Operator-Discretized Representation for Temporal Neural Networks
Authors: Yasunao Katayama
Venue: NeurIPS 2022 Submitted (2022)
Keywords: artificial neural network
spiking neural network
operator algebra
diagrammatic category theory

Abstract: This paper proposes a new representation of artificial neural networks to efficiently track their temporal dynamics as sequences of operator-discretized events. Our approach takes advantage of diagrammatic notions in category theory and operator algebra, which are known mathematical frameworks to abstract and discretize high-dimensional quantum systems, and adjusts the state space for classical signal activation in neural systems. The states for nonstationary neural signals are prepared at presynaptic systems with ingress creation operators and are transformed via synaptic weights to attenuated superpositions. The outcomes at postsynaptic systems are observed as the effects with egress annihilation operators (each adjoint to the corresponding creation operator) for efficient coarse-grained detection. The follow-on signals are generated at neurons via individual activation functions for amplitude and timing. The proposed representation attributes the different generations of neural networks, such as analog neural networks (ANNs) and spiking neural networks (SNNs), to the different choices of operators and signal encoding. As a result, temporally-coded SNNs can be emulated at competitive accuracy and throughput by exploiting proven models and toolchains for ANNs. 

18 Modern neural networks are expected to solve demanding Al problems with datastreams in ex- tremely high dimensions. Under avdely-available computing infrastructure, the situation is becom- 20 ing even more challenging. when the neural dynamics for data processing inherently temporal and online in the biological systems [1-4]. An appropriate neural network representation for Operator-Discretized Representation for Temporal Neural Networks
Anonymous Author(s)
Affiliation Address
email
Abstract
This paper proposes a new representation of artificial neural networks to efficiently track their temporal dynamics as sequences of deerrtordissretizeed events. Our approach takes advantage of diagrammatic notions in category theory and opera- tor algebra, which are known mathematical frameworks to abstract and discretize high- digh-nimmenional quantum systems, and adjusts the state space for classical signal activation in neural systems. The states for nonstationary neural signals are pre- pared at presynaptic systems with ingress creation operators, and are transformed via synaptic weights to attenuated superpositions. The outcomes postsynaptic systems are observed as the effects with egress annihilation operators (each adjoint to the corresponding creation operator) for efficient coarse-grainet detection. The follow-on signals are generated at neurons via individual activation functions for amplitude and timing. The proposed representation attributes the different gen- erations neural networks, such as analog neural networks (ANNs) and spiking neural networks (SNNs), to the different choices of operators and signal encoding. As a result, temporally-coded SNNs can be emulated at competitive accuracy and throughput by exploiting proven models and toolchains for ANNs.
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Introduction
22 natively handling sequences of timestamped events should significantly improve computational ef- ficiencies. When event sequences processed with artificial neural networks, known techniques 24 typically compute layer-wise outputs synchronously at every discretized time step to align their data and computing wavefront, as seen in recent investigations
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SNNs [5-7] time series forecast- 26 ing [8-11]. Though algorithms may sometimes be given in event-driven manners, their execution in SW has resort to fine-grained synchronous discretization [12-15] closed-form approximations
28 of temporal dynamics that require exact temporal ordering of the events [14, 16]. As a result, accu- racies competitive to ANNs have only been obtained an expense of throughput and scalability.
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30 In temporally executing neural networks in commercial systems, the period T. of the global clock is typically chosen small enough compared with the characteristic time of the neural dynamics to:
T. < to.
32 to precisely track the temporal dynamics, for example, the membrane potential changes to determine the next firing timing of SNNs. This is sharp contrast to how the biological brain operates with 34 low-frequency brain waves [17] closer to our behavioral time seale:
Te to.
Submitted to 36th Conference on Neural Information Processing Systems (NeurIPS 2022). Do not distribute.

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35
Energy and functional efficiencies can be significantly improved if a new representation can avoid synchronously computing the temporal dynamics at every small time step by better decoupling dif- ferent time scales, It is tempting for those with some physics background to apply techniques being developed for quantum systems since they are naturally asynchronous events in extremely high di- mensions. Indeed, operator algebra has been applied to Hopfield networks [18] as well as other classical systems [19-21]. However, since operators are used for stationary neuron states out of spins and charges rather than those for nonstationary neural signals traveling over axon-synapse- dendrite networks, its full potential has not been extracted for modern temporal workloads.
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Here in this paper, we propose a new representation of neural networks that can efficiently compute their dynamics as coarse-graines sequences of operator-discretizer events. Our approach takes ad- vantage of existing mathematical frameworks that have been originally developed to abstract and discretize high-dimensional quantum systems. These techniques are, with necessary modifications, applied to neural networks that are also high dimensional, but inherently are classical. Different generations of neural networks, such as ANNs and SNNs, are attributed to different choices of op- SNNs with
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erators and signal encoding. Our formulation can efficiently emulate temporally-codee fully exploiting existing assets, such as models and toolchains for ANNs. It should be noted that the scope of this paper is on classical neural networks, though the proposed representation may bring us 52 a new perspective on Al and quantum computing (QC) [22].
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2
Logical representation
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Let us start with the logical aspects. Figure I presents diagrammatic representations for quantum and neural networks. In short, once the state spaces are respectively defined, they look surprisingly similar, in particular when we regard qubits nonstationary and flying [23] well.
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2.1 Logical abstraction and state space
58
The operation of neural networks is to be abstracted by exploiting diagrammatic notions of categor- ical theories [24-26]. These techniques have been applied both to quantum and classical systems and their processes without much referring to actual physies inside [27]. Here, we will consider pure
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60
61 states only (i.c., wave function vectors rather than density matrices) for quantum, since our purpose explicitly compare quantum and classical networks.
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63 A known categorical diagram for a quantum network is examplified in Fig. I (a). It consists of three major blocks: the states, the proessestransformations, and the effect, for preparation, operation, 65 and observation of quantum systems, respectively. Without operation, the inner product of the state
66 Ip) represented by tensor product of each qubit (p:) state prepared at quantum system 5??
Ip) (pi) @ @ 1pm).
(3)
and the effect (al represented by a tensor product of each effect (ail at quantum system R.
(al =<ail @ .... (axl,
(4)
68 can compute the conditional probability P(alpp) as
(alp) 12 II (asle) F= II P(adl:)) = P((ap))
(5)
69 In general. the probabilities cannot be factorized this way other than for the slices, providing rich
70 set of non-classical computing power, such as with entanglement, to quantum networks.
71 The corresponding diagram for a classical neural network is proposed in Fig.1 (b). The states for
72 neural signals are prepared at presynaptic systems. They are transformed into weighted sums via synaptic networks. The outcomes are observed at postsynaptic systems as the effects to generate the
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74 follow-on states and signals. As the quantum case, we assume that the transformations in axon- synapse-dendrite networks are linear. We define, in analogy to the qubit, the cubit, which stands for 76 the abbreviation of classical universal bit, for neural signals. Though the definition is informational, rather than physical, we inherit Dirac notation but with double bras and kets, indicating that the

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[a]
Encodinn
Detection (ayl E EIT(R)
[b)
Encoding
Detection
(pt) E St(S)
Slice
lipu)) SE(S))
Slice
product
Cartesian product
Effect
Effect
State
Process/ Transformation
Process/ tranfooraottooaooonnn
Only signals from the same sources can Interfere
Signals from different sources interfere
(c)
(d)
Kolla,
47
of (lo))
a0la
Figure 1:
Diagramatic comparison of quantum and neural networks: (a) Quantum network sisting of states, processes/transformaiion and effects; (b) Corresponding diagram for a neural network; (c) Operator representation for creation, scattering, and annihilation of quantum wave packets; (d) Operator representation for creation, weighted sum, and annihilation neural signals. Note that weight matrix way (d) corresponds to scattering matrix Sij in (c).
states consist of macroscopic ensembles of qubits
Multiple types of logical cubits are defined:
Normalized full cubit Normalized half cubit
(c) |0]) + [|])), |d + |c| =1
U(1) SO(2)
c)) = [1) < |c|2 S
E U(())R
Unnormalized full cubit
(c) = {0)) + 1))
E R$ or
Unnormalized half cubit.
[[c) = c|]))
ER
(6)
The information encoded to cubits is assumed to be real for simplicity but be complex for 80 complex- neural networks [28].
81 A set of cubits (pp)) can compactly be represented by Cartesian product (or coproduct in category
82 theory terminology) of each cubit (lp))) axon 5 as
||e)) [l:))) @ ... Ilpx))
They to be detected by effect ((al) consisting of ((aull via dendrite R
((all ((ail) || o @ ((amll
(8)
84 Based on an argument for the linear systems in [29]. the norm p for cubits is expected to be eitherl
85 or 2, Euclidean norm (p 2). which is also found wireless communication and signal processing
86
literature [30] (e.g., ||]}) and |ІO}) for I and Q), makes sense to represent wave-like dynamics [31- 11-
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34] in complex-valued state spaces. while Manhatten norm (p 1) is for ordinary real-valued state
spaces typically assumed for classical probabilistic computing [29]. Under the linear weighted sum transformations in Cartesian- -rroucct state spaces, the log encoding [35] can consistently relate the summation of the inner product for each cubit to the mutiplication of the corresponding probabilities
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for the product event via bias thresholds P,'s and Protat = 11 P. as
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P(adlps P.
n
P(alp) P(aulp)) log Piotal
((all))) Σ ((osllps))
log
log II
P.
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2.2
Operators as neural computing primitives
83 Operator algebra is a well-established technique to systematically compute quantum physics prob- 34 lems in high-dimensional tensor-product spaces (or Fock for indistinguishable particles). Interac-
Further investigation the relation between qubits and cubits from physics point of view desired.

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95 tions between states are represented by scattering matrices (S (S-matrices) [36] exemplified in Fig.
96 (c), Here, we develop an operator formalism cartesian-produse state spaces for neural networks
97 in Fig. (d).
A neural signal at S is selectively activated in the entire state space spanned as,
I00) = ([01)) @ ... @ (|0,)) and [[+))) = | @ @ |1,) @ ... @ I(0,)).
(10)
39 States for concurrently activating multiple neural signals can be given, by specifically noting the activated systems and
11) = ((01)) e +
|| @ @ ||,;)) @... ... e ((0.))
(11)
Thus, [1, can mean single cubit state for S. only or a multiple cubit state in which only S is fully activated, depending on the context.
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The auduonty-adjoint creation and annihilation operators on these states, a and al are defined as 103
[1))) = a! [00)} and |[0)) = a, ||1))
(12)
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Multiple signals can be activated in different systems, for example, by
||,+)) = a. [10)).
(13)
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Depending on whether = j allowed each T, or not, they are superficially treated like Bosons for rate-coded SNNs (rSNNs) or like Fermions for temporally-coded SNNs (tSNNs).
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The transformation Ti from sender system Sy to receiver system R; is described as:
Tif = wiyafay.
(14)
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Noted that wy works as the scattering matrix. Cartesian product state space, rather than tensor- product, can incorporate the weighted sum naturally the superposition of incoming neural signals from different sources. Higher-order interactions are possible, for example as,
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T/ = w.j "
(15)
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However, in that case our original assumption of linear synaptic networks is not valid anymore.
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The logical neuron model in the operator representation is defined as effects for detecting incoming fragment of signal energies from presynaptic neurons to generate states for the follow-on neural signals. The signal detection process corresponds to the projective measurement in QC, leading 115 more advanced detection strategies than simple threshold detection strategies. When the fully
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activated state [|p,)) = al [10)) is detected by the effect (faal = ((0)) a, via Tij.
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I ((ayl) Tay (lp;)) 1" (30) [b, (wyyb(ay) a51 10)) === |way|" log P(oslp;)-
(16)
!17
Nonlinear binary operations such AND/OR are possible using appropriate activation functions with different thresholds, as those in perceptrons [37].
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Physical representation
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The proposed physical representation of neural networks is outlined in Fig. 2. It introduces explicit temporal dependences for operators and neural signals The operators for ingress and egress paths create and annihilate nonstationary neural signals over elastic physical media, i.e., axons (8 's)s and dendrites (R('s).
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3.1
Operators for eigenmodes
125 First, the physical representation of the creation and annihilation operators for stationary neural sig- nals af and a; are constructed in accordance with the quantum creation and annihilation operators of and a; in the one-dimensional transmission line (TL. model in circuit QED [38]. Circuit QED is one
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128 of the established baseline theories in QC, which bridges classical circuit dynamics and quantum.
The Hamiltonian H4 for TL creating consisting of N identical capacitors of the capacitance Co

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TL model for axon and dendrite
Neuron model
x=0
Σ mijafflf + dr Ti)
9000
Neuron
Co
Qn
Wi dy
T3)
Neuron
xaje (e 4/1101)
Rolla
On time:
da T,
Tj
Ingress a
Wip
Ti
4 du
Neuron &
Egress
4 d,
Delayed: E, dj T2 0
av
Synaptic network model
[0llā,(r -T) > xta 1>) Miy x/ diy -T)
((ar)) = ((0))a;( - -4t1)87001100 and du = aijoon + affmm
Figure 2: Physical representation of operator-discretized neural networks with explicit local time dependency with respect global time T. The creation and annihilation operators for ingress and egress paths represent nonstationary neural signal dynamics across axon-synapse-dendrit networks. LC TL models are used for axons and dendrites instead of RC cable models. The neuron model consists of different activation functions for signal amplitude and timing,
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(each containing the charge Qn) and N identical inductors of the inductance Lo (each containing the flux T -,
1), is given by
H. Σ 2C,
Q- + 2Lo
1 (
D 2nn))
Σ hww (kim Wm Wa (kim Wm).
(17)
where is the eigen mode index for given boundary condition. The lossless LC-based model can better transmit energy and information than the dissipative RC-based biological cable model [39]
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We define af (k:, w) and a; (k: w) as the classical counterpart of a (k; w) and az (, w). The following simple linear dispersion for constant velocity e are assumed in the range of interest:
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135
Owm
Wm const.
1 = 0km
km
Vm.
(18)
Consequently, af (kww) a! (w), a (k,w) w) a (w). Note that for neural signals is much slower than v for electrical signals in ordinary TL's [31, 32]. Though our focus is on artificial neural net- works, biological implications of the present approach will be further discussed in Appendix.
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3.2
Operators for nonstationary neural signals
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Second, the operators basis changed from (k, w) to (xx t). For ingress signals
af (ax, 1) = Σ a (kim; wm) (kim, wm) =(
(19)
(,( Σ a Wm (Kmm Win ) (tem s--umm)
For egress signals
(x(x, t) = Σ a! (Kim, Wm) (kim wm )k wmi))
(20)
(x(x,t) = Σ as (kims Wm ) (kim Wm ) i(kmm wwm!
5

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142 They represent creation and annihilation of neural signals centered at x = 0. and = O. and sent
143 or received at neuron i. To be more specific, for example, neural signal moving out of neuron created at the start of the TL. of a length I is given as
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af (t) (1o)) = a0 t) [10)..
(21)
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annihilates at the end of the TL after the geometrically-defied delay = 1/0 < To as
(1 d) (1) [10)) a, (0)
(22)
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3.3
Incorporating physical interaction at synapses
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When multiple neurons are interconnected via synaptic networks, physical interactions with explicit temporal dependences should be incorporated in addition to the free dynamics described above. We consider here primarily Tiy one-body potential scattering via an elastic scattering center as
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Tij = wija (1- - T; + agendd )å (t-11 - dig
(23)
where difon and ang are the delays in axon and dendrite between neurons i ande j, respectively. 150
151
3.4
Neuron model with activation functions for amplitude and timing
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The proposed representation of neural networks allows for more advanced detection strategies than threshold detection, for example, in LIF neurons usually found in the literature [39] s. This is
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somewhat inspired by the advancement in detection strategies in communication or storage channels
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[40]. Let us first consider a simple case when half-cubit neural signal of the peak amplitude zy from a presynaptic neuron j is generated at = tú by applying a creation operator as
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Ilp, = z t,) [10)).
(24)
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and observed by a postsynaptic neuron i at T; directly without a synapse.
((ox(1)) = [(0) (t T:),
(25)
158 In general, the state preparation (())) at 1y and the observation ((a;(1)] at 7; are not temporally alighed, so by using ingress-egress correlation function f(Att,)) ((0)) ā, at)a (10)),
(()())))))))) ((01) (t (,)=,,ā(( - f,) (10)) f, diy T;)
(26)
for 15 dig T; O, where dij- dojon ddgodd We should note that for Are Ale + Ala
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s(Ah) (AAss) f(0) = 1.
(27)
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With interactions at synapses, the state preparation and observation between neurons pair and j
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provides
((aa())) Tr (t))) = ((0) át TT) - (10)) = wiyzzf(t + du- - Ti). (28)
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Thus, the aggregated signal detected at neuron is
Σ (a)l( (I)(()))) = Σ (Ko)) a, - T.)C- -t) (10)) = Σ Euyazftt + dig T.).
(29)
This inner- mmner-product-basee detection in neural systems corresponds the projection measurement in quantum systems and is the key to enable efficient coarse gaass--rraied detection without tracking the membrane potential fine-grained time steps. For a given waveform defined by creation and annihilation operators, [' (() can extract temporally-codedd information. Alternatively, the right operator pair can be defined meet given f (()t,)). The latter approach is be taken when applying the present idea to efficient emulation of temporallycooded SNNs.
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By using appropriate activation functions 0r and 02 for the amplitute and the event firing time, resepetively, the detected signal can be converted to the follow-on signal in neuron i as
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J; = 0 Σ S1 + dip T.)), 1, = T₃ + 02 Σ Wig + dig T.)).
(30)
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Various nonlienarities can be incorporated via 01 and 0g if necessary.

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3.5 Learning algorithms with operators
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The weight update Awyy for unsupervised algorithms, such as Hebbian and STDP for SNNs, is 175 asynchronously (i.e., without explicit dependency on T;) related to ingress-ingred correlartion
Awy ((0)) ā, t.)a
di,)) (10))
((0 ( (t - aa)a -t d())
(31)
9(4 (T) 11 - d1y) =9(4= - daz)-
176 Even and odd functions are chosen for Hebbian and STDP, respectively.
The proposed representation support various supervised learning algorithms and toolchains, when temporal dynamics synchronously regulated by coarse-grain global clock n cycles as
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T:th nTe vi.
(32)
179 Fine- ceme-grained temporal correlations. such as coincidence, can be passed on to the operator correla- tions by defining global variable x(() zg(") f(t!"). Then
1n+ 1) oi Σ x()))
es ((+1)) = ((+t)) + +02 L w
(33)
The backward calculation can be performed by using the following relation:
ax!u+t) ax(1-1) On (n) ax(n+t) an! = (f(1! (f(((+t)) of + t (n++1) f f'ob)as
ax(") = an(x+1) ox(m)) + (x(+t1) wx
(34)
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Let us go through how this works further with a specific example in the next section.
Application to temporally-coded SNN
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The relation between ANNs and rate-coded SNNs (rSNNs) has been known [41]. Here, first theoretically prove that under the proposed representation, temporally-coded SNNs (tSNNs) can be cquivalently transformed into ANNs by appropriately assigning the operator via f and encoding via 1 and 02. Then we demonstrate practical benefits of doing so by running some benchmarks.
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4.1
New perspective on ANN-SNN equivalence
Proposition 1: When driven by global clock of T nT'.. operator-descritzeed neural networks defined by Eqs. 28 and 30 for the neural events (xi, t4) with the followng setting consistute ANNs.
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ANN: a (x) = σ2 0, and ((x)
(35)
The neural signals stay constant at x(()) = ()( for T.T)) = nTT. The operators become arbitrarily picked single-mode (k: w) ones. Perceptrons are constructed with binary inputs and Heaviside step function for a1-
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Proposition 2: When driven by a global clock of T(()) = nT.. operator-descritized neural networks defined by Eqs. 28 and 30 for the neural events (2,, ts) with the following setting constitute tSNNs.
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ISNN: 01 (x) = and 02(x))
(36)
The (SNN signals for x(!)) = ft, (n) -r!") take specific spike waveforms defined by nonstationary operators which spread into multiple modes in the (k, w) basis. The cut-off Xmin is defined as
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T&()) 5in5) < TTS²) + Te + 12 x(m) 2 Xmii ((T.).
(37)
198 Theorem 1: ISNN Proposition with I' (x)x)0((x) runs equivalently in forward and backward to ANN in Proposition with a (x) z (x Xmin) ) for Xmin f(T.) > 0.
7

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(a)
(b)
(c)
Inhibitory
Excitatory
Tz
Throughput (s.).)
accuracy
T
3D
of neural
Xmin
ANN
323
15NN (operator)
ESNN (Euler)
80
Epoch
Figure 3: (a) Activation function for operator-discretizer tSNNs with excitatory and inhibitory
neurons; (b) MNIST benchmark results for ANN, seerator-discretizel iSNN, and diser-dis
ISNN; (c) Realtive comparison of the number of neural signals and throughput.
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Proof. In forward, weighted sum tSNN reduces that of ANN as
(ZJ)ANN (f(e) TS))NNN, and (WgJANN (wigf(-Te diy))SSNN.
(38)
201
This is because for ISNN,
Σ Eung S( + -d -)(tt))= = Σ (T Σ Tuns(-TTedds -T.).
(39)
202
In backward,
(ox(nt1) oxxmm) (0xp3)) 1- + dij)
= (wij)ANN = (way (- +
ANN
ISN N
(40)
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Thus we can emulate tSNN using ANN by renormalizing wi with the constant f(-To + diy).
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Example 1: We can set ISNN as
f(x) = 3 02(z) = log, x and T. = diy -ee.f(-Te+d.)) =
(41)
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9 works as a base constant to carry or borrow across fine mme-grained unit time interval. The logarith- mic conversion works as a ReLU activation function ANN since the conversion is only valid for
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X minn 0. Bipolar neural signals are represented by combining excitatory and inhibitory neurons as shown Fa). This setting can also support rSNNS by allowing multiple spikes within T..
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Building blocks in modern ANN models, such as convolution, max pooling, and batch normaliza- tion, have be translated those SNNs. The translation is straightforward long as they are linear transformations. However, batch normalization blocks may require some attention, since they involve nonlinear operations to control both the number and the delay distribution of neural signals.
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Once the translations of building blocks are completed, the proposed representation for SNNs can support not only specific models and learning algorithms but wide variety of them. Under the -psretoo-discreetzeed representation. the inference paths of SNNs can be translated those of the corresponding ANNs. Thus the standard autograd learning strategy [42] for ANNs equally works
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without using costly strategies specific to SNNs. The instability associated with differentiating the spike activation function be avoided by substituting adjoint computation [43] to the operators rather than using arbitrary surrogate functions [6.44].
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4.2
Evaluation
Figure 3(b) compares MNIST benchmark results for ANN, Fuler-discretized tSNN, and operator- discretized iSNN. We used a stand-alone computing environment without GPU to minimize unde- sired throughput variations. The code for ANN straightforwardly follows reference implementations
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(a)
Test accuracy (a.u.) (b)
- CIFARID CIFARI00
GIFR1O
DIFAROO
- -IIIARRD -+- CIFAR10D
Test accuracy (a.u.)
Throughput (a.u.)
Throughput (a.u.
0.8
0.8
0.1
0.2
02
0.1
0.2
0.1
0.2
Xii
Xmin
Xmin
Xmin
Figure 4: Relative test accuracy and throughput as a function of Xmin for CIFAR 10& 100 with resnet18: (a) With batch normalization; (b) Without batch normalization. Xmin = 0 is for ANN.
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and default parameter settings under python 3.8.5 and PyTorch 1.9.1. Ir 0.001 with Adam op- timizer and is multiplied by 0.9 after every 10 epochs. To accommodate Euler-discretized ISNN, simple architecture of 784-350-10 is chosen. The Euler discretization algorithm follows the one in [12], There, forward and backward paths were calculated manually in 30 AT steps in each Ta SNN fully takes advantage of the existing toolchain capabilities of ANN, including autograd. For -iiseeeiddd ISNN, we used the conversion as stated in Example with Xmi = 0.1. In short, the result for operator-discretized ISNN achieves a significantly better throughput, than Euler-discretized one, demonstrating competitive accuracy and throughput to those of ANN.
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period. On the other hand., aperator-discretiin
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Figure 3(c) compares the number of neural signals and throughput for ISNN, Euler-discretized ISNN, and operator-discretizdd tSNN. In the Euler-discretized ISNN, the throughput severely affected despite the reduction of the number of spikes, Since information is encoded in time rather
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than in amplitude, naive discretization using fine-grained AT steps not very efficient terms of both accuracy and throughput. Indeed, the computing complexity proportionally increases as the number of AT steps, rather than as the number of neural signals. In contrast, both the number of spikes and throughput are comparable to those of ANN in operator-discretizze
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tSNN. The proposed emulation strategy meets computing efficiency without washing out actual neural signal waveforms by embedding fine-grained temporal dynamics into crosscorrelations of operators.
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The proposed emulation strategy is expected to be as scalable to larger workloads as ANNs. To validate this assumption, our emulation approach was applied to larger data sets and architectures. Figure& 4 summarises the benchmark results for CIFARIO& 100 and resnet18. This time, we used SGD with Ir = 0.1 with batch normalization and Ir 0.05 without batch normalization for better convergence. The learning rates were reduced by x 10 after every 30 epochs for a total of 90 epochs. Again, the ANN code follows reference implementations and default parameter settings in py PyTorch documentation, The programs were executed in x86 internal clusters for higher throughput (at an expense of throughput variations due to other jobs) with python 3.6.9 and PyTorch 1.2.0, but again without GPUs We used multicores
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single node since the conversion between ANN and ISNN is local i.e., not affected by the node configuration. The result confirms that both accuracy and throughput are similarly competitive to ANNs for larger datasets and models. We performed multiple runs for 10 different seeds. The standard deviations were S 1 % and S 10 % for accuracy and throughput, respectively.
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Conclusion
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This paper proposed new representation of neural networks that can efficiently compute their dy- namics as sequences of pperator-discretizee events. Our approach takes ad advantage of existing math- ematica frameworks that have been originally developed to abstract and discretize nigh-dimensional quantum systems with necessary modifications to handle neural networks. Different generations of neural networks, such as ANN and SNN, were attributed to different selections for operators and encoding. Our formulation, when applied to tSNNs, led to more computationally efficient SW em- ulation with fully exploiting existing ANN assets. Presently, learning not perfectly asynchronous because of Eq. 32. However, this limitation makes sense considering that the biological brains also
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use slow brain waves to efficiently regulate their operations without much affecting online tracking.

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Checklist
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The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes], [No], or [N/A]- - You are strongly encouraged to include justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example:
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Did you include the license to the code and datasets? [Yes] See Section ??.
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Did you include the license to the code and datasets? [No] The code and the data are proprietary.
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Did you include the license to the code and datasets? [N/A]
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Please do not modify the questions and only use the provided macros for your answers. Note that the
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Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
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1. For all authors...
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] To be summarized in the sup- plementary material
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(c) Did you discuss any potential negative societal impacts of your work? [N/A] This paper is on representation and is mostly thcoretical.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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(a) Did you state the full set of assumptions of all theoretical results? [Yes]
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(b) Did you include complete proofs of all theoretical results? [Yes]
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(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, applicable? [N/A]
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(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
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Reviews:
[{"review_id": "xXLca0vOeuA", "reviewer": "Reviewer_8i1a", "summary": "The authors propose a mathematical framework based on methods in quantum physics and category theory to interpret analog and spiking neural networks. I did not understand the motivation behind the study nor its contribution to the field. I am entering a brief review with low confidence score because the techniques employed in this paper are well outside my area of expertise. ", "questions": "None.", "limitations": "I don't believe there are any negative societal impacts or ethical concerns. My concerns regarding limitations are summarized above.", "rating": 3, "confidence": 1, "soundness": 1, "presentation": 1, "contribution": 1, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I will not comment on the technical details of this paper -- I have asked that the area chairs consider replacing me with another reviewer with more domain knowledge in quantum physics.\n\nNevertheless, I will comment briefly on the appeal of this paper to a general machine learning audience. In short, the paper is not well-written to appeal to this audience and on these grounds alone it may be justified to reject this paper. The abstract and introduction do not explain the basic ideas that will be employed and instead drops hints and jargon to those with a strong physics background (\"It is tempting for those with some physics background to apply techniques...\" and \"operator algebra has been applied to Hopfield networks [18]\"... without defining operator algebra.) I expect the technical details of this paper would not be easily digested to the average NeurIPS reader (e.g. Bra–ket notation is used without any explanation).\n\nPerhaps most importantly for a NeurIPS audience, the practical demonstrations of this work feel weak and lacking. The authors only show a brief proof-of-principle on MNIST. If this result is surprising or significant it is lost on me and should be better explained.\n\nFinally, I also want to note that several comments in the paper referring to biological networks strike me as concerning. For instance, \"the biological brain operates with low-frequency brain waves closer to our behavioral time scale\" -- is a dubious motivation for this study since (a) other neural oscillations are much faster than the theta frequency cited by the authors here, and (b) the biological function of all neural oscillations is still poorly understood and it is particularly controversial to suggest that these oscillations operate like a \"clock\"", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "poGT58ZLp0V", "reviewer": "Reviewer_DDpm", "summary": "The authors used the linear algebra which is well established in the quantum theory to rewrite the transmission line model, which is as fine as the cable model. By adding some assumptions, they built the tSNN (operator) model which likes a rate model including the time delay information. The keys to simplify the cable-level model to the rate-level model are the use of the constant velocity assumption which can eliminate the distance x and the use of the inner-product-based detection which can eliminate the time t.", "questions": "Overall, to me, this work does not bring any really new contribution to neuroscience and AI (see the above). Please clarify it. ", "limitations": "The authors should address the limitations of their model in detail, in particular, those assumptions used which are not biologically plausible.", "rating": 3, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1) The work makes a connection between a cable-level model to rate-level model, though there are some unclear issues (see below).\n\nWeaknesses:\n1) From the perspective of neuroscience, from the start, the motivation of setting the global clock T_c closer to our behavioral time scale is wrong. At the circuit level (not the function level), the neural dynamics is continuous. The signal detection process (Eq.16) is far from reality. There is a gap between the aggregated signal (Eq.29) and the event firing time (Eq.30), which needs to clarified by the authors. This gap leads to another question that all neurons in the model share the same time clock (Eq.32), which is not true in reality. \n2) From the perspective of artificial intelligence, the author gave a model (Eq.30 or Eq.33) which is equivalent to a feedforward network in some cases. However, it seems that the connection matrix in the equivalent feedforward network have to remain the same between layers. Though the author claims that their tSNN can encode the time delay information, the model (Eq.33) has just two variables which is not interesting in AI. In fact, we can intuitively get the ultimate model (Eq.33) without the complex linear algebra.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "OemuTBnfJt", "reviewer": "Reviewer_p6e7", "summary": "The paper applies a formalism from quantum theory to ANNs and SNNs that offers to deal with time discrete and asynchronous events. The theory is applied to temporal spiking neuronal networks (tSNNs) that are shown to be equivalent under certain conditions to ANNs. The coding is applied to CIFAR 10 & 100, and it is shown that the suggested operator-discretized tSNN have a better “throughput” than the Euler. discretization of tSNN, and that operator-discretized tSNN are comparable in “throughput” with ANN.", "questions": "(1)\tThe equivalence between tSNNs and ANNs is only shown with some restrictions that are not so clearly spelled out. In general, tSNNs are treated as ANNs in the literature mainly for time-to-first-spike (TTFS) coding, assuming an overall clock that resets the state of the system at discrete times, and from there on considers time as the analog variable of the ANN. This special case does not include the processing of multiple spiking signals, emitted from each single neuron, in real and continuous time, but never synchronized with an external clock or with other neurons of the network. Does you theorem contribute to this more general case? In what sense does it go beyond the well studied TTFS coding?\n\n(2)\tEvent-based simulations of spiking neurons has a long tradition and is shown in many papers and simulation platforms to dramatically outperform naïve Euler-discretized simulations of spiking networks. In what sense does your approach go beyond these event-based simulations of SNNs?\n\n(3)\tIf the aim is to contribute to the simulation of tSNNs, then the comparison in the simulations should be done with state-of-the art methods to simulate tSNNs. Can you show such results?\n\n(4)\tSo far the paper simply shows that the performance is comparable to ANNs, supporting the theorem, but begging the question whether the technique can really be used for the analysis of SNN on behavioral time scales, as explained in the Introduction. It is said that at present, the networks are not fully asynchronous, and this is justified by the slow waves in the brain. But slow waves are seen on the level of field potentials, and are far away from globally imposing spike times in individual neurons, as it is assumed in the paper. Can you give a hint how the technique will overcome this?\n", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "It is certainly always interesting to transfer successful methods from other fields. Yet, in the current example, it does not become clear, what one really gains over existing techniques in simulating ANNs, or tSNNs transformed to ANNs.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "0PA5i0usmj", "reviewer": "Reviewer_k9YU", "summary": "The authors propose an alternative representation of artificial neural networks through the use of machinery usually used to analyze complex quantum systems.  Through formalisms from operator algebra, they are able to link analog neural networks (ANNs) and spiking neural networks (SNNs) through different choices of operators.   For example, operator representations typically applied to quantum wave packets are ported to artificial neural networks under the formalism outlined by the authors.  Their proposed framework also allows for more advanced detection strategies in comparison to simple threshold detection commonly used.  They also show the practical benefits of their representation.  \n\nPast this, they they develop physical representations for the creation and annihilation operators for neural networks through considering transmission line models commonly used in quantum electrodynamics.  Their operator formalism then also allows them to consider the case of non-stationary neural signals, defining appropriate time dependent operators through re-use of operators they defined in the stationary case.", "questions": "Could the authors clarify -- when reading the introduction of the paper, the explanation of clock timings and comparing them to membrane time constants makes me think the point trying to be made there was to say that it is not the dynamics following much lower time scales that is important correct? but rather it is the dynamics that follow much larger time scales?  If this is the case, after that analogy what I am failing to see is how the proposed representation takes advantage of this fact; I think it would be extremely helpful to hammer away at the relation throughout the paper since it seems like such an important part of the introduction.  \n\nI also think more clarification related to Fig. 2 is required.  I do not think I have the necessary requisite knowledge to fully understand the LC model for signals traveling between axons/dendrites -- but there is a great deal going on in the figure.  I am curious as well, although the LC model has the advantage as you say that it can better transmit information, does this mean it is not possible to create an equivalent RC model?", "limitations": "The greatest limitations of the paper in my opinion are the fact that it requires a great deal of requisite knowledge that many will not have.  Fleshing out the appendix here would help a lot — and I think the authors really have to guide the reader a great deal more.  For example, even jumping into the modified bra and ket notation for the newly introduced ‘cubits’ was a bit overwhelming and I think not necessarily well motivated.  ", "rating": 3, "confidence": 2, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The main strength of the paper in my opinion is attempting to come up with a general framework to describe neural computation using notions from quantum computing.  As the authors note, this kind of theory goes both way — i.e. it can aid in certain aspects of quantum computing, but it may also provide another means to understand neural computation.  In addition to that, the authors also try to show the practical consequences/benefits of their framework by showing how it facilitates the conversion from temporally coded SNNs to ANNs.  As hardware implementations of neural computing advance in parallel to quantum computing over time, I believe that drawing these analogies and coming up with a cohesive framework can be of great benefit.\n\n\nHowever, I find much of the paper difficult to read both conceptually and theoretically.  In my opinion the paper assumes working knowledge of many different fields/concepts that a great deal of readers would struggle to grasp, i.e. quantum electrodynamics, categorical theories, operator theories, and so on.  I would be okay with this if the references provided at least served as good introductions to these concepts and were presented in a way that I would be at least able to follow high level ideas presented in the paper.  Greatly expanding the appendix in this case would not just be a benefit to the common reader, but a necessity in this case.  I appreciate what the authors are trying to do, but the presentation in my opinion needs a great deal of work.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Reject