Cortex-A53, 1
+GB DRAM for performance evaluation on CPU. We use
+the aarch64-OS Raspberry Pi OS. During evaluation, the
+frequency for CPU is 1.2 GHz.
+Our code and library
+MKLDNN are compiled on Raspberry Pi with GCC 10.2.
+Our code and baseline are implemented with MKLDNN
+v2.6.
+Before the evaluation for every test case, we warm up the
+device by running the test once. Then we repeat the test 10
+times and report the average value for latency, etc.
+D.4.3
+Desktop
+We use a desktop PC with Intel 11900KF CPU, 32 GB
+DRAM and RTX 3090 Ti GPU for perforamce evaluation
+on both desktop CPU and desktop GPU. We use x86 64-
+OS Ubuntu 20.04. During evaluation, the frequencies for
+CPU and GPU are 4.7 GHz and 2.0 GHz respectively. Our
+code is compiled with GCC 9.4.0. MKLDNN is compiled
+by Anaconda (tag omp h13be974 0). CUDA and CUDNN
+are compiled by NVIDIA. For CPU evaluations, our code
+and baseline are implemented with MKLDNN v2.6. For
+GPU evaluations, our code and baseline are implemented
+with CUDA 11.7 and CUDNN 8.2.1.
+16
+
+Pretrain: ADE20K Finetune: VOC12Aug
+UPerNet
+#Layers
+GFLOPs
+mIoU
+mAcc
+PSPNet-M
+#Layers
+GFLOPs
+mIoU
+mAcc
+DLV3-M
+#Layers
+GFLOPs
+mIoU
+mAcc
+Calibration
+0
+0
+37.66
+50.03
+Calibration
+0
+0
+30.93
+52.01
+Calibration
+0
+0
+35.28
+56.98
+Vanilla BP
+All
+541.0
+67.23[0.24]
+79.79[0.45]
+Vanilla BP
+All
+42.41
+53.51[0.27]
+67.01[0.19]
+Vanilla BP
+All
+54.35
+60.78[0.21]
+74.10[0.40]
+5
+503.9
+72.01[0.09]
+81.97[0.30]
+5
+12.22
+48.88[0.11]
+62.67[0.31]
+5
+14.77
+51.51[0.09]
+66.08[0.44]
+10
+507.6
+72.01[0.19]
+81.83[0.44]
+10
+22.46
+53.71[0.29]
+67.93[0.32]
+10
+33.10
+57.63[0.10]
+71.93[0.41]
+Ours
+5
+1.97
+71.76[0.11]
+81.57[0.07]
+Ours
+5
+0.11
+48.59[0.08]
+62.28[0.30]
+Ours
+5
+0.26
+49.40[0.00]
+64.13[0.54]
+10
+2.22
+71.78[0.23]
+81.55[0.38]
+10
+0.76
+52.77[0.37]
+66.82[0.47]
+10
+1.40
+55.14[0.15]
+69.48[0.26]
+Pretrain: ADE20K Finetune: Cityscapes
+UPerNet
+#Layers
+GFLOPs
+mIoU
+mAcc
+PSPNet-M
+#Layers
+GFLOPs
+mIoU
+mAcc
+DLV3-M
+#Layers
+GFLOPs
+mIoU
+mAcc
+Calibration
+0
+0
+34.15
+42.45
+Calibration
+0
+0
+28.83
+34.85
+Calibration
+0
+0
+41.33
+48.65
+Vanilla BP
+All
+1082.1
+73.02[0.14]
+81.01[0.20]
+Vanilla BP
+All
+84.82
+60.21[0.40]
+67.72[0.68]
+Vanilla BP
+All
+108.7
+71.12[0.14]
+79.81[0.04]
+5
+1007.7
+62.46[0.19]
+72.62[0.27]
+5
+24.43
+42.09[0.43]
+48.70[0.49]
+5
+29.5
+51.00[0.05]
+59.20[0.03]
+10
+1015.3
+64.01[0.21]
+73.11[0.32]
+10
+44.90
+54.03[0.24]
+61.48[0.10]
+10
+66.2
+61.02[0.14]
+69.80[0.06]
+Ours
+5
+3.94
+60.58[0.25]
+70.67[0.32]
+Ours
+5
+0.22
+41.59[0.38]
+48.10[0.41]
+Ours
+5
+0.50
+48.83[0.07]
+56.87[0.08]
+10
+4.43
+62.14[0.24]
+71.41[0.27]
+10
+1.51
+49.10[0.49]
+56.93[1.43]
+10
+2.74
+50.22[1.01]
+59.99[0.31]
+Table 9.
+Experimental results for semantic segmentation task for UPerNet, DeepLabV3-MobileNetV2 (DLV3-M) and PSPNet-
+MobileNetV2 (PSPNet-M). Models are pretrained on ADE20K dataset and finetuned on augmentated Pascal VOC12 dataset and Cityscapes
+dataset respectively. “#Layers” is short for “the number of active convolutional layers” that are trained. Strategy “Calibration” shows the
+accuracy when only the classifier and normalization statistics are updated to adapt differences (e.g. different number of classes) between
+pretraining dataset and finetuning dataset.
+No.
+#Input Channel
+#Output Channel
+Input Width
+Input Height
+0
+128
+128
+120
+160
+1
+256
+256
+60
+80
+2
+512
+512
+30
+40
+3
+512
+512
+14
+14
+4
+256
+256
+14
+14
+5
+128
+128
+28
+28
+6
+64
+64
+56
+56
+Table 10. Layer configuration for test cases in Figure 6 in Section
+5.5 in the paper.
+Before the evaluation for every test case, we warm up the
+device by running the 10 times. Then we repeat the test 200
+times and report the average value for latency, etc.
+D.4.4
+Test Case Configurations
+Table 10 lists the configurations for test cases shown in Fig-
+ure 6 in the paper. In addition to the parameters shown in
+the table, for all test cases, we set the batch size to 32, kernel
+size to 3 × 3, padding and stride to 1.
+E. More Results for Semantic Segmentation
+In this section, we extend the experimental results shown
+in Section 5.3 (Table 3). Table 9 shows the experimental re-
+sults for UPerNet, PSPNet-MobileNetV2 (PSPNet-M) and
+DeepLabV3-MobileNetV2 (DLV3-M) on two pairs of pre-
+traing and finetuning datasets. These results further show
+the effectiveness of our method on a dense prediction task.
+F. More Results for CIFAR10/100 with Differ-
+ent Hyper-Parameter Selections
+In this section, we extend the experimental results shown
+in Section 5.4 (Figure 4). Table 11 (page 18) shows the ex-
+perimental results for ResNet18 and ResNet34 on CIFAR
+datasets. For every model, we test our method with differ-
+ent patch sizes for gradient filtering and different numbers
+of active convolutional layers (#Layers in Table 11, e.g., if
+#Layers equals to 2, the last two convolutional layers are
+trained while other layers are frozen). These results further
+support the qualitative findings in Section 5.4.
+G. Results for Combining Gradient Filtering
+with Gradient Quantization
+In this section, we provide experimental results for com-
+bining our method, i.e.
+gradient filtering, with gradient
+quantization. Table 12 (page 19) shows experimental re-
+sults for ResNet18 and ResNet32 with gradient quantiza-
+tion methods PTQ [4] and PSQ [7] and different hyper-
+parameters. Both forward propagation and backward prop-
+agation are quantized to INT8. These results support the
+wide applicability of our method.
+H. More Results for On-device Performance
+Evaluation
+In this section, we extend the experimental results shown
+in Section 5.5. Figure 10 shows the energy savings and
+overhead of our method. For most test cases with patch
+4 × 4, we achieve over 80× energy savings with less than
+20% overhead on both CPU and GPU. Moreover, for the
+test case 1 on Raspberry Pi CPU, the forward propagation
+is even faster when applied our method (which results in
+negtive overheads).
+These results further show that our
+method is practical for the real deployment of both high-
+performance and IoT applications.
+17
+
+CIFAR10
+CIFAR100
+ResNet18
+#Layers
+ACC[%]
+FLOPs
+ResNet34
+#Layers
+ACC[%]
+FLOPs
+ResNet18
+#Layers
+ACC[%]
+FLOPs
+ResNet34
+#Layers
+ACC[%]
+FLOPs
+Vanilla
+BP
+1
+91.7
+128.25M
+Vanilla
+BP
+1
+94.2
+128.25M
+Vanilla
+BP
+1
+73.8
+128.39M
+Vanilla
+BP
+1
+76.9
+128.39M
+2
+93.6
+487.68M
+2
+96.6
+487.68M
+2
+77.6
+487.82M
+2
+82.0
+487.82M
+3
+93.7
+847.15M
+3
+96.6
+847.13M
+3
+77.6
+847.29M
+3
+82.1
+847.27M
+4
+94.4
+1.14G
+4
+96.8
+1.21G
+4
+78.0
+1.14G
+4
+83.0
+1.21G
++Gradient
+Filter
+R2
+1
+91.5
+8.18M
++Gradient
+Filter
+R2
+1
+94.2
+8.18M
++Gradient
+Filter
+R2
+1
+73.7
+8.31M
++Gradient
+Filter
+R2
+1
+77.0
+8.31M
+2
+92.7
+26.80M
+2
+96.6
+26.80M
+2
+75.6
+26.94M
+2
+81.1
+26.94M
+3
+92.8
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+3
+96.5
+45.44M
+3
+75.6
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+3
+81.1
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+4
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+4
+96.6
+64.07M
+4
+76.4
+60.15M
+4
+82.0
+64.21M
++Gradient
+Filter
+R4
+1
+91.4
+1.88M
++Gradient
+Filter
+R4
+1
+94.3
+1.88M
++Gradient
+Filter
+R4
+1
+73.7
+2.02M
++Gradient
+Filter
+R4
+1
+76.9
+2.02M
+2
+92.7
+7.93M
+2
+96.4
+7.93M
+2
+74.9
+8.07M
+2
+80.4
+8.07M
+3
+92.8
+13.99M
+3
+96.4
+13.98M
+3
+74.9
+14.12M
+3
+80.4
+14.12M
+4
+93.3
+19.12M
+4
+96.1
+20.04M
+4
+75.2
+19.26M
+4
+80.5
+20.17M
++Gradient
+Filter
+R7
+1
+91.5
+303.10K
++Gradient
+Filter
+R7
+1
+94.2
+303.10K
++Gradient
+Filter
+R7
+1
+73.7
+441.34K
++Gradient
+Filter
+R7
+1
+76.9
+441.34K
+2
+91.5
+3.21M
+2
+95.8
+3.21M
+2
+74.1
+3.35M
+2
+80.4
+3.35M
+3
+91.7
+6.12M
+3
+96.0
+6.12M
+3
+74.1
+6.26M
+3
+80.3
+6.26M
+4
+92.6
+8.90M
+4
+96.0
+9.03M
+4
+75.4
+9.04M
+4
+80.3
+9.17M
+Table 11. Experimental results on CIFAR10 and CIFAR100 datasets for ResNet18 and ResNet34 with different hyper-parameter selections.
+“ACC” is short for accuracy. “#Layers” is short for “the number of active convolution layers”. For example. #Layers equals to 2 means that
+only the last two convolutional layers are trained. “Gradient Filter R2/4/7” use proposed gradient filtering method with patch size 2 × 2,
+4 × 4 and 7 × 7, respectively.
+0
+1
+2
+3
+4
+5
+6
+0×
+20×
+40×
+60×
+80×
+100×
+120×
+Energy Savings [×times]
+CPU Energy Savings
+Jetson-R2
+Jetson-R4
+0
+1
+2
+3
+4
+5
+6
+0×
+20×
+40×
+60×
+80×
+100×
+GPU Energy Savings
+Jetson-R2
+Jetson-R4
+0
+1
+2
+3
+4
+5
+6
+Test Case - Baseline: MKLDNN
+0
+25
+50
+75
+100
+Percentage [%]
+Forward Cost
+20% Overhead
+Normalized CPU Overhead
+Jetson-R2
+Jetson-R4
+11900KF-R2
+11900KF-R4
+RPi3-R2
+RPi3-R4
+0
+1
+2
+3
+4
+5
+6
+Test Case - Baseline: CUDNN
+0
+20
+40
+60
+80
+100
+Forward Cost
+20% Overhead
+Normalized GPU Overhead
+Jetson-R2
+Jetson-R4
+RTX3090Ti-R2
+RTX3090Ti-R4
+Figure 10. Energy savings and overhead resuls on multiple CPUs and GPUs under different test cases (i.e., different input sizes, number of
+channels, etc..). For test case 4 and 5 with patch size 4 × 4 (Jetson-R4) on GPU, the latency of our method is too small to be captured by
+the power meter with a 20 ms sample rate so the energy savings data is not available. For most test cases with patch size 4 × 4, our method
+achieves over 80× energy savings with less than 20% overhead.
+18
+
+CIFAR10
+CIFAR100
+ResNet18
+ResNet34
+ResNet18
+ResNet34
+Strategy
+#Layers
+ACC[%]
+#OPs
+Strategy
+#Layers
+ACC[%]
+#OPs
+Strategy
+#Layers
+ACC[%]
+#OPs
+Strategy
+#Layers
+ACC[%]
+#OPs
+PTQ
+1
+91.6
+128.25M
+PTQ
+1
+93.6
+128.25M
+PTQ
+1
+74.0
+128.39M
+PTQ
+1
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+4
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+4
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+PTQ
++Gradient
+Filter
+R2
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+8.18M
+PTQ
++Gradient
+Filter
+R2
+1
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+8.18M
+PTQ
++Gradient
+Filter
+R2
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++Gradient
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++Gradient
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++Gradient
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+PTQ
++Gradient
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++Gradient
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++Gradient
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+9.03M
+4
+75.5
+9.04M
+4
+79.6
+9.17M
+Table 12. Experimental results for ResNet18 and ResNet34 with different gradient quantization methods (i.e., PTQ [4] and PSQ [7]) and
+hyper-parameter selections on CIFAR10/100. Feature map, activation, weight and gradient are quantized to INT8. “ACC” is short for
+accuracy. “#Layers” is short for “the number of active convolution layers”. For example. #Layers equals to 2 means that the last two
+convolutional layers are trained. “Gradient Filter R2/4/7” use proposed gradient filtering method with patch size 2 × 2, 4 × 4 and 7 × 7,
+respectively.
+19
+

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+Asymptotic decay function of the stationary tail probabilities along
+an arbitrary direction in a two-dimensional discrete-time QBD
+process
+Toshihisa Ozawa
+Faculty of Business Administration, Komazawa University
+1-23-1 Komazawa, Setagaya-ku, Tokyo 154-8525, Japan
+E-mail: [email protected]
+Abstract
+We deal with a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD pro-
+cess for short) on Z2
++ ×S0, where S0 is a finite set, and consider a topic remaining unresolved in
+our previous paper. In that paper, the asymptotic decay rate of the stationary tail probabilities
+along an arbitrary direction has been obtained. It has also been clarified that if the asymptotic
+decay rate ξc, where c is a direction vector in N2, is less than a certain value θmax
+c
+, the sequence
+of the stationary tail probabilities along the direction c geometrically decays without power
+terms, asymptotically. In this article, we give the function that the sequence asymptotically
+decays according to when ξc = θmax
+c
+, but it contains an unknown parameter. To determine the
+value of the parameter is a next challenge.
+Keywards: quasi-birth-and-death process, Markov modulated reflecting random walk, Markov
+additive process, asymptotic decay rate, asymptotic decay function, stationary distribution, ma-
+trix analytic method
+Mathematics Subject Classification: 60J10, 60K25
+1
+Introduction
+We deal with a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for
+short) {Y n} = {(Xn, Jn)} on Z2
++ × S0, where S0 is a finite set. This model is a Markov modulated
+reflecting random walk (MMRRW for short) whose transitions are skip free, and the MMRRW is
+a kind of reflecting random walk (RRW for short) with a background process, where the transition
+probabilities of the RRW vary depending on the state of the background process. One-dimensional
+QBD processes have been introduced by Macel Neuts and studied in the literature as one of the
+essential stochastic models in the queueing theory (see, for example, [1, 5, 7, 8]). The 2d-QBD
+process is a two-dimensional version of one-dimensional QBD process, and it enable us to analyze,
+for example, two-node queueing networks and two-node polling models.
+Assume the 2d-QBD process {Y n} is positive recurrent and denote by ν = (ν(x,j); (x, j) ∈
+Z2
++ × S0) the stationary distribution, where ν(x,j) is the stationary probability that the process
+is in the state (x, j). Our interest is asymptotics of the stationary distribution ν, especially, tail
+asymptotics in an arbitrary direction. Let an integer vector c = (c1, c2) be nonzero and nonnegative.
+Two typical objects of our study are the asymptotic decay rate ξc and asymptotic decay function
+hc(k) defined as, for j ∈ S0,
+ξc = − lim
+k→∞
+1
+k log ν(kc,j),
+lim
+k→∞
+ν(kc,j)
+hc(k) = gj,
+1
+arXiv:2301.02434v1  [math.PR]  6 Jan 2023
+
+where gj is a positive constant.
+Under a certain condition, the asymptotic decay rate of the
+probability sequence {νx+kc,j; k ≥ 0} does not depend on x and j if it exists, see Proposition 2.3
+of Ozawa [14]. In the case where c = (1, 0) or c = (0, 1), the asymptotic decay rate ξc has been
+obtained in Ozawa [10], see Corollary 4.3 of [14], and the asymptotic decay function hc(k) in Ozawa
+and Kobayashi [11], see Theorem 2.1 of [11]. The results in the case where c = (c, 0) or c = (0, c)
+for c ≥ 2 are automatically obtained from those in [10, 11]. In the case where c = (c1, c2) ≥ (1, 1),
+the asymptotic decay rate ξc has been obtained in Ozawa [14], see Theorem 3.2 of [14]. A condition
+ensuring the asymptotic decay function is given by hc(k) = e−ξck, an exponential function without
+a power term, has also been given in the theorem.
+In this article, we give the expression of the asymptotic decay function hc(k) when c = (c1, c2) ≥
+(1, 1).
+To this end, we clarify the analytic properties of the vector generating function of the
+stationary probabilities along the direction c, ϕc(z).
+The point z = eξc is a singular point of
+the vector function ϕc(z), and if ξc is equal to a certain value θmax
+c
+, z = eθmax
+c
+is a branch point
+of ϕc(z) with order one. From this result, we obtain the expression of hc(k), but it contains an
+unknown parameter.
+To determine the value of the parameter, it suffices to prove that ϕc(z)
+diverges elementwise at z = eθmax
+c
+. It seems to be a hard work and we leave it as a next challenge.
+We also generalize a part of existing results. One crucial point in analyzing the asymptotic decay
+function is how to analytically extend the G-matrix function appeared in the vector generating
+function of the stationary probabilities. In [11], it has been done under the assumption that all
+the eigenvalues of the G-matrix function are distinct, see Assumption 4.1 and Lemma 4.5 of [11].
+This assumption is not easy to verify in general. We, therefore, remove the assumption and give a
+general formula of the Jordan decomposition of the G-matrix function, see Section 3.1.
+The rest of the article is organized as follows. In Section 2, we describe the 2d-QBD process in
+detail and state assumptions and main results. In Section 3, an analytic extension of the G-matrix
+function is given in a general setting. The definition of G-matrix in the reverse direction and its
+properties are also given in the same section. They are used in the following section. The proof
+of the main results is given in Sections 4, where we demonstrate that the vector function ϕc(z)
+is elementwise analytic in the open disk with radius eξc + ε for some ε > 0, except for the point
+z = eξc, and clarify its singularity at the point z = eξc. The asymptotic decay function is obtained
+from those results. The paper concludes with some remarks in Section 5.
+2
+Model description and main results
+2.1
+Model description
+We consider the same model as that described in [14] and use the same notation.
+Denote by I2 the set of all the subsets of {1, 2}, i.e., I2 = {∅, {1}, {2}, {1, 2}}, and we use it
+as an index set. Divide Z2
++ into 22 = 4 exclusive subsets defined as
+Bα = {x = (x1, x2) ∈ Z2
++; xi > 0 for i ∈ α, xi = 0 for i ∈ {1, 2} \ α}, α ∈ I2.
+Let {Y n} = {(Xn, Jn)} be a 2d-QBD process on S = Z2
++ × S0, where S0 = {1, 2, ..., s0}. Let P be
+the transition probability matrix of {Y n} and represent it in block form as P =
+�
+Px,x′; x, x′ ∈ Z2
++
+�
+,
+where Px,x′ = (p(x,j),(x′,j′); j, j′ ∈ S0) and p(x,j),(x′,j′) = P(Y 1 = (x′, j′) | Y 0 = (x, j)). For α ∈ I2
+and i1, i2 ∈ {−1, 0, 1}, let Aα
+i1,i2 be a one-step transition probability block from a state in Bα, where
+we assume the blocks corresponding to impossible transitions are zero (see Fig. 1). Since the level
+process is skip free, for every x, x′ ∈ Z2
++, Px,x′ is given by
+Px,x′ =
+� Aα
+x′−x,
+if x ∈ Bα for some α ∈ I2 and x′ − x ∈ {−1, 0, 1}2,
+O,
+otherwise.
+(2.1)
+We assume the following condition throughout the paper.
+2
+
+Figure 1: Transition probability blocks
+Assumption 2.1. The 2d-QBD process {Y n} is irreducible and aperiodic.
+Next, we define several Markov chains derived from the 2d-QBD process. For a nonempty set
+α ∈ I2, let {Y α
+n} = {(Xα
+n, Jα
+n )} be a process derived from the 2d-QBD process {Y n} by removing
+the boundaries that are orthogonal to the xi-axis for each i ∈ α. The process {Y {1}
+n } is a Markov
+chain on Z × Z+ × S0 whose transition probability matrix P {1} = (P {1}
+x,x′; x, x′ ∈ Z × Z+) is given
+as
+P {1}
+x,x′ =
+�
+�
+�
+�
+�
+A{1}
+x′−x,
+if x ∈ Z × {0} and x′ − x ∈ {−1, 0, 1} × {0, 1},
+A{1,2}
+x′−x,
+if x ∈ Z × N and x′ − x ∈ {−1, 0, 1}2,
+O,
+otherwise,
+(2.2)
+where N is the set of all positive integers. The process {Y {2}
+n } on Z+ × Z × S0 and its transition
+probability matrix P {2} = (P {2}
+x,x′; x, x′ ∈ Z+ × Z) are analogously defined. The process {Y {1,2}
+n
+}
+is a Markov chain on Z2 × S0, whose transition probability matrix P {1,2} = (P {1,2}
+x,x′ ; x, x′ ∈ Z2) is
+given as
+P {1,2}
+x,x′ =
+�
+A{1,2}
+x′−x,
+if x′ − x ∈ {−1, 0, 1}2,
+O,
+otherwise.
+(2.3)
+Regarding X{1}
+1,n as the additive part, we see that the process {Y {1}
+n } = {(X{1}
+1,n , (X{1}
+2,n , J{1}
+n
+))} is
+a Markov additive process (MA-process for short) with the background state (X{1}
+2,n , J{1}
+n
+) (with
+respect to MA-processes, see, for example, Ney and Nummelin [9]).
+The process {Y {2}
+n } =
+{(X{2}
+2,n , (X{2}
+1,n , J{2}
+n
+))} is also an MA-process, where X{2}
+2,n is the additive part and (X{2}
+1,n , J{2}
+n
+) the
+background state, and {Y {1,2}
+n
+} = {(X{1,2}
+1,n
+, X{1,2}
+2,n
+), J{1,2}
+n
+)} an MA-process, where (X{1,2}
+1,n
+, X{1,2}
+2,n
+)
+the additive part and J{1,2}
+n
+the background state. We call them the induced MA-processes de-
+rived from the original 2d-QBD process. Let { ¯A{1}
+i
+; i ∈ {−1, 0, 1}} be the Markov additive kernel
+(MA-kernel for short) of the induced MA-process {Y {1}
+n }, which is the set of transition probability
+blocks and defined as, for i ∈ {−1, 0, 1},
+¯A{1}
+i
+=
+�
+¯A{1}
+i,(x2,x′
+2); x2, x′
+2 ∈ Z+
+�
+,
+¯A{1}
+i,(x2,x′
+2) =
+�
+�
+�
+�
+�
+A{1}
+i,x′
+2−x2,
+if x2 = 0 and x′
+2 − x2 ∈ {0, 1},
+A{1,2}
+i,x′
+2−x2,
+if x2 ≥ 1 and x′
+2 − x2 ∈ {−1, 0, 1},
+O,
+otherwise.
+3
+
+X2 ^
+B(2]
+B(1,2]
+[2]
+[1,2]
+12
+i1,i2
+17
+,12
+B(1)
+Bo
+0
+x1Let { ¯A{2}
+i
+; i ∈ {−1, 0, 1}} be the MA-kernel of {Y {2}
+n }, defined in the same manner. With respect to
+{Y {1,2}
+n
+}, the MA-kernel is given by {A{1,2}
+i1,i2 ; i1, i2 ∈ {−1, 0, 1}}. We assume the following condition
+throughout the paper.
+Assumption 2.2. The induced MA-processes {Y {1}
+n }, {Y {2}
+n } and {Y {1,2}
+n
+} are irreducible and
+aperiodic.
+According to [14], we assume several other technical conditions for the induced MA-process
+{Y {1,2}
+n
+}, concerning irreducibility and aperiodicity on subspaces. Let {Y +
+n } = {(X+
+n , J+
+n )} be a
+lossy Markov chain derived from the induced MA-process {Y {1,2}
+n
+} by restricting the state space
+of the additive part to N2. The process {Y +
+n } is a Markov chain on N2 × S0 whose transition
+probability matrix P + is given as P + = (P {1,2}
+x,x′ ; x, x′ ∈ N2), where P + is strictly substochastic.
+The process {Y +
+n } is also a lossy Markov chain derived from the original 2d-QBD process {Y n}
+by restricting the state space of the level to N2. We assume the following condition throughout the
+paper.
+Assumption 2.3. {Y +
+n } is irreducible and aperiodic.
+For k ∈ Z, let Z≤k and Z≥k be the set of integers less than or equal to k and that of integers
+greater than or equal to k, respectively. We also assume the following condition throughout the
+paper. For what this assumption implies, see Remark 3.1 of [14].
+Assumption 2.4.
+(i) The lossy Markov chain derived from the induced MA-process {Y {1,2}
+n
+} by
+restricting the state space to Z≤0 × Z≥0 × S0 is irreducible and aperiodic.
+(ii) The lossy Markov chain derived from {Y {1,2}
+n
+} by restricting the state space to Z≥0×Z≤0×S0
+is irreducible and aperiodic.
+The stability condition of the 2d-QBD process has already been obtained in [12]. Let a{1}, a{2}
+and a{1,2} = (a{1,2}
+1
+, a{1,2}
+2
+) be the mean drifts of the additive part in the induced MA-processes
+{Y {1}
+n }, {Y {2}
+n } and {Y {1,2}
+n
+}, respectively. By Corollary 3.1 of [12], the stability condition of the
+2d-QBD process {Y n} is given as follows:
+Lemma 2.1.
+(i) In the case where a{1,2}
+1
+< 0 and a{1,2}
+2
+< 0, the 2d-QBD process {Y n} is
+positive recurrent if a{1} < 0 and a{2} < 0, and it is transient if either a{1} > 0 or a{2} > 0.
+(ii) In the case where a{1,2}
+1
+≥ 0 and a{1,2}
+2
+< 0, {Y n} is positive recurrent if a{1} < 0, and it is
+transient if a{1} > 0.
+(iii) In the case where a{1,2}
+1
+< 0 and a{1,2}
+2
+≥ 0, {Y n} is positive recurrent if a{2} < 0, and it is
+transient if a{2} > 0.
+(iv) If one of a{1,2}
+1
+and a{1,2}
+2
+is positive and the other is non-negative, then {Y n} is transient.
+For the explicit expression of the mean drifts, see Section 3.1 of [12] and its related parts. We
+assume the following condition throughout the paper.
+Assumption 2.5. The condition in Lemma 2.1 that ensures the 2d-QBD process {Y n} is positive
+recurrent holds.
+Denote by ν the stationary distribution of {Y n}, where ν = (νx, x ∈ Z2
++), νx = (ν(x,j), j ∈ S0)
+and ν(x,j) is the stationary probability that the 2d-QBD process is in the state (x, j).
+4
+
+Figure 2: Domains Γ{1,2}, Γ{1} and Γ{2}
+2.2
+Main results
+Let ¯A{1}
+∗
+(z) and ¯A{2}
+∗
+(z) be the matrix generating functions of the MA-kernels of {Y {1}
+n } and
+{Y {2}
+n }, respectively, defined as
+¯A{1}
+∗
+(z) =
+�
+i∈{−1,0,1}
+zi ¯A{1}
+i
+,
+¯A{2}
+∗
+(z) =
+�
+i∈{−1,0,1}
+zi ¯A{2}
+i
+.
+The matrix generating function of the MA-kernel of {Y {1,2}
+n
+} is given by A{1,2}
+∗,∗
+(z1, z2), defined as
+A{1,2}
+∗,∗
+(z1, z2) =
+�
+i1,i2∈{−1,0,1}
+zi1
+1 zi2
+2 A{1,2}
+i1,i2 .
+Let Γ{1}, Γ{2} and Γ{1,2} be regions in which the convergence parameters of ¯A{1}
+∗
+(eθ1), ¯A{2}
+∗
+(eθ2)
+and A{1,2}
+∗,∗
+(eθ1, eθ2) are greater than 1, respectively, i.e.,
+Γ{1} = {(θ1, θ2) ∈ R2; cp( ¯A{1}
+∗
+(eθ1)) > 1},
+Γ{2} = {(θ1, θ2) ∈ R2; cp( ¯A{2}
+∗
+(eθ2)) > 1},
+Γ{1,2} = {(θ1, θ2) ∈ R2; cp(A{1,2}
+∗,∗
+(eθ1, eθ2)) > 1},
+where, for a nonnegative square matrix A with a finite or countable dimension, cp(A) denote the
+convergence parameter of A, i.e., cp(A) = sup{r ∈ R+; �∞
+n=0 rnAn < ∞, entry-wise}. We have
+cp(A{1,2}
+∗,∗
+(eθ1, eθ2)) = spr(A{1,2}
+∗,∗
+(eθ1, eθ2))−1, where for a square complex matrix A, spr(A) is the
+spectral radius of A. By Lemma A.1 of Ozawa [13], cp( ¯A{1}
+∗
+(eθ))−1 and cp( ¯A{2}
+∗
+(eθ))−1 are log-
+convex in θ, and the closures of Γ{1} and Γ{2} are convex sets; spr( ¯A{1,2}
+∗
+(eθ1, eθ2)) is also log-convex
+in (θ1, θ2), and the closure of Γ{1,2} is a convex set. Furthermore, by Proposition B.1 of Ozawa
+[13], Γ{1,2} is bounded under Assumption 2.2. We depict an example of the domains Γ{1,2}, Γ{1}
+and Γ{2} in Fig. 2.
+We define several extreme values and several functions with respect to the domains. For i ∈
+{1, 2}, define θmin
+i
+and θmax
+i
+as
+θmin
+i
+= inf{θi ∈ R : (θ1, θ2) ∈ Γ{1,2}},
+θmax
+i
+= sup{θi ∈ R : (θ1, θ2) ∈ Γ{1,2}},
+and for a direction vector c = (c1, c2) ∈ N2, θmax
+c
+as
+θmax
+c
+= sup{c1θ1 + c2θ2 : (θ1, θ2) ∈ Γ{1,2}}.
+For θ1 ∈ [θmin
+1
+, θmax
+1
+], there exist two real solutions to equation spr(A{1,2}
+∗,∗
+(eθ1, eθ2)) = 1, counting
+multiplicity. Denote them by θ2 = η2(θ1) and θ2 = ¯η2(θ1), respectively, where η2(θ1) ≤ ¯η2(θ1). For
+5
+
+C101 + C202 = 0max
+02 个
+01 = 0
+02 = 02
+r(1)
+spr
+amax
+n2 (0)
+r(2]
+r(1,2]
+n2(0)
+>
+0
+0
+1
+0
+01
+0Figure 3: Classification
+θ2 ∈ [θmin
+2
+, θmax
+2
+], also denote by θ1 = η1(θ2) and θ1 = ¯η1(θ2) the two real solutions to the equation
+spr(A{1,2}
+∗,∗
+(eθ1, eθ2)) = 1, where η1(θ2) ≤ ¯η1(θ2). For i ∈ {1, 2}, define θ∗
+i as
+θ∗
+i = sup{θi ∈ R : (θ1, θ2) ∈ Γ{i}}.
+For another characterization of θ∗
+i , see Proposition 3.7 of Ozawa [10], where θ∗
+i is denoted by z0.
+In terms of these points and functions, we geometrically classify the model into four types
+according to Section 4.1 of [14]. Define two points Q1 and Q2 as Q1 = (θ∗
+1, ¯η2(θ∗
+1)) and Q2 =
+(¯η1(θ∗
+2), θ∗
+2), respectively. Using these points, we define the following classification (see Fig. 3).
+Type 1: θ∗
+1 ≥ ¯η1(θ∗
+2) and ¯η2(θ∗
+1) ≤ θ∗
+2,
+Type 2: θ∗
+1 < ¯η1(θ∗
+2) and ¯η2(θ∗
+1) > θ∗
+2,
+Type 3: θ∗
+1 ≥ ¯η1(θ∗
+2) and ¯η2(θ∗
+1) > θ∗
+2,
+Type 4: θ∗
+1 < ¯η1(θ∗
+2) and ¯η2(θ∗
+1) ≤ θ∗
+2.
+By Proposition 2.3 of [14], for any direction vector c = (c1, c2) ∈ N2, the asymptotic decay
+rate in the direction c is space homogeneous. Hence, we denote it by ξc, which satisfies, for any
+(x, j) ∈ Z2
++ × S0,
+ξc = − lim
+k→∞
+1
+n log ν(x+kc.j).
+(2.4)
+The asymptotic decay rate ξc has already been obtained in [14], and as described in Section 4.1 of
+[14], it is given as follows.
+Theorem 2.1. Let c = (c1, c2) be an arbitrary direction vector in N2.
+Type 1:
+ξc =
+�
+�
+�
+c1θ∗
+1 + c2¯η2(θ∗
+1)
+if − c1
+c2 < ¯η′
+2(θ∗
+1),
+θmax
+c
+if ¯η′
+2(θ∗
+1) ≤ − c1
+c2 ≤ ¯η′
+1(θ∗
+2)−1,
+c1¯η1(θ∗
+2) + c2θ∗
+2
+if − c1
+c2 > ¯η′
+1(θ∗
+2)−1,
+where ¯η′
+2(x) =
+d
+dx ¯η2(x) and ¯η′
+1(x) =
+d
+dx ¯η1(x).
+Type 2:
+ξc =
+�
+�
+�
+c1θ∗
+1 + c2¯η2(θ∗
+1)
+if − c1
+c2 ≤ θ∗
+2−¯η2(θ∗
+1)
+¯η1(θ∗
+2)−θ∗
+1 ,
+c1¯η1(θ∗
+2) + c2θ∗
+2
+if − c1
+c2 > θ∗
+2−¯η2(θ∗
+1)
+¯η1(θ∗
+2)−θ∗
+1 .
+6
+
+A.
+0*
+02
+r(1,2]
+>
+01
+0* 1
+1
+Type 1
+Type 2
+Type 3
+Type 4Type 3: ξc = c1¯η1(θ∗
+2) + c2θ∗
+2.
+Type 4: ξc = c1θ∗
+1 + c2¯η2(θ∗
+1).
+The asymptotic decay function hc(k) in the direction c is defined as the function that satisfies,
+for some positive vector gc,
+lim
+k→∞
+νkc
+hc(k) = gc.
+(2.5)
+It is given as follows.
+Theorem 2.2. Let c be an arbitrary direction vector in N2.
+hc(k) =
+�
+k− 1
+2 (2l−1)e−ξck
+if ¯η′
+2(θ∗
+1) < − c1
+c2 < ¯η′
+1(θ∗
+2)−1 in Type 1,
+e−ξck
+otherwise,
+as k → ∞.
+(2.6)
+where l is some positive integer.
+Except for the case where ¯η′
+2(θ∗
+1) ≤ − c1
+c2 ≤ ¯η′
+1(θ∗
+2)−1 in Type 1, Theorem 2.2 has already been
+proved in [14], see Theorem 3.2 of [14].
+Hence, to this end, it suffices to prove the following
+proposition.
+Proposition 2.1. Assume Type 1 and set c = (c1, c2) = (1, 1).
+Then, the asymptotic decay
+function hc(k) is given as
+hc(k) =
+�
+k− 1
+2 (2l−1)e−θmax
+c
+k
+if ¯η′
+2(θ∗
+1) < − c1
+c2 = −1 < ¯η′
+1(θ∗
+2)−1,
+e−θmax
+c
+k
+if ¯η′
+2(θ∗
+1) = −1 or ¯η′
+1(θ∗
+2) = −1,
+(2.7)
+where l is some positive integer.
+From this proposition, we can obtain the same result for a general direction vector c ∈ N2, by
+using the block state process derived from the original 2d-QBD process; See Section 3.3 of [14].
+We, therefore, prove the proposition in Section 4.
+Remark 2.1. From the corresponding results for a 2d-RRW without a background process obtained
+in Malyshev [6], it is expected that the value of l in Theorem 2.2 is one, i.e., hc(k) = k− 1
+2 e−ξck.
+3
+Preliminaries
+Let z and w be complex valuables unless otherwise stated. For a positive number r, denote by ∆r
+the open disk of center 0 and radius r on the complex plain, and ∂∆r the circle of the same center
+and radius. We denote by ¯∆r the closure of ∆r. For a, b ∈ R+ such that a < b, let ∆a,b be an open
+annular domain on C defined as ∆a,b = {z ∈ C : a < |z| < b}. We denote by ¯∆a,b the closure of
+∆a,b. For r > 0, ε > 0 and θ ∈ [0, π/2), define
+˜∆r(ε, θ) = {z ∈ C : |z| < r + ε, z ̸= r, | arg(z − r)| > θ}.
+For r > 0, we denote by “ ˜∆r ∋ z → r” that ˜∆r(ε, θ) ∋ z → r for some ε > 0 and some θ ∈ [0, π/2).
+In the rest of the paper, instead of proving that a function f(z) is analytic in ˜∆r(ε, θ) for some ε > 0
+and θ ∈ [0, π/2), we often demonstrate that the function f(z) is analytic in ∆r and on ∂∆r \ {r}.
+In order to give general results, this section is described independently from other parts of the
+article.
+7
+
+3.1
+Analytic extension of a G-matrix function
+First, we define a G-matrix function according to Ozawa and Kobayashi [11]. For i, j ∈ {−1, 0, 1},
+let Ai,j be a substochastic matrix with a finite dimension s0, and define the following matrix
+functions:
+A∗,j(z) =
+�
+i∈{−1,0,1}
+ziAi,j, j = −1, 0, 1,
+A∗,∗(z, w) =
+�
+i,j∈{−1,0,1}
+ziwjAi,j.
+We assume the following condition throughout this subsection.
+Assumption 3.1. A∗,∗(1, 1) is stochastic.
+Let χ(z, w) be the spectral radius of A∗.∗(z, w), i.e., χ(z, w) = spr(A∗,∗(z, w)), and Γ be a
+domain on R2 defined as
+Γ = {(θ1, θ2) ∈ R2 : χ(eθ1, eθ2) < 1}.
+We assume the following condition throughout this subsection.
+Assumption 3.2. The Markov modulated random walk on Z2 × {1, 2, ..., s0} that is governed by
+{Ai,j; i, j ∈ {−1, 0, 1}} is irreducible and aperiodic.
+Under this assumption, A∗,∗(1, 1) is also irreducible and aperiodic. Furthermore, by Lemma 2.2
+of [11], Γ is bounded. Since χ(eθ1, eθ2) is convex in (θ1, θ2) ∈ R2, the closure of Γ is a convex set.
+Define extreme points θmin
+1
+and θmax
+2
+as follows:
+θmin
+1
+=
+inf
+(θ1,θ2)∈Γ θ1,
+θmax
+1
+=
+sup
+(θ1,θ2)∈Γ
+θ1.
+For θ1 ∈ [θmin
+1
+, θmax
+1
+], let θ2(θ1) and ¯θ2(θ1) be the two real solutions to equation χ(eθ1, eθ2) = 1,
+counting multiplicity, where θ2(θ1) ≤ ¯θ2(θ1). We set zmin
+1
+= eθmin
+1
+and zmax
+1
+= eθmax
+1
+. For n ≥ 1,
+define the following set of index sequences:
+In =
+�
+i(n) ∈ {−1, 0, 1}n :
+k
+�
+l=1
+il ≥ 0 for k ∈ {1, 2, ..., n − 1} and
+n
+�
+l=1
+il = −1
+�
+,
+where i(n) = (i1, i2, ..., in), and define the following matrix function:
+Dn(z) =
+�
+i(n)∈In
+A∗,i1(z)A∗,i2(z) · · · A∗,in(z).
+Define a matrix function G(z) as
+G(z) =
+∞
+�
+n=1
+Dn(z).
+By Lemma 4.1 of [11], this matrix series absolutely converges entry-wise in z ∈ ¯∆zmin
+1
+,zmax
+1
+. We call
+this G(z) the G-matrix function generated from {Ai,j; i, j ∈ {−1, 0, 1}}. For z ∈ ¯∆zmin
+1
+,zmax
+1
+, G(z)
+satisfies the inequality |G(z)| ≤ G(|z|) and the following matrix quadratic equation:
+A∗,−1(z) + A∗,0(z)G(z) + A∗,1(z)G(z)2 = G(z).
+(3.1)
+Furthermore, for z ∈ [zmin
+1
+, zmax
+1
+], it is the minimum nonnegative solution to equation (3.1). Hence,
+G(z) is an extension of a usual G-matrix in the queueing theory; see, for example, [7]. By Proposi-
+tion 2.5 of [11], we see that, for z ∈ [zmin
+1
+, zmax
+1
+], the Perron-Frobenius eigenvalue of G(z) is given
+by eθ2(log z), i.e., spr(G(z)) = eθ2(log z). By Lemma 4.1 of [11], G(z) satisfies
+I − A∗,∗(z, w) = w−1 (I − A∗,0(z) − wA∗,1(z) + A∗,1(z)G(z)) (wI − G(z)).
+(3.2)
+By Lemma 4.2 of [11], the following property holds true for G(z).
+8
+
+Lemma 3.1. G(z) is entry-wise analytic in the open annular domain ∆zmin
+1
+,zmax
+1
+.
+We give the eigenvalues of G(z) according to [11]. Note that our final aim in this subsection is
+to give an analytic extension of G(z) through its Jordan canonical form without assuming all the
+eigenvalues of G(z) are distinct. On the other hand, in [11], the eigenvalues were assumed to be
+distinct. Define a matrix function L(z, w) as
+L(z, w) = zw(I − A∗,∗(z, w)).
+Each entry of L(z, w) is a polynomial in z and w with at most degree 2 for each variable. We use
+a notation Ξ, defined as follows. Let f(z, w) be an irreducible polynomial in z and w and assume
+its degree with respect to w is m ≥ 1. Let a(z) be the coefficient of wm in f(z, w). Define a point
+set Ξ(f) as
+Ξ(f) = {z ∈ C : a(z) = 0 or (f(z, w) = 0 and fw(z, w) = 0 for some w ∈ C)},
+where fw(z, w) = (∂/∂w)f(z, w). Each point in Ξ(f) is an algebraic singularity of the algebraic
+function w = α(z) defined by polynomial equation f(z, w) = 0. For each point z ∈ C \ Ξ(f),
+f(z, w) = 0 has just m distinct solutions, which correspond to the m branches of the algebraic
+function. Let φ(z, w) be a polynomial in z and w defined as
+φ(z, w) = det L(z, w)
+and mφ its degree with respect to w, where s0 ≤ mφ ≤ 2s0. Let α1(z), α2(z), ..., αmφ(z) be the
+mφ branches of the algebraic function w = α(z) defined by the polynomial equation φ(z, w) = 0,
+counting multiplicity. We number the brunches so that they satisfy the following:
+(1) For every z ∈ ¯∆zmin
+1
+,zmax
+1
+and for every k ∈ {1, 2, ..., s0}, |αk(z)| ≤ eθ2(log |z|).
+(2) For every z ∈ ¯∆zmin
+1
+,zmax
+1
+and for every k ∈ {s0 + 1, s0 + 2, ..., mφ}, |αk(z)| ≥ e¯θ2(log |z|).
+(3) For every z ∈ [zmin
+1
+, zmax
+1
+], αs0(z) = eθ2(log z) and αs0+1(z) = e¯θ2(log z).
+This is possible by Lemma 4.3 of [11]. By Lemmas 4.3 and 4.4 of [11], the G-matrix function of
+G(z) satisfies the following property.
+Lemma 3.2. For every z ∈ ¯∆zmin
+1
+,zmax
+1
+, the eigenvalues of G(z) are given by α1(z), α2(z), ...,
+αs0(z).
+Without loss of generality, we assume that, for some nφ ∈ N and l1, l2, ..., lnφ ∈ N, the polynomial
+φ(z, w) is factorized as
+φ(z, w) = f1(z, w)l1f2(z, w)l2 · · · fnφ(z, w)lnφ,
+(3.3)
+where fk(z, w), k = 1, 2, ..., nφ, are irreducible polynomials in z and w and they are relatively
+prime. Since the field of coefficients of polynomials is C, this factorization is unique. For every
+k ∈ {1, 2, ..., mφ}, αk(z) is a branch of the algebraic function w = α(z) defined by the polynomial
+equation fn(z, w) = 0 for some n ∈ {1, 2, ..., nφ}. We denote such n by q(k), i.e., fq(k)(z, αk(z)) = 0.
+Since αs0(z) is the Perron-Frobenius eigenvalue of G(z) when z ∈ [zmin
+1
+, zmax
+1
+], the multiplicity of
+αs0(z) is one and we have lq(s0) = 1. Define a point set E1 as
+E1 =
+nφ
+�
+n=1
+Ξ(fn).
+9
+
+Since, for every n, the polynomial fn(z, w) is irreducible and not identically zero, the point set E1
+is finite. Every branch αk(z) is analytic in C \ E1. Define a point set E2 as
+E2 = {z ∈ C \ E1 : fn(z, w) = fn′(z, w) = 0
+for some n, n′ ∈ {1, 2, ..., nφ} such that n ̸= n′ and for some w ∈ C}.
+Since, for any n, n′ such that n ̸= n′, fn(z, w) and fn′(z, w) are relatively prime, the point set E2
+is finite. Note that every branch αk(z) is analytic in a neighborhood of any z0 ∈ E2. For every
+k ∈ {1, 2, ..., mφ} and for every z ∈ C \ (E1 ∪ E2), the multiplicity of αk(z) as a zero of det L(z, w)
+is equal to lq(k). This means that, for every z ∈ ¯∆zmin
+1
+,zmax
+1
+\ (E1 ∪ E2), the multiplicity of the
+eigenvalue αk(z) of G(z) is lq(k), which does not depend on z. Define a positive integer m0 as
+m0 =
+s0
+�
+k=1
+1
+lq(k)
+.
+This m0 is the number of different branches in {αi(z) : i = 1, 2, ..., s0} when z ∈ C \ (E1 ∪ E2).
+Denote the different branches by ˇαk(z), k = 1, 2, ..., m0, so that ˇαm0(z) = αs0(z). Instead of using
+q(k), we define a function ˇq(k) so that lˇq(k) indicates the multiplicity of ˇαk(z) when z ∈ C\(E1∪E2).
+We always have lˇq(m0) = 1.
+We give the Jordan normal form of G(z). Define a domain Ω as Ω = ∆zmin
+1
+,zmax
+1
+\ (E1 ∪ E2). For
+k ∈ {1, 2, ..., m0} and for i ∈ {1, 2, ..., lˇq(k)}, define a positive integer tk,i as
+tk,i = min
+z∈Ω dim Ker (ˇαk(z)I − G(z))i
+and a point set Gk,i as
+Gk,i = {z ∈ Ω : dim Ker (ˇαk(z)I − G(z))i > tk,i}.
+Since ˇαk(z) and G(z) are analytic in Ω, we see from the proof of Theorem S6.1 of [3] that each Gk,i
+is an empty set or a set of discrete complex numbers. For k ∈ {1, 2, ..., m0} and i ∈ {1, 2, ..., lˇq(k)},
+define a nonnegative integer sk,i as
+sk,i = 2tk,i − tk,i+1 − tk,i−1,
+where tk,0 = 0 and tk,lˇq(k)+1 = lˇq(k). For k ∈ {1, 2, ..., m0}, define a positive integer mk,0 and point
+set EG
+k as
+mk,0 = tk,1,
+EG
+k =
+lˇq(k)
+�
+i=1
+Gk,i.
+When z ∈ Ω \ EG
+k , this mk,0 is the number of Jordan blocks of G(z) with respect to the eigenvalue
+ˇαk(z) and, for i ∈ {1, 2, ..., lˇq(k)}, sk,i is the number of Jordan blocks whose dimension is i. Hence,
+the Jordan normal form of G(z) takes a common form in z ∈ Ω \ �m0
+k=1 EG
+k . For k ∈ {1, 2, ..., m0}
+and for i ∈ {1, 2, ..., mk,0}, denote by mk,i the dimension of the i-th Jordan block of G(z) with
+respect to the eigenvalue ˇαk(z), where we number the Jordan blocks so that if i ≤ i′, mk,i ≥ mk,i′.
+For each k ∈ {1, 2, ..., m0}, they satisfy �mk,0
+i=1 mk,i = lˇq(k). Denote by Jn(λ) the n-dimensional
+Jordan block of eigenvalue λ. For z ∈ Ω \ �m0
+k=1 EG
+k , the Jordan normal form of G(z), JG(z), is
+given by
+JG(z) = diag(Jmk,i(ˇαk(z)), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0),
+(3.4)
+where mm0,0 = 1 and Jmm0,1(ˇαm0(z)) = αs0(z). Note that the matrix function JG(z) is defined on
+C and analytic in C \ E1. An analytic extension of G(z) is given by the following theorem.
+10
+
+Theorem 3.1. There exist vector functions:
+ˇvL
+k,i,j(z), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i,
+such that they are analytic in C \ E1 and satisfy for every z ∈ ∆zmin
+1
+,zmax
+1
+\ (E1 ∪ E0) that
+G(z) = T L(z)JG(z)(T L(z))−1,
+(3.5)
+where E0 is a set of discrete complex numbers and matrix function T L(z) is defined as
+T L(z) =
+�ˇvL
+k,i,j(z), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i
+�
+.
+Since the proof of Theorem 3.1 is elementary and very lengthy, we give it in Appendix A. In
+Theorem 3.1, {ˇvL
+k,i,j(z)} is the set of the generalized eigenvectors of G(z), but we denote them with
+superscript L since they are generated from the matrix function L(z, w); see Appendix A. Define a
+point set EL
+T as
+EL
+T = {z ∈ C \ E1 : det T L(z) = 0},
+which is an empty set or a set of discrete complex numbers since det T L(z) is not identically zero.
+Define a matrix function ˇG(z) as
+ˇG(z) = T L(z)JG(z)(T L(z))−1 = T L(z)JG(z) adj(T L(z))
+det(T L(z))
+.
+(3.6)
+Then, it is entry-wise analytic in C\(E1∪EL
+T ). By Theorem 3.1 and the identity theorem for analytic
+functions, this ˇG(z) is an analytic extension of the matrix function G(z). Hence, we denote ˇG(z)
+by G(z). By Lemma 3.1, G(z) is entry-wise analytic in ∆zmin
+1
+,zmax
+1
+. The following corollary asserts
+that G(z) is also analytic on the outside boundary of ∆zmin
+1
+,zmax
+1
+except for the point z = zmax
+1
+.
+Corollary 3.1. The extended G-matrix function G(z) is entry-wise analytic on ∂∆zmax
+1
+\ {zmax
+1
+}.
+Since this corollary can be proved in a manner similar to that used in the proof of Lemma 4.7
+of [11], we omit it.
+Denote by ˇuL
+m0,1,1(z) the last row of the matrix function (T L(z))−1, and define a diagonal
+matrix function Js0(z) as Js0(z) = diag
+�
+0
+· · ·
+0
+αs0(z)
+�
+, where αs0(z) = ˇαm0(z). Then, since
+mm0,0 = 1 and mm0,1 = 1, we obtain the following decomposition of G(z) from (3.6):
+G(z) = G†(z) + αs0(z)ˇvL
+m0,1,1(z)ˇuL
+m0,1,1(z),
+(3.7)
+where
+G†(z) = T L(z)(JG(z) − Js0(z))(T L(z))−1.
+By the definition, G(z) satisfies, for n ≥ 1,
+G(z)n = G†(z)n + αs0(z)nˇvL
+m0,1,1(z)ˇuL
+m0,1,1(z),
+(3.8)
+and G†(z), for z ∈ ¯∆zmin
+1
+,zmax
+1
+, spr(G†(z)) ≤ spr(G†(|z|) < spr(G(|z|)) = αs0(|z|). Furthermore,
+in a neighborhood of z = zmax
+1
+, we have spr(G†(z)) < αs0(zmax
+1
+). Since the point z = zmax
+1
+is a
+branch point of ˇαm0(z) (= αs0(z)), there exists a function ˜αs0(ζ) being analytic in a neighborhood
+of ζ = 0 and satisfying
+ˇαm0(z) = αs0(z) = ˜αs0((zmax
+1
+− z)
+1
+2 ).
+11
+
+Let ˜vs0(ζ) be a vector function satisfying
+L(zmax
+1
+− ζ2, ˜αs0(ζ))˜vs0(ζ) = 0,
+where ˜vs0(ζ) is elementwise analytic in a neighborhood of ζ = 0.
+Denote by ˜T(ζ) the matrix
+function given by replacing the last column of T L(zmax
+1
+− ζ2) with ˜vs0(ζ) and by ˜us0(ζ) the last
+row of ˜T(ζ)−1. By the definition, ˜T(ζ) as well as ˜us0(ζ) is entry-wise analytic in a neighborhood
+of ζ = 0. Define a diagonal matrix function ˜Js0(ζ) as ˜Js0(ζ) = diag
+�
+0
+· · ·
+0
+˜αs0(ζ)
+�
+. For later
+use, we give the following lemma.
+Lemma 3.3. There exists a matrix function ˜G(ζ) being entry-wise analytic in a neighborhood of
+ζ = 0 and satisfying G(z) = ˜G((zmax
+1
+−z)
+1
+2 ) in a neighborhood of z = zmax
+1
+. This ˜G(ζ) is represented
+as
+˜G(ζ) = ˜G†(ζ) + ˜αs0(ζ)˜vs0(ζ)˜us0(ζ),
+(3.9)
+where ˜G†(ζ) is a matrix function being entry-wise analytic in a neighborhood of ζ = 0 and satisfying
+G†(z) = ˜G†((zmax
+1
+− z)
+1
+2 ) in a neighborhood of z = zmax
+1
+. In a neighborhood of ζ = 0, spr( ˜G†(ζ)) <
+˜αs0(0) = αs0(zmax
+1
+).
+Proof. Give ˜G†(ζ) as
+˜G†(ζ) = ˜T(ζ)(JG(zmax
+1
+− ζ2) − Js0(zmax
+1
+− ζ2)) ˜T(ζ)−1.
+Then, by (3.7), we obtain the results of the lemma.
+The following limit with respect to αs0(z) (= ˇαm0(z)) is given by Proposition 5.5 of [11] (also
+see Lemma 10 of [4]).
+Lemma 3.4.
+lim
+˜∆zmax
+1
+∋z→zmax
+1
+αs0(zmax
+1
+) − αs0(z)
+(zmax
+1
+− z)
+1
+2
+= −αs0,1 =
+√
+2
+�
+−¯ζ1,w2(ζ2(zmax
+1
+))
+> 0,
+(3.10)
+where z = ¯ζ1(w) is the larger one of two real solutions to equation χ(z, w) = 1 and ¯ζ1,w2(w) =
+(d2/dw2) ¯ζ1(w).
+Let R(z) be the rate matrix function generated from {Ai,j; i, j = −1, 0, 1}; for the definition of
+R(z), see Section 4.1 of [11]. Define a matrix function N(z) as
+N(z) = (I − A∗,0(z) − A∗,1(z)G(z))−1.
+N(z) is well defined for every z ∈ ¯∆zmin
+1
+,zmax
+1
+. The extended G(z) satisfies the following property.
+Lemma 3.5.
+lim
+˜∆zmax
+1
+∋z→zmax
+1
+G(zmax
+1
+) − G(z)
+(zmax
+1
+− z)
+1
+2
+= −G1
+= −αs0,1N(zmax
+1
+)vR(zmax
+1
+)uG
+s0(zmax
+1
+) ≥ O, ̸= O,
+(3.11)
+where uG
+s0(zmax
+1
+) is the left eigenvector of G(zmax
+1
+) with respect to the eigenvalue eθ2(log zmax
+1
+) =
+αs0(zmax
+1
+), vR(zmax
+1
+) the right eigenvector of R(zmax
+1
+) with respect to the eigenvalue e−¯θ2(log zmax
+1
+) =
+e−θ2(log zmax
+1
+) and they satisfy uG
+s0(zmax
+1
+)N(zmax
+1
+)vR(zmax
+1
+) = 1.
+Since this lemma can be proved in a manner similar to that used in the proof of Proposition
+5.6 of [11], we omit it.
+12
+
+3.2
+G-matrix in the reverse direction and its properties
+Let A−1, A0 and A1 be square nonnegative matrices with a finite dimension. Define a matrix
+function A∗(z) and matrix Q as
+A∗(z) = z−1A−1 + A0 + zA1,
+(3.12)
+Q =
+�
+�
+�
+�
+�
+A0
+A1
+A−1
+A0
+A1
+A−1
+A0
+A1
+...
+...
+...
+�
+�
+�
+�
+� .
+(3.13)
+We assume:
+(a1) Q is irreducible.
+(a2) The infimum of the maximum eigenvalue of A∗(eθ) in θ ∈ R is less than or equal to 1, i.e.,
+infθ∈R spr(A∗(eθ)) ≤ 1.
+Then, there are two real solutions to equation cp(A∗(eθ)) = 1, counting multiplicity, see comments
+to Condition 2.6 in [13]. We denote the solutions by θ and ¯θ, where θ ≤ ¯θ. The rate matrix
+and G-matrix generated from the triplet {A−1, A0, A1} also exist; we denote them by R and G,
+respectively. R and G are the minimal nonnegative solutions to the following matrix quadratic
+equations:
+R = R2A−1 + RA0 + A1,
+(3.14)
+G = A−1 + A0G + A1G2.
+(3.15)
+We have
+I − A∗(z) = (I − zR)(I − H)(I − z−1G),
+(3.16)
+spr(R) = e−¯θ,
+spr(G) = eθ,
+(3.17)
+where H = A0 + A1G; see, for example, Lemma 2.2 of [13]. We define a rate matrix and G-matrix
+in the reverse direction generated from the triplet {A−1, A0, A1}, denoted by Rr and Gr, as the
+minimal nonnegative solutions to the following matrix quadratic equations:
+Rr = (Rr)2A1 + RrA0 + A−1,
+(3.18)
+Gr = A1 + A0Gr + A−1(Gr)2.
+(3.19)
+In other words, Rr and Gr are, respectively, the rate matrix and G-matrix generated from the
+triplet by exchanging A−1 and A1. Since z−1A1 + A0 + zA−1 = A∗(z−1), we obtain by (3.16) and
+(3.17) that
+I − A∗(z−1) = (I − zRr)(I − Hr)(I − z−1Gr),
+(3.20)
+spr(Rr) = eθ,
+spr(Gr) = e−¯θ,
+(3.21)
+where Hr = A0 + A−1Gr. We use the following property in the proof of Proposition 4.5.
+Lemma 3.6. Let v be the right eigenvector of G with respect to the eigenvalue eθ and vr that of
+Gr with respect to the eigenvalue e−¯θ, i.e., Gv = eθv and Grvr = e−¯θvr. If θ = ¯θ, we have v = vr,
+up to multiplication by a positive constant.
+Proof. By (3.16) and (3.20), we obtain
+A∗(eθ)v = v,
+A∗(e
+¯θ)vr = A∗(eθ)vr = vr.
+Since spr(A∗(eθ)) = 1 and A∗(eθ) is irreducible, the right eigenvector of A∗(eθ) with respect to the
+eigenvalue of 1 is unique, up to multiplication by a positive constant. This implies v = vr.
+13
+
+4
+Proof of Proposition 2.1
+4.1
+Methodology and outline of the proof
+Define the vector generating function of the stationary probabilities in direction c ∈ N2, ϕc(z), as
+ϕc(z) =
+∞
+�
+k=0
+zkνkc.
+Also define zmin
+c
+and zmax
+c
+as zmin
+c
+= eθmin
+c
+and zmax
+c
+= eθmax
+c
+, respectively.
+Hereafter, we set
+c = (1, 1). In order to obtain the asymptotic function of the stationary tail probability in the
+direction c = (1, 1), we apply the following lemma to the vector generating function ϕc(z).
+Lemma 4.1 (Theorem VI.4 of Flajolet and Sedgewick [2]). Let f be a generating function of a
+sequence of real numbers {an, n ∈ Z+}, i.e., f(z) = �∞
+n=0 anzn. If f(z) is singular at z = z0 > 0
+and analytic in ˜∆z0(ε, θ) for some ε > 0 and some θ ∈ [0, π/2) and if it satisfies
+lim
+˜∆z0∋z→z0
+(z0 − z)αf(z) = c0
+(4.1)
+for α ∈ R \ {0, −1, −2, ...} and some nonzero constant c0 ∈ R, then
+lim
+n→∞
+�nα−1
+Γ(α) z−n
+0
+�−1
+an = c
+(4.2)
+for some real number c, where Γ(z) is the gamma function. This means that the asymptotic function
+of the sequence {an} is given by nα−1z−n
+0 .
+For the purpose, we prove the following propositions in Section 4.2.
+Proposition 4.1. Assume Type 1. If ¯η′
+1(θ∗
+2) ≤ −c1/c2 = −1 ≤ 1/¯η′
+2(θ∗
+1), the vector function ϕc(z)
+is elementwise analytic in ˜∆zmax
+c
+(ε, θ) for some ε > 0 and some θ ∈ [0, π/2).
+Proposition 4.2. Assume Type 1. If ¯η′
+1(θ∗
+2) ≤ −c1/c2 = −1 ≤ 1/¯η′
+2(θ∗
+1), there exist a vector
+function ˜ϕc(ζ) being meromorphic in a neighborhood of ζ = 0 and satisfying ϕc(z) = ˜ϕc((zmax
+c
+−
+z)
+1
+2 ) in a neighborhood of z = zmax
+c
+. If ¯η′
+1(θ∗
+2) < −1 < 1/¯η′
+2(θ∗
+1), the point ζ = 0 is a pole of ˜ϕc(ζ)
+with at most order one; if ¯η′
+1(θ∗
+2) = −1 or ¯η′
+2(θ∗
+1) = −1, it is a pole of ˜ϕc(ζ) with at most order two.
+By Proposition 4.2, if ¯η′
+1(θ∗
+2) < −1 < 1/¯η′
+2(θ∗
+1), the Puiseux series of ϕc(z) is represented as
+ϕc(z) =
+∞
+�
+k=−1
+ϕc
+1,k(zmax
+c
+− z)
+k
+2 ,
+(4.3)
+where {ϕc
+1,k} is a series of coefficient vectors; if ¯η′
+1(θ∗
+2) = −1 or ¯η′
+2(θ∗
+1) = −1, it is represented as
+ϕc(z) =
+∞
+�
+k=−2
+ϕc
+2,k(zmax
+c
+− z)
+n
+2 ,
+(4.4)
+where {ϕc
+2,k} is a series of coefficient vectors. Let l be a positive integer such that ϕc
+1,l−2 ̸= 0 and
+ϕc
+1,k−2 = 0 for all positive integer k less than l. Then, applying Lemma 4.1 to (4.3), we obtain
+hc(k) = k− 1
+2 (2l−1)(zmax
+c
+)−k = k− 1
+2 (2l−1)e−θmax
+c
+k.
+This completes the former half of the proof of Proposition 2.1. If ¯η′
+1(θ∗
+2) = −1 or ¯η′
+2(θ∗
+1) = −1,
+ϕc(z) satisfies the following property, which will be proved in Section 4.2.
+14
+
+Proposition 4.3. Assume Type 1. Then, we have, for some positive vectors uc
+1 and uc
+2,
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)ϕc(z) =
+�
+�
+�
+uc
+1
+if ¯η′
+1(θ∗
+2) = −1 and ¯η′
+2(θ∗
+1) < −1,
+uc
+2
+if ¯η′
+1(θ∗
+2) < −1 and ¯η′
+2(θ∗
+1) = −1,
+uc
+1 + uc
+2
+if ¯η′
+1(θ∗
+2) = ¯η′
+2(θ∗
+1) = −1.
+(4.5)
+Hence, ϕc
+2,−2 is positive, and by Lemma 4.1, we obtain
+hc(k) = (zmax
+c
+)−k = e−θmax
+c
+k.
+This completes the latter half of the proof of Proposition 2.1.
+Remark 4.1. Assume Type 1 and ¯η′
+1(θ∗
+2) < −c1/c2 = −1 < 1/¯η′
+2(θ∗
+1). If the vector function ϕc(z)
+diverges at z = zmax
+c
+, the coefficient vector ϕc
+1,−1 in (4.3) must be nonzero and , by Lemma 4.1, we
+have
+hc(k) = k− 1
+2 (zmax
+c
+)−k = k− 1
+2 e−θmax
+c
+k.
+4.2
+Proof of Propositions 4.1, 4.2 and 4.3
+Recall that the direction vector c is set as c = (1, 1). Notation of this subsection follows [14].
+Denote by Φ{1,2} = (Φ{1,2}
+x,x′ ; x, x′ ∈ Z2) the fundamental matrix (potential matrix) of P {1,2}, i.e.,
+Φ{1,2} = �∞
+n=0(P {1,2})n, where P {1,2} = (P {1,2}
+x,x′ ; x, x′ ∈ Z2) is the transition probability matrix of
+the induced MA-process {Y {1,2}
+n
+}. For x ∈ Z2, define the matrix generating function of the blocks
+of Φ{1,2} in direction c, Φc
+x,∗(z), as
+Φc
+x,∗(z) =
+∞
+�
+k=−∞
+zkΦ{1,2}
+x,kc .
+According to equation (3.3) of [14], we divide ϕc(z) into three parts as follows:
+ϕc(z) = ϕc
+0(z) + ϕc
+1(z) + ϕc
+2(z),
+(4.6)
+where
+ϕc
+0(z) =
+�
+i1,i2∈{−1,0,1}
+ν(0,0)(A∅
+i1,i2 − A{1,2}
+i1,i2 )Φc
+(i1,i2),∗(z),
+(4.7)
+ϕc
+1(z) =
+∞
+�
+k=1
+�
+i1,i2∈{−1,0,1}
+ν(k,0)(A{1}
+i1,i2 − A{1,2}
+i1,i2 )Φc
+(k+i1,i2),∗(z),
+(4.8)
+ϕc
+2(z) =
+∞
+�
+k=1
+�
+i1,i2∈{−1,0,1}
+ν(0,k)(A{2}
+i1,i2 − A{1,2}
+i1,i2 )Φc
+(i1,k+i2),∗(z).
+(4.9)
+According to [14], we focus on ϕc
+2(z) and consider another skip-free MA-process generated from
+{Y {1,2}
+n
+}. The MA-process is { ˆY n} = {( ˆXn, ˆJn)} = {( ˆX1,n, ˆX2,n), ( ˆRn, ˆJn)}, where ˆX1,n = X{1,2}
+1,n
+,
+ˆX2,n and ˆRn are the quotient and remainder of X{1,2}
+2,n
+− X{1,2}
+1,n
+divided by 2, respectively, and
+ˆJn = J{1,2}
+n
+. The state space of { ˆY n} is Z2 × {0, 1} × S0 and the additive part { ˆXn} is skip
+free. From the definition, if ˆXn = (x1, x2) and ˆRn = r in the new MA-process, it follows that
+X{1,2}
+1,n
+= x1, X{1,2}
+2,n
+= x1 + 2x2 + r in the original MA-process. Hence, ˆY n = (k, 0, 0, j) means
+15
+
+Y {1,2}
+n
+= (k, k, j). Denote by ˆP = ( ˆPx,x′; x, x′ ∈ Z2) the transition probability matrix of { ˆY n},
+which is given as
+ˆPx,x′ =
+�
+ˆA{1,2}
+x′−x,
+if x′ − x ∈ {−1, 0, 1}2,
+O,
+otherwise,
+where
+ˆA{1,2}
+−1,1 =
+�
+A{1,2}
+−1,1
+O
+A{1,2}
+−1,0
+A{1,2}
+−1,1
+�
+,
+ˆA{1,2}
+0,1
+=
+�
+O
+O
+A{1,2}
+0,1
+O
+�
+,
+ˆA{1,2}
+1,1
+=
+�O
+O
+O
+O
+�
+,
+ˆA{1,2}
+−1,0 =
+�
+A{1,2}
+−1,−1
+A{1,2}
+−1,0
+O
+A{1,2}
+−1,−1
+�
+,
+ˆA{1,2}
+0,0
+=
+�
+A{1,2}
+0,0
+A{1,2}
+0,1
+A{1,2}
+0,−1
+A{1,2}
+0,0
+�
+,
+ˆA{1,2}
+1,0
+=
+�
+A{1,2}
+1,1
+O
+A{1,2}
+1,0
+A{1,2}
+1,1
+�
+,
+ˆA{1,2}
+−1,−1 =
+�O
+O
+O
+O
+�
+,
+ˆA{1,2}
+0,−1 =
+�
+O
+A{1,2}
+0,−1
+O
+O
+�
+,
+ˆA{1,2}
+1,−1 =
+�
+A{1,2}
+1,−1
+A{1,2}
+1,0
+O
+A{1,2}
+1,−1
+�
+.
+Denote by ˆΦ = (ˆΦx,x′; x, x′ ∈ Z2) the fundamental matrix of ˆP, i.e., ˆΦ = �∞
+n=0( ˆP)n, and for
+x = (x1, x2) ∈ Z2, define a matrix generating function ˆΦx,∗(z) as
+ˆΦx,∗(z) =
+∞
+�
+k=−∞
+zk ˆΦx,(k,0) =
+�
+Φc
+(x1,x1+2x2),∗(z)
+Φc
+(x1,x1+2x2−1),∗(z)
+Φc
+(x1,x1+2x2+1),∗(z)
+Φc
+(x1,x1+2x2),∗(z)
+�
+.
+(4.10)
+We consider analytic properties of the matrix function Φc
+(x1,x1+2x2),∗(z) through ˆΦ(x1,x2),∗(z). Define
+blocks ˆA{2}
+i1,i2, i1, i2 ∈ {−1, 0, 1}, as ˆA{2}
+−1,1 = ˆA{2}
+−1,0 = ˆA{2}
+−1,−1 = O and
+ˆA{2}
+0,1 =
+�
+O
+O
+A{2}
+0,1
+O
+�
+,
+ˆA{2}
+0,0 =
+�
+A{2}
+0,0
+A{2}
+0,1
+A{2}
+0,−1
+A{2}
+0,0
+�
+,
+ˆA{2}
+0,−1 =
+�
+O
+A{2}
+0,−1
+O
+O
+�
+,
+ˆA{2}
+1,1 =
+�O
+O
+O
+O
+�
+,
+ˆA{2}
+1,0 =
+�
+A{2}
+1,1
+O
+A{2}
+1,0
+A{2}
+1,1
+�
+,
+ˆA{2}
+1,−1 =
+�
+A{2}
+1,−1
+A{2}
+1,0
+O
+A{2}
+1,−1
+�
+.
+For i1, i2 ∈ {−1, 0, 1}, define the following matrix generating functions:
+ˆA{1,2}
+∗,i2 (z) =
+�
+i∈{−1,0,1}
+zi ˆA{1,2}
+i,i2 ,
+ˆA{1,2}
+i1,∗ (z) =
+�
+i∈{−1,0,1}
+zi ˆA{1,2}
+i1,i ,
+ˆA{2}
+∗,i2(z) =
+�
+i∈{0,1}
+zi ˆA{2}
+i,i2,
+ˆA{2}
+i1,∗(z) =
+�
+i∈{−1,0,1}
+zi ˆA{2}
+i1,i.
+Define a vector generating function ˆϕ2(z) as
+ˆϕ2(z) =
+�ˆϕ2,1(z)
+ˆϕ2,2(z)
+�
+=
+∞
+�
+k=1
+�
+i1,i2∈{−1,0,1}
+ˆν(0,k)( ˆA{2}
+i1,i2 − ˆA{1,2}
+i1,i2 )ˆΦ(i1,k+i2),∗(z),
+(4.11)
+where, for x = (x1, x2) ∈ Z2
++, ˆνx =
+�
+ν(x1,x1+2x2)
+ν(x1,x1+2x2+1)
+�
+and hence, for k ≥ 0,
+ˆν(0,k) =
+�
+ν(0,2k)
+ν(0,2k+1)
+�
+.
+By equation (3.9) of [14], ϕc
+2(z) is represented as
+ϕc
+2(z) = ˆϕ2,1(z) +
+�
+i1,i2∈{−1,0,1}
+ν(0,1)(A{2}
+i1,i2 − A{1,2}
+i1,i2 )Φc
+(i1,i2+1),∗(z).
+(4.12)
+16
+
+Hence, we consider analytic properties of the vector function ϕc
+2(z) through ˆϕc
+2(z) and ˆΦx,∗(z).
+Let ˆG0,∗(z) be the G-matrix function generated from the triplet { ˆA{1,2}
+∗,−1 (z), ˆA{1,2}
+∗,0
+(z), ˆA{1,2}
+∗,1
+(z)}.
+By equations (3.11) and (3.13) of [14], we have, for x2 ≥ 0,
+ˆΦ(x1,x2),∗(z) = zx1 ˆG0,∗(z)x2 ˆΦ(0,0),∗(z),
+(4.13)
+and this leads us to
+ˆϕ2(z) =
+∞
+�
+k=1
+�
+i2∈{−1,0,1}
+ˆν(0,k)( ˆA{2}
+∗,i2(z) − ˆA{1,2}
+∗,i2 (z)) ˆG0,∗(z)k+i2 ˆΦ(0,0),∗(z).
+(4.14)
+Hence, analytic properties of the vector function ˆϕ2(z) as well as the matrix function ˆΦx,∗(z) can
+be clarified through ˆG0,∗(z) and ˆΦ(0,0),∗(z).
+By (4.14), ˆϕ2(z) is represented as
+ˆϕ2(z) = ˆa(z, ˆG0,∗(z))ˆΦ(0,0),∗(z),
+(4.15)
+where
+ˆa(z, w) =
+∞
+�
+k=1
+ˆν(0,k) ˆD(z, ˆG0,∗(z))wk−1,
+ˆD(z, w) = ˆA{2}
+∗,−1(z) + ˆA{2}
+∗,0 (z)w + ˆA{2}
+∗,1 (z)w2 − Iw.
+First, we consider ˆΦ(0,0),∗(z). Let ˆGr
+0,∗(z) be the G-matrix function in the reverse direction generated
+from the triplet { ˆA{1,2}
+∗,−1 (z), ˆA{1,2}
+∗,0
+(z), ˆA{1,2}
+∗,1
+(z)}, which means that ˆGr
+0,∗(z) is the G-matrix function
+generated from the triplet by exchanging ˆA{1,2}
+∗,−1 (z) and ˆA{1,2}
+∗,1
+(z); see Section 3.2. Define a matrix
+function ˆU(z) as
+ˆU(z) = ˆA{1,2}
+∗,−1 (z) ˆGr
+0,∗(z) + ˆA{1,2}
+∗,0
+(z) + ˆA{1,2}
+∗,1
+(z) ˆG0,∗(z).
+(4.16)
+Then, ˆΦ(0,0),∗(z) in (4.15) is given as
+ˆΦ(0,0),∗(z) =
+∞
+�
+n=0
+ˆU(z)n = (I − ˆU(z))−1 = adj(I − ˆU(z))
+det(I − ˆU(z))
+.
+(4.17)
+Recall that zmin
+c
+= eθmin
+c
+and zmax
+c
+= eθmax
+c
+.
+For θ ∈ [θmin
+c
+, θmax
+c
+], let (ηR
+c,1(θ), ηR
+c,2(θ)) and
+(ηL
+c,1(θ), ηL
+c,2(θ)) be the two real roots of the simultaneous equations:
+spr(A{1,2}
+∗,∗
+(eθ1, eθ2)) = 1,
+θ1 + θ2 = θ,
+(4.18)
+counting multiplicity, where ηL
+c,1(θ) ≤ ηR
+c,1(θ) and ηL
+c,2(θ)) ≥ ηR
+c,2(θ).
+Note that ηL
+c,1(θmax
+c
+) =
+ηR
+c,1(θmax
+c
+) and ηL
+c,2(θmax
+c
+) = ηR
+c,2(θmax
+c
+). By equations (3.18) and (3.32) of [14], we have
+spr( ˆG0,∗(eθ)) = e2ηR
+c,2(θ).
+(4.19)
+Since the eigenvalues of ˆGr
+0,∗(z) are coincide with those of the rate matrix function generated from
+the same triplet { ˆA{1,2}
+∗,−1 (z), ˆA{1,2}
+∗,0
+(z), ˆA{1,2}
+∗,1
+(z)}, we have
+spr( ˆGr
+0,∗(eθ)) = e−2ηL
+c,2(θ).
+(4.20)
+By Lemmas 3.1 and 3.3 and Corollary 3.1, ˆG0,∗(z) and ˆGr
+0,∗(z) satisfy the following properties.
+17
+
+Proposition 4.4.
+(1) The extended G-matrix functions ˆG0,∗(z) and ˆGr
+0,∗(z) are entry-wise an-
+alytic in ∆zmin
+c
+,zmax
+c
+∪ ∂∆zmax
+c
+\ {zmax
+c
+}. The point z = zmax
+c
+is a common branch point of
+ˆG0,∗(z) and ˆGr
+0,∗(z) with order one.
+(2) There exist matrix functions ˜G0,∗(ζ) and ˜Gr
+0,∗(ζ) being analytic in a neighborhood of ζ = 0
+and satisfying ˆG0,∗(z) = ˜G0,∗((zmax
+c
+− z)
+1
+2 ) and ˆGr
+0,∗(z) = ˜Gr
+0,∗((zmax
+c
+− z)
+1
+2 ), respectively, in
+a neighborhood of z = zmax
+c
+.
+In order to investigate singularity of ˆΦ(0,0),∗(z) at z = zmax
+c
+, we give the following proposition.
+Proposition 4.5. The maximum eigenvalue of ˆU(zmax
+c
+) is 1, and it is simple.
+Proof. By equation (3.30) of [14], we have spr( ˆA{1,2}
+∗,∗
+(zmax
+c
+, e2ηR
+c,2(θmax
+c
+))) = 1. Let v be the right
+eigenvector of ˆA{1,2}
+∗,∗
+(zmax
+c
+, e2ηR
+c,2(θmax
+c
+)) with respect to eigenvalue 1.
+Since spr( ˆG0,∗(zmax
+c
+)) =
+e2ηR
+c,2(θmax
+c
+) and spr( ˆGr
+0,∗(zmax
+c
+)) = e−2ηL
+c,2(θmax
+c
+) = e−2ηR
+c,2(θmax
+c
+), we have, by Lemma 3.6,
+ˆG0,∗(zmax
+c
+)v = e2ηR
+c,2(θmax
+c
+)v,
+ˆGr
+0,∗(zmax
+c
+)v = e−2ηR
+c,2(θmax
+c
+)v,
+Hence,
+ˆU(zmax
+c
+)v = ˆA{1,2}
+∗,∗
+(zmax
+c
+, e2ηR
+c,2(θmax
+c
+))v = 1.
+This means that the value of 1 is an eigenvalue of ˆU(zmax
+c
+), and we obtain spr( ˆU(zmax
+c
+)) ≥ 1.
+Suppose spr( ˆU(zmax
+c
+)) > 1. Then, since spr( ˆU(eθ)) is convex in θ ∈ R, there exist a positive
+θ0 < θmax
+c
+such that spr( ˆU(eθ0)) = 1. For this θ0, ˆΦ(0,0),∗(z) diverges at z = eθ0 < zmax
+c
+. This
+contradicts Proposition 3.1 of [14], which asserts that ˆΦ(0,0),∗(z) absolutely convergent in z ∈
+∆zmin
+c
+,zmax
+c
+. Hence, spr( ˆU(zmax
+c
+)) ≤ 1, and this implies the maximum eigenvalue of ˆU(zmax
+c
+) is 1.
+Since ˆU(zmax
+c
+) is irreducible, it is simple.
+Let ˆλU(z) be the eigenvalue of ˆU(z) satisfying ˆλU(z) = spr( ˆU(z)) for z ∈ [zmin
+c
+, zmax
+c
+]. Let
+ˆuU(z) and ˆvU(z) be the left and right eigenvectors of ˆU(z) with respect to the eigenvalue ˆλU(z),
+respectively, satisfying ˆuU(z)ˆvU(z) = 1. Define a matrix function ˜U(ζ) as
+˜U(ζ) = ˆA{1,2}
+∗,−1 (zmax
+c
+− ζ2) ˜Gr
+0,∗(ζ) + ˆA{1,2}
+∗,0
+(zmax
+c
+− ζ2) + ˆA{1,2}
+∗,1
+(zmax
+c
+− ζ2) ˜G0,∗(ζ).
+By Proposition 4.4, ˜U(ζ) is entry-wise analytic in a neighborhood of ζ = 0 and satisfies ˆU(z) =
+˜U((zmax
+c
+− z)
+1
+2 ) in a neighborhood of z = zmax
+c
+. Define a matrix function ˜Φ(0,0),∗(ζ) as
+˜Φ(0,0),∗(ζ) = (I − ˜U(ζ))−1 = adj(I − ˜U(ζ))
+det(I − ˜U(ζ))
+.
+(4.21)
+ˆΦ(0,0),∗(z) and ˜Φ(0,0),∗(ζ) satisfy the following properties.
+Proposition 4.6.
+(1) The matrix function ˆΦ(0,0),∗(z) is entry-wise analytic in ∆zmin
+c
+,zmax
+c
+∪∂∆zmax
+c
+\
+{zmax
+c
+}.
+(2) ˜Φ(0,0),∗(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0, and the point ζ = 0 is a pole
+of ˜Φ(0,0),∗(ζ) with order one. ˆΦ(0,0),∗(z) is represented as ˆΦ(0,0),∗(z) = ˜Φ(0,0),∗((zmax
+c
+− z)
+1
+2 ) in
+a neighborhood of z = zmax
+c
+.
+18
+
+(3) ˆΦ(0,0),∗(z) satisfies
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)
+1
+2 ˆΦ(0,0),∗(z) = ˆgΦˆvU(zmax
+c
+)ˆuU(zmax
+c
+) > O,
+(4.22)
+where both ˆvU(zmax
+c
+) and ˆuU(zmax
+c
+) are positive,
+ˆgΦ = −
+�
+ˆuU(zmax
+c
+)( ˆA{1,2}
+∗,−1 (zmax
+c
+) ˆGr
+0,∗,1 + ˆA{1,2}
+∗,1
+(zmax
+c
+) ˆG0,∗,1)ˆvU(zmax
+c
+)
+�−1
+> 0,
+(4.23)
+and ˆGr
+0,∗,1 and ˆG0,∗,1 are the limits of ˆGr
+0,∗(z) and ˆG0,∗(z), respectively, given by Lemma 3.5.
+Proof. By (4.16) and Proposition 4.4, ˆU(z) is entry-wise analytic in ∆zmin
+c
+,zmax
+c
+∪ ∂∆zmax
+c
+\ {zmax
+c
+}.
+Hence, by (4.17), ˆΦ(0,0),∗(z) is entry-wise meromorphic in the same domain. Recall that, under
+Assumption 2.2, the induced MA-process {Y {1,2}
+n
+} is irreducible and aperiodic. Hence, in a manner
+similar to that used in the proof of Proposition 5.2 of [11], we obtain by Proposition 4.5 that, for
+every z ∈ ∆zmin
+c
+,zmax
+c
+∪ ∂∆zmax
+c
+\ {zmax
+c
+},
+spr( ˆU(z)) < spr( ˆU(|z|)) < spr( ˆU(zmax
+c
+)) = 1,
+and this leads us to det(I − ˆU(z)) ̸= 0. This completes the proof of statement (1).
+By (4.21), ˜Φ(0,0),∗(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0. Since ˜U(0) =
+ˆU(zmax
+c
+), we see by Proposition 4.5 that det(I − ˜U(0)) = 0 and the multiplicity of zero of det(I −
+˜U(ζ)) at ζ = 0 is one. Hence, by the identity theorem for analytic functions, det(I − ˜U(ζ)) is
+nonzero in a neighborhood of ζ = 0 except for the point ζ = 0 and the point ζ = 0 is a pole of
+˜Φ(0,0),∗(ζ) with order one. This completes the proof of statement (2) since the representation of
+ˆΦ(0,0),∗(z) is obvious.
+Define a function f(λ, z) as
+f(λ, z) = det(λI − ˆU(z)).
+By Corollary 2 of Seneta [15] and Proposition 4.5 (also see Proposition 5.11 of [11]),
+adj(I − ˆU(zmax
+c
+)) = fλ(1, zmax
+c
+)ˆvU(zmax
+c
+)ˆuU(zmax
+c
+),
+(4.24)
+where fλ(λ, z) =
+∂
+∂λf(λ, z) and both ˆvU(zmax
+c
+) and ˆuU(zmax
+c
+) are positive since ˆU(zmax
+c
+) is irre-
+ducible. Furthermore, in a manner similar to that used in the proof of Proposition 5.9 of [11], we
+obtain
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)− 1
+2 f(1, z) = −c0fλ(1, zmax
+c
+),
+(4.25)
+where c0 = ˆuU(zmax
+c
+)( ˆA{1,2}
+∗,−1 (zmax
+c
+) ˆGr
+0,∗,1 + ˆA{1,2}
+∗,1
+(zmax
+c
+) ˆG0,∗,1)ˆvU(zmax
+c
+) < 0 since, by Lemma 3.5,
+both ˆG0,∗,1 and ˆGr
+0,∗,1 are nonzero and nonpositive. By (4.21), this completes the proof of statement
+(3).
+Let αs0(z) be the eigenvalue of ˆG0,∗(z) that satisfies, for z ∈ [zmin
+c
+, zmax
+c
+], αs0(z) = spr( ˆG0,∗(z)) =
+e2ηR
+c,2(log z). Let ˆuG(z) and ˆvG(z) be the left and right eigenvectors of ˆG0,∗(z) with respect to the
+eigenvalue αs0(z), satisfying ˆuG(z)ˆvG(z) = 1. By Lemma 3.3, ˜G0,∗(ζ) in Proposition 4.4 satisfies
+the following property.
+19
+
+Proposition 4.7. There exists a matrix function ˜G†
+0,∗(ζ) entry-wise analytic in a neighborhood of
+ζ = 0 such that ˜G0,∗(ζ) is represented as
+˜G0,∗(ζ) = ˜G†
+0,∗(ζ) + ˜αs0(ζ)˜vG(ζ)˜uG(ζ),
+(4.26)
+where function ˜αs0(ζ), row vector function ˜uG(ζ) and column vector ˜vG(ζ) are elementwise analytic
+in a neighborhood of ζ = 0 and satisfying αs0(z) = ˜αs0((zmax
+c
+− z)
+1
+2 ), ˆuG(z) = ˜uG((zmax
+c
+− z)
+1
+2 )
+and ˆvG(z) = ˜vG((zmax
+c
+− z)
+1
+2 ), respectively, in a neighborhood of z = zmax
+c
+. In a neighborhood of
+ζ = 0, ˜G†
+0,∗(ζ) satisfies spr( ˜G†
+0,∗(ζ)) < αs0(zmax
+c
+) = e2ηR
+c,2(θmax
+c
+). Furthermore, ˜G0,∗(ζ) satisfies, for
+n ≥ 1,
+˜G0,∗(ζ)n = ˜G†
+0,∗(ζ)n + ˜αs0(ζ)n˜vG(ζ)˜uG(ζ).
+(4.27)
+Let ˆν(0,∗)(z) be the generating function of {ˆν(0,k)} defined as ˆν(0,∗)(z) = �∞
+k=1 zkˆν(0,k). Define
+a matrix function ˆU2(z) as
+ˆU2(z) = ˆA{2}
+0,∗ (z) + ˆA{2}
+1,∗ (z) ˆG∗,0(z),
+and let ˆuU
+2 (z) and ˆvU
+2 (z) be the left and right eigenvectors of ˆU2(z) with respect to the maximum
+eigenvalue of ˆU2(z), satisfying ˆuU
+2 (z)ˆvU
+2 (z) = 1. By Lemma 5.3 of [11] (also see Proposition 3.5 of
+[14]), ˆν(0,∗)(z) satisfies the following properties.
+Proposition 4.8. Assume Type 1.
+(1) The vector function ˆν(0,∗)(z) is elementwise analytic in ¯∆e2θ∗
+2 \ {e2θ∗
+2}.
+(2) If θ∗
+2 < θmax
+2
+, ˆν(0,∗)(z) is elementwise meromorphic in a neighborhood of z = e2θ∗
+2 and the
+point z = e2θ∗
+2 is a pole of ˆν(0,∗)(z) with order one. It satisfies, for some positive constant ˆg2,
+lim
+˜∆
+e2θ∗
+2 ∋z→e2θ∗
+2
+(e2θ∗
+2 − z)ˆϕ2(z) = ˆg2ˆuU
+2 (e2θ∗
+2),
+(4.28)
+where ˆuU
+2 (e2θ∗
+2) is positive.
+Define a vector function ˜a(ζ, w) as
+˜a(ζ, w) =
+∞
+�
+k=1
+ˆν(0,k) ˆD(zmax
+c
+− ζ2, ˜G0,∗(ζ))wk−1.
+(4.29)
+Then, the vector functions ˆa(z, ˆG0,∗(z)) in (4.15) and ˜a(ζ, ˜G0,∗(ζ)) satisfy the following properties.
+Proposition 4.9. Assume Type 1.
+(1) If ¯η′
+1(θ∗
+2) ≤ −c1/c2 = −1, the vector function ˆa(z, ˆG0,∗(z)) is elementwise analytic in ∆zmin
+c
+,zmax
+c
+∪
+∂∆zmax
+c
+\ {zmax
+c
+}.
+(2) If ¯η′
+1(θ∗
+2) < −1, ˜a(ζ, ˜G0,∗(ζ)) is elementwise analytic in a neighborhood of ζ = 0; if ¯η′
+1(θ∗
+2) =
+−1, it is elementwise meromorphic in a neighborhood of ζ = 0 and the point ζ = 0 is a
+pole of it with order one. The vector function ˆa(z, ˆG0,∗(z)) is represented as ˆa(z, ˜G0,∗(z)) =
+˜a((zmax
+c
+− z)
+1
+2 , ˜G0,∗((zmax
+c
+− z)
+1
+2 )) in a neighborhood of z = zmax
+c
+.
+20
+
+(3) If ¯η′
+1(θ∗
+2) = −1, ˆa(z, ˆG0,∗(z)) satisfies, for a positive constant ˆga
+2,
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)
+1
+2 ˆa(z, ˆG0,∗(z)) = ˆga
+2 ˆuG(zmax
+c
+) ≥ 0⊤, ̸= 0⊤.
+(4.30)
+Proof. By Proposition 4.2 of [11], if ¯η′
+1(θ∗
+2) ≤ −1, we have for z ∈ ∆zmin
+c
+,zmax
+c
+∪ ∂∆zmax
+c
+\ {zmax
+c
+}
+that |αs0(z)| < αs0(zmax
+c
+) = e2ηR
+c,2(θmax
+c
+) ≤ e2θ∗
+2, and this implies spr( ˆG0,∗(z)) < e2θ∗
+2. Hence, by
+Lemma 3.2 of [11] and Proposition 4.8, vector series ˆa(z, ˆG0,∗(z)) elementwise converges absolutely
+in ∆zmin
+c
+,zmax
+c
+∪ ∂∆zmax
+c
+\ {zmax
+c
+}. This completes the proof of statement (1).
+By Proposition 4.7, we have
+˜a(ζ, ˜G0,∗(ζ)) = ˜a(ζ, ˜G†
+0,∗(ζ))
++ (˜αs0(ζ)−1ˆν(0,∗)(˜αs0(ζ)) − ˆν(0,1)) ˆD(zmax
+c
+− ζ2, ˜αs0(ζ))˜vG(ζ)˜uG(ζ).
+(4.31)
+If ¯η′
+1(θ∗
+2) ≤ −1, spr( ˜G†
+0,∗(ζ)) < e2ηR
+c,2(θmax
+c
+) ≤ e2θ∗
+2 in a neighborhood of ζ = 0.
+Hence, vec-
+tor series ˜a(ζ, ˜G†
+0,∗(ζ)) is elementwise convergent absolutely and analytic in a neighborhood of
+ζ = 0. If ¯η′
+1(θ∗
+2) < −1, ˜αs0(0) = αs0(zmax
+c
+) = e2ηR
+c,2(θmax
+c
+) < e2θ∗
+2, and this implies |˜αs0(ζ)| < e2θ∗
+2
+in a neighborhood of ζ = 0.
+Hence, by Proposition 4.8, the vector function ˜a(ζ, ˜G0,∗(ζ)) as
+well as ˆν(0,∗)(˜αs0(ζ)) is elementwise analytic in a neighborhood of ζ = 0.
+If ¯η′
+1(θ∗
+2) = −1,
+˜αs0(0) = e2ηR
+c,2(θmax
+c
+) = e2θ∗
+2. Hence, by Proposition 4.8, the vector function ˜a(ζ, ˜G0,∗(ζ)) as well as
+ˆν(0,∗)(˜αs0(ζ)) is meromorphic in a neighborhood of ζ = 0 and the point ζ = 0 is a pole of it with
+order one. This completes the proof of statement (2).
+If ¯η′
+1(θ∗
+2) = −1, αs0(zmax
+c
+) = e2ηR
+c,2(θmax
+c
+) = e2θ∗
+2. Hence, by Lemma 3.4 and Proposition 4.8, we
+have
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)
+1
+2 ˆν(0,∗)(αs0(z))
+=
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)
+1
+2
+αs0(zmax
+c
+) − αs0(z)(αs0(zmax
+c
+) − αs0(z))ˆν(0,∗)(αs0(z))
+= (−αs0,1)−1ˆg2ˆuU
+2 (e2θ∗
+2),
+where αs0,1 is the limit of αs0(z) given by (3.10) and it is negative. This leads us to
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)
+1
+2 ˆa(z, ˆG0,∗(z))
+= (−αs0,1)−1ˆg2e−2θ∗
+2 ˆuU
+2 (e2θ∗
+2)D(zmax
+c
+, e2θ∗
+2)ˆvG(zmax
+c
+)ˆuG(zmax
+c
+).
+(4.32)
+From this, we see that ˆga
+2 ˆuG(zmax
+c
+) in (4.30) is given by the right-hand side of (4.32). Since ˆuU
+2 (e2θ∗
+2)
+is positive, ˆga
+2 is also positive. This completes the proof of statement (3).
+Finally, we give the proof of Propositions 4.1, 4.2 and 4.3.
+Proof of Proposition 4.1. Assume Type 1 and ¯η′
+1(θ∗
+2) ≤ −c1/c2 = −1 ≤ 1/¯η′
+2(θ∗
+1). Since ϕc(z) is a
+probability vector generating function, it is automatically analytic elementwise in ∆zmax
+c
+. Hence,
+we prove it is elementwise analytic on ∂∆zmax
+c
+\ {zmax
+c
+}. For the purpose, we use equations (4.6),
+(4.7), (4.10), (4.12), (4.13) and (4.15).
+By Propositions 4.4, 4.6 and 4.9, ˆG0,∗(z), ˆa(z, ˆG0,∗(z)) and ˆΦ(0,0),∗(z) are elementwise analytic
+on ∂∆zmax
+c
+\ {zmax
+c
+}. Hence, by (4.13) and (4.15), ˆΦx,∗(z) and ˆϕ2(z) are also analytic elementwise
+on ∂∆zmax
+c
+\ {zmax
+c
+}. By (4.10), the analytic property of ˆΦx,∗(z) implies that Φc
+x,∗(z) is entry-wise
+21
+
+analytic on ∂∆zmax
+c
+\{zmax
+c
+}. Hence, by (4.12), ϕc
+2(z) is elementwise analytic on ∂∆zmax
+c
+\{zmax
+c
+}. In
+the same way, we can see that if ¯η′
+2(θ∗
+1) ≤ −1, ϕc
+1(z) is elementwise analytic on ∂∆zmax
+c
+\{zmax
+c
+}. By
+(4.7), the analytic property of Φc
+x,∗(z) implies that ϕc
+0(z) is elementwise analytic on ∂∆zmax
+c
+\{zmax
+c
+}.
+As a result, we see by (4.6) that ϕc(z) is elementwise analytic on ∂∆zmax
+c
+\ {zmax
+c
+}. This completes
+the proof.
+Proof of Proposition 4.2. Assuming Type 1 and ¯η′
+1(θ∗
+2) ≤ −c1/c2 = −1 ≤ 1/¯η′
+2(θ∗
+1), we also use
+equations (4.6), (4.7), (4.10),(4.12), (4.13) and (4.15).
+First, we consider about Φc
+x,∗(z) and ϕc
+0(z), where x = (x1, x2) ∈ Z2. Define ˜Φ(x1,x2),∗(ζ) as
+˜Φ(x1,x2),∗(ζ) = (zmax
+c
+− ζ2)x1 ˜G0,∗(ζ)x2 ˜Φ(0,0),∗(ζ).
+Then, by Propositions 4.4 and 4.6, the matrix function ˜Φx,∗(ζ) is entry-wise meromorphic in a
+neighborhood of ζ = 0 and satisfies ˆΦx,∗(z) = ˜Φ(x1,x2),∗((zmax
+c
+− z)
+1
+2 ) in a neighborhood of z =
+zmax
+c
+.
+The point ζ = 0 is a pole of ˜Φx,∗(ζ) with order one.
+Hence, by (4.10), there exists a
+matrix function ˜Φc
+x,∗(ζ) being entry-wise meromorphic in a neighborhood of ζ = 0 and satisfying
+Φc
+x,∗(z) = ˜Φc
+x,∗((zmax
+c
+− z)
+1
+2 ) in a neighborhood of z = zmax
+c
+. The point ζ = 0 is a pole of ˜Φc
+x,∗(ζ)
+with order one. Define ˜ϕc
+0(z) as
+˜ϕc
+0(ζ) =
+�
+i1,i2∈{−1,0,1}
+ν(0,0)(A∅
+i1,i2 − A{1,2}
+i1,i2 )˜Φc
+(i1,i2),∗(ζ),
+which satisfies the same analytic property as ˜Φc
+x,∗(ζ). It also satisfies ϕc
+0(z) = ˜ϕc
+0((zmax
+c
+− z)
+1
+2 ) in
+a neighborhood of z = zmax
+c
+.
+Next, we consider about ϕc
+2(z). Define ˜ϕ2(ζ) as
+˜ϕ2(ζ) = ˜a(ζ, ˜G0,∗(ζ))˜Φ(0,0),∗(ζ)
+By Propositions 4.6 and 4.9 and (4.15), ˜ϕ2(ζ) is entry-wise meromorphic in a neighborhood of
+ζ = 0 and satisfying ˆϕ2(z) = ˜ϕ2((zmax
+c
+− z)
+1
+2 ) in a neighborhood of z = zmax
+c
+. If ¯η′
+1(θ∗
+2) < −1, the
+point ζ = 0 is a pole of ˜ϕ2(ζ) with at most order one; if ¯η′
+1(θ∗
+2) = −1, it is a pole of ˜ϕ2(ζ) with at
+most order two. Represent ˜ϕ2(ζ) in block form as ˜ϕ2(ζ) =
+�˜ϕ2,1(ζ)
+˜ϕ2,2(ζ)
+�
+and define ˜ϕc
+2(ζ) as
+˜ϕc
+2(ζ) = ˜ϕ2,1(ζ) +
+�
+i1,i2∈{−1,0,1}
+ν(0,1)(A{2}
+i1,i2 − A{1,2}
+i1,i2 )˜Φc
+(i1,i2+1),∗(ζ).
+Then, the vector function ˜ϕc
+2(ζ) is elementwise meromorphic in a neighborhood of ζ = 0, and by
+(4.12), it satisfies ϕc
+2(z) = ˜ϕc
+2((zmax
+c
+− z)
+1
+2 ) in a neighborhood of z = zmax
+c
+. If ¯η′
+1(θ∗
+2) < −1, the
+point ζ = 0 is a pole of ˜ϕc
+2(ζ) with at most order one; if ¯η′
+1(θ∗
+2) = −1, it is a pole of ˜ϕc
+2(ζ) with at
+most order two.
+Finally, we consider about ϕc(z). In the same way as that used for ϕc
+2(z), we can see that
+there exists a vector function ˜ϕc
+1(ζ) being elementwise meromorphic in a neighborhood of ζ = 0
+and satisfying ϕc
+1(z) = ˜ϕc
+1((zmax
+c
+− z)
+1
+2 ) in a neighborhood of z = zmax
+c
+. If ¯η′
+2(θ∗
+1) < −1, the point
+ζ = 0 is a pole of ˜ϕc
+1(ζ) with at most order one; if ¯η′
+2(θ∗
+1) = −1, it is a pole of ˜ϕc
+1(ζ) with at most
+order two. Define ˜ϕc(ζ) as
+˜ϕc(ζ) = ˜ϕc
+0(ζ) + ˜ϕc
+1(ζ) + ˜ϕc
+2(ζ).
+Then, the vector function ˜ϕc(ζ) is elementwise meromorphic in a neighborhood of ζ = 0, and by
+(4.6), it satisfies ϕc(z) = ˜ϕc((zmax
+c
+− z)
+1
+2 ) in a neighborhood of z = zmax
+c
+. If ˜η′
+1(θ∗
+2) < −c1/c2 =
+−1 < 1/˜η′
+2(θ∗
+1), the point ζ = 0 is a pole of ˜ϕc(ζ) with at most order one; if ˜η′
+1(θ∗
+2) = −1 or
+˜η′
+2(θ∗
+1) = −1, it is a pole of ˜ϕc(ζ) with at most order two. This completes the proof.
+22
+
+Proof of Proposition 4.3. Assume Type 1. By Proposition 4.6 and equations (4.10) and (4.13),
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)Φc
+x,∗(z) = O.
+(4.33)
+Hence, by (4.7),
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)ϕc
+0(z) = 0⊤.
+(4.34)
+If ¯η′
+1(θ∗
+2) = −1, by Propositions 4.6 and 4.9 and equations (4.12) and (4.34), representing ˆuU(zmax
+c
+)
+in block form as ˆuU(zmax
+c
+) =
+�
+ˆuU
+1 (zmax
+c
+)
+ˆuU
+2 (zmax
+c
+)
+�
+, we obtain
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)ϕc
+2(z) = uc
+2 = ˆga
+2ˆgΦˆuG(zmax
+c
+)ˆvU(zmax
+c
+)ˆuU
+1 (zmax
+c
+) > 0⊤,
+(4.35)
+where ˆuG(zmax
+c
+) is nonzero and nonnegative and other terms on the right-hand side of the equation
+are positive; if ¯η′
+1(θ∗
+2) < −1, we have
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)ϕc
+2(z) = 0⊤.
+(4.36)
+In a manner similar to that used for ϕc
+2(z), we can see that if ¯η′
+2(θ∗
+1) = −1, then for some positive
+vector uc
+1,
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)ϕc
+1(z) = uc
+1,
+(4.37)
+and if ¯η′
+1(θ∗
+2) < −1,
+lim
+˜∆zmax
+c
+∋z→zmax
+c
+(zmax
+c
+− z)ϕc
+1(z) = 0⊤.
+(4.38)
+As a result, by (4.6), (4.35), (4.36), (4.37) and (4.38), we obtain (4.5) in Proposition 4.3.
+5
+Concluding remarks
+We consider another topic, which relates to the singularity of the vector generating function ϕc(z)
+at z = zmax
+c
+= eθmax
+c
+, where c ∈ N2.
+Recall that P {1,2} = (P {1,2}
+x,x′ ; x, x′ ∈ Z2) is the transition probability matrix of the induced
+MA-process {Y {1,2}
+n
+} and Φ{1,2} = (Φ{1,2}
+x,x′ ; x, x′ ∈ Z2) the fundamental matrix (potential matrix)
+of P {1,2}. Let hΦ
+c (k) be the asymptotic decay function of the matrix sequence {Φ{1,2}
+x,kc ; k ∈ N}, i.e.,
+for some positive matrix C,
+lim
+k→∞ Φ{1,2}
+x,kc /hΦ
+c (k) = C.
+(5.1)
+By Proposition 4.6, we obtain
+hΦ
+c (k) = k− 1
+2 e−θmax
+c
+k.
+(5.2)
+Furthermore, recall that P + is a partial matrix of P {1,2} given by restricting the state space of
+the level to the positive quadrant, i.e., P + = (P {1,2}
+x,x′ ; x, x′ ∈ N2). P + is also a partial matrix of
+the transition probability matrix of the original 2d-QBD process, P = (Px,x′; x, x′ ∈ Z2
++), i.e.,
+P + = (Px,x′; x, x′ ∈ N2). Let ˜Q = ( ˜Qx,x′; x, x′ ∈ N2) be the fundamental matrix of P +, i.e.,
+23
+
+˜Q = �∞
+n=0(P +)n. For j, j′ ∈ S0, denote by ˜q(x,j),(x′,j′) the (j, j′)-entry of ˜Qx,x′. The entries of ˜Q
+are called an occupation measure in [13]. By Theorem 5.1 of [13], the asymptotic decay rate of the
+matrix sequence { ˜Qx,kc; k ∈ N} is given by eθmax
+c
+, i.e.,
+− lim
+k→∞
+1
+k log ˜q(x,j),(kc,j′) = θmax
+c
+,
+(5.3)
+which coincides with that of the matrix sequence {Φ{1,2}
+x,kc ; k ∈ N}. One question, therefore, arises:
+Does the asymptotic decay function of the matrix sequence { ˜Qx,kc; k ∈ N} coincide with that of
+the matrix sequence {Φ{1,2}
+x,kc ; k ∈ N}? If the answer to the question is yes, we can indicate that the
+vector generating function ϕc(z) diverges at z = eθmax
+c
+.
+References
+[1] Bini, D.A., Latouche, G. and Meini, B., Oxford University Press, Oxford (2005).
+[2] Flajolet, P. and Sedgewick, R., Analytic Combinatorics, Cambridge University Press, Cam-
+bridge (2009).
+[3] Gohberg, I., Lancaster, P. and Rodman, L., Matrix Polynomials, SIAM, Philadelphia (2009).
+[4] Kobayashi, M. and Miyazawa, M., Revisit to the tail asymptotics of the double QBD process:
+Refinement and complete solutions for the coordinate and diagonal directions, Matrix-Analytic
+Methods in Stochastic Models (2013), 145-185.
+[5] Latouche, G. and Ramaswami, V., Introduction to Matrix Analytic Methods in Stochastic
+Modeling, SIAM, Philadelphia (1999).
+[6] Malyshev, V.A., Asymptotic behavior of the stationary probabilities for two-dimensional pos-
+itive random walks, Siberian Mathematical Journal 14(1) (1973), 109–118.
+[7] Neuts, M.F., Matrix-Geometric Solutions in Stochastic Models, Dover Publications, New York
+(1994).
+[8] Neuts, M.F., Structured stochastic matrices of M/G/1 type and their applications, Marcel
+Dekker, New York (1989).
+[9] Ney, P. and Nummelin, E., Markov additive processes I. Eigenvalue properties and limit the-
+orems, The Annals of Probability 15(2) (1987), 561–592.
+[10] Ozawa, T., Asymptotics for the stationary distribution in a discrete-time two-dimensional
+quasi-birth-and-death process, Queueing Systems 74 (2013), 109–149.
+[11] Ozawa, T. and Kobayashi, M., Exact asymptotic formulae of the stationary distribution of a
+discrete-time two-dimensional QBD process, Queueing Systems 90 (2018), 351-403.
+[12] Ozawa, T., Stability condition of a two-dimensional QBD process and its application to esti-
+mation of efficiency for two-queue models, Performance Evaluation 130 (2019), 101–118.
+[13] Ozawa,
+T.,
+Asymptotic properties of the occupation measure in a multidimensional
+skip-free
+Markov
+modulated
+random
+walk,
+Queueing
+Systems
+97
+(2021),
+125–161.
+(DOI:10.1007/s11134-020-09673-9)
+24
+
+[14] Ozawa,
+T.,
+Tail
+Asymptotics
+in
+any
+direction
+of
+the
+stationary
+distribution
+in
+a
+two-dimensional discrete-time QBD process,
+Queueing Systems
+102 (2022),
+227–267.
+(DOI:10.1007/s11134-022-09860-w)
+[15] E. Seneta: Non-negative Matrices and Markov Chains, revised printing. Springer-Verlag, New
+York (2006).
+A
+Proof of Theorem 3.1
+First, we give the generalized eigenvectors of G(z) for z ∈ ∆zmain
+1
+,zmax
+1
+\E1, then analytically extend
+them to z ∈ C \ E1.
+For each k ∈ {1, 2, ..., m0} and for each z ∈ Ω \ �m0
+k=1 EG
+k , since the Jordan normal form of G(z)
+is given by (3.4), there exist linearly independent vectors called the generalized eigenvectors of G(z)
+with respect to the eigenvalue ˇαk(z), ˇvk,i,j(z), i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i, satisfying
+(ˇαk(z)I − G(z))ˇvk,i,j(z) = ˇvk,i,j+1(z),
+(A.1)
+where ˇvk,i,mk,i+1(z) = 0.
+For each i, ˇvk,i,j(z), j = 1, 2, ..., mk,i, are called a Jordan sequence
+of the generalized eigenvectors.
+Using the Jordan sequences, we define lˇq(k) × 1 block vectors,
+vk,i,j(z), i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i, as
+vk,i,j(z) = vec
+�ˇvk,i,j(z)
+ˇvk,i,j+1(z)
+· · ·
+ˇvk,i,mk,i(z)
+0
+· · ·
+0�
+,
+where, for a matrix A =
+�
+a1
+a2
+· · ·
+an
+�
+, vec(A) is the column vector given by
+vec(A) =
+�
+�
+�
+�
+�
+a1
+a2
+...
+an
+�
+�
+�
+�
+� .
+We also define a vector space VG
+k (z) as
+VG
+k (z) = span {vk,i,j(z) : i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i}.
+Note that the generalized eigenvectors ˇvk,i,j(z) are not unique but VG
+k (z) is. Since the generalized
+eigenvectors are linearly independent, vk,i,j(z) are also linearly independent and we have
+dim VG
+k (z) =
+mk,0
+�
+i=1
+mk,i = lˇq(k).
+For k ∈ {1, 2, ..., m0}, define an lˇq(k) × lˇq(k) block matrix function ΛG
+k (z) as
+ΛG
+k (z) =
+�
+�
+�
+�
+�
+�
+�
+ˇαk(z)I − G(z)
+−I
+ˇαk(z)I − G(z)
+−I
+...
+...
+ˇαk(z)I − G(z)
+−I
+ˇαk(z)I − G(z)
+�
+�
+�
+�
+�
+�
+�
+.
+We give the following proposition.
+Proposition A.1. For each k ∈ {1, 2, ..., m0} and for each z ∈ Ω \ �m0
+k=1 EG
+k ,
+Ker ΛG
+k (z) = VG
+k (z).
+(A.2)
+25
+
+Proof. Assume v ∈ VG
+k (z).
+Then, by the definition of VG
+k (z), we have ΛG
+k (z)v = 0 and v ∈
+Ker ΛG
+k (z). For v = vec
+�v1
+v2
+· · ·
+vlˇq(k)
+�
+, assume ΛG
+k (z)v = 0. If there exists an index i such
+that vi = 0, then by the assumption, for every j such that i ≤ j ≤ lˇq(k), we have vj = 0, and this
+implies v ∈ VG
+k (z).
+By Theorem S6.1 of [3], since the matrix function ΛG
+k (z) is entry-wise analytic in ∆zmin
+1
+,zmax
+1
+\E1,
+there exist lˇq(k) vector functions vG
+k,i(z), i = 1, 2, ..., lˇq(k), that are elementwise analytic and linearly
+independent in ∆zmin
+1
+,zmax
+1
+\ E1 and satisfy
+ΛG
+k (z)vG
+k,i(z) = 0, i = 1, 2, ..., lˇq(k).
+Hence, for each z ∈ Ω \ �m0
+k=1 EG
+k , vG
+k,i(z) ∈ VG
+k (z). We select the vectors composed of the Jordan
+sequences from {vG
+k,i(z), i = 1, 2, ..., lˇq(k)}. Represent each vG
+k,i(z) in block form as
+vG
+k,i(z) = vec
+�
+vG
+k,i,1(z)
+vG
+k,i,2(z)
+· · ·
+vG
+k,i,lˇq(k)(z)
+�
+.
+From the proof of Proposition A.1, we see that, for every i ∈ {1, 2, ..., lˇq(k)}, there exists a positive
+integer µk,i such that vG
+k,i,j(z) ̸= 0 for every j ∈ {1, 2, ..., µk,i} and vG
+k,i,j(z) = 0 for every j ∈
+{µk,i + 1, µk,i + 2, ..., lˇq(k)}. Renumber the elements of {vG
+k,i(z)} so that if i ≤ i′, then µk,i ≥ µk,i′.
+Define a set of vector functions, ˇVk, according to the following procedure.
+(S1) Set ˇVk = ∅ and i = 1.
+(S2) If vG
+k,i,µk,i(z) is linearly independent of {vG
+k,i′,µk,i′(z) : vG
+k,i′(z) ∈ ˇVk}, append vG
+k,i(z) to ˇVk.
+(S3) If i = lˇq(k), stop the procedure; otherwise add 1 to i and go to (S2).
+Proposition A.2. For k ∈ {1, 2, ..., m0}, the number of elements of ˇVk is mk,0.
+Proof. Since, for every i ∈ {1, 2, ..., lˇq(k)}, (ˇαk(z)I−G(z))vG
+k,i,µk,i = 0 and dim Ker (ˇαk(z)I−G(z)) =
+mk,0, the number of elements of ˇVk is less than or equal to mk,0. If it is strictly less than mk,0, we
+have
+dim Ker ΛG
+k (z) = dim span {vG
+k,i(z), i = 1, 2, ..., lˇq(k)} < dim VG
+k (z).
+This contradicts (A.2), and we see that the number of elements of ˇVk is just mk,0.
+Denote by ˇvG
+k,1(z), ˇvG
+k,2(z), ..., ˇvG
+k,mk,0(z) the elements of ˇVk. For i ∈ {1, 2, ...mk,0}, define ˇµk,i in
+a manner similar to that used for defining µk,i. We assume ˇvG
+k,i(z), i = 1, 2, ..., mk,0, are numbered
+so that if i ≤ i′, then ˇµk,i ≥ ˇµk,i′.
+Proposition A.3. For k ∈ {1, 2, ..., m0} and for i ∈ {1, 2, ..., mk,0}, ˇµk,i = mk,i
+Proof. For each i ∈ {1, 2, ..., mk,0}, {ˇvG
+k,i,1(z), ˇvG
+k,i,2(z), ..., ˇvG
+k,i,µk,i(z)} is a Jordan sequence of the
+generalized eigenvectors of G(z) with respect to the eigenvalue ˇαk(z).
+Hence, considering the
+procedure defining ˇvG
+k,i(z), we see that, for every i ∈ {1, 2, ..., mk,0}, ˇµk,i ≤ mk,i. Suppose there
+exists some i0 ∈ {1, 2, ..., mk,0} such that ˇµk,i = mk,i for every i ∈ {1, 2, ..., i0 −1} and ˇµk,i0 < mk,i0.
+Then, there exists a vector v = vec
+�v1
+v2
+· · ·
+vmk,i0
+0
+· · ·
+0�
+in VG
+k (z) such that vi ̸= 0
+for every i ∈ {1, 2, ..., mk,i0} and v is linearly independent of {vG
+k,i(z), i = 1, 2, ..., lˇq(k)}. By the
+same reason as that used in the proof of Proposition A.2, this contradicts (A.2) and, for every
+i ∈ {1, 2, ..., mk,0}, ˇµk,i must be mk,i.
+26
+
+From this proposition, we see that, for z ∈ Ω \ �m0
+k=1 EG
+k , {ˇvG
+k,i,j(z) : k = 1, 2, ..., m0, i =
+1, 2, ..., mk,0, j = 1, 2, ..., mk,i} is the set of generalized eigenvectors corresponding to the Jordan
+normal form (3.4). Define a matrix function T G(z) as
+T G(z) =
+�ˇvG
+k,i,j(z), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i
+�
+,
+which is entry-wise analytic in ∆zmin
+1
+,zmax
+2
+\ E1. Define a point set EG
+T as
+EG
+T = {z ∈ ∆zmin
+1
+,zmax
+2
+\ E1 : det T G(z) = 0},
+which is an empty set or a set of discrete complex numbers. Then, for z ∈ Ω \ (�m0
+k=1 EG
+k ∪ EG
+T ), we
+obtain the Jordan decomposition of G(z) as
+G(z) = T G(z)JG(z)(T G(z))−1.
+(A.3)
+Since G(z) is entry-wise analytic in ∆zmin
+1
+,zmax
+1
+, we see by the identity theorem for analytic functions
+that the right hand side of (A.3) is also entry-wise analytic in the same domain.
+Next, we analytically extend ˇvG
+k,i,j(z), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i. Define
+matrix functions F1(z, w) and F2(z) as
+F1(z, w) = z(I − A∗,0(z) − 2wA∗,1(z)),
+F2(z) = zA∗,1(z),
+where F1(z, w) is entry-wise analytic on C2 and F2(z) on C. By (3.2), we have
+L(z, w) = F1(z, w)(wI − G(z)) + F2(z)(wI − G(z))2.
+(A.4)
+For k ∈ {1, 2, ..., m0}, define a lˇq(k) × lˇq(k) block matrix function ΛL
+k,n(z) as
+ΛL
+k (z) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+L(z, ˇαk(z))
+−F1(z, ˇαk(z))
+−F2(z)
+L(z, ˇαk(z))
+−F1(z, ˇαk(z))
+−F2(z)
+...
+...
+...
+L(z, ˇαk(z))
+−F1(z, ˇαk(z))
+−F2(z)
+L(z, ˇαk(z))
+−F1(z, ˇαk(z))
+L(z, ˇαk(z))
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+which is entry-wise analytic in C \ E1.
+Proposition A.4. For every k ∈ {1, 2, ..., m0} and for every z ∈ ¯∆zmin
+1
+,zmax
+1
+,
+Ker ΛL
+k (z) = Ker ΛG
+k (z).
+(A.5)
+Before proving this proposition, we give another one.
+Proposition A.5. For every k ∈ {1, 2, ..., s0} and z ∈ ∆zmin
+1
+,zmax
+1
+,
+F1(z, αk(z)) + F2(z)(αk(z)I − G(z)) = z (I − A∗,0(z) − αk(z)A∗,1(z) + A∗,1(z)G(z))
+is regular (invertible).
+Proof. Let R(z) be the rate matrix function generated from {Ai,j; i, j = −1, 0, 1}; for the definition
+of R(z), see Subsection 4.1 of [11]. By Lemma 4.3 of [11], nonzero eigenvalues of R(z) are given by
+αk(z)−1, k = s0 + 1, s0 + 2, ..., mφ. Since, for every k ∈ {1, 2, ..., s0}, k′ ∈ {s0 + 1, s0 + 2, ..., mφ}
+and z ∈ ∆zmin
+1
+,zmax
+1
+, |αk(z)| ≤ αs0(|z|) < |αk′(z)|, I − αk(z)R(z) is regular.
+Define a matrix
+function H(z) as H(z) = A∗,0(z) + A∗,1(z)G(z), then by Corollary 4.1 of [11], I − H(z) is regular
+in ∆zmin
+1
+,zmax
+1
+. By Lemma 4.1 of [11], we have
+I − A∗,0(z) − αk(z)A∗,1(z) − A∗,1(z)G(z) = (I − αk(z)R(z))(I − H(z)),
+(A.6)
+and this implies the assertion of the proposition.
+27
+
+Proof of Proposition A.4. Assume a vector v = vec
+�v1
+v2
+· · ·
+vlˇq(k)
+�
+satisfies ΛL
+k (z)v = 0.
+Then, we have for i ∈ {1, 2, ..., lˇq(k)} that
+L(z, ˇαk(z))vi = F1(z, ˇαk(z))vi+1 + F2(z)vi+2,
+(A.7)
+where vlˇq(k)+1 = vlˇq(k)+2 = 0. We prove by induction that this v satisfies, for every i ∈ {1, 2, ..., lˇq(k)},
+(ˇαk(z)I − G(z))vi = vi+1. Let i0 be the maximum integer less than or equal to lˇq(k) that satisfies,
+for every i ∈ {i0 + 1, i0 + 2, ..., lˇq(k)}, vi = 0. Then, we have L(z, ˇαk(z))vi0 = 0. By (A.4), we have
+L(z, ˇαk(z)) = (F1(z, ˇαk(z)) + F2(z)(ˇαk(z)I − G(z)))(ˇαk(z)I − G(z)).
+(A.8)
+Hence, by Proposition A.5, we obtain (ˇαk(z)I − G(z))vi0 = 0 = vi0+1. Assume the assumption of
+induction holds for a positive integer i less than or equal to i0. Then,
+L(z, ˇαk(z))vi−1 = F1(z, ˇαk(z))vi + F2(z)vi+1
+= (F1(z, ˇαk(z)) + F2(z)(ˇαk(z)I − G(z)))vi,
+(A.9)
+and by (A.8), (A.9) and Proposition A.5, we obtain (ˇαk(z)I − G(z))vi−1 = vi. Hence, v satisfies,
+for every i ∈ {1, 2, ..., lˇq(k)}, (ˇαk(z)I − G(z))vi = vi+1, and this leads us to ΛG
+k (z)v = 0.
+Next, assume a vector v = vec
+�v1
+v2
+· · ·
+vlˇq(k)
+�
+satisfies ΛG
+k (z)v = 0. Then, we have for
+i ∈ {1, 2, ..., lˇq(k)} that (ˇαk(z)I − G(z))vi = vi+1, where vlˇq(k)+1 = 0. By (A.8), this v satisfies, for
+every i ∈ {1, 2, ..., lˇq(k)},
+L(z, ˇαk(z))vi = F1(z, ˇαk(z))vi+1 + F2(z)(ˇαk(z)I − G(z))vi+1
+= F1(z, ˇαk(z))vi+1 + F2(z)vi+2,
+(A.10)
+and this implies ΛL
+k (z)v = 0.
+Let k be an arbitrary integer in {1, 2, ..., m0}. By Propositions A.1 and A.4, we have
+dim Ker ΛL
+k (z) = lˇq(k),
+except for some discrete points in C. Hence, by Theorem S6.1 of [3], since the matrix function
+ΛL
+k (z) is entry-wise analytic in C\E1, there exist lˇq(k) vector functions vL
+k,i(z), i = 1, 2, ..., lˇq(k), that
+are elementwise analytic and linearly independent in C \ E1 and satisfy
+ΛL
+k (z)vL
+k,i(z) = 0, i = 1, 2, ..., lˇq(k).
+By Proposition A.4, for each i, vL
+k,i(z) also satisfies ΛG
+k (z)vL
+k,i(z) = 0 for every z ∈ ∆zmin
+1
+,zmax
+1
+\ E1.
+Hence, by the identity theorem, we see that vL
+k,i(z) is an analytic extension of vG
+k,i(z). By the same
+procedure as that used for selecting {ˇvG
+k,i(z), i = 1, 2, ..., mk,0} from {vG
+k,i(z), i = 1, 2, ..., lˇq(k)}, we
+select mk,0 vectors from {vL
+k,i(z), i = 1, 2, ..., lˇq(k)} and denote them by {ˇvL
+k,i(z), i = 1, 2, ..., mk,0}.
+For each i, ˇvL
+k,i(z) is represented in block form as
+ˇvL
+k,i(z) = vec
+�
+ˇvL
+k,i,1(z)
+ˇvL
+k,i,2(z)
+· · ·
+ˇvL
+k,i,mk,i(z)
+0
+· · ·
+0
+�
+.
+Define a matrix function T L(z) as
+T L(z) =
+�ˇvL
+k,i,j(z), k = 1, 2, ..., m0, i = 1, 2, ..., mk,0, j = 1, 2, ..., mk,i
+�
+,
+which is entry-wise analytic in C \ E1. Since each ˇvL
+k,i,j(z) is an analytic extension of ˇvG
+k,i,j(z), we
+have for z ∈ Ω \ (�m0
+k=1 EG
+k ∪ EG
+T ) that
+G(z) = T L(z)JG(z)(T L(z))−1,
+which is (3.5). Set E0 as E0 = E2 ∪ (�m0
+k=1 EG
+k ) ∪ EG
+T , then E0 is a set of discrete complex numbers
+and we have Ω\(�m0
+k=1 EG
+k ∪EG
+T ) = ∆zmin
+1
+,zmax
+1
+\(E1 ∪E0). This completes the proof of Theorem 3.1.
+28
+

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@@ -0,0 +1,1020 @@
+Reversal of quantised Hall drifts at non-interacting and interacting topological
+boundaries
+Zijie Zhu,∗ Marius G¨achter,∗ Anne-Sophie Walter, Konrad Viebahn,† and Tilman Esslinger
+Institute for Quantum Electronics & Quantum Center, ETH Zurich, 8093 Zurich, Switzerland
+The transport properties of gapless edge modes at boundaries between topologically distinct
+domains are of fundamental and technological importance. Therefore, it is crucial to gain a better
+understanding of topological edge states and their response to interparticle interactions.
+Here,
+we experimentally study long-distance quantised Hall drifts in a harmonically confined topological
+pump of non-interacting and interacting ultracold fermionic atoms. We find that quantised drifts
+halt and reverse their direction when the atoms reach a critical slope of the confining potential,
+revealing the presence of a topological boundary. The drift reversal corresponds to a band transfer
+between a band with Chern number C = +1 and a band with C = −1 via a gapless edge mode,
+in agreement with the bulk-edge correspondence for non-interacting particles. We establish that a
+non-zero repulsive Hubbard interaction leads to the emergence of an additional edge in the system,
+relying on a purely interaction-induced mechanism, in which pairs of fermions are split.
+The existence of individual edge modes at topo-
+logical boundaries plays a crucial role in quantum Hall
+physics. More specifically, a non-trivial topology in the
+bulk of a material ensures that its edge modes are gapless
+and chiral. Gaplessness is related to the bulk-edge corre-
+spondence, stating that the number of topological edge
+modes is equal to the difference in Chern number across
+an interface [1].
+Consequently, a gapless mode should
+allow an adiabatic transfer from one band to another,
+resulting in a reflection of transverse bulk currents in
+the opposite direction if the two bands feature opposite
+Chern numbers. However, the coherence time in most
+electronic materials is not sufficient to observe this effect,
+and edges are generally probed spectroscopically [1–4].
+Moreover, studies of edge physics in engineered quantum
+systems, such as ultracold atoms and photonics, have so
+far been focussed on chirality [5–9] and localisation [10–
+13]. A boundary reflection has not been detected [14, 15],
+or it was disregarded [16, 17], and to our knowledge it
+has never been studied for variable interaction strength.
+Here, we observe the reversal of quantised bulk drifts due
+to harmonic trapping in a topological Thouless pump,
+the temporal analogue of the quantum Hall effect [18–
+20].
+The reflection is a fundamental manifestation of
+confined topological matter and directly shows the gap-
+less nature of topological edge modes. Going beyond the
+non-interacting regime, we discover the emergence of a
+second edge for repulsive Hubbard U.
+The experiments are performed
+with
+ultracold
+fermionic potassium-40 atoms, which are loaded into the
+potential of a generalised optical lattice formed by a com-
+bination of standing and running waves of wavelength
+λ = 1064 nm [22]. This creates an array of decoupled,
+one-dimensional tubes.
+Along the tube direction, the
+periodically modulated Rice-Mele-Hubbard Hamiltonian
+with harmonic confinement is realised,
+ˆH(τ) = −
+�
+j,σ
+�
+t + (−1)jδ(τ)
+� �
+ˆc†
+jσˆcj+1σ + h.c.
+�
+(1)
++ ∆(τ)
+�
+j,σ
+(−1)jˆc†
+jσˆcjσ + U
+�
+j
+ˆc†
+j↑ˆcj↑ˆc†
+j↓ˆcj↓
++
+�
+j,σ
+Vjˆc†
+jσˆcjσ ,
+where ˆcjσ is the fermionic annihilation operator for spin
+σ ∈ {↑, ↓} on site j, and t denotes the average tun-
+nelling.
+An adiabatic modulation of bond dimerisa-
+tion δ(τ) = δ0 cos(2πτ/T) and sublattice offset ∆(τ) =
+∆0 sin(2πτ/T) traces a closed trajectory in the δ–∆ plane
+around the origin, referred to as critical point. There-
+fore, an insulator or homogeneously filled band at U = 0
+describes a topological pump [18, 20] with T being the
+pump period.
+Experimentally, the topological pump
+manifests itself as a quantised drift of the atom position
+by one unit cell per pump cycle [23–27].
+The harmonic confinement is characterised by the
+trap frequency ν, entering Eq. 1 as Vj = 1
+2m(2πνaj)2 ≡
+V0j2 (a = λ/2, lattice spacing; m, atomic mass). Due to
+the confinement, the atoms are initially localised at the
+centre of the trap. Topological pumping then leads to a
+quantised drift of atoms against the confining potential
+(Fig. 1A). Our measurements show that the quantised
+drift changes its direction at a certain distance from the
+trap centre. We will demonstrate that this happens when
+the gradient of the confinement overcomes the band gap
+and a boundary between topological and trivial regions
+emerges. For repulsive interactions, we observe another
+reflection, closer to the trap centre, while a part of the
+atoms keeps drifting in the original direction (Fig. 1A).
+In the following, we develop a description of the re-
+flection in terms of gapless edge modes and the bulk-
+edge correspondence within the framework of the Harper-
+Hofstadter-Hatsugai (HHH) model with one real (x) and
+one synthetic (n) dimension. The model features bulk
+arXiv:2301.03583v1  [cond-mat.quant-gas]  9 Jan 2023
+
+2
+Floquet
+Energy
+0
+-π
+π
+Quasimomentum      
+Gradient
+Interactions
+A
+B
+C
+FIG. 1. Reflection of quantised Hall drifts off a topo-
+logical interface. (A) Topological Thouless pumping in the
+presence of confining potentials. In the non-interacting case
+(top), a harmonic trap gives rise to topological trivial (C = 0)
+and non-trivial (C ̸= 0) regions, separated by a topological
+interface. The atoms exhibit a quantised drift until they are
+reflected at the interface. With repulsive on-site interactions
+(bottom), the reflection happens closer to the centre, accom-
+panied by atoms still drifting in the original direction. (B)
+Using Floquet theory, the 1D Rice-Mele pump can be mapped
+to a 2D Harper-Hofstadter-Hatsugai model with a linear gra-
+dient along the synthetic dimension n which represents the
+photon number. The magnetic flux per plaquette is Φ = 1/2
+in units of the magnetic flux quantum [21]. The gradient along
+n leads to a transverse Hall drift along x (red arrows) due to
+the nontrivial topology of the bands.
+(C) Schematic spec-
+trum of the mapped 2D Hofstadter model in a semi-infinite
+geometry. The lowest two bands have C = ±1, respectively.
+The linear gradient induces Bloch oscillations in the synthetic
+reciprocal space (dashed arrows). A gapless edge mode (solid
+arrow) appears at the topological interface. The reflection of
+the Hall drift can be understood as atoms being transported
+from the lower band (C = 1) to the higher band (C = −1)
+via the topological edge mode.
+Chern bands with C = +1 and C = −1.
+An ex-
+act mapping between the non-interacting 1D Rice-Mele
+Hamiltonian (Eq. 1) and the two-dimensional (2D) HHH
+model can be obtained using Floquet theory, illustrated
+in Fig. 1B (for derivation see, e.g., refs. [19] and [21]). A
+linear gradient along the synthetic dimension n appears
+in the mapping since the state with n photons acquires
+an energy of −nℏω, where ω = 2π/T is the pump fre-
+quency. The gradient along n or, equivalently, an exter-
+nal force causes Bloch oscillations along the synthetic re-
+ciprocal dimension kn which, in turn, lead to a Hall drift
+or ‘anomalous velocity’ along the transverse real direc-
+tion x [14, 15]. The bulk Hall drift along x corresponds
+exactly to the quantised displacement measured in the
+topological pump. The trap induces a boundary between
+topological (Ccentre = 1) and trivial (Cright = 0) regions
+and a single gapless edge mode emerges, according to the
+bulk-edge correspondence: Ccentre −Cright = 1. The edge
+modes connects two bands of opposite Chern invariant,
+as shown in Fig. 1C. Thus, a Bloch oscillation transfers
+the atoms from the ground to the first excited band via
+that edge mode. Since the first excited band has Chern
+number −1 the atoms are now moving ‘backwards’, re-
+sulting in a reversal of the quantised Hall drift.
+Fig. 2 shows experimental in-situ images of the
+atomic cloud as a function of time τ at U = 0.
+The
+data shows a quantised drift of 1.00(1) × 2a/T up to
+about 60 T, which confirms the long coherence time of
+Bloch oscillations which induce the transverse drift. At
+τ ≃ 75 T the atoms change their drift direction, which
+is a key observation of this work. The expected topo-
+logical boundary (red dashed line) represents the posi-
+tion at which the local tilt from the external harmonic
+potential ∆ext (j) ≡
+1
+2 |Vj − Vj−1| = V0
+��j − 1
+2
+�� equals
+the maximum sublattice offset ∆0, thus, xedge/ (2a) ≃
+1
+2∆0/V0 = 92(7).
+Beyond this position the total sub-
+lattice offset ceases to change sign, rendering the region
+outside xedge topologically trivial. The boundary caused
+by the harmonic confinement is not infinitely sharp, but
+smoothened over several lattice sites.
+This leads to a
+small T–dependence of the reflection position (Fig. S1),
+compared to its absolute value, and the calculation above
+should be understood as the outermost point of the re-
+flective region. The reflected atoms exhibit a quantised
+drift of −0.99(3) × 2a/T in the opposite direction, in
+agreement with a transfer to the first excited band with
+C = −1.
+The linear relation between the position of
+topological boundary xedge and the maximum sublattice
+offset ∆0 is further confirmed by measuring the reflection
+in different lattices (Fig. S1). The reflection is observed
+under all parameter settings tested in this work, high-
+lighting that the existence of the topological boundary is
+robust.
+In addition to the reflection, we also observe a cloud
+of atoms temporarily remaining at the boundary before
+gradually dissolving. This process can be understood via
+the presence of topologically trivial edge states, which
+hybridise with the gapless edge modes. To simplify the
+picture, let us consider a sharp domain wall between
+C = 1 and C = 0 (Fig. 2C). According to the bulk-
+edge correspondence, the topologically nontrivial region
+contributes exactly one gapless mode whereas the trivial
+region can contribute gapped edge modes. Due to tunnel
+coupling at the interface, hybridisation takes place [34]
+and gaps on the order of the pump frequency 2π/T
+emerge. Bloch oscillations along kn can now lead to non-
+adiabatic ‘Landau-Zener’ transfers between topological
+and trivial edge modes, causing an incomplete transfer
+to the higher band, and atoms remaining at the bound-
+ary.
+Subsequent Bloch oscillations will transfer atoms
+back into the topological domain, leading to the dissolu-
+
+3
+quasimomentum
+energy
+trivial
+nontrivial
+0
+36
+108
+72
+144
+time τ (T)
+1st
+2nd
+2nd
+2nd
+2nd
+Brillouin zone
+Dimerisation
+Site ofset
+A
+C
+B
+FIG. 2. Measuring the reversal of a quantised Hall drift. (A) The time-trace of atomic in-situ images shows a quantised
+drift along x before the atoms are reflected at the topological boundary. Each density image is averaged over three individual
+measurements with the parameters V0 = 0.0148(9)t, ∆0 = 2.7(1)t, and T = 3 ms = 12.8(2)ℏ/t. The red dashed line indicates
+the topological boundary xedge/ (2a) ≈
+1
+2∆0/V0. The white dashed lines are linear fits to the atom drift, yielding slopes of
+1.00(1) × 2a/T before, and −0.99(3) × 2a/T after the reflection. Cloud positions, averaged over the transverse direction, are
+fitted using Gaussians. The experimental Chern marker (lower panel, points) is determined by the velocity of the right-moving
+cloud at different positions. The theoretical Chern marker (line) is calculated in a staggered potential Vj = V0 (−1)j j which
+has the same local tilt ∆ext(j) = V0
+��j − 1
+2
+�� as the harmonic trap [21]. In a local density approximation picture, the local tilt
+∆ext shifts the δ–∆ pump trajectory upwards. Depending on whether or not the trajectory encloses the critical point, the pump
+is rendered topological or trivial. (B) Measured band populations as a function of time τ. Each density image is averaged over
+six individual measurements with the parameters V0 = 0.0191(6)t, ∆0 = 3.2(2)t and T = 3 ms = 10.7(2)ℏ/t. The total atom
+number remains constant, within error bars, throughout the experiment. Due to the underlying honeycomb lattice geometry
+in the x–z plane, the first Brillouin zone has a diamond shape [21]. The band population is inverted when the bulk current
+is reflected off the topological interface, manifesting the gapless nature of the topological edge mode. (C) Hybridisation of
+the edge modes at the topological interface due to tunnelling. Bloch oscillations along kn can lead to Landau-Zener transfers
+between topologically trivial and nontrivial edge modes. The population of trivial edge modes explains the atoms being left at
+the boundary. Hybridisation never changes the total number of gapless edge modes at the boundary.
+tion of the cloud at the boundary. While the harmonic
+confinement leads to a more complex level structure [21],
+the underlying process remains qualitatively the same.
+We support the in-situ images with measurements
+of band population before, during, and after the reflec-
+tion, as shown in Fig. 2B [21]. Before the reflection, we
+find a filled ground band, which is consistent with the
+observation of a quantised Hall drift. At the reflection
+(τ ≃ 72 T), we observe an inversion of the population to
+the first excited band. After the reflection, the inverted
+population remains almost unchanged while the atoms
+are travelling back, highlighting the absence of incoher-
+ent relaxation to the ground band, even after more than
+a hundred Bloch oscillations.
+We further explore the effect of attractive and repul-
+sive interactions on the topological boundary. For attrac-
+tive Hubbard U = −3.48(7)t = 1.27(7)∆0 (Fig. 3A), the
+quantised Hall drift is reversed at the same position as
+in the non-interacting situation. This can be explained
+in terms of the Rice-Mele model in which fermions in the
+strongly attractive regime approach the limit of hard-core
+bosons
+[22], and the condition for the emergence of a
+topological boundary ∆ext (j) = ∆0 remains unchanged.
+For repulsive Hubbard U = 3.48(7)t (Fig. 3B), we
+observe a second reflection in addition to the original one.
+Compared to the non-interacting case, this reflection ap-
+pears much closer to the trap centre.
+The zoomed-in
+image (Fig. 3C) shows that a proportion of the atoms
+start to move backwards after about 12 cycles. In con-
+
+4
+Dimerisation
+Site ofset
+OD
+time т (T)
+B
+D
+E
+C
+A
+Time τ (   )
+T
+τ0
+τ0 + 0.5
+τ0 + 1
+FIG. 3.
+Reflection of quantised Hall drifts from an interacting topological edge.
+(A) An attractive Hubbard
+interaction of U = −3.48(7)t leads to the same reflection behaviour as observed for U = 0 (measurement parameters otherwise
+identical to Fig. 2A). (B) Repulsive Hubbard interactions of U = 3.48(7)t lead to the emergence of a second reflection, closer
+to the trap centre, which we attribute to an interacting topological boundary. A zoom-in (C) shows that the early reflection
+happens after about twelve cycles. The white dashed lines are guides to the eye, calculated as linear fits to the cloud position,
+extracted as the sum of a skewed and a regular Gaussian. (D) Microscopic description of the interaction-induced reflection for
+repulsive Hubbard U. When the maximum energy offset between two neighbouring sites 2 (∆0 − ∆ext) becomes smaller than
+the Hubbard interaction, formation of double occupancies is prohibited and one atom is left in the higher-energy site of a unit
+cell, which then drifts in the opposite direction. (E) The critical point in the δ–∆ plane is split into two in the presence of
+repulsive Hubbard U [35]. When the pump trajectory encloses both critical points, a quantised drift is expected, as in the
+non-interacting system. The local tilt given by the external potential ∆ext shifts the trajectory along the ∆–axis, eventually
+enclosing just one critical point. Since a single split critical point features half the topological charge of the orignal one, the
+material’s topology changes and a boundary emerges.
+trast to the drift reversal in the non-interacting system, a
+large fraction of the atom cloud still undergoes quantised
+drifting in the original direction.
+In the following, we
+develop a microscopic picture of the interaction-induced
+partial reflection in the limiting case of two isolated spins
+(↑, ↓), which approximates our initial state in a unit cell
+(Fig. 3D). As long as the maximum energy offset between
+two neighbouring sites 2 (∆0 − ∆ext) is larger than the
+Hubbard U, the formation of a double occupancy is en-
+ergetically allowed and the quantised drift persists [22],
+even in the presence of an external potential. However,
+when ∆ext becomes larger, the energy offset between two
+neighbouring sites remains always smaller than U, dou-
+ble occupancy formation becomes prohibited. In the lat-
+ter case, one atom of a singlet pair is transferred to the
+energetically excited site of a unit cell, which will subse-
+quently drift in the opposite direction. The other atom,
+in the lower-energy site, will move onwards because on-
+site interactions become irrelevant if there is only one
+atom per unit cell.
+Since the underlying Hamiltonian
+(Eq. 1) is SU(2)–symmetric, spin–↑ and spin–↓ have equal
+probability of being reflected and they should remain cor-
+related after the splitting process.
+The full many-body description of the interaction-
+induced reversal requires the development of suitable
+topological invariants for smooth confinements and
+strong interactions, which goes beyond the scope of this
+work. Nevertheless, we obtain an intuition of the bound-
+ary’s topological origins using again the idea of shifted
+pump trajectories in the δ–∆ plane with a staggered po-
+tential (c.f. Fig. 2A). Numerical simulations have shown
+that a repulsive Hubbard U can split the critical point at
+the origin into two separate ones [35], each retaining half
+the original topological charge.
+The distance between
+the new critical points is approximately U up to a cor-
+rection on the order of the tunnelling t [36]. When the
+trajectory encloses both critical points, quantised drift of
+two spins (↑, ↓) is expected, as in the non-interacting sys-
+tem. As the position-dependent local tilt ∆ext(j) shifts
+the trajectory upwards, it will enclose only one of the
+critical points beyond certain position along x (Fig. 3E).
+This indicates a transition of topological properties and
+the emergence of an interacting topological edge in real
+space.
+The estimation of the interacting boundary at
+∆ext(j) ≃ ∆0 − U/2 lies close to the centre and agrees
+with the microscopic picture discussed above. Similar to
+the non-interacting case, this boundary should be con-
+sidered as the outermost position of the reflective region.
+In conclusion, we have experimentally observed a re-
+versal of quantised Hall drifts at a topological boundary
+in a harmonic potential. The reflection is a direct mani-
+festation of the gapless nature of topological edge modes
+between Chern bands of opposite sign. We explore the
+effect of Hubbard interactions, both attractive and re-
+
+5
+pulsive, and find an asymmetric behavior with respect
+to U = 0. While on the attractive side the topological
+boundary is unaffected, repulsive interactions lead to the
+emergence of a second interface, featuring a splitting of
+quantised drifts. As a result, our experiments could en-
+able the realisation of circular current patterns for con-
+structing novel many-body phases [37].
+More broadly,
+our work allows the exploration of the bulk-edge corre-
+spondence in the presence of interactions [38], as well as
+the investigation of edge reconstruction [39] in the quan-
+tum Hall effect and in interacting topological insulators.
+ACKNOWLEDGMENTS
+We would like to thank Jason Ho, Gian-Michele Graf,
+Thomas Ihn, Fabian Grusdt, Fabian Heidrich-Meisner,
+Armando Aligia, and Eric Bertok for valuable discus-
+sions. We also thank Julian L´eonard and Nur ¨Unal for
+comments on an earlier version of the manuscript. We
+would like to thank Alexander Frank for his contribu-
+tions to the electronic part of the experimental setup.
+We acknowledge funding by the Swiss National Science
+Foundation (Grant Nos. 182650, 212168, NCCR-QSIT,
+and TMAG-2 209376) and European Research Council
+advanced grant TransQ (Grant No. 742579).
+∗ These authors contributed equally.
+† [email protected]
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+
+7
+SUPPLEMENTAL MATERIALS
+Dependence of the drift reversal on experimental
+parameters
+The expected drift reversal happens when the maxi-
+mum local site offset over one pump-cycle ∆0 is equal
+to the local tilt given by the harmonic trap. This po-
+sition given by xedge ≃ ∆0a/V0 with a = λ/2 and
+V0 = 1
+2m(2πνa)2. By measuring the reflection point in
+lattices with different ∆0, we verify the relevant scaling
+xedge ∝ ∆0, as shown in Fig. S1. The blue line in Fig.
+S1A marks the theoretically expected xedge with the un-
+certainty propagated from the uncertainty of the trap
+frequency ν. The disagreement between theory and ex-
+periment for larger values of ∆0 can be explained by the
+finite waist of the lattice beams. In order to explore the
+edge in our system, the atoms are pumped by almost 100
+unit cells (∼ 100 µm). Due to the Gaussian envelope of
+the transverse beams, which are essential to realise the
+pump, the lattice is effectively shallower far away from
+the centre.
+Thus, ∆0 decreases towards the edge and
+atoms are reflected sooner.
+We also find a small dependence of the reflection point
+on the pump period, compared to its absolute value,
+which spans roughly 10 unit cells when changing T from
+2 ms to 10 ms (Fig. S1B). This can be understood by con-
+sidering the energy spectrum of the Harper-Hofstadter-
+Hatsugai (HHH) model in a harmonic potential, which
+will be discussed below.
+Experimental sequence
+We start by preparing a degenerate cloud of fermionic
+40K in a crossed dipole trap. We have a spin mixture
+of mF = {−9/2, −7/2} except for the measurements in
+Fig. 2B and Fig. S1, where a spin-polarised cloud in the
+magnetic state F = 9/2, mF = −9/2 is used. The spin-
+polarised cloud is directly loaded into the pumping lat-
+tice, while the spin mixture is first loaded into an inter-
+mediate chequerboard lattice with strongly attractive in-
+teractions. The two-step loading precludes the presence
+of atoms in the higher band and gives a larger fraction
+of atoms in doubly occupied unit cells [22].
+After pumping the system for varying times, we either
+take a in-situ absorption image to measure the density
+or detect the band population with band-mapping. The
+latter is implemented with an exponential ramp to switch
+off the lattice beam in 500 µs, followed by a time-of-flight
+expansion of 25 ms before absorption imaging.
+A
+B
+FIG. S1.
+Experimental dependence of the reflection
+position. The reflection point is expected to depend linearly
+on the maximal site-offset per pump cycle ∆0 which is experi-
+mentally verified in (A). Deviations for large values of ∆0 can
+be explained by the finite waist of the laser beams forming
+the lattice. (B) shows the period dependence of the reflec-
+tion position. Changing the pump period T over an order of
+magnitude only changes the reflection point by about 10 unit
+cells, which is a result of the smooth boundary of a harmonic
+potential.
+Realisation of a Thouless pump in the Rice-Mele
+model
+The lattice setup is comprised of non-interfering stand-
+ing waves in x, y, and z directions, together with in-
+terfering laser beams in the x–z plane. All the lattice
+beams come from a single laser source at wavelength
+λ = 1064 nm. These potential combine to form a honey-
+comb lattice in the x–z plane, which can be considered
+as isolated tubes of one-dimensional superlattices along
+x in the limit of deep transverse lattices.
+In each 1D
+tube the potential can be modeled by a one-dimensional
+superlattice with two sites per unit cell. With this setup,
+we realise the Rice-Mele model [22]. In the tight-binding
+limit, the Rice-Mele model can be described with three
+numbers: the offset energy ∆ between the two sites of
+a unit cell, the averaged nearest-neighbour tunneling t
+and the bond dimerisation δ which gives half the differ-
+ence between the inter- and intra-dimer tunnellings. By
+having a dynamical control of the relative phase ϕ be-
+tween the laser beams generating the interfering and the
+non-interfering lattice, we manage to shift the two with
+respect to each other. This shift modulates ∆ and δ pe-
+riodically, which can be depicted as a closed trajectory
+in the ∆-δ coordinate (Fig. S2). In the adiabatic limit,
+this realises a Thouless pump with its hallmark quan-
+
+8
+tised transport. In this case, the atomic displacement is
+given by the number of revolutions around the origin of
+the ∆-δ plane.
+φ = 0
+φ = ∏/2
+φ = ∏
+φ = 3∏/2
+x
+E
+x
+E
+x
+E
+x
+E
+FIG. S2. Realisation of a Thouless pump in the Rice-
+Mele model. In our system the 1D lattice potential can be
+modelled by a superlattice with two sites per unit cell, which
+is depicted in the bottom part of this figure as a function of
+relative phase ϕ. The local site offset ∆ as well as the bond
+dimerisation δ is depicted in the ∆-δ plots corresponding to
+the respective potentials. The resulting pump displacement
+corresponds to the number of revolutions around the origin
+in the ∆-δ plane.
+Mapping a 1D Thouless pump to a 2D Hofstadter
+Model with quantum Hall response
+A 1D Thouless pump with a period of T, as realised
+in our experiment, can be mapped to a 2D topological
+tight-binding (HHH) model with an applied electric field
+E = 2πℏ
+qT where q can be thought of as a fictitious charge
+of netural atoms. Due to the topological bandstructure,
+this electric field leads to a transverse current Itrans = q
+T
+of one atom per period, when considering a fully occupied
+band. The 2D model therefore has a quantised transverse
+conductance σtrans = Itrans/E =
+q2
+2πℏ analogous to the
+Hall conductance in the Quantum Hall Effect (QHE).
+The time-periodicity of the Hamiltonian in Eq. 1 with
+ˆH(τ) = ˆH(τ + T) allows us to use Floquet’s theorem.
+Solutions of the time-dependent Schr¨odinger equation
+iℏ∂τ |Ψ(τ)⟩ = H(τ) |Ψ(τ)⟩
+(S1)
+can thus be written as
+|Ψ(τ)⟩ = e−iϵτ/ℏ |u(τ)⟩
+(S2)
+with |u(τ + T)⟩ = |u(τ)⟩ and ϵ ∈ R. Due to the time-
+periodicity of u(τ) we expand it as a Fourier series,
+|u(τ)⟩ =
+�
+n
+e−iωnτ |un⟩ ,
+(S3)
+where ω = 2π/T is the pump frequency.
+The change
+from the time-domain into the Fourier-domain is the key
+ingredient to map the 1D Thouless pump to a 2D tight-
+binding model.
+The index n is also called the photon
+number of the mode |un⟩.
+Using a multi-index α = (j, σ) we write the T-periodic
+1D Hamiltonian for U = 0 in the Fourier-basis:
+ˆH(τ) =
+�
+α,β
+hαβ(τ) |α⟩ ⟨β|
+(S4)
+=
+�
+α,β,m
+e−imωτhm
+αβ |α⟩ ⟨β|
+with hm
+αβ =
+1
+T
+� T
+0 eimωτhαβ(τ)dτ and |α⟩ corresponding
+to an atom localised on site j with spin σ.
+Likewise,
+we use Fourier decomposition to express the solutions to
+Eq. S1 as
+|Ψ(τ)⟩ = e−iϵτ/ℏ �
+n,α
+e−inωτun,α |α⟩ .
+(S5)
+where un,α = ⟨α|un⟩. As a result, we obtain an eigen-
+value equation for un,α:
+ϵun,α = −nℏωun,α +
+�
+β,m
+hm
+αβun−m,β ∀n, α
+(S6)
+which
+can
+be
+understood
+as
+a
+time
+independent
+Schr¨odinger equation of a 2D tight-binding model with
+a tilted potential energy along one axis.
+By explicitly
+evaluating the hm
+αβ, we get
+H2D = Hreal + Hsynth + Hdiag + HV + Htilt,
+(S7)
+with
+Hreal = −t
+�
+j,n,σ
+(ˆc†
+j,n,σˆcj+1,n,σ + h.c.),
+(S8)
+Hdiag = −δ0
+2
+�
+j,n,σ
+e−iπj(ˆc†
+j,n,σˆcj+1,n+1,σ
++ ˆc†
+j,n,σˆcj+1,n−1,σ + h.c.),
+Hsynth = −∆0
+2
+�
+j,n,σ
+e−iπj(iˆc†
+j,n,σˆcj,n+1,σ + h.c.),
+HV =
+�
+j,n,σ
+V (j)ˆc†
+j,n,σˆcj,n,σ,
+Htilt = −
+�
+j,n,σ
+ℏωnˆc†
+j,n,σˆcj,n,σ .
+Hreal and Hsynth describe tunneling along the real (x)
+and synthetic (n) dimension, respectively.
+The diagonal tunnelling terms in Hdiag are crucial be-
+cause they open a bandgap between the ground band and
+the first excited band, characterised by the topological
+Chern number C which is further related to the quantised
+Hall conductance via σtrans =
+q2
+2πℏC. The terms in HV
+
+9
+describe the external potential along the real-space direc-
+tion. Htilt corresponds to a linear tilt in potential energy
+along the synthetic dimension which can be thought of as
+originating from an electric field E = 2πℏ
+qT pointing along
+n.
+Edge modes and their reflection properties
+To illustrate the topological edge modes in the presence
+of an external potential, we evaluate the spectrum of H2D
+in the adiabatic limit (ω → 0) for different potentials
+V (j) = 1
+2m(2πνa)2jκ
+(S9)
+with m being the mass of 40K, trap frequency ν = 134 Hz,
+lattice spacing a = 532 nm, and lattice site j. The pa-
+rameter κ, an even integer, characterises steepness of the
+trap; the limit κ → ∞ corresponds to the textbook case
+of infinitely sharp walls [28]. Fig. S3A-C shows the nu-
+merically calculated energy spectra, omitting states on
+localised to the left edge for clarity.
+Fig. S3A shows the spectrum for a box-like potential
+with κ = 24. In this case there is a family of topologi-
+cal edge states, marked in red, which connect the lower
+and the upper band (black), separated from topologically
+trivial states above 5 kHz (also in black). All red states
+are localised along the right edge in x-direction.
+The
+lower and upper band have Chern number 1 and −1, re-
+spectively. Considering the dynamics in this model, an
+applied electric field along n as defined in Htilt leads to
+Bloch oscillations with a period T along kn. At the same
+time, the center-of-mass of the atoms moves by one unit
+cell per Bloch oscillation period along x, which corre-
+sponds to the quantised bulk Hall drift [14, 15]. This drift
+can be evaluated in the numerics by following the eigen-
+states in Fig. S3A in real space. Since the red edge states
+are gapped from the next higher-lying trivial (black)
+states, the atoms ‘Bloch-oscillate’ from the ground to
+the excited band via the red-marked edge modes over
+several periods. Once they are in the excited band they
+are transported backwards along x.
+Fig. S3B shows the situation for κ = 10. It behaves
+similarly to Fig. S3A, except that there are more lo-
+calised states marked in red, compared to κ = 2. Like-
+wise, these states are transported along x as they undergo
+Bloch oscillations. As before, this family of edge states
+is gapped from trivial states and connects right-moving
+to left-moving states, which leads to the reflection phe-
+nomenon.
+The experimentally relevant case is a harmonic trap
+with κ = 2 (see also refs. [23, 29–31]). Fig. S3C shows
+that for κ = 2 the number of localised sates outside of
+the bands is even larger than for κ = 10.
+As before,
+we adiabatically follow these localised states along kn
+A
+B
+C
+D
+FIG. S3. Energy spectra for different confining poten-
+tials. States localised to the left edge are omitted for clarity.
+Energy spectra of H2D in the limit ω → 0 for κ = 24 (A),
+κ = 10 (B), and κ = 2 (C). Topological edge modes which
+connect the two bands with Chern number 1 and −1 in (A)
+and (B) are marked in red. The upper inset in (C) marks
+the topological boundary where the reflection is observed as
+described in the main text. The inset in the center of (C)
+shows the tiny avoided crossings which can lead to a slight
+period dependence of the observed reflection point (Fig. S1).
+(D) Energy spectrum for a linearly increasing staggered po-
+tential. The gapless, topological edge mode is marked in red.
+
+10
+and evaluate their centre-of-mass along x.
+By numer-
+ically observing these drifts we confirm that the states
+describe quantised drifting in a large region, which man-
+ifests their nontrivial topological nature. Thus, the κ = 2
+case is ideal to observe the reflection after long-distance
+quantised Hall drifts. However, the gaps between topo-
+logical (right-moving and left-moving) and trivial (sta-
+tionary) states become smaller, compared to the κ = 24
+and κ = 10 cases, as shown in the insets of Fig. S3C
+(κ = 2). As a result, the reflection point for κ = 2 is
+spread out over several unit cells but the reflection itself
+remains intact. Faster pumping leads to non-adiabatic
+crossings of the energy gaps between right-moving and
+left-moving states, causing the reflection to happen later
+in time and further up in energy. We confirm this depen-
+dence experimentally in Fig. S1B, which shows a later
+reversal for smaller pump periods.
+Fig. S3D shows the spectrum for the linearly increasing
+staggered potential, described in the following sections.
+This potential allows a straightforward identification of
+the gapless edge mode (red line). The states correspond-
+ing to this gapless edge mode are localised around the
+topological boundary.
+x
+E
+FIG. S4. Linearly increasing staggered potential. To
+elucidate the topology in our system, a linearly increasing
+staggered potential is considered (blue): V (j) = jV0(−1)j
+with V0 = 1/2 × m(2πνa)2, as before. It is chosen such that
+the local tilt always equals the tilt from the harmonic poten-
+tial (orange), but alternates in sign. The staggered potential
+allows a simple pictorial representation of the emergence of
+the topological boundary. In a local density approximation
+the trap linearly shifts the pump trajectory upwards in the
+∆-δ plane, as depicted in the upper part of the figure. As
+soon as the trajectory ceases to enclose the critical point, a
+topological–trivial boundary develops.
+Staggered potential
+Another possibility to identify the topological bound-
+ary in our system makes use of a staggered potential.
+First, we consider a potential with uniform staggering,
+given by Vc(j) = V (−1)j, where j indexes the lattice-site
+and 2V corresponds to the energy difference between ad-
+jacent sites. Adding such a potential to the Rice-Mele
+Hamiltonian (Eq. 1) changes its trajectory in the ∆-δ
+plane. The onsite energy in such a system is given by
+(∆(τ) + V )(−1)j, which ranges from −∆ + V to ∆ + V .
+The tunnellings are unchanged. Therefore, the trajectory
+remains circular and it is simply shifted upwards by an
+amount V .
+A topological boundary emerges for a linearly increas-
+ing staggered potential, given by V (j) = jV0(−1)j, with
+V0 =
+1
+2m(2πνa)2 as before.
+V (j) is chosen such that
+it has the same local tilt as the harmonic trap in the
+experiment. Within the local density approximation we
+assign a ∆-δ trajectory locally to each unit cell.
+The
+trajectories are thus linearly shifted upwards as func-
+tion of j (Fig. S4), describing a change of topology in
+real space. We expect the local density approximation
+to be valid since the atomic eigenstates in the exper-
+iment are strongly localised.
+Similar models with lin-
+early increasing staggered potential have been studied in
+refs. [29, 32, 38].
+Local Chern marker
+The mathematical formulation of the Chern number as
+a topological invariant requires translational invariance,
+which does not apply to realistic experiments. Instead,
+we use a local quantity, known as Chern marker [8, 33].
+The local Chern marker depends on the real-space posi-
+tion and it is defined by:
+c(rγ) = −4π
+Ac
+Im
+�
+s=A,B
+⟨rγs| ˆP ˆx ˆQˆy ˆP |rγs⟩ ,
+(S10)
+where rγ is the position of the unit cell γ with sub-lattice-
+sites at positions rγA and rγB, |rγs⟩ = c†
+γs |0⟩ is the state
+localised on the corresponding lattice site , Ac is the area
+of a real-space unit cell, ˆQ = 1− ˆP and ˆP is the projector
+onto the ground band. Defining ˆP is not unambiguously
+possible in our system (Eq. S7) because of the energy
+shift from the harmonic confinement.
+Instead, we use
+a linearly increasing staggered potential, as described in
+the previous paragraph. This model leaves the bands in-
+tact and a ground band can be unambiguously defined.
+Experimentally, a local probe of the band topology is the
+velocity of the Hall drift, plotted in Fig. 2A. Theory and
+experiment agree approximately with one another. The
+local velocity is extracted from the atomic positions by
+fitting linear functions to groups of three adjacent dat-
+apoints in ten pump cycles. The resulting velocities are
+plotted against position and smoothed through a running
+average of width three (ten cycles).
+

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+1 
+ 
+Theory of Edge Effects and Conductunce for Applications in Graphene-
+based Nanoantennas 
+Tomer Berghau   , Touvia Milo  , Oded Gottlie   and Gregory Ya. Slepya    
+ 
+ 1
+1School of Mechanical Engineering, Tel Aviv University, Tel Aviv 69978, Israel 
+2 Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa, 3200003, Israel  
+3  School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel 
+*Correspondence: [email protected] (T.B); [email protected] (G.S.) 
+ 
+Abstract: In this paper, we develop a theory of edge effects in graphene for its applications 
+to nanoantennas in the THz, infrared, and visible frequency ranges. Its characteristic feature 
+is self-consistence reached due the formulation in terms of dynamical conductance instead of 
+ordinary used surface conductivity. The physical model of edge effects is based on using the 
+concept of Dirac fermions. The surface conductance is considered as a general susceptibility 
+and is calculated via the Kubo approach. In contrast with earlier models, the surface 
+conductance becomes non-homogeneous and non-local. The spatial behavior of the surface 
+conductance depends on the length of the sheet and the electrochemical potential. Results of 
+numerical simulations are presented for lengths in the range of 2.1 - 800nm and 
+electrochemical potentials ranging between 0.1 – 1.0 eV. It is shown that if the length 
+exceeded 800 nm, our model agrees with the classical Drude conductivity model with a 
+relatively high degree of accuracy. For rather short lengths, the conductance usually exhibits 
+spatial oscillations, which absent in conductivity and strongly affect the properties of 
+graphene based antennas. The period and amplitude of such spatial oscillations, strongly 
+depend on the electrochemical potential. The new theory opens the way for realizing 
+electrically controlled nanoantennas by changing the electrochemical potential may of the 
+gate voltage. The obtained results may be applicable for the design of carbon based 
+nanodevices in modern quantum technologies. 
+ 
+Keywords: graphene; edge effects; optical conductance; nanoantennas 
+ 
+ 
+ 
+1. Introduction 
+ 
+Innovative electromagnetic nanoantennas, which generally function at wide range of 
+frequencies (from THz until video frequencies), play a vital role in the emerging field of 
+photonics and plasmonics [1-6]. These antennas can be used as promising tools for 
+transforming near-field light into far-field and vice versa. Their excellent capabilities are 
+usually utilized in a wide scope of applications.  Among them, are traditional applications for 
+basic elements in electronics, high-speed communications, informatics, and quantum 
+computing [7] (in particular quantum nanomechanical qubit on carbon nanotube [8]). As far 
+as commercial applications of carbon-based nanostructures [9], one can list some novel 
+applications such as: i) high-resolved spectroscopy [10]; ii) high-speed communication [11]; 
+iii) light emission and detection; iv) identification of biomolecules and medical diagnostic 
+applications [12,13]. The design of the devices mentioned above, requires taking into account 
+edge effects and their correct physical description. It should cover a wide frequency range 
+from microwave until optical. These recent innovations have stimulated huge interest in the 
+theory of nanoantennas. Microwave standards become invalid starting from the THz region, 
+as the size of the antennas is miniaturized to micrometer scale [6]. Therefore, the common 
+
+2 
+ 
+model of perfect electric conductor, which is widely used in microwaves, is not suitable in 
+the range from THz to optical frequencies. By manipulating the different types of finite-size 
+(edge) effects [14-16], the conventional antenna configurations can be also used in the range 
+from microwave to optical frequencies [1-6]. However, antenna devices which operate in the 
+terahertz range (0.1–10 THz), have some fundamental limitations on their applicability in 
+practical devices [6]. Recent studies have pointed out that graphene is one of the best 
+candidates for overcoming the limitations mentioned above [6]. 
+A nanoantenna in its theoretical analysis, is generally considered as a system 
+terminated by a well-defined edge. The edge geometry is one of the widely used idealizations 
+in modern physics. For example, in all branches of classical mechanics and physics (e.g., 
+elasticity, hydrodynamics, acoustics, electrodynamics), the edge has a configurational 
+character. It corresponds to a point or contour at the surface of the body on which the normal 
+vector is usually undefined (half-plane, wedge, cone, etc.). Taking the edge position into 
+consideration, allows us to simplify the boundary conditions and employ certain analytical 
+techniques such as separation of variables, using for example special orthogonal coordinate 
+systems [17] or the Wiener-Hopf method [18]. Such analytical solutions are especially 
+attractive for the qualitative analysis of novel types of problems involving interacting fields 
+of different physical origin (for example, plasmonics and optoelectrofluidics [19]). However, 
+such idealization may lead to the loss of uniqueness of the problem statement. The reason for 
+it is that the placement of an arbitrary point singularity at the edge, generally does not violate 
+the governing wave equation as well as the boundary and  radiation conditions. The analytic 
+solutions thus obtained, are correct from the mathematical point of view. However, they may 
+turn to be physically incorrect because due to lack of uniqueness, they may  correspond to 
+another source of the field. In order to obtain the corresponding of unique solution, one must 
+enforce  the condition of finite (bounded)  energy over an arbitrary area (finite extend)  of 
+space (Meixner condition) [20]. Such an approach usually leads to the creation of field 
+singularities at the vicinity of the edge. Note, that the edge-like configuration does not change 
+the constitutive properties of the medium (for example, such as a perfect conductor or a 
+perfect insulator in classical electrodynamics).  
+ 
+The extensive recent progress in nanotechnologies, lead to the discovery of  novel 
+artificial types of condensed matter, such as graphene [21], carbon nanotubes (CNTs) [22], 
+Weyl and Dirac semimetals [23] and topological insulators [24,25]. Consequently, a great 
+number of fundamental physical problems were reconsidered and in particular, but 
+nevertheless most of them failed  to consider the important edge concept. The edge for the 
+different types of nanostructures corresponds to the spatial area in the vicinity of the sample 
+boundary which size is rather large compared with the inter-atomic distance. However, the 
+mechanism of electronic transport at the edge is dramatically different from the 
+corresponding properties of the bulk region. One of the reasons for this disparity is the 
+existence of new types of quantum states strongly confined to the vicinity of the edge, which 
+are able to produce novel physical properties such as topological order (so called, edge states 
+[16,21,25]). Such transport mechanisms manifest themselves in the special optical and 
+optomechanical properties of different types of metamaterials (e.g., carbon based 
+nanostructures).  
+ 
+The electrical properties of macroscopic structures may be described both in terms of 
+conductivity and conductance [16], which are equivalent and coupled via the simple constant 
+coefficient defined by the size values of the sample. As it was noted in [16], the conductivity 
+for nanomaterials (including graphene) is a well-defined value only when the sample is 
+enough large. It is clear from intuitive point of view, that the electric current becomes 
+homogeneous and insensitive to boundaries of the sample. When the sample’s size is 
+reduced, the current becomes non-homogeneous and non-local with respect to the field 
+
+3 
+ 
+variations in the material. Then, the concept of conductivity loses its meaning. The behavior 
+of the electrons in nanoantennas becomes sensitive to the feeding lines, detectors, modulators 
+and the edges due to the quantum-mechanical interference. As we will show, exactly optical 
+conductance will be suitable parameter for formulation of the effective boundary conditions 
+for electromagnetic field in nanoantennas. In contrast with conductivity, it is non-
+homogeneous and not coupled via simple relation with optical conductivity (which is 
+homogeneous value). It is described by the Kubo approach instead of Boltzmann’s transport 
+equation.  
+The edge areas play the main role in forming the emission in classical radio-frequency 
+antennas [26] and optical nanoantennas [27]. One of the promising types of nanoantennas that 
+operate in the  THz and optical frequency range,  are based on carbon-based nanostructures 
+(graphene, CNTs) [28-31]. The physical mechanism of their radiation, is based on the 
+existence of a strongly retarded surface plasmon predicted in [28, 32], experimentally 
+observed in [33] and used for interpretation of the intriguing measurements of the THz 
+conductivity peak [34]. These works use different models for the charge transport [28, 32, 34, 
+35], but all of them are not self-consistent. In another words, the boundary-value problem for 
+the Maxwell equations is formulated for the real geometry of the object, while the value of 
+conductivity is introduced as a phenomenological parameter which is determined by using a  
+model corresponding to an infinitely large structure. Of course, such an approach appears to 
+be attractive since it allows to reach, simplification due to the separation of the EM-field and 
+charge transport equations. Such an approach keeps the self-consistent requirement for planar 
+structures with the Fresnel transmission-reflection condition of EM-fields (graphene films, 
+semitransparent mirrors, etc.) [37-40]. However, it may not be adequate for the detailed 
+description of EM-field scattering and antenna emission in the general case. One can clearly 
+expect that the error of such a non-consistent approach may be ignored as being very small, 
+providing the size of the system is rather large. However, it means, that the effect of the 
+Fresnel transmission-reflection dominates the field forming, whereas the antenna efficiency 
+decreases. Nevertheless, it is impossible to say a priori what is the validity bound of such a 
+simplification. 
+This problem becomes especially relevant for graphene-based structures, because of 
+the corresponding large geometrical size disparity for applications in different graphene 
+devices. The advanced graphene synthesis methods make it possible to grow graphene 
+samples from 2.1nm to few centimeters [6,41-46]. The detail description of optical graphene 
+conductivity requires a self-consistent analysis. Such a self-consistent approach determines 
+the field acting on electrons produced over their motion, by taking into consideration the 
+finite-size configuration and using an appropriate microscopic model for the edge.  The 
+formulation of such a self-consistent analysis is one of the main contributions of this paper. 
+One of the important results of this paper, is analyzing the effect of the edges (shape of finite 
+extend) and determine the corresponding error when self-consistency is not prevailed. It is 
+important to note; that edge effects do not only add up to the quantitative differences in the 
+value of conductivity. Edge effects are also able to dramatically affect the special physical 
+mechanism of charge transport. It makes required the formulation of the theory in terms of 
+optical conductance, which is important for applications in antenna design.        
+ 
+The paper is organized as follows:  models of the edge of a graphene ribbon based on 
+the Dirac-fermion concept, are first discussed in Section 2. Thereafter, based on using the 
+Kubo approach and the concept of general susceptibility [47], we analytically obtain an 
+expression for the surface conductance of a terminated (finite) graphene sheet.  Results of 
+numerical simulations together with a discussion are presented in Section 3. Conclusion and 
+outlook are finally given in Section 4. 
+
+4 
+ 
+2. Optical conductance of a terminated graphene sheet.  
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 1. Problem statement: (a) Geometry of a graphene sheet with zigzag edge configuration; (b) 
+Fermi-Dirac cones for the model of pseudospin dispersion. 
+ 
+2.1 Kubo approach for optical conductance of graphene 
+ 
+In the following we will use the Kubo approach for conductance calculation as a universal 
+technique which couples the generalized forces of arbitrary physical origin with the responses 
+of correspondent origin via general susceptibilities [47]. Towards this goal it is convenient to 
+define the forces as tangential components of electric field and responses as components of 
+current densities. Thus, the general susceptibilities are defined by a 2
+2
+
+matrix where  
+ 
+ 
+
+
+ 
+,
+,
+1,2
+; ,
+a
+ab
+b
+b
+j
+K
+E
+d
+
+
+
+
+
+
+
+  
+x
+x x
+x
+x  (1) 
+and                                           
+
+
+
+
+
+
+
+
+
+
+2
+0
+; ,
+ˆ
+ˆ
+ˆ
+ˆ
+,
+0,
+0,
+,
+ab
+i t
+a
+b
+b
+a
+K
+e
+e
+x
+t
+x
+x
+x
+t
+dt
+
+
+
+
+ 
+
+
+
+
+x x
+x
+x
+x
+x
+                          (2) 
+ 
+ 
+Here
+ 
+ 
+ˆ
+a
+a
+j
+j
+
+x
+x
+, 
+
+
+
+
+ˆ
+ˆ
+,
+,
+a
+a
+j
+t
+i ex
+t
+
+
+x
+x ,
+ 
+ˆ
+aj
+x
+ represents the observable current 
+density, 
+
+
+ˆ
+,
+ax
+t x is an operator of charge displacement per unit area. The general 
+susceptibility tensor
+
+
+; ,
+ab
+K
+
+
+x x depends on the geometry of the sample as well as the 
+electronic properties of the medium. Therefore, exactly it has a physical meaning of non-local 
+optical conductance (in the units
+2 /
+e
+) similar to the dc conductance of graphene in [16] .  
+ 
+The next step is the transformation of Equation (2) to a form that is convenient for 
+applications in graphene electrodynamics. Thus one can write 
+(b) 
+(a) 
+
+Energy
+Valley K'
+Valley KA-atoms
+B-atomsremoved
+B-atoms
+1
+A-atoms removed5 
+ 
+
+
+
+
+0
+ˆ
+ˆ
+; ,
+i t
+ab
+ab
+ab
+ie
+K
+e
+A
+B
+dt
+
+
+
+  
+
+
+x x
+                                     (3) 
+ 
+where
+
+
+
+
+
+
+
+
+ˆ
+ˆ
+ˆ
+ˆ
+ˆ
+ˆ
+,
+0,
+,
+0,
+,
+ab
+ab
+a
+b
+b
+a
+A
+j
+t
+x
+B
+x
+j
+t
+
+
+
+
+x
+x
+x
+x
+and                                                     
+
+
+
+
+0
+0
+ˆ
+ˆ
+ˆ
+ˆ
+ab
+ab
+ab
+ss
+s
+ss
+A
+Tr A
+A
+
+
+
+ 
+                                                (4) 
+where 
+0ˆ  is the density matrix of free motion. 
+ The diagonal matrix elements of the operator ˆ ab
+A are defined by 
+  
+
+
+
+
+
+
+
+
+
+
+ˆ
+ˆ
+ˆ
+,
+0,
+ab
+a
+b
+s s
+ss
+s
+ss
+A
+j
+t
+x
+
+
+
+
+ 
+x
+x
+                                              (5) 
+ 
+and following Equation (4) one gets 
+ 
+
+
+
+
+
+
+
+
+0
+ˆ
+ˆ
+ˆ
+,
+0,
+ab
+a
+b
+ss
+s s
+s
+s
+ss
+A
+j
+t
+x
+
+
+
+
+
+
+
+
+x
+x
+                                         (6) 
+Here 
+
+
+,
+y
+s
+p k
+
+denotes the combinative discrete-continuous index (p is the discrete number 
+of the state and 
+yk  is its continuous wave-number over the y-axis).   
+ 
+The temporal behavior in the linear approximation can be expressed as  
+
+
+
+
+
+
+
+
+
+ /
+ˆ
+ˆ
+,
+0,
+s
+s
+i
+t
+a
+a
+ss
+ss
+j
+t
+j
+e
+
+ 
+
+
+
+
+
+x
+x
+ and as a result we obtain 
+
+
+
+
+
+
+
+
+
+
+0
+ˆ
+ˆ
+ˆ
+0,
+0,
+s
+s
+i
+t
+ab
+a
+b
+ss
+s s
+s
+s
+ss
+A
+j
+x
+e
+
+
+
+
+
+
+
+
+
+
+
+
+
+x
+x
+                                 (7) 
+A similar relation may be also obtained for ˆ ab
+B , namely 
+
+
+
+
+
+
+
+
+
+
+0
+ˆ
+ˆ
+ˆ
+0,
+0,
+s
+s
+i
+t
+ab
+b
+a
+ss
+s s
+s
+s
+ss
+B
+x
+j
+e
+
+
+
+
+
+
+
+
+
+
+
+
+x
+x
+                                     (8) 
+Combining Equations (7) and (8) with Equation (3) lead to 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+0
+0
+; ,
+4
+ˆ
+ˆ
+ˆ
+ˆ
+0,
+0,
+0,
+0,
+s
+s
+s
+s
+ab
+s
+s
+i
+i
+t
+t
+i t
+a
+b
+b
+a
+ss
+s s
+ss
+ss
+s s
+i
+K
+e
+j
+x
+e
+x
+j
+e
+dt
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+x x
+x
+x
+x
+x
+   
+  
+(9)  
+For shortness, we will omit the initial value at time t=0 in the matrix elements of 
+(
+
+
+
+
+ 
+
+
+'
+'
+ˆ
+ˆ
+0,
+a
+a
+ss
+ss
+j
+j
+
+x
+x
+ 
+, etc.). Using elementary integration yields  
+
+6 
+ 
+
+
+ 
+
+
+ 
+
+
+
+
+
+
+ 
+
+
+ 
+
+
+
+
+
+
+0
+ˆ
+ˆ
+ˆ
+ˆ
+; ,
+a
+b
+b
+a
+s s
+ss
+ss
+s s
+ab
+ss
+s
+s
+s
+s
+s
+s
+j
+x
+x
+j
+ie
+K
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+  
+
+
+
+
+
+
+
+
+
+
+
+
+x
+x
+x
+x
+x x
+        
+(10) 
+ 
+Introducing the following change 
+,
+s
+s s
+s
+ 
+
+  in the second term of Equation (10) and 
+using the relation 
+ 
+
+
+
+
+ 
+
+
+'
+'
+'
+ˆ
+ˆ
+/
+a
+s
+s
+a
+ss
+ss
+j
+ie
+x
+
+
+
+
+x
+x
+ ,  the latter  can be written as   
+
+
+ 
+
+
+ 
+
+
+
+
+
+ 
+ 
+
+0
+0
+ˆ
+ˆ
+1
+; ,
+a
+b
+ss
+s s
+ab
+ss
+s s
+s
+s
+s
+s
+s
+s
+j
+j
+K
+i
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+  
+
+
+
+
+
+x
+x
+x x
+                    
+(11) 
+where 
+
+
+
+
+/
+0
+( )
+1/
+1
+B
+k T
+ss
+f
+e
+ 
+
+
+
+
+
+
+ denotes the Fermi-distribution and  is the 
+electrochemical potential. The electrochemical potential is defined here by the concentration 
+of electrons/holes according to the  following relation; 
+
+
+2
+1
+0
+/
+F
+n
+v
+
+
+
+
+[37-40]. This 
+potential vanishes for a perfectly clean graphene at zero temperature [16] and may be 
+effectively controlled via the gate voltage [37-40]. 
+ 
+Because of the Hermittivity nature of the current operator, we have 
+ 
+
+
+ 
+
+
+ˆ
+ˆ
+b
+b
+s s
+ss
+j
+j
+
+
+
+x
+x
+ where the upper asterisk denotes complex conjugate. Therefore, one 
+can also write Equation (11) in the . compact form 
+
+
+ 
+ 
+
+
+; , '
+0
+ab
+ab
+ab
+i
+K
+
+
+
+ 
+
+ 
+x x
+                                         (12) 
+where  
+ 
+ 
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+'
+ˆ
+ˆ
+.
+a
+b
+ss
+s s
+ab
+s
+s
+s
+s
+s
+s
+j
+j
+f
+f
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+x
+x
+                       
+(13) 
+ 
+ 
+2.2.  Zigzag edges 
+ 
+In this Subsection we will apply the general result to the edge of a zigzag type. The electronic 
+properties of the ribbon are described by the model of Dirac fermions corresponding to a 
+tight-binding model on a two-dimensional honeycomb lattice [16,25,48].  We will use the 
+zigzag-type of boundary conditions for Dirac fermions on a terminated (finite) lattice derived 
+in [16,25,48], since it was demonstrated that this type of boundary condition can be generally 
+applied to a terminated honeycomb lattice in the case of electron-hole symmetry [25]). 
+The A and B atoms are coupled in every spinor modes. From physical point of view, 
+the interpretation of electronic transport may be considered as a motion from A atoms to B 
+ones and vice versa, while A-A and B-B motions are forbidden. The electron transport for a 
+zigzag edge in valleys K and K’ is independent and will be considered separately. The wave 
+function is expressed as a linear combination of the eigen 4D spinors 
+ 
+
+7 
+ 
+The pseudo-spinor modes for valley K have the following form; 
+ 
+
+
+
+
+
+
+ 
+ 
+,
+(
+)
+,
+,
+,
+,
+0
+0
+0
+0
+y
+s
+A s
+s
+i k y
+t
+B s
+s
+Ks
+t
+u
+x
+t
+v
+x
+t
+e
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+x
+x
+Ψ
+x
+                              (14a) 
+
+
+
+
+
+
+ 
+ 
+(
+)
+,
+,
+0
+0
+0
+0
+,
+,
+,
+y
+s
+i k y
+t
+K s
+A s
+s
+B s
+s
+t
+e
+t
+u
+x
+t
+v
+x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Ψ
+x
+x
+x
+                         
+(14b)
+ 
+ 
+ 
+The functions  
+ 
+,
+u x
+v x  in the valley K (and in K’ respectively) take the following form for 
+bulk and edge states 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+1
+sin
+2
+2
+1
+sinh
+2
+2
+1
+sin
+2
+2
+1
+sinh
+2
+2
+Bulk
+b
+s
+s
+p
+px
+Edge
+e
+s
+s
+p
+Bulk
+b
+s
+s
+p
+px
+Edge
+e
+s
+s
+p
+L
+u
+x
+u
+x
+i
+B
+x
+lL
+L
+u
+x
+u
+x
+B
+x
+lL
+L
+v
+x
+v
+x
+B
+x
+lL
+L
+v
+x
+v
+x
+i
+B
+x
+lL
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+ 
+
+ 
+
+
+
+
+
+
+ 
+
+ 
+
+
+
+ 
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+ 
+
+
+                                
+(14c)
+ 
+ 
+ 
+where, 
+ 
+
+
+1
+sin 2
+1
+2
+px
+b
+s
+px
+L
+B
+L
+
+
+
+
+
+
+
+
+
+
+
+
+                                             (14d) 
+ 
+ 
+
+
+1
+1
+2
+sinh 2
+1
+2
+L
+e
+s
+L
+B
+e
+L
+
+
+
+
+
+
+
+
+
+
+
+
+                                       (14e)
+ 
+ 
+ 
+ 
+where l represents the normalization length in the y-direction (which goes to infinity in the 
+final result), 
+sA is a  normalization constant, 
+,
+y
+xp
+k k
+are the wavenumbers which are defined 
+by the dispersive relation 
+
+
+2
+xp
+ik L
+y
+xp
+y
+xp
+k
+ik
+k
+ik
+e
+
+
+
+
+and 
+
+
+,
+y
+s
+p k
+
+is the combinative 
+discrete-continuous index (p is the discrete number with respect to the x-axis and 
+yk  is the 
+
+8 
+ 
+continuous wave-number over the y-axis).  This dispersion relation has an infinite number of 
+real roots with a single point of concentration at infinity (confined modes) and one imaginary 
+root that corresponds to the edge state [16]. Transformation to the edge state from confined 
+modes in Equations (14a)-(14e) and characteristic equation may be done via exchange 
+px
+
+
+
+.The two signs in (14) correspond to electrons and holes respectively. As one can see 
+Eq. (14) satisfies the Dirac equation and is subject to the following boundary conditions; 
+
+
+
+
+/2
+/2
+0
+s
+s
+u
+L
+v
+L
+
+
+
+[48].The components u(x) and v(x) are separately orthogonal, while 
+non-orthogonal mutually due to their coupling over electron motion between the atoms of A 
+and B sub-lattices.  Since the expression for the ac-conductivity with a zigzag edge is 
+isotropic, we have 
+
+
+
+
+; ,
+; ,
+ab
+ab
+K
+K
+
+
+
+
+
+
+x x
+x x ,where the general susceptibility is
+ 
+
+
+ 
+ 
+
+
+1
+; , '
+0
+K
+i
+
+
+
+
+ 
+
+ 
+x x
+,
+ 
+ 
+with 
+ 
+ 
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+'
+ˆ
+ˆ
+ss
+s s
+s
+s
+s
+s
+s
+s
+f
+f
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+j x
+j x
+                                (15) 
+Here represents the matrix element of the current operator, σ denotes the xy-vector of the Pauli 
+matrices,  
+2
+2
+s
+F
+xp
+y
+v
+k
+k
+  
+
+is the energy of the s-th state and ,
+F
+e v  correspond to the 
+electron charge and Fermi velocity respectively.  
+ 
+The general susceptibility may be presented here as the sum of two components of 
+different 
+origin 
+
+
+
+
+
+
+; , '
+; , '
+; , '
+Inter
+Intra
+K
+K
+K
+
+
+
+
+
+x x
+x x
+x x
+, 
+where 
+
+
+ 
+1
+; , '
+Inter
+K
+i
+
+
+
+
+ 
+
+x x
+ corresponds to the interband motion which  may be ignored 
+omitted for rather low (THz) frequencies. The second term 
+
+
+ 
+1
+; , '
+0
+Intra
+K
+i
+
+
+
+
+x x
+corresponds to the intraband motion and leads to the common conductivity law 
+ 
+   
+Intra
+i
+x
+
+ 
+j x
+E x , where the surface conductance is found by substituting the 
+pseudospinor modes (14) into (15), which renders  
+ 
+
+
+
+
+ 
+ 
+
+
+2
+2
+2
+2
+( )
+2
+0
+s
+n
+y
+Intra
+F
+s
+s
+y
+n
+k
+ie v
+f
+x
+u
+x
+v
+x
+dk
+i
+ 
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+                
+(16)
+ 
+(for details of deriving Equation (16) see Appendix A). The explicit expression for the 
+conductivity given in (16) is spatially inhomogeneous (x-depended) and is a manifestation of 
+edge effect and incorporating a self-consistent description.  
+2.3. Approximation for zero temperature. 
+Let us next consider the important case of the conductance at zero temperature, which opens 
+the way for further simplification of Equation (16). In this case the Fermi distribution may be 
+replaced by a step function and its derivative in (16) can be transformed into a  Dirac delta 
+function 
+
+
+/
+f
+
+ 
+
+
+  
+
+, which allows an integration of (16). Performing the integration 
+in 
+ 
+
+9 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 2. Illustration with respect to the zero-temperature approximation. Black lines correspond to 
+confined modes, red line corresponds to the edge mode, and blue line corresponds to the first confined 
+mode. The black point corresponds to the critical wavenumber in which the mutual transformation of 
+edge mode to the first confined mode takes place. Dotted horizontal line corresponds to the given 
+electrochemical potential. The values 
+yn
+k
+denote the roots of Equation (17). 
+ 
+possible in terms of the discrete number of  roots of the following transcendental equation
+
+
+yk
+
+
+
+( qualitatively depicted in Figure 2), which may be  also written as   
+
+
+2
+2
+2
+2
+tg
+yn
+yn
+yn
+k
+k
+L
+k
+
+
+
+
+
+                                                       (17)                                                     
+The value 
+/
+F
+v
+
+
+
+means normalizing the electrochemical potential by the electron energy. 
+The  corresponding edge mode may be obtained by the formal exchange 
+yn
+k
+ik
+
+(such root 
+exists only under  special conditions). For the integration over 
+yk we use the following 
+property of the Dirac delta-function  
+
+
+
+ 
+
+
+
+
+
+yn
+y
+y
+y
+n
+yn
+F k
+k
+F k
+dk
+k
+ 
+
+
+
+
+
+
+
+
+
+                                             
+(18)
+ 
+where 
+
+
+yk
+
+denotes the y-component of the group velocity of pseudospin (prime means the 
+derivative). The final explicit result for the conductivity can be written as 
+ 
+
+
+
+
+2
+1
+2
+2
+2
+2
+2
+2
+1
+2
+2
+sin
+sin
+2
+2
+0
+N
+n
+yn
+yn
+n
+yn
+Intra
+B
+L
+L
+k
+x
+k
+x
+L
+k
+e
+x
+i
+i
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+         
+(19)
+ 
+kynL 
+kynL 
+ 
+ 
+
+20
+18
+16
+14
+12
+10
+8
+6
+4
+2
+0
+-25
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+k,L10 
+ 
+The values of 
+yn
+k  depend on the electrochemical potential and satisfy the characteristic 
+Equation (17). The normalized coefficients  
+n
+B  in (19) are defined by  
+
+
+
+
+1
+2
+2
+2
+2
+2
+2
+sin 2
+1
+2
+2
+yn
+n
+n
+n
+yn
+yn
+k L
+B
+L
+k
+k L
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                                                  
+(20) 
+ Equation (19) may be finally transformed to  
+ 
+
+
+ 
+
+
+2
+1
+2
+2
+2
+2
+2
+2
+1
+2
+2
+1
+sin
+sin
+2
+2
+1
+2
+0
+N
+yn
+yn
+n
+yn
+Intra
+L
+L
+L
+k
+x
+k
+x
+k
+L
+e
+x
+i
+i
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    
+(21) 
+(for details see Appendix B). 
+2.4. The limit of an infinite sheet. 
+In this Subsection we show that our model reduces to the familiar Drude relation for the 
+conductivity in infinitely wide sheet. Taking the limit L   , and using the following 
+transformation
+
+ 
+
+
+ 
+1
+1
+2
+...
+2
+...
+x
+n
+L
+dk
+
+
+
+
+
+
+
+
+.  Equation (16) yields 
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+2
+2
+2
+0
+( )
+1
+1
+1
+cos
+2
+cos
+2
+2
+2
+s
+n
+y
+Intra
+F
+x
+x
+x
+y
+k
+ie v
+x
+i
+f
+k
+x
+L
+k
+x
+L
+dk dk
+ 
+
+
+
+
+
+
+
+
+
+
+ 
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+ 
+(22) 
+Introducing the polar variables
+sin
+xk
+
+
+, 
+cos
+yk
+
+
+, we obtain 
+x
+y
+dk dk
+d d
+ 
+
+ . Using the 
+zero temperature approximation, we exchange the Fermi-distribution derivative by the  Dirac 
+delta-function and integrate over  ,  so that  Equation (22) becomes 
+ 
+
+
+
+
+
+
+
+
+
+
+2
+2
+2
+2
+0
+2
+0
+1
+1
+1
+cos
+2
+cos
+cos
+2
+cos
+2
+2
+Intra
+ie
+x
+i
+x
+L
+x
+L
+d
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+                (23) 
+Furthermore, using the addtion theorem for Bessel functions [49] 
+ 
+cos
+i
+m
+im
+m
+m
+e
+i
+J
+e
+
+
+
+
+
+
+
+ 
+                                                           (24) 
+we integrate  (23) and obtain  
+
+11 
+ 
+ 
+
+
+
+
+
+
+
+
+
+
+2
+0
+0
+2
+1
+1
+1
+2
+2
+0
+2
+2
+Intra
+ie
+x
+J
+x
+L
+J
+x
+L
+i
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+            (25) 
+If the observation point x (of under the assumption of large L), is placed rather far from the 
+edges, the arguments of  Bessel functions become large, so that the asymptotic relations [49] . 
+In this case, the last two terms in (25) become indefinitely small. The first term, which is 
+exactly equal to the Drude conductivity 
+
+
+2
+2
+/
+0
+Drude
+G
+ie
+i
+ 
+
+
+
+, becomes dominate over the 
+whole area of the ribbon excluding the narrow vicinities of the edges. Thus, it is 
+demonstrated that in this limit our model asymptotically reduces to the classical expression 
+for the Drude conductivity.    
+2.5. Optical conductance for the edge with infinite-mass boundary condition 
+ 
+This problem is of special interest in connection with a suspended sheet. The interaction 
+between the edges of the sheet with the electrodes, results in the creation of electrostatic 
+potential.  Following [13], the electron-hole symmetry, which is generally restricted for 
+boundary conditions of a zigzag or armchair types is broken. It may be considered as a 
+manifestation of a staggered potential at zigzag boundaries, which may change the nature of 
+the boundary condition. For, infinitely large value of potential it leads to an “infinite-mass” 
+boundary condition, which may be written as
+
+
+
+
+/2
+/2
+s
+s
+u
+L
+v
+L
+
+ 
+
+. The corresponding 
+eigen pseudospins are then given by Equation (14) with 
+ 
+ 
+
+
+2
+2
+1
+1
+2 2
+s
+s
+xn
+xn
+i k
+x
+i k
+x
+n
+su
+x
+e
+e
+L
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+                                          (26) 
+ 
+
+
+2
+2
+1
+1
+2 2
+s
+s
+xn
+xn
+i k
+x
+i k
+x
+n
+sv
+x
+e
+e
+L
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+                                           (27) 
+where  
+/
+xn
+k
+n
+L
+
+
+and 
+2
+2
+s
+y
+xn
+i
+y
+xn
+k
+ik
+e
+k
+k
+
+
+
+
+. 
+In order to calculate the conductance, we use the Kubo approach with the pseudo spinors 
+defined in (26) and (27). The difference from the zigzag edge formulation, is due to the lack 
+of orthogonality inf (26) and (27), which leads to a non-local conductance (spatial 
+dispersion). The conductance operator does not add up to the convolution form because of the 
+non-homogeneity of the finite-length structure.  In the case of a rather wide sheet the non-
+local component becomes relatively small and may be usually ignored. In this particular case 
+the conductance relates to Equation (16) with the pseudospins given given in (26) and (27).   
+ 
+3. Numerical results and discussion 
+The optical properties of graphene are generally defined by the geometrical size of the sample 
+and the value of the electrochemical potential. These parameters may vary over a wide range 
+and thus are of practical interest to nanoantenna design. The recent advances in graphene 
+technology make it possible to produce graphene samples from few nanometers to few 
+centimeters [6,7,21]. The electrochemical potential may also vary over the interval 0
+1.0
+
+
+
+eV, by doping the sample or by applying gate voltage  [6,7,21]. In this Section we will 
+
+12 
+ 
+present some numerical results of conductivity simulations for a wide range of physical 
+parameters, based on the theory developed above. 
+ 
+One of the main results of the present study according to Equations (16) and (19), is 
+that the non-homogeneity of conductance is mainly due to the edge effects. The conductance 
+distribution is controlled by the parameter
+/
+F
+L
+L
+v
+
+
+
+, which defines the number of modes 
+supported by the conductance value. Figures 3-5 present the normalized conductance 
+distribution for a rather large length 
+800
+L 
+nm and for different values of the 
+electrochemical potential (increasing  corresponds to increasing number of modes). The 
+qualitative behavior of the conductance is the same in all these Figures. The conductance 
+oscillates with respect to the spatial variable and decreases near the edges. The amplitude and 
+period of oscillations decrease with increasing of the electrochemical potential  . The 
+average value of the conductance (to a high degree of accuracy), corresponds to the classical 
+Drude model of conductivity, as discussed in Section 2.4. One can anticipate that such small 
+oscillations are unable to manifest themselves in the scattering of electromagnetic waves, due 
+to the smallness of their period compared with the wavelength. Thus, we may conclude that 
+the Drude model can be applied for this range of parameters.  
+The considered scenario changes dramatically with shortening of the sheet, as shown 
+in Figures 6-8 (each Figure corresponds to a different value of the length for the same value 
+of electrochemical potential). One can clearly see the enhancement of oscillations, which 
+makes the Drude model invalid for such values of parameters. In fact, it becomes impossible 
+to introduce the conductivity concept in its ordinary meaning. As it was mentioned above, the 
+value defined by Equation (16) has the meaning of optical conductance, which strongly 
+depend on the geometrical size of the sheet. It may be coupled with Drude conductivity by 
+the Relation 
+ 
+         
+ 
+ 
+1
+Drude
+Intra
+x
+L
+x
+G
+
+
+
+                                               
+(28) 
+where  
+ 
+ 
+ 
+
+
+2
+1
+2
+2
+2
+2
+2
+2
+1
+1
+sin
+sin
+2
+2
+1
+N
+yn
+yn
+n
+yn
+L
+L
+x
+k
+x
+k
+x
+k
+L
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+         
+(29) 
+ 
+ 
+is x-dependent coefficient, in which the configuration of the sample manifests itself.  The 
+situation is rather similar to the electron transport in graphene in dc field (the concepts of 
+conductance and conductivity discussed and compared in [16,50]).  
+The physical mechanism for the conductivity oscillations is the interference between 
+the pseudospin modes due to the reflection from the sheet boundaries. The important features 
+are demonstrated in the single-mode conductance (Figures 6 (a),(b)). For the rather small 
+electrochemical potential the active mode is an edge type.  The sign of the conductance is 
+negative, which corresponds to its inductive origin. The increase in the electrochemical 
+potential leads to the transformation from inductive to capacitive one (sign exchange)  due to 
+the transformation from the edge mode to the bulk one.  
+As one can see, the conductivity of a graphene sheet changes its qualitative behavior 
+for rather small values of length. However, graphene antennas generally exhibit a resonate 
+behavior at much lower frequencies as well as their metallic counterparts, which is 
+experimentally implemented in the THz range [43,45,46,51]. Thus one can effectively exploit 
+the electrically tunable conductance of graphene exactly for such small sizes, where the 
+conventional models of conductivity become invalid due to the importance of edge effects. In 
+
+13 
+ 
+summary, it is important to note that including edge effects in the physical modeling, opens a 
+new way for electrical controlling of resonant graphene antennas via the overturned 
+electrochemical potential by means of the gate voltage. In some cases where the 
+electrochemical potential varies adiabatically slow in time, it will also produce a modulation 
+of the THz emission. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 3. The spatial distribution of the conductance in the units 
+/
+Drude
+G
+Ll . L=800nm,  =0.1eV. 
+ 
+The concept of the optical conductance developed in this paper allows formulating the 
+effective boundary conditions for electromagnetic field at the surface of graphene sheet in the 
+form  
+ 
+
+
+ 
+
+0
+0
+,
+,
+,
+Drude
+z
+z
+G
+x
+
+
+
+
+
+
+
+
+
+n H
+H
+n
+n E                                        (30) 
+ 
+Their using leads to modification of integral equations of antenna theory and methods of their 
+solution. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+?10~3
+2.5
+www
+2
+0.5
+o.Drudemodel
+0
+0.5
+-0.4
+0.3
+-0.2
+-0.1
+0.1
+0.2
+0.3
+0.4
+0.5
+X/L14 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 4. The spatial distribution of the conductance in the units 
+/
+Drude
+G
+Ll . L=800nm,  =0.5eV.  
+ 
+ 
+   
+ 
+ 
+ 
+ 
+ 
+Figure 5. The spatial distribution of the conductance in the units 
+/
+Drude
+G
+Ll . L=800nm,  =1.0eV. 
+ 
+The considered scenario changes dramatically with shortening of the sheet, as shown 
+in Figures 6-8 (each Figure corresponds to a different value of the length for the same value 
+of electrochemical potential). One can clearly see the enhancement of oscillations, which 
+makes the Drude model invalid for such values of parameters. In fact, it becomes impossible 
+to introduce the conductivity concept in its ordinary meaning. The value defined by Equation 
+(16), has the meaning of an “effective” conductivity, which strongly depend on the 
+geometrical size of the sheet. The situation is rather similar to attempt to describe the optical 
+properties of semiconductor quantum dots via the dielectric function in the limit of weak 
+conferment [50] (the “effective” dielectric function strongly depends on the sample 
+configuration). The physical mechanism for the conductivity oscillations, is the interference 
+between the pseudospin modes due to the reflection from the sheet boundaries. The important 
+features are demonstrated in the single- mode conductivity (Figures 6 (a), (b)). For the rather 
+small electrochemical potential the active mode is an edge type.  The sign of the conductivity 
+is negative, which corresponds to its inductive origin. The increase in the electrochemical 
+potential leads to the transformation from inductive to capacitive one (sign exchange), due to 
+the transformation from the edge mode to the bulk one. 
+
+12 ×10-3
+10
+6
+Drudemodel
+5
+3
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+x/ L2.02
+0.018
+0.016
+0.012
+0.01
+0.008
+0.006
+0.004
+.0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0.1
+0.2
+0.3
+0.4
+0.5
+x/L15 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 6. The spatial distribution of the conductance in the units 
+/
+Drude
+G
+Ll  for different values of 
+length and  =0.1 eV; (a) L=2nm (single-mode regime; edge mode); (b) L=10nm (single-mode 
+regime; bulk mode); (c) L=50nm (three-mode regime; all modes are of bulk type).  
+(a) 
+(b) 
+(c) 
+
+2.5×10-3
+XX
+Drude model
+2
+1.5
+0.5
+-0.5
+-0.4
+-0.3
+-0.2
+0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+X/L0.005
+0
+-0.005
+-0.01
+-0.015
+-0.02
+XX
+o.- Drude model
+-0.025
+-0.03
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+x/L2.7×10~3
+2.6
+2.5
+2.4
+2.3
+2.2
+xX
+2.1
+e--Drude model
+2
+1.9
+1.8
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+X/L16 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 7. The spatial distribution of the conductance in the units 
+/
+Drude
+G
+Ll  for different values of 
+length.  =0.3 eV; (a) L=2nm; (b) L=10nm; (c) L=50nm.  
+ 
+(a) 
+(b) 
+(c) 
+
+0.015
+0.014
+Drude model
+0.013
+0.012
+0.011
+0.01
+0.009
+0.008
+0.007
+0.006
+0.005
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+x/ L9×10-3
+8
+e--Drudemodel
+7
+6
+5
+4
+3
+1
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+X/ L8
+6
+5
+3
+2
+model
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+X/L17 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 8. The spatial distribution of the conductance in the units 
+/
+Drude
+G
+Ll  for different values of 
+length.  =1.0 eV; (a) L=2nm; (b) L=10nm; (c) L=50nm. 
+ 
+(a) 
+(b) 
+(c) 
+
+0.12
+Drude model
+0.1
+0.08
+ 0.06
+0.04
+0.02.
+0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+X/ L0.03
+0.025
+0.02
+0.01
+0.005
+Drude model
+0.
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+X/L0.03
+0.025
+0.02
+ 0.015
+0.01
+0.005
+Drude
+%.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+X/ L18 
+ 
+4. Conclusion and outlook 
+The main results of the paper can be summarized as follows: 
+ 
+1) We have developed a new theory of interaction of electromagnetic field with graphene 
+sheet for nanoantenna applications in the THz, infrared and optical frequency ranges. The 
+main characteristic feature of our theory is accounting for edge effects in a self-consistent 
+manner. It is based on the concept of optical conductance considered as a general 
+susceptibility and calculated by Kubo approach. The model is based on the concept of 
+Dirac pseudo-spins founded via solving  the boundary-value problem for the Dirac 
+equation with the appropriate  boundary conditions satisfying  the physical model, 
+including  edge  effects  of the sheet; 
+2) The main manifestation of the importance of edge effects is demonstrated by the  
+inhomogeneity of the optical conductance. The amplitude and period of its oscillations 
+depend on the length of the sheet and on the electrochemical potential. It is defined by the 
+number of pseudo-spin modes supporting  the conductance; 
+3) The developed theory is applied for the simulation of the sheet conductance in a  wide 
+range of  sample parameters ( length 2.1nm – 800nm and electrochemical potential 0.1 – 
+1.0 eV). It is shown, that for a length exceeding 800nm our model and the widely used 
+Drude model of conductivity agree to a high degree of accuracy.  However, for small 
+geometric sizes (i.e., smaller than 50nm), the physical picture of conductivity with respect 
+to the Drude model changes dramatically due to the influence of edge effects. This 
+circumstance should be accounted for in the design of graphene-based resonant THz 
+antennas and other types of photonic and plasmonic nanodevices;  
+4) It is shown, that the qualitative distribution of the conductivity along the sheet strongly 
+depends on the electrochemical potential. Thus, it is possible to control the conductivity 
+and performance of graphene nanoantennas, by means of varying  the gate voltage; 
+Our theory allows reformulation of the effective boundary conditions for the electromagnetic 
+field at the surface of the graphene sheet with accounting of the edge effects. It requires the 
+modification of integral equations of antenna theory and the methods of their solution. This 
+should be one of the subjects of future research activity as well as their application to 
+nanoantennas and other nanodevices. 
+Author Contributions: Developments of the physical models, derivation of the basis equations, 
+interpretation of the physical results and righting the paper have been done by T.B., T.M., O.G. and 
+G.S. jointly. The numerical simulations were produced by T.B.  
+Funding: This research was funded by NATO grant number NATO SPS-G5860 and by 
+H2020,project TERASSE 823878. 
+Institutional Review Board Statement: Not applicable. 
+Informed Consent Statement: Not applicable. 
+Data Availability Statement: Not applicable. 
+Conflicts of Interest: The authors declare no conflict of interest 
+ 
+ 
+ 
+ 
+
+19 
+ 
+Appendix A. Derivation of Equation (16) 
+ 
+In this Appendix we   discuss the boundary-value problem for pseudo-spin defined by 
+Equation (14). As it was mentioned above, the pseudo-spin satisfies  the Dirac equation with 
+the following boundary conditions; 
+
+
+
+
+/2
+/2
+0
+s
+s
+u
+L
+v
+L
+
+
+
+ [25].  It may be also 
+transformed into the Helmholtz equation with two special sets of boundary conditions. We 
+have the Dirichlet condition at the left-hand side and the impedance condition 
+
+
+/2
+0
+y
+x
+x L
+k
+u
+
+ 
+
+at the right-hand side for the components u(x)(determined by the  Dirac 
+Equation). The situation is precisely inverted for the second type, namely a Dirichlet 
+condition at the right-hand side and an impedance condition  
+
+/2
+0
+y
+x
+x
+L
+k
+v
+
+ 
+
+at the left-
+hand side. These problems are Hermitian, whereby the eigenmodes form a complete basis. 
+The components u(x) and v(x) are both separately orthogonal, but are mutually non-
+orthogonal, due to their coupling over the electron motion between the atoms of A and B 
+sublattices. The property of orthogonality is shown at Appendix C. Using completeness, 
+orthogonality and normalization conditions 
+ 
+ 
+/2
+/2
+2
+2
+/2
+/2
+1
+L
+L
+p
+p
+L
+L
+u
+x
+dx
+v
+x
+dx
+
+
+
+
+
+
+ , we obtain  
+ 
+ 
+
+
+2
+p
+p
+p
+u
+x u
+x
+x
+x
+
+
+
+
+
+
+
+
+
+
+                                                   
+(A.1) 
+ 
+ 
+
+
+2
+p
+p
+p
+v
+x v
+x
+x
+x
+
+
+
+
+
+
+
+
+
+
+                                                    
+(A.2) 
+ Starting from the xx-component and using the basis relation (11) for a=x, b=x, the matrix 
+element of the current density operator, one gets  
+ 
+ 
+
+
+
+
+ 
+ 
+ 
+ 
+
+
+
+
+1
+ˆ
+,0
+y
+y
+i k
+k
+y
+x
+F
+n
+n
+n
+n
+ss
+j
+ev
+u
+x v
+x
+v
+x u
+x
+e
+l
+
+
+
+
+ 
+
+x
+                           
+(A.3) 
+Next we examine the limit of
+ 
+l  by making the exchange 
+
+
+1
+1
+2
+s
+n
+l
+
+
+
+
+
+
+
+
+
+ 
+and 
+the same for 
+,
+s n
+
+ . Summing over s,s’ and  taking into account both electrons and holes 
+ 
+
+
+in 
+ 
+nv
+x
+
+as well and the charge carriers in two valleys K and K’, leads to.  
+ 
+
+
+
+
+
+
+
+
+ 
+
+2
+2
+2
+( ,
+; )
+4
+0
+( )
+;
+y
+y
+n
+y
+i k
+k
+y y
+F
+xx
+y
+y
+xx
+nn
+y
+n
+n
+k
+ie v
+K
+dk dk
+e
+i
+f
+k
+ 
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+ 
+
+
+x x
+x,x
+    
+                         
+(A.4) 
+where 
+ 
+
+20 
+ 
+
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+,
+;
+xx
+pp
+y
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+x x k
+v
+x v
+x
+u
+x u
+x
+u
+x u
+x
+v
+x v
+x
+u
+x v
+x
+v
+x u
+x
+v
+x u
+x
+u
+x v
+x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+ 
+
+ 
+
+
+
+
+
+
+
+                             
+(A.5) 
+ 
+Note, that the summation in (A.5), means including the contribution from both electrons and 
+holes and the valleys K, K’. The sum over electrons and holes may be transformed to the sum 
+over the electron states  by means of their doubling. The  sum of the two other terms is zero 
+due to the opposite sign of 
+ 
+nv
+x  (subject to the same
+ 
+nu
+x ). The sums over the electron 
+states are decomposed into the components over the two valleys, implying that 
+ 
+ 
+ 
+ 
+...
+...
+...
+2
+...
+K
+K
+K
+n
+n
+n
+n
+
+
+
+
+
+
+
+
+                                            (A.6) 
+ Finally 
+invoking 
+these 
+transformations 
+and 
+using 
+the 
+well-known 
+identity 
+
+
+ 
+1
+2
+ihy
+e dh
+y
+
+
+
+
+
+
+
+ , we obtain  
+
+
+
+ 
+
+
+
+ 
+ 
+
+
+2
+2
+2
+2
+0( )
+;
+'
+'
+2
+0
+n
+y
+F
+xx
+y
+n
+n
+n
+k
+f
+ie v
+K
+dk
+u
+x
+v
+x
+i
+ 
+
+
+
+ 
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+x,x
+x
+x
+  
+(A.7) 
+The above equation relates to the only existing spin-state (a real physical spin rather than a 
+pseudospin). Therefore, the total conductivity must be doubled, which corresponds to the 
+value of the conductivity given by Equation (16). Other components of the conductivity 
+tensor may be obtained in a similar way.  For example, we have 
+
+
+
+
+; ,
+; ,
+yy
+xx
+K
+K
+
+
+
+
+
+x x
+x x and 
+
+
+
+
+; ,
+; ,
+0
+xy
+yx
+K
+K
+
+
+
+
+
+
+x x
+x x
+. 
+ 
+Appendix B: Group velocity of pseudospins in graphene sheet 
+ 
+ Here we calculate the normalized group velocity of the pseudospins defined as 
+
+
+/
+g
+y
+y
+v
+k
+k
+
+
+
+
+ 
+
+,based on the characteristic equation
+
+
+
+
+1
+tg
+=
+x
+x
+y
+x
+k L
+f k
+k
+k
+
+
+. The y-
+component of the wavevector is considered as an independent variable . By taking derivative 
+with respect to
+yk , one gets   
+
+21 
+ 
+ 
+2
+1
+x
+y
+x
+y
+y
+f
+f
+k
+k
+k
+k
+k
+
+
+
+
+
+ 
+
+
+
+                                                        (A.8) 
+ 
+And by using the relation 
+
+
+2
+2
+x
+y
+y
+k
+k
+k
+
+
+
+, we obtain 
+
+
+
+
+2
+2
+2
+2
+y
+y
+g
+y
+x
+y
+y
+y
+y
+y
+k
+k
+v
+k
+k
+k
+k
+k
+k
+k
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                                                        (A.9) 
+For the derivative 
+/
+x
+f
+k
+
+
+we have 
+
+
+
+
+2
+2
+tg
+cos
+x
+x
+x
+x
+x
+k L
+k L
+k L
+f
+k
+k
+
+
+
+
+                                                   (A.10) 
+The trigonometric functions may be expressed through the algebraic ones using the 
+characteristic equation which gives  
+
+
+2
+2
+y
+y
+x
+y
+x
+k
+L
+k
+f
+k
+k k
+
+
+
+
+
+                                                       (A.11) 
+Expressing the group velocity from (A.9) and using (A.10), (A.11), we obtain 
+
+
+
+
+
+
+
+
+
+2
+1
+y
+y
+g
+y
+y
+y
+k
+k L
+v
+k
+k
+L
+k
+
+
+
+
+
+                                                 (A.12) 
+ 
+In order to determine the renormalization  coefficient  
+n
+B  , we can make a  similar 
+transformation: express 
+
+
+sin 2
+xk L thought 
+
+
+tg
+xk L and apply the characteristic Equation (17), 
+which renders by virtue of Equation (20), 
+
+
+
+
+1
+2
+1
+y
+yn
+yn
+n
+y
+yn
+k
+k
+k
+B
+k
+k L
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                                                 (A.13) 
+ with  the conductivity  given explicitly  in Equation (21). 
+Appendix C: Orthogonality  of pseudo-spin modes 
+The Equations for pseudo-spin mode with number s has the form 
+,
+s
+y
+s
+s
+s
+s
+y
+s
+s
+s
+v
+k v
+u
+x
+u
+k u
+v
+x
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+,                                                     (A.14) 
+The similar Relation may be formulated for another mode with the number s: 
+
+22 
+ 
+,
+s
+y
+s
+s
+s
+s
+y
+s
+s
+s
+v
+k v
+u
+x
+u
+k u
+v
+x
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+                                                    (A.15) 
+Let us multiply first Equation (A.14) to 
+su and second Equation multiply to
+sv  . Summarize 
+them and integrate over the interval 
+/2
+/2
+L
+x
+L
+
+
+
+ . Using the boundary conditions, we 
+obtain 
+ 
+ 
+ 
+ 
+/ 2
+/ 2
+/ 2
+/ 2
+0
+L
+L
+s
+s
+s
+s
+s
+s
+L
+L
+u
+x u
+x dx
+v
+x v
+x dx
+
+
+
+
+
+
+
+
+
+
+
+                           (A.16) 
+The similar Relation may be obtained from (A.16) by rearranging the indexes s and s. We 
+have  
+ 
+ 
+ 
+ 
+/ 2
+/ 2
+/ 2
+/ 2
+0
+L
+L
+s
+s
+s
+s
+s
+s
+L
+L
+u
+x u
+x dx
+v
+x v
+x dx
+
+
+
+
+
+
+
+
+
+
+
+                           (A.17) 
+The eigenvalues with the different indexes are non-degenerate. The pair of Equations 
+(A.16),(A.17) may be considered as a system of linear algebraic equations with respect to the 
+integrals. The determinant of this system 
+s
+s
+s
+s
+
+
+
+
+
+
+
+
+is non-zero. It means, that for s
+s
+
+ 
+ 
+ 
+ 
+ 
+/ 2
+/ 2
+/ 2
+/ 2
+0
+L
+L
+s
+s
+s
+s
+L
+L
+u
+x u
+x dx
+v
+x v
+x dx
+
+
+
+
+
+
+
+
+                          (A.18) 
+which gives the orthogonality relation in the required form.  
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+ 
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+An investigation of challenges encountered when
+specifying training data and runtime monitors
+for safety critical ML applications⋆.
+Hans-Martin Heyn1,2[0000−0002−2427−6875], Eric Knauss1,2[0000−0002−6631−872X],
+Iswarya Malleswaran1, and Shruthi Dinakaran1
+1 Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
+2 University of Gothenburg, SE-405 30 Gothenburg, Sweden
+Abstract. [Context and motivation] The development and opera-
+tion of critical software that contains machine learning (ML) models
+requires diligence and established processes. Especially the training data
+used during the development of ML models have major influences on
+the later behaviour of the system. Runtime monitors are used to pro-
+vide guarantees for that behaviour. [Question / problem] We see ma-
+jor uncertainty in how to specify training data and runtime monitoring
+for critical ML models and by this specifying the final functionality of
+the system. In this interview-based study we investigate the underlying
+challenges for these difficulties. [Principal ideas/results] Based on ten
+interviews with practitioners who develop ML models for critical appli-
+cations in the automotive and telecommunication sector, we identified 17
+underlying challenges in 6 challenge groups that relate to the challenge of
+specifying training data and runtime monitoring. [Contribution] The
+article provides a list of the identified underlying challenges related to
+the difficulties practitioners experience when specifying training data
+and runtime monitoring for ML models. Furthermore, interconnection
+between the challenges were found and based on these connections rec-
+ommendation proposed to overcome the root causes for the challenges.
+Keywords: artificial intelligence · context · data requirements · machine
+learning · requirements engineering · runtime monitoring
+1
+Introduction
+With constant regularity, unexpected and undesirable behaviour of machine
+learning (ML) models are reported in academia [9,24,26,51,52], the press, and
+by NGOs3. These problems become especially apparent, and reported upon,
+when ML models violate ethical principles. Racial, religious, or gender biases
+are introduced through a lack of insight into the (sometimes immensely large
+⋆ This project has received funding from the European Union’s Horizon 2020 research
+and innovation program under grant agreement No 957197.
+3 non-governmental organisations, e.g., https://algorithmwatch.org/en/stories/
+arXiv:2301.13476v1  [cs.SE]  31 Jan 2023
+
+2
+H.-M. Heyn et al.
+set of) training data and missing runtime checks for example in large language
+models such as GPT-3 [1], or facial recognition software based on deep learn-
+ing [36]. Unfortunately, improving the performance of deep learning models often
+requires an exponential growth in training data [3]. Data requirements can help
+in preventing unnecessarily large and biased datasets [48]. Due to changes in the
+environment, ML models can become “stale”, i.e., the context changes so signif-
+icantly that the performance of the model decreases below acceptable levels [5].
+Runtime monitors collect performance data and indicate the need for re-training
+of the model with updated training data. However, these monitors need to be
+specified at design time. Data requirements can support the specification of run-
+time monitor [7]. The lack of specifications becomes specifically apparent with
+ML models that are part of critical software 4 because it is not possible to estab-
+lish traceability from system requirements (e.g., functional safety requirements)
+to requirements set on the training data and the runtime monitoring [35].
+Challenges of specifying 
+training data (RQ 1)
+5 Unclear design 
+domain
+3 Missing guidelines for 
+data selection
+6 Unsuitable safety 
+standards
+Challenges of specifying 
+runtime monitoring (RQ 2)
+1 Lack of explainability 
+about ML decisions
+2 Missing conditions for 
+runtime checks
+4 Overhead for 
+monitoring solution
+Fig. 1: Overview of identified challenge categories
+Scope and research questions
+The purpose of this study is to highlight current challenges experienced by prac-
+titioners in specifying training data and runtime monitoring for ML in safety
+critical software.
+The paper contributes a practitioner’s point of view on the challenges re-
+ported in academic literature. The aim is to identify starting-points for a future
+engineering research on the use of runtime monitors for critical ML systems. The
+following research questions guided this study:
+RQ1: What are challenges encountered by practitioners when specifying train-
+ing data for ML models in safety critical software?
+RQ2: What are challenges encountered by practitioners when specifying run-
+time monitors especially in relation to fulfilling safety requirements?
+4 We define critical software as software that is safety, privacy, ethically, and/or mission
+critical, i.e., a failure in the software can cause significant injury or the loss of life,
+invasion of personal privacy, violation of human rights, and/or significant economic
+or environmental consequences [31].
+
+Challenges when specifying data and runtime monitors
+3
+Figure 1 shows the main themes we found in answering the research ques-
+tions. Concerning RQ1, the interviewees reported on several problems: the data
+selection process is nontransparent and guidelines especially towards defining
+suitable measures for data variety are missing. There are no clear context def-
+initions that help in defining data needs, and current safety standards provide
+little guidance. Concerning RQ2, we found that the problem of defining suitable
+metrics and the lack of guidance from safety standards also inhibits the ability to
+specify runtime monitors. Furthermore, practitioners reported on challenges re-
+garding explainability of ML decisions, and the processing and memory overhead
+caused by runtime monitors in safety critical embedded systems.
+The remaining sections of this paper are structured as follows: Section 2
+outlines and argues for the research methods of this study; Section 3 presents the
+results amd answers to the research questions; Section 4 discusses the findings,
+provides recommendations to practitioners and for further research, identifies
+related literature, elaborates on threats to validity, and provides a conclusion.
+2
+Research Method
+We applied a qualitative interview-based survey with open-ended semi-structured
+interviews for data collection. Following the suggestions of Creswell and Creswell
+[13] the qualitative study was conducted in four steps: Preparation of interviews,
+data collection through interviews, data analysis, and result validation.
+Preparations of interviews Based on the a-priori formulated research ques-
+tions, two of the researchers of this study created an interview guide5 which was
+validated and improved by the remaining two researchers. The interview guide
+contains four sections of questions: the first section includes questions about
+the interviewees’ current role, background and previous experiences. The second
+section focuses on questions that try to understand challenges when specifying
+and selecting training data for ML models and how training data affect the per-
+formance of these models. The third section investigates challenges when ML
+models are incorporated in critical systems and how they affect the ability to
+specify training data. The fourth section concentrates on the run time monitor-
+ing aspect of the ML model and contains questions on challenges when specifying
+runtime monitors.
+Sampling strategy: We chose the participants for this study purposefully using
+a maximum variation strategy [14]. We were able to recruit interviewees from
+five different companies, ranging from a local start-up to a multinational world
+leading communication company. An overview is given in Table 1.
+A selection criteria for the company was that they must work with safety-critical
+systems and ML. Within the companies we tried to find interview candidates
+with different roles and work experiences to obtain a view beyond the developers’
+perspective. Besides function developers and ML model developers, we were
+5 The interview guide is available at https://doi.org/10.7910/DVN/WJ8TKY.
+
+4
+H.-M. Heyn et al.
+Table 1: Companies participating in the study
+Company
+Area of operations
+Employees Countries
+1
+Telecommunication networks
+> 10.000
+World
+2
+Automotive OEM
+> 10.000
+World
+3
+Automatic Driving
+> 1.000
+Europe
+4
+Industrial camera systems
+> 1000
+USA
+5
+Deep Learning optimisation for IoT > 100
+Sweden
+Table 2: Participants of the study
+Inter-
+viewee
+Role
+Experience
+A
+Researcher (Academic)
+Functional Safety for ADAS
+B
+Function developer
+Sensor and perception systems
+C
+Principal engineer
+ML model integration
+D
+ML model developer
+Distributed and edge systems
+E
+Function owner
+ADAS perception functions
+F
+Function developers
+and test engineer
+Automatic driving systems
+G
+Data Scientist
+Distributed systems
+H
+Requirement Engineer
+Perception systems
+I
+Researcher (Academic)
+Neural Network development
+J
+Functional Safety Manager Sensor systems
+ADAS: Advanced Driver Assistance Systems
+interested in interviewing requirement engineers and product / function owners
+because they represent key roles in deriving system or function specifications.
+We provided the companies with a list of roles that we identified beforehand as
+interesting for interviewing6. Additionally, we interviewed two researchers from
+academia who participate in a joint industry EU Horizon 2020 project called
+VEDLIoT7. Both researchers worked also with ML models in industry before.
+Therefore, they could provide insights into both the academic and the industry
+perspective. A list of the ten interviewees for this study is provided in Table 2.
+Data collection through interviews All interviews were conducted remotely
+using either the conference software Zoom or Microsoft Teams and took between
+60 - 90 minutes. The a-priori defined interview guide was only available to the
+interviewers and was not distributed to the participants beforehand. Each par-
+ticipant was interviewed by two interviewers who alternated in asking questions
+and observing. At the start of each interview, the interviewers provided some
+background information about the study’s purpose. Then, the interview guide
+was followed. However, as we encouraged discussions with the interviewees, we
+allowed deviations from the interview guide by asking additional questions, or
+changing the order of the questions when it was appropriate [30]. All interviews
+were recorded and semi-automatically transcribed. The interviewers manually
+checked and anonymised the results.
+6 The list included functional safety experts, requirement engineers, product owners
+or function owners, function or model developers, and data engineers.
+7 Very efficient deep learning in the Internet of Things
+
+Challenges when specifying data and runtime monitors
+5
+Data analysis The data analysis followed suggestions by Saldana [41] and
+consisted of two cycles of coding and validation of the themes through a workshop
+and member checking.
+First coding cycle: Attribute coding was used to extract information about the
+participants’ role and previous experiences. Afterwards, the two interviewers
+independently applied structural coding to collect phrases in the interviews that
+represent topics relevant to answering the research questions. The researchers
+compared the individually assigned codes and applied descriptive coding with the
+aim of identifying phrases that describe common themes across the interviews.
+Theme validation: In a focus group, the identified themes were presented and dis-
+cussed. Thirteen researchers from both industry and academia in the VEDLIoT
+project participated. Three of the participants also were interviewed for this
+study. The aim of the focus group was to reduce bias in the selection of themes
+and to identify any additional themes that the researchers might have missed.
+Second coding cycle: After the themes were identified and validated, the second
+coding cycle was used to map the statements of the interviewees to the themes,
+and consequently identify the answers to the research questions. The second
+cycle was conducted by the two researchers who did not conduct the first cycle
+coding in order to reduce confirmation bias. The mapping was then confirmed
+and agreed upon by all involved researchers.
+Result validation Member checking, as described in [14, Ch. 9] was used to
+validate the identified themes that answer RQ 1 and RQ 2. Additionally, we
+presented the results in a 60 minute focus group to an industry partner and
+allowed for feedback and comments on the conclusions we drew from the data.
+3
+Results
+During the first coding cycle, structural coding resulted in 117 statements for
+RQ1 and 77 statements for RQ2. Through descriptive coding preliminary themes
+were found. The statements and preliminary themes were discussed during a
+workshop. Based on the feedback from the workshop, 117 statements for RQ1
+were categorised into eight final challenge themes and three challenge categories
+relating to the challenge of specifying training data. Similar, the 77 original
+statements for RQ2 were categorised into 13 final challenge themes in five chal-
+lenge categories relating to the challenge of specifying runtime monitoring. A
+total of six challenge categories emerged for both RQs, out of which two cate-
+gories contain challenges relating to both training data and runtime monitoring
+specification, and three challenge themes base on statements from both RQs.
+The categories and final challenge themes are listed in Table 3.
+3.1
+Answer to RQ1: Challenges practitioners experience when
+specifying training data
+The interviewees were asked to share their experiences in selecting training data,
+the influence of the selection of training data on the system’s performance and
+
+6
+H.-M. Heyn et al.
+Table 3: Challenge groups (bold) and themes found in the interview data. Data.:
+Challenges related to specifying training data (RQ1). Monitor.: Challenges re-
+lated to specifying runtime monitoring (RQ2).
+Relates to
+Related
+ID
+Challenge Theme
+Data. Monitor. Literature
+1
+Lack of explainability about ML decisions
+✓
+1.1 No access to inner states of ML models
+✓
+[18]
+1.2 No failure models for ML models
+✓
+[51]
+1.3 IP protection
+✓
+2
+Missing conditions for runtime checks
+✓
+2.1 Unclear metrics and/or boundary conditions
+✓
+[11,21,43]
+2.2 Unclear measure of confidence
+✓
+[17,34]
+3
+Missing guidelines for data selection
+✓
+✓
+3.1 Disconnection from requirements
+✓
+[16,42]
+3.2 Grown data selection habits
+✓
+[20,33]
+3.3 Unclear completeness criteria
+✓
+[49]
+3.4 Unclear measure of variety
+✓
+✓
+[45,50]
+4
+Overhead for monitoring solution
+✓
+4.1 Limited resources in embedded systems
+✓
+[38]
+4.2 Meeting timing requirements
+✓
+4.3 Reduction of true positive rate
+✓
+5
+Unclear design domain
+✓
+5.1 Design domain depends on available data
+✓
+[6]
+5.2 Uncertainty in context
+✓
+[22]
+6
+Unsuitable safety standards
+✓
+✓
+6.1 Focus on processes instead of technical solution
+✓
+✓
+[10]
+6.2 No guidelines for probabilistic effects in software
+✓
+[28,43]
+6.3 Safety case only through monitoring solution
+✓
+[31,46]
+safety, and any experiences and thoughts on defining specifications for training
+data for ML. Based on the interview data, we identified three challenge groups
+related to specifying training data: missing guidelines for data selection, unclear
+design domain, and unsuitable safety standards
+Missing guidelines for data selection Four interviewees reported on a lack
+of guidelines and processes related to the selection of training data. A reason
+can be that data selection bases on “grown habits” that are not properly doc-
+umented. Unlike conventional software development, the training of ML is an
+iterative process of discovering the necessary training data based on experience
+and experimentation. Requirements set on the data are described as discon-
+nected and unclear for the data selection process. For example, one interviewee
+stated that if a requirements is set that images shall contain a road, it remains
+unclear what specific properties this road should have. Six interviewees described
+missing requirements on the data variety and missing completeness criteria as a
+reason for the disconnection of requirements from data selection.
+“How much of it (the data) should be in darkness? How much in rainy conditions,
+and how much should be in snowy situations?” - Interview F
+“For example, we said that we shall collect data under varying weather conditions.
+What does that mean?” - Interview B
+Another interviewee stated that it is not clear how to measure variety, which
+could be a reason why it is difficult to define requirements on data variety.
+
+Challenges when specifying data and runtime monitors
+7
+“What [is] include[d] in variety of data? Is there a good measure of variety?” -
+Interview A
+Unclear design domain Three interviewees describe uncertainty in the design
+domain as a reason for why it is difficult to specify training data. If the design
+domain is unclear, it will be challenging to specify the necessary training data.
+“We need to understand for what context the training data can be used.” - Interview J
+“ODD [(Operational Design Domain)]? Yes, of course it translates into data require-
+ments.” - Interview F
+Unsuitable safety standards Because we were specifically investigating ML
+in safety critical applications, we asked the participants if they find guidance in
+safety standards towards specifying training data. Five interviewees stated that
+current safety standards used in their companies do not provide suitable guid-
+ance for the development of ML models. While for example ISO 26262 provides
+guidance on how to handle probabilistic effects in hardware, no such guidance is
+provided for software related probabilistic faults.
+“The ISO 26262 gives guidance on the hardware design; [...] how many faults per
+hour [are acceptable] and how you achieve that. For the software side, it doesn’t
+give any failure rates or anything like that. It takes a completely process oriented
+approach [...].” - Interview C
+One interviewee mentioned that safety standards should emphasise more the
+data selection to prevent faults in the ML model due to insufficient training.
+“To understand that you have the right data and that the data is representative,
+ISO 26262 is not covering that right now which is a challenge.” - Interview H
+3.2
+Answer to RQ2: Challenges practitioners experience when
+specifying runtime monitors
+We asked the interviewees on the role of runtime monitoring for the systems they
+develop, their experience with specifying runtime monitoring, and the relation of
+runtime monitoring to fulfilling safety requirements on the system. We identified
+five challenge groups related to runtime monitoring: lack of explainability about
+ML decisions, missing conditions for runtime checks, missing guidelines for data
+selection, overhead for monitoring solution, and unsuitable safety standards.
+Lack of explainability about ML A reason why it is difficult to specify
+runtime monitors for ML models is the inability to produce failure models for
+ML. In normal software development, causal cascades describe how a fault in a
+software components propagates trough the systems and eventually leads to a
+failure. This requires the ability to break down the ML model into smaller com-
+ponents and analyse their potential failure behaviour. Four interviewees however
+reported that they can only see the ML model as a “black-box” with no access
+to the inner states of the ML model. As a consequence, there is no insight into
+the failure behaviour of the ML model.
+
+8
+H.-M. Heyn et al.
+“[Our insight is] limited because it’s a black box. We can only see what goes in
+and then what comes out to the other side. And so if there is some error in the
+behavior, then we don’t know if it’s because [of a] classification error, planning
+error, execution error?” - Interview F
+The reason for opacity of ML models is not necessarily due to technology limita-
+tions, but also due to constraints from protection of intellectual property (IP).
+“Why is it a black box? That’s not our choice. That’s because we have a supplier
+and they don’t want to tell us [all details].” - Interview F
+Missing conditions for runtime checks A problem of specifying runtime
+monitors is the need for finding suitable monitoring conditions. This requires
+the definition of metrics, goals and boundary conditions. Five interviewees report
+that they face challenges when defining these metrics for ML models.
+“What is like a confidence score, accuracy score, something like that? Which score
+do you need to ensure [that you] classified [correctly]?” - Interview F
+Especially the relation between correct behaviour of the ML model and measure
+of confidence is unclear, and therefore impede runtime monitoring specification.
+“We say confidence, that’s really important. But what can actually go wrong here?”
+- Interview J
+Missing guidelines for data selection The inability to specify the meaning
+of data variety also relates to missing conditions for runtime checks. For example,
+runtime monitors can be used to collect additional training data, but without a
+measure of data variety it is difficult to find the required data points.
+Overhead for monitoring solution An often overlooked problem seems to
+be the induced (processing) overhead from a monitoring solution. Especially in
+the automotive sector, many software components run on embedded computer
+devices with limited resources.
+“You don’t have that much compute power in the car, so the monitoring needs to
+be very light in its memory and compute footprint on the device, maybe even a
+separate device that sits next to the device.” - Interview F
+And due to the limited resources in embedded systems, monitoring solutions can
+compromise timing requirements of the system. Additionally, one interviewee
+reported concerns regarding the reduction of the ML model’s performance.
+“[. . . ] the true positive rate is actually decreasing when you have to pass it through
+this second opinion goal. It’s good from a coverage and safety point of view, but it
+reduces the overall system performance.” - Interview F
+
+Challenges when specifying data and runtime monitors
+9
+Unsuitable safety standards Safety standards are mostly not suitable for be-
+ing applied to ML model development. Therefore, safety is often ensured through
+non-ML monitoring solutions. Interviewees reported that it is not a good solution
+to rely only on the monitors for safety criticality:
+“[. . . ] so the safety is now moved from the model to the monitor instead, and it
+shouldn’t be. It should be the combination of the two that makes up safety.” -
+Interview B
+One reason is that freedom of inference between a non-safety critical component
+(the ML model), and a safety critical component (the monitor) must be ensured
+which can complicate the system design.
+“And then especially if you have mixed critical systems [it] means you have ASIL
+[(Automotive Safety Integrity Level)] and QM [(Quality Management)] components
+in your design [and] you want to achieve freedom from interference in your system.
+You have to think about safe communication and memory protection.” - Interview J
+4
+Discussion and Conclusion
+The results reveal connections between the challenges. Not all theme groups re-
+late exclusively to one of the two challenges. For example, themes in the groups
+unsuitable safety standards and missing guidelines for data selection relate to
+both challenges of specifying training data and runtime monitoring. Further-
+more, we identified cause-effect relations between different themes and across
+different group of themes. For example IP protection is a cause for the inability
+of accessing the inner states and for creating failure models for ML model. We
+based this assessment on a semantic analyses of the words used in the statements
+relating to these themes. For example, Interviewee F stated that:
+“That neural network is something [of a] black box in itself. You don’t know why it
+do[es] things. Well, you cannot say anything about its inner behavior” - Interview F
+Later in the interview, the same participants states:
+“Why is it a black box? That’s not our choice. That’s because we have a supplier
+and they don’t want to tell us [all details].” - Interview F
+Figure 2 illustrates the identified cause-effect relations, relations between the
+themes, and how the different themes relate to the challenges.
+Recommendations to practitioners and for further research The iden-
+tified root causes of the challenges described by the participants allowed us to
+formulate recommendations listed in Table 4. Because these recommendations
+try to solve root causes described by the participants of the interview study,
+we think they are a useful first step towards solving the challenges related to
+specifying training data and runtime monitoring.
+
+10
+H.-M. Heyn et al.
+5 Unclear design
+domain
+3 Missing guidelines for
+data selection
+2 Missing conditions for
+runtime checks
+2.2 Unclear mea-
+sure of confidence
+6 Unsuitable safety
+standards
+Challenges of
+specifying
+runtime
+monitoring
+Challenges of
+specifying
+training data
+3.1 Disconnection
+from requirements
+3.3 Unclear comple-
+teness criteria
+5.1 Data require-
+ments depend on
+design domain 
+5.2 Uncertainty in
+context
+1 Lack of explainability
+about ML decisions
+1.1 No access to
+inner states
+6.1 Focus on
+processes instead of
+technical solutions
+6.2 No guidelines for
+probabilistic effects
+in software
+6.3 Safety case only
+through monitoring
+solution
+2.1 Unclear  
+metrics / bound-
+ary conditions
+4 Overhead for
+monitoring solution
+4.2 Meeting timing
+requirements
+4.1 Limited
+resources in
+embedded
+systems
+3.2 Grown data
+selection habits
+1.2 No failure
+models
+1.3 IP protection
+4.3 Reduction of
+true positive rate
+3.4 Unclear
+measure of variety
+Fig. 2: Connection between the identified challenge themes. Enclosed themes
+have been identified as causes for the surrounding themes. Furthermore, dotted
+lines indicate relations between different themes.
+4.1
+Related Literature
+The problem of finding the “right” data: For acquiring data, data scientists have
+to rely on data mining with little to no quality checking and potential biases [4].
+Biased datasets are a common cause for erroneous or unexpected behaviour of
+ML models in critical environments, such as in medical diagnostic [8], in the
+juridical system [19,37], or in safety-critical applications [15,46].
+There are attempts to create “unbiased” datasets. One approach is to curate
+manually the dataset, such as in the FairFace dataset [29], the CASIA-SURF
+CeFaA dataset [32], or Fairbatch [40]. An alternative road is to use data augmen-
+tation techniques to “rebalance” the dataset [27,45]. However, it was discovered
+that it is not sufficient for avoiding bias to use an assumed balanced datasets
+during training [20,49,50] because it is often unclear which features in the data
+need to be balanced. Approaches for curating or manipulating the dataset re-
+quire information on the target domain, i.e., one needs to set requirements on
+the dataset depending on the desired operational context [6,16,22]. But deriving
+a data specification for ML is not common practise [25,33,42].
+
+Challenges when specifying data and runtime monitors
+11
+Table 4: Recommendations for practitioners and suggestions for further research
+ID
+Recommendation
+I
+Avoid restrictive IP protection. IP protection is a cause for the inability of accessing the
+inner states of the ML models (black-box model). This causes a nontransparent measure of
+confidence, and an inability to formulate failure models. To our knowledge, no studies have yet
+been performed on the consequences of IP protection of ML models on the ability to monitor
+and reason (e.g., in a safety case) for the correctness of ML model decisions.
+II Relate measures of confidence to actual performance metrics. For runtime monitoring,
+the measure of confidence is often used to evaluate the reliability of the ML model’s results.
+But without understanding and relating that measure to clearly defined performance metrics of
+the ML model first, the measure of confidence provides little insight for runtime monitoring. In
+general, defining suitable metrics and boundary conditions should become an integral part of
+RE for machine learning as it affects both the ability to define data requirements and runtime
+monitoring requirements.
+III Overcome grown data selection habits. Grown data selection habits have been mentioned
+as a reason for a lack of clear completeness criteria and a disconnection from requirements.
+Based on our results, we argue that more systematic data selection processes need to be es-
+tablished in companies. This would allow for a better connection of the data selection process
+to requirement engineering and it creates a traceability between system requirements, com-
+pleteness criteria and data requirements. Additionally, it might also reduce the amount of data
+needed for training, and therefore cost of development.
+IV Balance hardware limitation in embedded systems. Runtime monitoring causes a pro-
+cessing and memory overhead that can compromise timing requirements and reduce the ML
+model’s performance. Today, safety criticality of systems with ML is mostly ensured through
+monitoring solutions. By decomposing the safety requirements instead onto both the monitor-
+ing and the ML model, the monitors might become more resource efficient, faster, and less
+constraining in regards to the decisions of the ML model. However, safety requirements on the
+ML models might trigger requirements on the training data.
+The problem of finding the “right” runtime monitor: Through clever test strate-
+gies, some uncertainty can be eliminated in regards to the behaviour of the
+model [11]. However, ML components are often part of systems of systems and
+their behaviour is hard to predict and analyse at design time [47]. DevOps prin-
+ciples from software engineering give promising ideas on how to tackle remaining
+uncertainty at runtime [34]. As part of the operation of the model, runtime mod-
+els that “augment information available at design-time with information moni-
+tored at runtime” help in detecting deviations from the expected behaviour [17].
+These runtime models for ML can take the form of model assertions, i.e., check-
+ing of explicitly defined attributes of the model at runtime [28]. However, the
+authors state that “bias in training sets are out of scope for model assertion”. An-
+other model based approach can be the creation of neuron activation patterns for
+runtime monitoring [12]. Other approaches treat the ML model as “black-box”,
+and only check for anomalies and drifts in the input data [39] the output [43],
+or both [18]. However, similar to the aforementioned challenges when specifying
+data for ML, runtime monitoring needs an understanding on how to “define, re-
+fine, and measure quality of ML solutions” [23], i.e., in relation to non-functional
+requirements one needs to understand which quality aspects are relevant, and
+how to measure them [21]. Most commonly applied safety standards emphasise
+processes and traceability to mitigate systematic mistakes during the develop-
+ment of critical systems. Therefore, if the training data and runtime monitoring
+cannot be specified, a traceability between safety goals and the deployed system
+cannot be established [10].
+
+12
+H.-M. Heyn et al.
+For many researchers and practitioners, runtime verification and monitoring
+is a promising road to assuring safety and robustness for ML in critical soft-
+ware [2,11]. However, runtime monitoring also creates a processing and memory
+overhead that needs to be considered especially in resource-limited environments
+such as embedded devices [38].
+The related work has been mapped to the challenges identified in the inter-
+view study in Table 3.
+4.2
+Threats to validity
+A lack of rigour (i.e., degree of control) in the study design can cause confound-
+ing which can manifest bias in the results [44]. The following mechanisms in this
+study tried to reduce confounding: The interview guide was peer-reviewed by an
+independent researcher, and a test session of the interview was conducted. To
+reduce personal bias, at least two authors were present during all interviews, and
+the authors took turn in leading the interviews. To confirm the initial findings
+from the interview study and reduce the risk of researchers’ bias, a workshop was
+organised which was also visited by participants who were not part of the inter-
+view study. Another potential bias can arise from the sampling of participants.
+Although we applied purposeful sampling, we still had to rely on the contact
+persons of the companies to provide us with suitable interview candidates. We
+could not directly see a list of employees and choose the candidates ourselves.
+Regarding generalisability of the findings, the limited number of companies in-
+volved in the study can pose a threat to external validity. However, two of the
+companies are world-leading companies in their fields, which, in our opinion,
+gives them a deep understanding and experience of the discussed problems. Fur-
+thermore, we included companies from a variety of different fields to establish
+better generalisability. Furthermore, our data includes only results valid for the
+development of safety-critical ML models. We assume that the findings are ap-
+plicable also to other forms of criticality, such as privacy-critical, but we cannot
+conclude on that generalisability based on the available data.
+4.3
+Conclusion
+This paper reported on a interview-based study that identified challenges related
+to specifying training data needs and runtime monitoring for safety critical ML
+models. Through interviews conducted at five companies we identified 17 chal-
+lenges in six groups. Furthermore, we performed a semantic analysis to identify
+the underlying root-causes. We saw that several underlying challenges affect
+both the ability to specify training data and runtime monitoring. For example,
+we concluded that restrictive IP protection can cause an inability to access and
+understand the inner states of a ML model. Without insight into the ML model’s
+state, the measure of confidence cannot be related to actual performance metrics.
+Without clear performance metrics, it is difficult to define the necessary degree of
+variety in the training data. Furthermore, grown data selection impedes proper
+requirement engineering for training data. Finally, safety requirements should be
+
+Challenges when specifying data and runtime monitors
+13
+distributed on both the ML model which can cause requirements on the training
+data, and on runtime monitors which can reduce the overhead by the moni-
+toring solution. These recommendations will serve as starting point for further
+engineering research.
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+

LdE4T4oBgHgl3EQfJww5/content/tmp_files/2301.04923v1.pdf.txt

@@ -0,0 +1,2046 @@
+Semi-Lagrangian Finite-Element Exterior
+Calculus for Incompressible Flows
+Wouter Tonnon*
+Ralf Hiptmair†
+January 13, 2023
+1
+Incompressible Navier-Stokes Equations
+We consider the incompressible Navier-Stokes equations—a standard hydrodynamic
+model for the motion of an incompressible, potentially-viscous fluid—in a container
+with rigid walls, where we impose so-called “free boundary conditions” in the par-
+lance of [31, p. 346] and [43, p. 502], see the latter article for further references. We
+search the fluid velocity field u(t, x) and the pressure p(t, x) as functions of time t and
+space x on a bounded, Lipschitz domain Ω ⊂ Rd such that they solve the evolution
+boundary-value problem
+∂tu + u · ∇u − ε∆u + ∇p = f,
+on ]0, T[×Ω,
+(1a)
+∇ · u = 0,
+on ]0, T[×Ω,
+(1b)
+u · n = 0,
+on ]0, T[×∂Ω,
+(1c)
+εn × ∇ × u = 0,
+on ]0, T[×∂Ω,
+(1d)
+u = u0,
+on {0} × Ω,
+(1e)
+where ε ≥ 0 denotes a (non-dimensional) viscosity, f a given source term, T > 0 the
+final time, ∂Ω the boundary of Ω, and n(x) the outward normal vector at x ∈ ∂Ω. The
+initial condition u0 is to satisfy ∇ · u0 = 0 in Ω and u0 · n = 0, εn × ∇ × u0 = 0 on
+∂Ω. Based on the variational description of the Navier-Stokes equations as described
+in [5], u can be interpreted as a differential 1-form [33] and we can recast system (1)
+in the following way. Let Λk(Ω) for k ∈ N denote the space of differential k-forms on
+*SAM, ETH Zürich, CH-8092 Zürich, [email protected]
+†SAM, ETH Zürich, CH-8092 Zürich, [email protected]
+1
+arXiv:2301.04923v1  [math.NA]  12 Jan 2023
+
+Ω. Then we search ω ∈ Λ1(Ω) and p ∈ Λ0(Ω) such that
+Duω + εδdω + dp = f,
+on ]0, T[×Ω,
+(2a)
+δω = 0,
+on ]0, T[×Ω,
+(2b)
+tr ⋆ω = 0,
+on ]0, T[×∂Ω,
+(2c)
+ε tr ⋆dω = 0,
+on ]0, T[×∂Ω,
+(2d)
+ω = ω0,
+on{0} × Ω,
+(2e)
+where Duω denotes the material derivative of ω with respect to u, d : Λk−1(Ω) �→
+Λk(Ω) the exterior derivative, δ : Λk(Ω) �→ Λk−1(Ω) the exterior coderivative, and the
+trace tr is the pullback under the embedding ∂Ω ⊂ ¯Ω. Here, u is related to ω through
+ω := uZ, i.e. u is the vector proxy of ω w.r.t. the Euclidean metric. Similarly, we have
+that Λ1(Ω) ∋ ω0 := u0Z and Λ1(Ω) ∋ f := fZ. Note that (2) can be derived through
+classical vector calculus for vector proxies as shown in Appendix A for d = 3.
+As shown in [18, 19], sufficiently-smooth solutions ω :]0, T[�→ Λ1(Ω) of the in-
+compressible Navier-Stokes equations as given in system (2) satisfy an energy relation,
+that is,
+dE
+dt (t) := d
+dt
+1
+2
+�
+Ω
+ω(t) ∧ ⋆ω(t) = −ε
+�
+Ω
+dω(t) ∧ ⋆dω(t).
+(3)
+This relation implies energy conservation for ε = 0. In the case of ε = 0, we also have
+helicity conservation, that is,
+ε = 0 =⇒ dH
+dt (t) := d
+dt
+�
+Ω
+dω(t) ∧ ω(t) = 0.
+(4)
+Note that the Onsager conjecture tells us that in the case ε = 0 the solutions need to
+be at least Hölder regular with exponent 1
+3 for energy conservation to hold [28].
+Remark 1 We acknowledge that the boundary condition (1d) is non-standard. This
+boundary condition was chosen because it is the natural boundary condition associ-
+ated to system (2). To enforce the standard no-slip boundary conditions, (1d) could
+be replaced by εu × n = u on ]0, T[×∂Ω, that is, we impose an essential instead
+of natural boundary condition to system (2). Unfortunately, in this case, the scheme
+presented in this work leads to an ill-posed system. In the case ε = 0, the only imposed
+boundary condition (1c) is standard.
+Remark 2 Boundary conditions (1c),(1d) can be interpreted as slip boundary con-
+ditions. However, on smooth domains Ω, they are only equivalent to Navier’s slip
+boundary conditions if the Weingarten map related to ∂Ω vanishes [31, section 2].
+2
+Outline and Related Work
+We propose a semi-Lagrangian approach to the discretization of the reformulated
+Navier-Stokes boundary value problem (2). This method revolves around the dis-
+cretization of the material derivative Duω in the framework of a finite-element Galerkin
+2
+
+discretization on a fixed spatial mesh. The main idea is to approximate Duω by back-
+ward difference quotients involving transported snapshots of the 1-form ω, which can
+be computed via the pullback induced by the flow of the velocity vector field u.
+Semi-Lagrangian methods for transient transport equations like (2) are well-es-
+tablished for the linear case when u is a given Lipschitz-continuous velocity field.
+In particular, for ω a 0-form, that is, a plain scalar-valued function, plenty of semi-
+Lagrangian approaches have been proposed and investigated, see, for instance, [8, 7,
+20, 21, 23, 35, 39, 40, 6, 12, 45]. We refer to [24, Chapter 5] for a comprehensive
+pre-2013 literature review on the analysis of general semi-Lagrangian schemes. Most
+of these methods focus on mapping point values under the flow, with the exception
+of a particularly interesting class of semi-Lagrangian methods known as Lagrange-
+Galerkin methods.
+Lagrange-Galerkin methods do not transport point values, but
+rather triangles (in 2D) or tetrahedra (in 3D). Refer to [10] for a review of those meth-
+ods.
+Meanwhile semi-Lagrangian methods for transport problems for differential forms
+of any order have been developed [26, 25, 24]. The next section will review these
+semi-Lagrangian methods for linear transport problems with emphasis on 1-forms.
+We will also introduce a new scheme which is second-order in space and time based
+on so-called “small edges”, see section 3.1.2 for details.
+Semi-Lagrangian schemes for the incompressible Navier-Stokes equations are also
+well-documented in literature, with emphasis on the Lagrange-Galerkin method [10,
+14, 15, 30, 1, 11, 9]. A survey of the application of Lagrange-Galerkin methods to the
+incompressible Navier-Stokes equations is given in [10]. It is important to note that
+these methods require the evaluation of integrals of transported quantities and, in case
+these integrals cannot be computed exactly, instabilities can occur [12, 32]. A possi-
+ble remedy is to add an additional stabilization term that includes artificial diffusion
+[10]. Other semi-Lagrangian methods for incompressible Navier-Stokes equations di-
+rectly transport point values with the nodes of a mesh instead of evaluating integrals
+of transported quantities, see [34, 29, 47, 46, 13] and [16], where the last work makes
+use of exponential integrators [17]. Most authors employ spectral elements for the
+discretization in space [34, 29], but any type of finite-element space with degrees-of-
+freedom relying on point evaluations can be used. The methods proposed in [47, 46]
+are also based on finite-element spaces with degrees-of-freedom on nodes, but em-
+ploy backward-difference approximations for the material derivative. The work [13]
+proposes an explicit semi-Lagrangian method still built around the transport of point
+values in the nodes of the mesh. The diffusion term is also taken into account in a
+semi-Lagrangian fashion and the incompressibility constraint is enforced by means of
+a Chorin projection. Also [13] proposes an explicit semi-Lagrangian scheme using the
+same principles, but based on the vorticity-streamfunction form of the incompressible
+Navier-Stokes equations.
+All the mentioned semi-Lagrangian schemes rely on the transport of point val-
+ues of continuous vector fields, which is the perspective embraced in formulation (1).
+However, we believe that, in particular in the case of free boundary conditions (1c)
+3
+
+and (1d), the semi-Lagrangian method based on (2) offers benefits similar to the bene-
+fits bestowed by the use of discrete differential forms (finite-element exterior calculus,
+FEEC [3, 4]) for the discretization of electromagnetic fields. Section 4 will convey that
+the boundary conditions (2c), (2d), and the incompressiblity constraint can very natu-
+rally be incorporated into a variational formulation of (2) posed in spaces of 1-forms.
+This has been the main motivation for pursuing the new idea of a semi-Lagrangian
+method for (2) that employs discrete 1-forms. Another motivation has been the ex-
+pected excellent robustness of the semi-Lagrangian discretization in the limit ε �→ 0.
+Numerical tests reported in section 5 will confirm this.
+Two more aspects of our method are worth noting: Firstly, a discrete 1-form ωh will
+not immediately spawn a continuous velocity field uh = ωZ
+h, However, continuity is
+essential for defining a meaningful flow. We need an additional averaging step, which
+we present in section 4.1. Secondly, since semi-Lagrangian methods fail to respect the
+decay/conservation laws (3) and (4) exactly, we present a way how to enforce them in
+section 4.3.
+3
+Semi-Lagrangian Advection of differential forms
+3.1
+Discrete differential forms
+We start from a simplicial triangulation Kh(Ω) of Ω, which may rely on a piecewise
+linear approximation of ∂Ω so that it covers a slightly perturbed domain.
+3.1.1
+Lowest-order case: Whitney forms
+For Λ0(Ω)—the space of 0-forms on Ω, which is just a space of real-valued func-
+tions—the usual (Lagrange) finite-element space of continuous, piecewise-linear, poly-
+nomial functions provides the space Λ0
+h,1(Ω) of lowest-order discrete 0-forms.
+Let d ∈ {2, 3}, K a d-simplex with edges {e1, .., e3(d−1)}. To construct lowest-
+order discrete 1-forms on K, we associate to every edge ei a local shape function
+wei. Let the edge ei be directed from vertex v1
+i to v2
+i , then the local shape function
+wei ∈ Λ1(K) associated with edge ei is
+wei := λv1
+i dλv2
+i − λv2
+i dλv1
+i ,
+(5)
+where λv represents the barycentric coordinate associated with vertex v. We define
+the lowest-order, local space of discrete 1-forms
+Λ1
+h,1(K) := span{we; e an edge of K}.
+(6)
+Using these local spaces, we can define the global space of lowest-order, discrete 1-
+forms
+Λ1
+h,1(Ω) := {ω ∈ Λ1(Ω); ∀K ∈ Kh(Ω) : ω|K ∈ Λ1
+h,1(K)},
+(7)
+4
+
+(0,0)
+(1,0)
+(0,1)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+(a) 9 small edges of a second-order element in 2D. All the edges between the different
+connection points are small edges. In 3D, we simply have all these small edges on
+the faces of the simplex.
+edge no.
+l.s.f.
+edge no.
+l.s.f.
+1
+[ x
+y ] �→
+�
+x(x+y−1)
+−(x−1)(x+y−1)
+�
+6
+[ x
+y ] �→
+�
+(y−1)(x+y−1)
+x(1−x−y)
+�
+2
+[ x
+y ] �→
+�
+−y2
+y(x−1)
+�
+7
+[ x
+y ] �→
+�
+−xy
+x(x−1)
+�
+3
+[ x
+y ] �→
+� −y2
+xy
+�
+8
+[ x
+y ] �→
+� y(1−y)
+xy
+�
+4
+[ x
+y ] �→
+� −xy
+x2
+�
+9
+[ x
+y ] �→
+�
+y(x+y−1)
+x(1−x−y)
+�
+5
+[ x
+y ] �→
+�
+x(1−y)
+x2
+�
+(b) Local shape functions (l.s.f.) for the unit triangle associated with second-order,
+discrete differential forms in 2D as proposed in [38]. Each shape function corre-
+sponds to the small edge in (a) with the same numbering.
+Figure 1: Illustration of small edges (a) and corresponding local shape functions (b)
+for the unit triangle.
+5
+
+where Λ1(Ω) again denotes the space of differential 1-forms on Ω. We demand that
+for every ω ∈ Λ1(Ω) integration along any smooth oriented path yields a unique value.
+Thus, the requirement ω ∈ Λ1(Ω) imposes tangential continuity on the vector proxy
+of ω.
+3.1.2
+Second-order discrete forms
+Similar to the lowest-order case, the space Λ0
+h,2(Ω) of second-order discrete 0-forms
+is spawned by the usual (Lagrange) finite-element space of continuous, piecewise-
+quadratic, polynomial functions.
+Let d ∈ {2, 3}, K a d-simplex with edges {e1, .., e3(d−1)} and vertices {v1, .., vd+1}.
+To construct second-order discrete 1-forms, we associate local shape functions to
+"small edges". We can construct 3(d + 1)(d − 1) small edges [38, Definition 3.2]
+by defining ∀i ∈ {1, .., d + 1} and ∀j ∈ {1, .., 3(d − 1)}
+{vi, ej} := {vi + 1
+2(x − vi); x ∈ ej},
+where {vi, ej} denotes the small edge. In Figure 1a we illustrate the 9 small edges of a
+2-simplex. For example, we see that small edge 9 can be written as {(0, 0), [(1, 0), (0, 1)]}.
+To make the difference between small edges and edges of the mesh explicit, we will
+sometimes refer to the latter as "big edges".
+The local shape function [38, Definition 3.3] associated with {vi, ej} is given by
+w{vi,ej} := λviwej,
+where wej denotes the Whitney 1-form associated with the big edge ej as defined in
+(5). In Figure 1b we give explicit expressions for the shape functions associated with
+the small edges in Figure 1a. Note that the local shape functions of the form w{v,e}
+associated with small edges in the interior (d = 2) or on the same face (d = 3) of
+the form {v, e} such that v /∈ ∂e (example: small edge 7, 8, and 9 in Figure 1a) are
+linearly dependent. We define the second-order, local space of discrete 1-forms [38,
+Definition 3.3]
+Λ1
+h,2(K) := span{w{v,e}; v a vertex of K, e a (big) edge of K}.
+(8)
+Using these local spaces, we can define the global space of second-order, discrete 1-
+forms
+Λ1
+h,2(Ω) := {ω ∈ Λ1(Ω); ∀K ∈ Kh(Ω) : ω|K ∈ Λ1
+h,2(K)},
+(9)
+where again we have tangential continuity by a similar argument as in section 3.1.1.
+3.1.3
+Projection operators
+We denote by Eh,p(Ω) the global set of big edges (p = 1) or small edges (p = 2)
+associated with Kh(Ω). We will define the projection operator Ih,p : Λ1(Ω) �→ Λ1
+h,p(Ω)
+6
+
+as the unique operator that maps ω ∈ Λ1(Ω) to ωh ∈ Λ1
+h,p(Ω) such that the mismatch
+�
+e∈Eh,p(Ω)
+��
+e
+ω −
+�
+e
+ωh
+�2
+(10)
+is minimized. Note that for p = 1, this mismatch can be made to vanish. In this case,
+Ih,1 agrees with the usual edge-based nodal projection operator [27, Eq. (3.11)].
+In practice, we can compute the projection locally as follows. Let K ∈ Kh(Ω) be
+a d-simplex, d ∈ {2, 3}, and let {s1, .., sNp,d} and {ws1, .., wsNp,d} denote the corre-
+sponding big (p = 1) or small (p = 2) edges and corresponding shape functions as
+introduced above. Specifically, we have N1,2 = 3, N1,3 = 6, N2,2 = 9, and N2,3 = 24.
+We can define the matrix
+(M)i,j =
+�
+si
+wsj,
+1 ≤ i, j ≤ Np,d.
+(11)
+We will say that there is an interaction from edge sj to si if (M)i,j ̸= 0. Note that
+for p = 1, M is the identity matrix. For p = 2 the local shape functions are linearly
+dependent and, thus, the above matrix is not invertible. However, we can decompose
+M into invertible and singular sub-matrices. For illustrative purposes we display for
+p = 2 and d = 2 the decomposition of M in Figure 2b. The three top-left sub-matrices
+in Figure 2b are invertible 2 × 2 matrices that describe the interaction between the
+two small edges that lie on the same big edge, that is, the blue, red, and green sub-
+matrix in Figure 2b correspond to the blue, red, and green small edges in Figure 2a,
+respectively. The orange sub-matrix in Figure 2b is a 3 × 3 matrix with rank 2 that
+describes the interaction between the three small edges that lie in the interior of the
+simplex in Figure 2a, that is, the orange small edges. The gray sub-matrix encodes the
+one-directional interaction from the the small edges that lie on a big edge to the small
+edges in the interior. Note that the decomposition of M as given in Figure 2b is not
+limited to d = 2. The idea can be extended to d = 3 by considering each face of a
+3-simplex as a 2-simplex. This is sufficient, since for d = 3 we have no small edges in
+the interior and there is no interaction between small edges that do not lie on the same
+face. We give the general structure of M in Figure 2c. Note that the small, purple
+sub-matrices represent invertible 2 × 2 matrices and the bigger, orange sub-matrices
+represent 3 × 3 matrices with rank 2.
+In order to find ωh
+��
+K ∈ Λ1
+h,p(K) such that ωh
+��
+K = Ih,pω
+��
+K, let ⃗ηK be a vector of
+coefficients η1
+K, .., η
+Np,d
+K
+such that
+ωh|K =
+Np,d
+�
+i=1
+ηi
+Kwsi.
+(12)
+We can then compute ⃗ηK as a least-squares solution of
+M⃗ηK =
+��
+si
+ω
+�
+1≤i≤Np,d
+.
+(13)
+7
+
+(a) 2-simplex K
+(b) matrix M (d = 2)
+(c) matrix M (d = 3)
+Figure 2: For p = 2 and d = 2 the matrix M corresponding to the 2-simplex K in (a)
+has the form given in (b). Each row and column in M is associated to a small edge
+in (a). Each sub-matrix in (b) describes the interactions between edges with the same
+color in (a). The gray sub-matrix is an exception as it describes the one-directional
+interaction between the small edges that lie on a big edge and the small edges that lie
+in the interior. For d = 3, we M has the structure as shown in Figure 2c, where the
+purple sub-matrixs are 2 × 2 invertible matrices and the orange sub-matrixs are 3 × 3
+matrices of rank 2.
+Without loss of generality we assume that M has the form as given in Figure 2c. Then,
+we solve (13) as follows:
+1. The local shape functions related to small edges that lie on a big edge of the
+simplex are linearly independent. We solve for their coefficients first, that is, we
+solve the system corresponding to the invertible blue sub-matrices in Figure 2c
+first.
+2. Using the results from step 1, we can move the gray sub-matrix in Figure 2c
+to the right-hand side. Then, we solve the matrix-system corresponding to the
+orange sub-matrices in Figure 2c in a least-squares sense.
+If we perform the above steps for all K ∈ Kh(Ω), we find ωh = Ih,pω ∈ Λ1
+h,p(Ω).
+Note that only the shape functions associated to small edges on a face contribute to
+the tangential fields on that face. Therefore, the above procedure yields tangential
+continuity.
+Remark 3 For p = 1, (13) reduces to
+ηi
+K =
+�
+si
+ω,
+∀i ∈ {1, .., 3(d − 1)}
+(14)
+with si a big edge of the 3-simplex K for all i ∈ {1, .., 3(d − 1)}. This yields the
+standard nodal interpolation operator of [27, Eq. (3.11)].
+8
+
+3.2
+Semi-Lagrangian material derivative
+The method described in this section is largely based on [24, 26]. Throughout this
+section, unless stated otherwise, we fix the stationary, Lipschitz-continuous velocity
+field u ∈ W 1,∞(Ω) with u · n = 0 on ∂Ω. This means that we consider a linear trans-
+port problem and our main concern will be the discretization of the material derivative
+Duω for a 1-form ω. We can define the flow ]0, T[×Ω ∋ (τ, x) �→ Xτ(x) ∈ Rd as the
+solution of the initial value problems
+∂
+∂τ Xt,t+τ(x) = u(Xt,t+τ(x)),
+Xt(x) = x.
+(15)
+Given that flow we can define the material derivative for a time-dependent differential
+1-form ω
+Duω(t) := ∂
+∂τ X∗
+t,t+τω(t + τ)
+����
+τ=0
+.
+(16)
+We employ a first- or second-order, backward-difference method to approximate the
+derivative. Writing X∗
+t,t−τ for the pullback of forms under the flow, we obtain for
+sufficiently-smooth t �→ ω(t) and a timestep 0 < τ → 0
+Duω(t) = 1
+τ
+�
+ω(t) − X∗
+t,t−τω(t − τ)
+�
++ O(τ 2)
+(17)
+or
+Duω(t) = 1
+2τ
+�
+3ω(t) − 4X∗
+t,t−τω(t − τ) + X∗
+t,t−2τω(t − 2τ)
+�
++ O(τ 3),
+(18)
+respectively. Note that both backward-difference methods are A-stable [41]. In the
+remainder of this section we restrict ourselves to (17), but exactly the same considera-
+tions apply to (18).
+Given a temporal mesh .. < tn < tn+1 < .., we approximate ω(tn, ·) ∈ Λ1(Ω)
+by a discrete differential form ωn
+h ∈ Λ1
+h,p(Ω) with p ∈ {1, 2}. Using the backward-
+difference quotient (17), we can define the discrete material derivative for fixed timestep
+τ > 0
+(Dβω)(tn) ≈ 1
+τ
+�
+ωn
+h − Ih,pX∗
+t,t−τωn−1
+h
+�
+∈ Λ1
+h,p(Ω),
+(19)
+where we need to use the projection operator Ih,p : Λ1(Ω) �→ Λh,p(Ω) since X∗
+t,t−τωn−1
+h
+/∈
+Λ1
+h,p(Ω) in general. Recall from section 3.1 that the degrees of freedom for discrete
+1-forms are associated to small (p = 2) or big (p = 1) edges. As discussed in sec-
+tion 3.1.3, evaluating the interpolation operator entails integrating X��
+t,t−τωn−1
+h
+over
+small (p = 2) or big (p = 1) edges. We can approximate these integrals as follows
+�
+e
+X∗
+t,t−τωn−1
+h
+=
+�
+Xt,t−τ(e)
+ωn−1
+h
+≈
+�
+¯
+Xt,t−τ(e)
+ωn−1
+h
+,
+(20)
+where e is a small or big edge and
+¯Xt,t−τ(e) =
+�
+(1 − ξ)Xt,t−τ(v1) + ξXt,t−τ(v2); 0 ≤ ξ ≤ 1
+�
+(21)
+9
+
+Figure 3: Edge e (in red) is transported using the flow β (in blue). The exact trans-
+ported edge Xτ(e) and the approximate transported edge ¯Xτ(e) are given in orange
+and green.
+with v1, v2 the vertices of e. Instead of transporting the edge e using the exact flow
+Xt,t−τ, we follow [12, 24, 26] and only transport the vertices of the small edges (p = 2)
+or big edges (p = 1) and obtain a piecewise linear (second-order) approximation
+¯Xt,t−τ(e) of the transported edge Xt,t−τ(e) as illustrated in Figure 3. We can approxi-
+mate the movement of the endpoints of e under the flow as defined by (15) by solving
+(15) using the explicit Euler method or Heun’s method for the first- and second-order
+case, respectively. We will elaborate on this further in section 4.1.
+In Figure 3, we can also see that the approximate transported edge may intersect
+several different elements of the mesh. When we evaluate the integral in (20), it can
+happen that there are discontinuities of ωn−1
+h
+along ¯Xt,t−τ(e). Therefore, we cannot
+apply a global quadrature rule to the entire integral. Instead, we split ¯Xt,t−τ(e) into
+several segments defined by the intersection of ¯Xt,t−τ(e) with cells of the mesh. In
+our implementation, for the sake of stability, we find the intersection points by trans-
+forming back to a reference element as visualised in Figure 4. Algorithm 1 gives all
+details. Note that we can forgo the treament of any special cases (e.g. intersection
+with vertices) without jeopardizing stability. After we split the transported edge into
+segments, we can evaluate the integrals over these individual pieces exactly, because
+we know that ωn−1
+h
+is of polynomial form when restricted to individual elements of the
+mesh (see section 3.1).
+When simulating the fluid model (2), we will not have access to an exact velocity
+field. Instead we only have access to an approximation of the velocity field. This ap-
+proximation may not satisfy exact vanishing normal boundary conditions. Therefore,
+a part of ¯Xt,t−τ(e) may end up outside the domain. This can also happen due to an
+approximation of the flow by explicit timestepping. Since ωn−1
+h
+is not defined outside
+10
+
+Algorithm 1 Splitting 1-simplex over mesh elements (see Figure 4 for illustration).
+Here, Kref denotes the reference simplex.
+Input: x0 ∈ K0 ∈ Kh(Ω) and x1 vertices of a 1-simplex e.
+Output: Number of elements N, elements {K0, .., KN−1} ∈ Kh(Ω)N.
+1: K ← K0
+2: Fold ← NULL
+3: Kold ← NULL
+4: N ← 1
+5: E ← {K}
+6: while x1 /∈ K do
+7:
+Find the isoparametric mapping ϕK : Kref �→ K
+8:
+Find face F ⊂ ∂K s.t. F ̸= Fold and ϕ−1
+K (e) ∩ ϕ−1
+K (F) ̸= ∅
+9:
+K ← K ∈ Kh(Ω) s.t. F ⊂ ∂K and K ̸= Kold (K on the other side of face F)
+10:
+Fold ← F
+11:
+N ← N + 1
+12:
+E ← E ∪ {K}
+13: end while
+ϕK : Kref �→ K
+x0
+e
+K
+ϕ−1
+K (e)
+Kref
+K0
+x1
+Figure 4: The red line indicates the line that spans multiple elements. On the left we
+see the reference triangle associated with the yellow element in the mesh on the right.
+11
+
+Figure 5: A coarse triangulation of Ω = {x ∈ R2; ||x|| < 1} with the velocity field
+u = [−y, x]T satisfying u · n = 0. Despite the vanishing normal components of the
+velocity, the blue edge gets transported out of the domain to the green edge.
+the domain, we set
+�
+¯
+Xt,t−τ(e)|Rd\Ω
+ωn−1
+h
+:=
+len
+�
+¯Xt,t−τ(e)
+��
+Rd\Ω
+�
+len
+�
+¯Xt,t−τ(e)
+�
+�
+e
+ωn−1
+h
+,
+(22)
+where ¯Xt,t−τ(e)
+��
+Rd\Ω is the part of ¯Xt,t−τ(e) that lies outside the domain Ω and len
+�
+·)
+gives the arclength of the argument. This is motivated by the situation displayed in
+Figure 5—a case where an edge gets transported out of the domain due to the use of
+approximate flow maps despite vanishing normal components of the velocity. If we
+set the value defined in (22) to zero in this case, it would be equivalent to applying
+vanishing tangential boundary conditions, which is inconsistent with (1). Instead, (22)
+just takes the tangential components from the previous timestep.
+We arrive at the following approximation of the material derivative
+(Dβω)(tn) ≈ 1
+τ
+�
+ωn
+h − Ih,p ¯X∗
+t,t−τωn−1
+h
+�
+,
+(23)
+where the only difference between (19) and (23) is that Xt,t−τ was replaced by ¯Xt,t−τ
+and Ih,p is implemented based on (22). Note that in our scheme ¯X∗
+t,t−τ is always
+evaluated in conjunction with Ih,p, which means that we need define ¯Xt,t−τ only on
+small (p = 2) or big (p = 1) edges. In fact, ¯Xt,t−τ is defined through (21) for all points
+that lie on small (p = 2) or big (p = 1) edges.
+Given a velocity field u ∈ W 1,∞(Ω) with u · n = 0, it was shown in [26, section
+4] that using a first-order backward difference scheme and lowest-order elements for
+the spatial discretization, we can approximate a smooth solution ω ∈ Λ1(Ω) of
+Duω = 0,
+(24)
+12
+
+with an L2-error of O(τ − 1
+2h), where h is the spatial meshwidth and τ > 0 is the
+timestep size. However, numerical experiments [26, section 6] performed with τ =
+O(h) show an error of O(h)—a slight improvement over the a-priori estimates. This
+motivates us to link the timestep to the mesh width as τ = O(h).
+4
+Semi-Lagrangian Advection applied to the Incom-
+pressible Navier-Stokes Equations
+Given a temporal mesh t0 < t1 < ... < tN−1 < tN, we elaborate a single timestep
+tn−1 �→ tn of size τ := tn − tn−1, n ≤ N. We assume that approximations ωk
+h ∈
+Λ1
+h,p(Ω) of ω(tk, ·) ∈ Λ1(Ω) are available for k < n with [0, T] �→ ω ∈ Λ1(Ω) a
+solution of (2).
+4.1
+Approximation of the flow map
+In the Navier-Stokes equations, the flow is induced by the unknown, time-dependent
+velocity field u(t, x). Therefore, (15) becomes
+∂
+∂τ Xt,t+τ(x) = u(t + τ, Xt,t+τ(x)),
+Xt(x) = x ,
+t ∈ (0, T).
+(25)
+The discretization of the material derivative requires us to approximate the flow map
+Xt,t−τ in order to evaluate (21).
+4.1.1
+A first-order scheme
+We use the explicit Euler method to approximate the (backward) flow according to
+Xtn,tn−τ(x) ≈ x − τu(tn − τ, x),
+(26)
+where tn is a node in the temporal mesh and τ denotes the timestep size. We only have
+access to an approximation un−1
+h
+:= (ωn−1
+h
+)\ of u at time tn−1, which gives
+Xtn,tn−τ(x) ≈ x − τun−1
+h
+(x).
+(27)
+Note that a direct application of the explicit Euler method would require an evaluation
+of the velocity field at tn. Instead, we perform a constant extrapolation and evaluate
+the velocity field at tn−1, that is, we use un−1
+h
+in (27).
+The approximation un−1
+h
+resides in the space of vector proxies for discrete dif-
+ferential 1-forms as discussed in section 3.1. This means that only tangential conti-
+nuity of un−1
+h
+across faces of elements of the mesh is guaranteed, while discontinu-
+ities may appear in the normal direction of the faces. Therefore, un−1
+h
+is not defined
+point-wise—even though (27) requires point-wise evaluation. For that reason, we will
+13
+
+replace un−1
+h
+by a globally-continuous, smoothened velocity field ¯un−1
+h
+that approxi-
+mates un−1
+h
+(see section 4.1.3 for the construction). We then have
+Xtn,tn−τ(x) ≈ x − τ ¯un−1
+h
+(x)
+(28)
+which yields a first-order-in-time approximation of Xtn,tn−τ(x), provided that ¯un−1
+h
+is
+a first-order approximation of u(tn−1, ·).
+4.1.2
+A second-order scheme
+A second-order approximation can be achieved by using Heun’s method [44] instead
+of explicit Euler. We find the following second-order in time approximations
+Xtn,tn−τ(x) ≈ x − τ
+2
+�
+u∗
+h(x) + un−1
+h
+(x − τu∗
+h(x))
+�
+,
+(29)
+Xtntn−2τ(x) ≈ x − τ
+�
+u∗
+h(x) + un−2
+h
+(x − 2τu∗
+h(x))
+�
+,
+(30)
+where we approximate the velocity field at tn by the linear extrapolation u∗
+h = 2un−1
+h
+−
+un−2
+h
+. As described in section 4.1.1, we replace the velocity fields by suitable smooth
+approximations. We obtain
+Xtn,tn−τ(x) ≈ ¯Xt−τ(x) := x − τ
+2
+�¯u∗
+h(x) + ¯un−1
+h
+(x − τ ¯u∗
+h(x))
+�
+,
+(31)
+Xtn,tn−2τ(x) ≈ ¯Xt−2τ(x) := x − τ
+�¯u∗
+h(x) + ¯un−2
+h
+(x − 2τ ¯u∗
+h(x))
+�
+,
+(32)
+where ¯u•
+h with • = ∗, n − 1, n − 2 denotes the smoothened version of u•
+h as it will be
+constructed in the next section.
+4.1.3
+Smooth reconstruction of the velocity field
+Given a discrete velocity field uZ
+h ∈ Λ1
+h,p(Ω), we can define a smoothened version ¯uh
+of uh that is
+• Lipschitz continuous to ensure stable evaluation of (28),
+• well-defined on every point of the meshed domain,
+• practically computable, and
+• second-order accurate.
+We introduce ¯uh as follows. Let hmin denote the length of the shortest edge of the
+mesh and (ui
+h)i=1,..,d the components of uh. Then,
+¯ui
+h(x) =
+1
+hmin
+� xi+ 1
+2 hmin
+xi− 1
+2 hmin
+ui
+h([x1, . . . , xi−1, ξ, xi+1, . . . , xd]T)dξ
+(33)
+14
+
+provides a second-order, Lipschitz-continuous approximation of uh. In the above def-
+inition, we can also replace hmin by a localized parameter that scales as O(h) with h
+the length of the edges "close" to x. Note that the above integral can be evaluated up
+to machine precision using the algorithm as described in section 3.2 for (20). The av-
+eraging (33) provides a second-order approximation of uh on every point in the mesh.
+4.2
+A first- and second-order SL scheme
+We are now ready to turn the ideas of section 3 into a concrete numerical scheme
+for the incompressible Navier-Stokes equations as given in (2). We cast (2a) and
+(2b) into weak form and, subsequently, do Galerkin discretization in space relying
+on those spaces of discrete differential forms introduced in section 3.1. For the first-
+order scheme, we have the following discrete variational formulation. Given ωn−1
+h
+∈
+Λ1
+h,1(Ω), we search pn
+h ∈ Λ0
+h,1(Ω), ωn
+h ∈ Λ1
+h,1(Ω) such that
+�1
+τ
+�
+ωn
+h − Ih,1 ¯X∗
+tn,tn−τωn−1
+h
+�
+, ηh
+�
+Ω
++ε (dωn
+h, dηh)Ω + (dpn
+h, ηh)Ω = (f n, ηh)Ω ,
+(34a)
+(ωn
+h, dψh)Ω = 0
+(34b)
+for all ηh ∈ Λ1
+h,1(Ω) and ψh ∈ Λ0
+h,1(Ω). Ih,p denotes the projection operator as defined
+in section 3.1. For the second-order scheme, we use second-order timestepping and
+second-order discrete differential forms. Given ωn−2
+h
+, ωn−1
+h
+∈ Λ1
+h,2(Ω), we search pn
+h ∈
+Λ0
+h,2(Ω), ωn
+h ∈ Λ1
+h,2(Ω) such that
+� 1
+2τ
+�
+3ωn
+h − 4Ih,2 ¯X∗
+tn,tn−τωn−1
+h
++ Ih,2 ¯X∗
+tn,tn−2τωn−2
+h
+�
+, ηh
+�
+Ω
++ε (dωn
+h, dηh)Ω + (dpn
+h, ηh)Ω = (f n, ηh)Ω ,
+(35a)
+(ωn
+h, dψh)Ω = 0
+(35b)
+for all ηh ∈ Λ1
+h,2(Ω) and ψh ∈ Λ0
+h,2(Ω). Numerical experiments reported in section 5
+give evidence that these schemes indeed do provide first- and second-order conver-
+gence for smooth solutions. Note that the schemes presented in this section only re-
+quire solving symmetric, linear systems of equations at every time-step.
+4.3
+Conservative SL schemes
+In order to enforce energy-tracking—the correct behavior of the total energy E(t) over
+time as expressed in (3)—we add a suitable constraint plus a Lagrange multiplier to
+15
+
+the discrete variational problems proposed in section 4.2. Given ωn−1
+h
+∈ Λ1
+h,1(Ω), we
+search pn
+h ∈ Λ0
+h,1(Ω), ωn
+h ∈ Λ1
+h,1(Ω), and µn ∈ R such that
+�1
+τ
+�
+ωn
+h − Ih,1 ¯X∗
+tn,tn−τωn−1
+h
+�
+, ηh
+�
+Ω
++ (dpn
+h, ηh)Ω + ε(dωn
+h,dηh)Ω
++µn [(ωn
+h, ηh)Ω + 2ετ(dωn
+h, dηh)Ω − τ(f n, ηh)Ω] = (f n, ηh)Ω ,
+(36a)
+(ωn
+h, dψh)Ω = 0,
+(36b)
+(ωn
+h, ωn
+h)Ω + 2ετ(dωn
+h, dωn
+h)Ω − τ(f n, ωn
+h)Ω =
+�
+ωn−1
+h
+, ωn−1
+h
+�
+Ω (36c)
+for all ηh ∈ Λ1
+h,1(Ω) and ψh ∈ Λ0
+h,1(Ω). Note that the last scalar equation enforces
+energy conservation for ε = 0 and f = 0. To solve the nonlinear system (36) for
+ωn
+h, pn
+h, µn, we propose the following iterative scheme. Assume that we have a se-
+quence (ωn
+h,k)k∈N with ωn
+h,k → ωn
+h(k → ∞). Then we can employ the Newton-like
+linearization
+(ωn
+h, ωn
+h)Ω ←
+�
+ωn
+h,k, ωn
+h,k
+�
+Ω
+=
+�
+ωn
+h,k−1, ωn
+h,k−1
+�
+Ω + 2
+�
+ωn
+h,k−1, ωn
+h,k − ωn
+h,k−1
+�
+Ω + O
+�
+||ωn
+h,k − ωn
+h,k−1||2
+Ω
+�
+.
+(37)
+We use the above expansion to replace the quadratic terms (ωn, ωn)Ω and (dωn, dωn)Ω
+and arrive at the following linear variational problem to be solved in every step of
+the inner iteration. Given ωn−1
+h
+, ωn
+h,k−1 ∈ Λ1
+h,1(Ω), we search pn
+h,k ∈ Λ0
+h,1(Ω), ωn
+h,k ∈
+Λ1
+h,1(Ω), and µn
+k ∈ R such that
+�1
+τ
+�
+ωn
+h,k − Ih,1 ¯X∗
+tn,tn−τωn−1
+h
+�
+, ηh
+�
+Ω
++
+�
+dpn
+h,k, ηh
+�
+Ω + ε(dωn
+h,k, dηh)Ω
++µn
+k
+�
+(ωn
+h,k−1, ηh)Ω + 2ετ(dωn
+h,k−1, dηh)Ω − τ(f n, ηh)Ω
+�
+= (f n, ηh)Ω ,
+(38a)
+�
+ωn
+h,k, dψh
+�
+Ω = 0,
+(38b)
+�
+ωn
+h,k−1, ωn
+h,k−1
+�
+Ω + 2
+�
+ωn
+h,k−1, ωn
+h,k − ωn
+h,k−1
+�
++2ετ[(dωn
+h,k−1, dωn
+h,k−1)Ω + 2(dωn
+h,k−1, dωn
+h,k − dωn
+h,k−1)]
+−τ(f n, ωn
+h,k−1)Ω =
+�
+ωn−1
+h
+, ωn−1
+h
+�
+Ω
+(38c)
+for all ηh ∈ Λ1
+h,1(Ω) and ψh ∈ Λ0
+h,1(Ω). This is a symmetric, linear system that
+is equivalent to the original system in the limit (ωn
+h,k, pn
+h,k, µn
+k) → (ωn
+h, pn
+h, µn). In
+numerical experiments we observe that it takes around 2-3 steps of the inner iteration
+to converge to machine precision using an initial guess ωn
+h,0 = ωn−1
+h
+. We can apply the
+same idea for energy-tracking to our second-order scheme as proposed in section 4.2.
+16
+
+Remark 4 For the case ε = 0 and f = 0, we can also enforce helicity conservation by
+adding a suitable Lagrange multiplier to the discrete system. Given ωn−1
+h
+∈ Λ1
+h,1(Ω),
+we search pn
+h ∈ Λ0
+h,1, ωn
+h ∈ Λ1
+h,1(Ω), and λn ∈ R such that
+�1
+τ
+�
+ωn
+h − Ih,1 ¯X∗
+tn,tn−τωn−1
+h
+�
+, ηh
+�
+Ω
++ (dpn
+h, ηh)Ω + ε(dωn
+h, dηh)Ω
++λn(ωn
+h, dηh)Ω + λn(dωn
+h, ηh)Ω + µn(ωn
+h, ηh)Ω = 0,
+(39a)
+(ωn
+h, dψh)Ω = 0,
+(39b)
+(ωn
+h, ωn
+h)Ω =
+�
+ωn−1
+h
+, ωn−1
+h
+�
+Ω ,
+(39c)
+(ωn
+h, dωn
+h)Ω =
+�
+ωn−1
+h
+, dωn−1
+h
+�
+Ω
+(39d)
+for all ηh ∈ Λ1
+h,1(Ω) and ψh ∈ Λ0
+h,1(Ω). By linearization as for energy-tracking, we
+obtain the following system. Given ωn−1, ωn
+h,k−1 ∈ Λ1
+h,1(Ω), we search pn
+h,k ∈ Λ0
+h,1(Ω),
+ωn
+h,k ∈ Λ1
+h,1(Ω), and λn
+k ∈ R such that
+�1
+τ
+�
+ωn
+h,k − Ih,1 ¯X∗
+tn,tn−τωn−1�
+, ηh
+�
+Ω
++
+�
+dpn
+h,k, ηh
+�
+Ω + ε(dωn
+h,k, dηh)Ω
++λn
+k(ωn
+h,k−1, dηh)Ω + λn
+k(dωn
+h,k−1, ηh)Ω + µn
+k(ωn
+h,k−1, ηh)Ω = 0,
+(40a)
+�
+ωn
+h,k, dψh
+�
+Ω = 0,
+(40b)
+�
+ωn
+h,k−1, ωn
+h,k−1
+�
+Ω + 2
+�
+ωn
+h,k−1, ωn
+h,k − ωn
+h,k−1
+�
+=
+�
+ωn−1
+h
+, ωn−1
+h
+�
+Ω , (40c)
+�
+ωn
+h,k−1, dωn
+h,k
+�
+Ω +
+�
+ωn
+h,k, dωn
+h,k−1
+�
+Ω −
+�
+ωn
+h,k−1, dωn
+h,k−1
+�
+Ω =
+�
+ωn−1
+h
+, dωn−1
+h
+�
+Ω (40d)
+for all ηh ∈ Λ1
+h,1(Ω) and ψh ∈ Λ0
+h,1(Ω). Again, this is a symmetric, linear system
+that is equivalent to the original system in the limit (ωn
+h,k, pn
+h,k, µn
+k) → (ωn
+h, pn
+h, µn).
+Also, numerical experiments hint that it takes around 2-3 steps of the inner iteration to
+converge to machine precision using an initial guess ωn
+h,0 = ωn−1
+h
+.
+5
+Numerical Results
+In this section, we present multiple numerical experiments to validate the new scheme.
+In the following, we will always consider schemes that include energy-tracking as in-
+troduced in section 4.3 unless stated otherwise. We only include helicity-conservation
+as introduced in Remark 4 for domains in R3 when ε = 0 and f = 0. The ex-
+periments are based on a C++ code that heavily relies on MFEM [2]. The source
+code is published under the GNU General Public License in the online code repository
+https://gitlab.com/WouterTonnon/semi-lagrangian-tools.
+17
+
+10−1
+mesh width h
+10−3
+10−2
+10−1
+L2 Error u
+first-order,
+ε = 0
+first-order,
+ε = 0.01π−2
+first-order,
+ε = 0.1π−2
+second-order,
+ε = 0
+second-order,
+ε = 0.01π−2
+second-order,
+ε = 0.1π−2
+Figure 6: Convergence results for Experiment 1 using the first- and second-order
+schemes on simplicial meshes with mesh width h, timestep τ = 0.065804h. We ob-
+serve first- and second-order algebraic convergence for all values of ε.
+5.1
+Experiment 1: Decaying Taylor-Green Vortex
+We consider the incompressible Navier-Stokes equations with Ω = [− 1
+2, 1
+2]2, T = 1,
+varying ε ≥ 0, f = 0, and vanishing boundary conditions. An exact, classical solution
+is the following Taylor-Green vortex [42]
+u(t, x) =
+� cos(πx1) sin(πx2)
+− sin(πx1) cos(πx2)
+�
+e−2π2εt.
+(41)
+We ran a h-convergence analysis for different values of ε ≥ 0 and summarize the
+results in Figure 6. We also track the energy for different values of ε and compare the
+energy to the exact solution in Figure 7.
+5.2
+Experiment 2: Taylor-Green Vortex
+We consider the incompressible Navier-Stokes equations with Ω = [−1, 1]2, T = 1,
+varying ε ≥ 0, f and the boundary conditions chosen such that
+u(t, x) =
+� cos(πx1) sin(πx2)
+− sin(πx1) cos(πx2)
+�
+(42)
+is an exact, classical solution. We ran a h-convergence analysis for all parameters
+and summarize the results in Figure 8. We observe first- and second-order algebraic
+convergence for the corresponding schemes. Note that the error of the scheme is stable
+18
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+time t
+0.58
+0.60
+0.62
+0.64
+0.66
+0.68
+0.70
+L2 Norm u
+exact,
+ε = 0
+first-order,
+ε = 0
+exact,
+ε = 0.01π−2
+first-order,
+ε = 0.01π−2
+exact,
+ε = 0.1π−2
+first-order,
+ε = 0.1π−2
+(a) first-order
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+time t
+0.58
+0.60
+0.62
+0.64
+0.66
+0.68
+0.70
+L2 Norm u
+exact,
+ε = 0
+second-order,
+ε = 0
+exact,
+ε = 0.01π−2
+second-order,
+ε = 0.01π−2
+exact,
+ε = 0.1π−2
+second-order,
+ε = 0.1π−2
+(b) second-order
+Figure 7: Energy of the discrete and exact solution for Experiment 1 using the first-
+and second-order, energy-tracking schemes on a simplicial mesh with mesh width h =
+0.0949795, timestep τ = 0.00625.
+19
+
+10−1
+100
+mesh width h
+10−2
+10−1
+100
+L2 Error u
+first-order,
+ε = 0
+first-order,
+ε = 0.01
+first-order,
+ε = 1
+second-order,
+ε = 0
+second-order,
+ε = 0.01
+second-order,
+ε = 1
+Figure 8: Convergence results for Experiment 2 using the first- and second-order,
+non-conservative schemes on simplicial meshes with mesh width h, timestep τ =
+0.032902h. As ε → 0 the error remains bounded.
+as ε → 0. This is in agreement with the analysis performed on the vectorial advection
+equations presented in [24]. This experiment thus suggests that this analysis can be
+extended to the scheme presented in this work.
+5.3
+Experiment 3: A rotating hump problem
+The Taylor-Green vortices provide exact solutions to the incompressible Navier-Stokes
+equations, but they are rather "static" solutions. In this experiment, we consider a more
+dynamic solution. Let us consider the incompressible Navier-Stokes equations with
+Ω = [− 1
+2, 1
+2]2, T = 1, ε = 0, f = 0, and vanishing normal boundary conditions. We
+consider the following initial condition
+u0(x) =
+�
+−πex1 cos(πx1) sin(πx2)
+πex1 sin(πx1) cos(πx2) − ex1 cos(πx1) cos(πx2)
+���
+.
+(43)
+The exact solution to this problem is unknown, so we compare the solution computed
+by our scheme to the solution produced by the incompressible Euler solver Gerris [37].
+The algorithm used in this solver is described in [36]. We computed the solution to this
+problem using the second-order, energy-tracking scheme presented in this work. Then,
+we plotted the magnitude of the computed velocity vector field for different mesh-sizes
+and time-steps at different time instances in Figures 10 to 13. Note that different visu-
+alisation tools were used to visualize the fields computed using the different solvers,
+but we observe that the solution computed by the semi-Lagrangian scheme comes vi-
+sually closer to the solution computed by Gerris as we decrease the mesh width and
+20
+
+10−1
+mesh width h
+10−2
+10−1
+100
+L2 Error u
+second-order,
+cons
+Figure 9: Convergence results for Experiment 2 using the second-order, conservative
+scheme on simplicial meshes with mesh width h, timestep τ = 0.06580h, and final
+time T = 1. The reference solution is a solution computed by Gerris [36]
+time step. This is confirmed by Figure 9, where we display the L2 error between the
+solution computed using the semi-Lagrangian scheme and the solution computed using
+Gerris. In Figure 15, we display the vector field as computed using the second-order,
+conservative semi-Lagrangian scheme.
+Also, in Figure 16 we display the values of the L2 norm over time of the solu-
+tions produced using our first- and second-, energy-tracking and non-energy-tracking
+schemes. Note that the energy-tracking schemes preserve the L2 norm as expected.
+The first-order, non-conservative scheme seems unstable at first, but in reality the or-
+dinate axis spans a very small range and it turns out that the L2 norm converges to a
+bounded value for longer run-times. Note that the helicity has no meaning in R2.
+5.4
+Experiment 4: Taylor-Green Vortex in 3D
+To observe conservation of helicity, we need to consider a problem in 3D. We con-
+sider the incompressible Navier-Stokes equations with Ω = [− 1
+2, 1
+2]3, T = 1, ε = 0,
+vanishing normal boundary conditions, and f chosen such that
+u(t, x) =
+�
+�
+cos(πx1) sin(πx2) sin(πx3)
+− 1
+2 sin(πx1) cos(πx2) sin(πx3)
+− 1
+2 sin(πx1) sin(πx2) cos(πx3)
+�
+�
+(44)
+is a solution. Note that since the solution is static, we can enforce helicity conser-
+vation despite f ̸= 0. We run several experiments using the first- and second-order,
+21
+
+(a) t = 0.25
+(b) t = 0.5
+(c) t = 0.75
+(d) t = 1
+Figure 10: Experiment 3: mesh width h = 0.379918 and time-step τ = 0.025.
+(a) t = 0.25
+(b) t = 0.5
+(c) t = 0.75
+(d) t = 1
+Figure 11: Experiment 3: mesh width h = 0.0949795 and time-step τ = 0.00625.
+(a) t = 0.25
+(b) t = 0.5
+(c) t = 0.75
+(d) t = 1
+Figure 12: Experiment 3: mesh width h = 0.023744875 and time-step τ = 0.0015625.
+(a) t = 0.25
+(b) t = 0.5
+(c) t = 0.75
+(d) t = 1
+Figure 13: Reference solution Experiment 3 computed using [37].
+0
+1
+2
+3
+4
+5
+Figure 14: Colorbar associated with Figures 10 to 13
+22
+
+(a) t = 0.25
+(b) t = 0.5
+(c) t = 0.75
+(d) t = 1
+0
+1
+2
+3
+4
+5
+Figure 15: Velocity field for Experiment 3 computed using the second-order, conser-
+vative semi-Lagrangian scheme on a simplicial mesh with mesh width h = 0.189959
+and time-step τ = 0.0125. The colors indicate the magnitude of the vector.
+23
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+time t
+2.350
+2.355
+2.360
+2.365
+2.370
+2.375
+2.380
+L2 Norm u
+second-order,
+cons
+first-order,
+cons
+second-order,
+non-cons
+first-order,
+non-cons
+Figure 16: The L2 norm of the computed solutions for Experiment 3 using differ-
+ent variants of the semi-Lagrangian scheme on a simplicial mesh with mesh width
+h = 0.04748975 and time-step τ = 0.003125. In the legend, ’cons’ is short for ’con-
+servative’ and refers to energy-tracking schemes. We use ε = 0 and f = 0.
+conservative semi-Lagrangian schemes. We summarize the results in Figure 17 and we
+observe first- and second-order algebraic convergence for the corresponding schemes.
+In Figure 18 and Figure 19 we plot the L2 norm and helicity over time of the discrete
+solution for both the first- and second-order scheme. We observe that both quantities
+are conserved up to machine precision.
+5.5
+Experiment 5: A transient solution in 3D
+To verify the scheme for transient solutions in 3D, we consider the incompressible
+Navier-Stokes equations with Ω = [− 1
+2, 1
+2]3, T = 1, ε = 0, vanishing normal boundary
+conditions and f is chosen such that
+u(t, x) =
+�
+�
+−x2π cos( t
+4 + πx2x3) cos(πx2)
+−x3π cos( t
+4 + πx1x3) cos(πx3)
+−x1π cos( t
+4 + πx1x2) cos(πx1)
+�
+�
+(45)
+is a solution. We ran a simulation for different mesh-sizes with time-steps determined
+by a suitable CFL condition. We summarize the results in Figure 20. We observe
+second-order convergence for the second-order scheme. The first-order scheme seems
+to achieve an order of convergence that is between first- and second-order, but this may
+be pre-asymptotic behaviour.
+24
+
+10−1
+2 × 10−1
+3 × 10−1
+4 × 10−1
+6 × 10−1
+mesh width h
+10−2
+10−1
+L2 Error u
+first-order
+second-order
+Figure 17: Convergence results for Experiment 4 using the first- and second-order,
+conservative schemes on simplicial meshes with mesh width h, timestep τ =
+1
+√
+2h.
+0.2
+0.4
+0.6
+0.8
+1.0
+time t
+0.430
+0.435
+0.440
+0.445
+0.450
+0.455
+L2 Norm u
+second-order,
+cons
+first-order,
+cons
+second-order,
+non-cons
+first-order,
+non-cons
+Figure 18: The L2 norm of the computed solutions for Experiment 4 using differ-
+ent variants of the semi-Lagrangian scheme on a simplicial mesh with mesh width
+h = 0.08838834764 and time-step τ = 0.0625. In the legend, ’cons’ is short for ’con-
+servative’ and refers to energy-tracking schemes and helicity-conserving schemes. We
+use ε = 0 and f = 0.
+25
+
+0.2
+0.4
+0.6
+0.8
+1.0
+time t
+−0.006
+−0.004
+−0.002
+0.000
+Helicity u
+second-order,
+cons
+first-order,
+cons
+second-order,
+non-cons
+first-order,
+non-cons
+Figure 19: The helicity of the computed solutions for Experiment 4 using differ-
+ent variants of the semi-Lagrangian scheme on a simplicial mesh with mesh width
+h = 0.08838834764 and time-step τ = 0.0625. In the legend, ’cons’ is short for
+’conservative’ and refers to energy-tracking and helicity-conserving schemes. We use
+ε = 0 and f = 0.
+10−1
+2 × 10−1
+3 × 10−1
+mesh width h
+10−1
+L2 Error u
+first-order
+second-order
+Figure 20: Convergence results for Experiment 5 using the first- and second-order
+schemes without energy-tracking and helicity-conservation on simplicial meshes with
+mesh width h, timestep τ =
+1
+√
+2h.
+26
+
+0
+0.5
+1
+1.5
+2
+2.5
+Figure 21: Velocity field at T = 7.93s of Experiment 6 computed using the second-
+order, non-conservative semi-Lagrangian scheme on a simplicial mesh with mesh
+width h = 0.189959 and τ = 0.01.
+5.6
+Experiment 6: Lid-driven cavity with slippery walls
+In this section, we simulate a situation that resembles a lid-driven cavity problem.
+Consider the incompressible Navier-Stokes with Ω = [− 1
+2, 1
+2]2, T = 7.93, ε = 0,
+vanishing normal boundary conditions and the initial velocity field is set equal to zero.
+Then, to simulate a moving lid at the top, we apply the force-field f(t, x) = [v(x), 0]T
+with
+v(x) =
+�
+exp
+�
+1 −
+1
+1−100(0.5−x2)2
+�
+,
+if 1 − 100(0.5 − x2)2 > 0,
+0,
+else.
+(46)
+This force field gives a strong force in the x1-direction close to the top lid, but quickly
+tapers off to zero as we go further from the top lid. In Figure 21, we display the
+computed velocity field. Note that, because we apply slip boundary conditions, we do
+not expect to observe vortices. The numerical solution reproduces this expectation.
+27
+
+5.7
+Experiment 7: A more complicated domain
+The numerical experiments given above, show the convergence and conservative prop-
+erties of the introduced schemes. However, these experiments are all performed on
+very simple, rectangular domains. In this experiment, we consider a more complicated
+domain and mesh (generated using [22]) as shown in Figure 22.
+We consider the case of the incompressible Navier-Stokes equations on the domain
+as given in Figure 22, T = 100, ε = 0, f = 0 and vanishing normal boundary condi-
+tions. We need to construct an initial condition that is divergence-free with vanishing
+normal boundary conditions. Following an approach close to a Chorin projection, we
+start with
+w(x, y) =
+�sin
+�
+2 cos(
+�
+x2 + y2) − atan2(y, x)
+�
+sin
+�
+cos(
+�
+x2 + y2) − 2 atan2(y, x)
+�
+�
+.
+(47)
+We use this definition to define a scalar function, ϕ, as
+∆ϕ = ∇ · w
+in Ω,
+(48)
+∇ϕ · ˆn = w · ˆn
+on ∂Ω.
+(49)
+We can define our initial condition, u0, as
+u0 = w − ∇ϕ
+(50)
+Note that u0 is divergence-free and has vanishing normal boundary conditions. The
+above system of equations can be solved using an appropriate finite-element imple-
+mentation.
+Note that in this experiment, the field outside the domain is unknown. This is
+well-defined on a continuous level, since vanishing boundary conditions imply that
+no particle will flow in from outside the domain. However, on the discrete level we
+cannot guarantee that the same will happen. It could happen that a part of a transported
+edge (as discussed in section 3.2) ends up outside the domain. In this case, we will
+assume that the average of the vector field along the part of the edge that lies outside
+the domain, will have the same value as the average of the corresponding edge in its
+original location (before transport) at the previous timestep.
+The first ten seconds were simulated and a video of the results can be found at
+https://youtu.be/Eica8XHLtxY. For the different schemes, we also tracked
+the energy in Figure 23.
+6
+Conclusion
+We have developed a mesh-based semi-Lagrangian discretization of the time-dependent
+incompressible Navier-Stokes equations with free boundary conditions recast as a non-
+linear transport problem for a momentum 1-form. A linearly implicit fully discrete
+version of the scheme enjoys excellent stability properties in the vanishing-viscosity
+28
+
+Figure 22: Domain and mesh associated with Experiment 7.
+0
+10
+20
+30
+40
+50
+60
+time t
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+L2 Norm u
+first-order,
+non-cons
+second-order,
+non-cons
+second-order,
+cons
+first-order,
+cons
+Figure 23: The L2 norm of the computed solutions for Experiment 7 using different
+variants of the semi-Lagrangian scheme on a simplicial mesh as given in Figure 22
+and time-step τ = 0.01. In the legend, ’cons’ is short for ’conservative’ and refers to
+energy-tracking schemes.
+29
+
+limit and is applicable to inviscid incompressible Euler flows. However, in this case
+conservation of energy and helicity have to be enforced separately. Making the reason-
+able choice of a time-step size proportional to the mesh width, the algorithm involves
+only local computations. Yet, these are significantly more expensive compared to those
+required for purely Eulerian finite-element and finite-volume methods. At this point
+the verdict on the competitiveness of our semi-Lagrangian scheme is still open.
+A
+Two formulations of the momentum equation
+Consider the momentum equation in (1)
+∂tu + u · ∇u − ε∆u + ∇p = 0.
+Note that we have by standard vector calculus identities
+∆u = ∇(∇ · u) − ∇ × ∇ × u,
+where we can use ∇ · u = 0 to obtain
+∆u = −∇ × ∇ × u.
+This allows us to rewrite the momentum equation as
+∂tu + u · ∇u + ε∇ × ∇ × u + ∇p = 0.
+Using the gradient of the dot-product, we find
+∇(u · u) = 2u · ∇u + 2u × (∇ × u).
+This identity allows us to rewrite the momentum equation to
+∂tu + ∇(u · u) − u × (∇ × u) + ε∇ × ∇ × u + ∇
+�
+−1
+2u · u + p
+�
+= 0.
+From [27, 24], we obtain the identity
+(Luω)\ = ∇(u · u) − u × (∇ × u)
+where ω is the differential 1-form such that u = ω\. Since the material derivative for
+this 1-form is
+Duω := ∂tω + Luω,
+we find that the momentum equation can be written as
+Duω + εδdω + d˜p = 0,
+where ˜p = − 1
+2u · u + p.
+30
+
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