Learning Without Clocks: Partition-Invariant Neural Network

Han Wang; Jingyuan Bao; Juntao Wu; Saiao Zhou; Yuman Nie

cs.LGOct 27, 2026

Learning Without Clocks: Partition-Invariant Neural Network

Authors: Han Wang, Jingyuan Bao, Juntao Wu, Saiao Zhou, Yuman Nie

Organizations: Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei 230031, China · Science Island Branch, Graduate School of University of Science and Technology of China, Hefei 230026, China

Abstract

Event cameras, spiking cochleas and similar sensors emit asynchronous events, yet most learning systems first bin them into frames or packets, so the output depends on how the stream was sliced. Networks whose state is updated exactly at every event and evolves in closed form between events are invariant to this slicing. We ask what more makes the output (i) independent of the processing order of simultaneous events and (ii) continuous in the time stamps, even when two events cross in time, while (iii) staying sensitive to temporal order. For arbitrary nonlinear event models we prove that (i) and (ii) are each equivalent to pairwise commutation of the event updates, and that once updates commute, all order dependence comes from their non-commutation with the inter-event dynamics and vanishes linearly in the time lag. This identifies a class of time-mediated non-commutative (TM) models and yields a resolution-robustness bound for any model. We construct the minimal TM layer, a learnable, provably Lipschitz generalization of the Hassenstein-Reichardt correlator; prove that no single additive layer can uniformly represent ordered pair interactions, whereas a three-dimensional TM state can; show that each TM layer adds two levels of temporal nesting versus one for additive layers; and give a single-layer higher-order construction (TM-k) and a stacking rule preserving all three properties. We instantiate these results as the Clock-Free Event (CFE) layer, linear in the number of events, give a topology-free variant for temporal point processes with closed-form likelihoods, and specify a preregistered clock-robustness protocol with a physical-time threat model for spike-retiming attacks. In preliminary development-set experiments, the TM state beats an otherwise identical additive state by 3.5-9.1 points on DVS128 Gesture and by 24.5 and 30.6 points on SHD and Speech Commands.

Figures & tables

Figure 1: Clocked versus clock-free processing. Top: binned pipelines slice the event stream into frames of width

\Delta

; the input tensor depends on

\Delta

and on the phase of the grid, and moving one event across a bin edge by an arbitrarily small

\varepsilon

changes a count ( Proposition 11 ). Bottom: a CFE network updates its state exactly at every event (jumps, coloured by event type) and evolves it in closed form between events (blue); queries (

\blacklozenge

) can be placed at arbitrary times, and the result does not depend on how the stream is cut into packets ( Proposition 3 ). Right: output of a single site as two events of types

a

and

b

cross in time, computed from the model equations with illustrative parameters. A gated update ( Sel ) jumps at the crossing and returns two different values at an exact tie, depending on the processing order; the TM layer is continuous and still distinguishes the two orders. Tie invariance and crossing continuity are each equivalent to commuting jumps ( Theorem 5 ), and order information then enters only through the non-commutation of the jumps with the drift ( Theorem 8 ).

Family	Update at an event	(P)	(T)	(C)	(O 2 )
Binned networks (frames, fixed time step; e.g. PLIF [ 4 ] )	counts added to the current bin	no	yes	no	yes, at bin resolution
Additive event SSMs (exact decay; NS )	$x\leftarrow x+b_{c}$	yes	yes	yes	no ( Theorem 15 )
Selective or input-gated affine updates ( Sel )	$x\leftarrow G_{c}x+b_{c}$ , generic $G_{c}$	yes	no	no	yes
Gated nonlinear updates with ODE drift [ 25 , 26 ]	$x\leftarrow\mathrm{GRU}(x,c)$	yes	no	no	yes
Online log-signature models [ 15 ]	interval composition, non-commuting generators	yes	excl. †	fixed order ‡	yes
TM (this work)	$u\leftarrow u+\alpha_{c}$ , $v\leftarrow v+g_{c}R_{c}s$	yes	yes	yes	yes

Table 1: Properties of event-model families under exact computation, for generic parameters. (P) partition invariance; (T) tie invariance; (C) continuity in the time stamps across crossings; (O 2 ) ordered pair interactions within one layer. Justifications are in Section A.13 .

Figure 2: The CFE network. An event stream (top left; ON and OFF events of a schematic rotating gesture), or a sampled signal converted into events by a send-on-delta encoder (bottom left, Proposition 22 ), drives the CFE layer. The sensor is tiled into sites; an event (red pixel) applies a jump to the state of every site in its

3\times 3

neighbourhood, with a letter given by its polarity and relative offset, and every site state drifts in closed form between events. TM site: events write the fast trace

u

and, through the mark-dependent gate

g_{c}(m)R_{c}

, read the delayed trace

s

into the accumulator

v

; no event writes a coordinate that events read, so all jumps commute ( Corollary 7 ). Exact parallel scan: each event acts as an affine map, so the states are computed exactly by a segmented parallel scan, and tie groups may be processed in any order. Queries read the site states

x(T_{q})

at arbitrary times; a readout shared by all arms (signed-log normalization with raw features, convolution over the site grid, anchor-based pooling, a memory across queries and a linear layer) gives a prediction at every query. For event sequences without topology, an intensity head with a closed-form compensator gives the exact likelihood ( Proposition 23 ). Bar heights are schematic.

Check	Result	Supports
Closed form vs. recursion (200 streams, 5–60 events, $\approx$ 30% ties)	max. relative error $2.5\times 10^{-15}$	Prop. 12
Shuffling within ties (100 streams, $\approx$ 60% ties)	max. $\|\Delta x\|$ : TM $1.7\times 10^{-13}$ , NS $1.4\times 10^{-14}$ , Sel-Merge $3.4\times 10^{-13}$ , Sel $43.5$	Thm. 5
Time-stamp perturbations $10^{-4}$ – $3.2$ ms, with reordering (300)	observed/bound $\leq 0.0089$	Thm. 14
Swap at lag $\delta\in[10^{-5},10^{-1}]$ ms	TM $3.84\times 10^{-3}\delta$ ; NS $1.88\times 10^{-3}\delta$ ; Sel , Sel-Merge 0.2526–0.2528	Cor. 6 , Thm. 8
NS fiber ( $d=5$ ; 50 $b$ -events)	state residual $\leq 6.7\times 10^{-16}$ while $y_{\mathrm{ord}}$ falls from 37.00 to 32.30	Thm. 15
TM represents $y_{\mathrm{ord}}$ (200 streams)	$\|K-\kappa\|\leq 1.1\times 10^{-16}$ ; all errors within bound; max. 6.4% (bound 9.5%)	Prop. 16

Table 2: Numerical checks of the theory (FP64, seed 23).

Figure 3: State difference when two events of different types swap places at lag

\delta

(FP64). TM and NS change linearly in

\delta

, below the Lipschitz bound of Theorem 14 . Selective updates jump by a constant amount at every lag, as Corollary 6 predicts.

Setting (clips)	Arm	200	400	600	800
V1, folds 0, 2 (198)	TM	170	175	177	176
	NS	152	165	169	169
	Zero	138	156	159	162
V1, folds 0–3 (495)	TM	407	419	440	440
V1, folds 0–3 (495)	best unordered	367	407	418	419
V2, folds 0, 2 (198)	TM	170	177	179	179

Table 3: DVS128 Gesture, development folds: correct clips at the 200/400/600/800 ms prefixes. V1:

8\times 8

-pixel sites; V2:

4\times 4

-pixel sites with augmentation. “Best unordered” is the better of NS and Zero at each prefix.

Figure 4: DVS128 Gesture, folds 0 and 2 (198 calibration clips): accuracy against observation window for TM, NS and the stateless arm ( Zero ). The benefit of the ordered state is largest early.

Dataset, configuration	Arm	10-point mean	final
SHD, augmented (A)	TM	81.80	87.40
	NS	64.31	62.88
	Zero	–	8.73
SHD, base	TM	73.25	76.45
	NS	51.59	47.37
	Zero	11.36	8.86

Table 4: Audio benchmarks, validation splits (accuracy, %). SHD: mean over 10 query times and at 1000 ms. Speech Commands (SC): 9,981 validation clips; TM and NS are still training (epoch 20).

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Variant	10-point mean	1000 ms
Base (no augmentation, 8 channels per site)	73.25	76.45
A: augmented, 8 channels per site	81.80	87.40
A with 2 channels per site	63.89	63.43
A with 4 channels per site (B)	75.83	82.27
B with channel-position letters (12 letters)	71.83	75.07
A with 16 channels per site	74.61	77.15

Appendix

Table 5: SHD validation accuracy (%) of TM variants.

Permutation-Invariant Learning

Permutation-invariant learning builds models whose outputs do not change when equivalent inputs are reordered.

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Collision Avoidance Event-Based Vision Robotic Perception State Space Models

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Robotic systems operating in dynamic environments require visual perception that evolves continuously with the incoming sensory stream. Event cameras provide microsecond temporal resolution and asynchronous sensing, but most learning-based methods accumulate events into frames or temporal bins, introducing an integration delay that can limit fast reaction. Here we propose REACT, a fully spiking state-space model for event-driven temporal perception that processes raw events one by one, without temporal accumulation. REACT uses a complex-valued spiking neuron, C-SiLIF, whose continuous-time dynamics are driven by the physical inter-event interval, allowing its internal state to evolve at the temporal resolution of individual events. We evaluate REACT on gesture recognition and time-to-collision (TTC) estimation from full-field event streams, without a target bounding box or localization input. On EvTTC, REACT achieves a 9.59% relative TTC error with 4.6 ms end-to-end inference latency, within 0.15 percentage points of the best learned method while requiring no target prior. At the dataset's mean approach speed, this latency corresponds to only 4 cm of vehicle motion, compared with 1 m for the fastest competing learned method. REACT further supports anytime TTC prediction, zero-shot transfer to a different driving sequence, and INT8 quantization, reducing the estimated energy consumption from 18.5 to 2.8 mJ per 32,768 events. These results show that event-driven spiking state-space dynamics can provide low-latency, continuously updated temporal perception for reactive robotic systems.

Abstract

Permutation-Invariant Learning

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