GC-LIF: Spiking Group-Equivariant Neural Network With Group-Coupled Membrane Dynamics

Nhan Trong Luu; Duong Trung Luu; Thang Cong Truong

cs.NEOct 29, 2026

GC-LIF: Spiking Group-Equivariant Neural Network With Group-Coupled Membrane Dynamics

Authors: Nhan Trong Luu, Duong Trung Luu, Thang Cong Truong

Organizations: College of ICT, Can Tho University, Can Tho, Vietnam. · Center for Digital Transformation and Communication, Can Tho University, Can Tho, Vietnam. · Department of Computer Science and Engineering, The University of Aizu, Japan.

Abstract

Spiking neural networks (SNNs) promise event-driven, low-power inference, but on vision tasks they still acquire rotation invariance only through data augmentation. Group-equivariant CNNs build such symmetries into the architecture, yet a faithful spiking analogue is missing: a leaky integrate-and-fire (LIF) neuron placed after a $G$ -equivariant convolution can break equivariance through its membrane recurrence. We give a necessary and sufficient condition for this not to happen. The layer is equivariant at every time step if and only if the neuron parameters $(β, v_{\mathrm{th}})$ are constant along the joint orbits of the group action on the (group, space) index. We solve it in closed form, and it covers any elementwise stateful neuron. Its sufficiency half specialises a known result for equivariant recurrent networks; its necessity half rules out the heterogeneous per-neuron parameters common in SNNs. The per-channel parameterisation that satisfies it, Group-Coupled LIF (GC-LIF), is combined with $p4$ and $p4m$ -equivariant convolutions and a spike-rate group max-pool into SGE-Net. On MNIST, KMNIST, Fashion-MNIST, CIFAR-10 and CIFAR-100, evaluating each checkpoint on the clean test set and all its group-transformed copies, SGE-Net trained without rotations is invariant under $C_4$ and $D_4$ and outperforms a plain SNN trained with rotations in nine of ten dataset-group pairs, whereas non-equivariant baselines lose up to $72$ percentage points in the worst case. Operator-level tests confirm the theory before and after training, and an analytical estimate places SGE-Net at $1.7\times$ the arithmetic energy of a standard CNN and $2.3\times$ below its ANN counterpart at eight time steps.

Figures & tables

Figure 1: Overview of SGE-Net : lifting and group convolutions, group BatchNorm and GC-LIF neurons, group spatial average pooling between blocks, and a readout that averages spike rates over time and max-pools over the group axis.

Models	MNIST	FMNIST	KMNIST	CIFAR10	CIFAR100
CNN	99.20 $\pm$ 0.32	91.13 $\pm$ 2.54	93.88 $\pm$ 0.51	79.93 $\pm$ 3.43	49.48 $\pm$ 2.10
SCNN	98.68 $\pm$ 0.13	89.84 $\pm$ 0.57	89.20 $\pm$ 5.53	75.77 $\pm$ 0.06	34.94 $\pm$ 5.59
GCNN	97.86 $\pm$ 1.59	91.37 $\pm$ 1.02	91.61 $\pm$ 0.70	79.81 $\pm$ 1.84	47.21 $\pm$ 6.04
SGE-Net	95.58 $\pm$ 1.46	87.78 $\pm$ 2.16	88.30 $\pm$ 0.10	73.58 $\pm$ 4.38	40.50 $\pm$ 3.05

Table 1: Clean test accuracy (%): training and testing without any transformation. Mean and

95\%

confidence interval over four seeds, from the checkpoint selected on the validation split (Section 4.1 ). Group convolutions are

p4

Table 2: Full-orbit test accuracy (%) for the cyclic groups (Equation 18 ). Aug.: trained on clean images ( ✗ ✗ , w/o train aug) or with random group transformations ( ✓ ✓ , w/ train aug). Mean and

95\%

confidence interval over four seeds. For GCNN and SGE-Net at

C_{4}

, the ✗ ✗ rows equal the clean accuracies of Table 1 .

Table 3: Full-orbit test accuracy (%) for the dihedral groups, with

|D_{4}|=8

and

|D_{8}|=16

. Layout as in Table 2 .

Figure 2: Relative equivariance error

\Delta_{\mathrm{eq}}

of the membrane potential of a three-block

\mathrm{Conv}_{G}\to\mathrm{BN}\to\textsc{GC-LIF}{}

stack at initialisation,

T=8

, averaged over all group elements. The shaded band is the float32 rounding floor. (a) By group and depth:

C_{4}

and

D_{4}

stay on the floor, while

C_{8}

and

D_{8}

accumulate the kernel sampling residual. (b) Error against the relative spread of

(\beta,v_{\mathrm{th}})

when they are untied along the group axis on the exact

C_{4}

grid, so that broken sharing is the only source of error. The leftmost point is tied GC-LIF .

Figure 3: Fraction of disagreeing spikes between

L(\rho(g)\,x)

and

\rho(g)\,L(x)

, averaged over

g\in G

, at each depth of the GC-LIF stack at initialisation. The

C_{4}

and

D_{4}

values are exactly zero and are marked by circles at the axis floor. The dashed line is the

C_{4}

stack with

(\beta,v_{\mathrm{th}})

untied along the group axis at

10\%

relative spread.

Figure 4: Relative membrane error per time step on the exact

C_{4}

grid, untying one LIF parameter at a time with

10\%

relative spread. An untied

v_{\mathrm{th}}

corrupts the output at

t=1

; an untied

\beta

is exact at

t=1

and then degrades, as predicted by Equation 44 .

Figure 5: Layer-level relative equivariance error of

\mathrm{Conv}_{G}\circ\mathrm{Lift}

against the group order

n

, for bilinear kernel resampling and for the steerable basis of Equation 12 . The grey curve is the resampling floor of the test harness itself, more than an order of magnitude below the measured errors.

Data	Group	$\mathbf{\Delta_{\mathrm{eq}}}$		Spike disagreement		Logit deviation	Prediction agreement
Data	Group	init	trained	init	trained	Logit deviation	Prediction agreement
KMNIST	$C_{4}$	$0$	$3.0\times 10^{-3}$	$0$	$2.7\times 10^{-6}$	$1.7\times 10^{-4}$	1.0000
KMNIST	$C_{8}$	n.d.	$8.8\times 10^{-1}$	$5.6\times 10^{-9}$	$1.9\times 10^{-1}$	$5.7\times 10^{-1}$	0.5770
KMNIST	$D_{4}$	$0$	$4.0\times 10^{-3}$	$0$	$3.9\times 10^{-6}$	$2.4\times 10^{-4}$	1.0000
KMNIST	$D_{8}$	$4.3\times 10^{-1}$	$8.8\times 10^{-1}$	$3.9\times 10^{-7}$	$1.5\times 10^{-1}$	$5.1\times 10^{-1}$	0.6062
CIFAR10	$C_{4}$	$0$	$1.4\times 10^{-2}$	$0$	$5.5\times 10^{-5}$	$4.6\times 10^{-4}$	1.0000

Table 4: Equivariance of SGE-Net at the deepest spiking block, averaged over the non-identity group elements: relative error

\Delta_{\mathrm{eq}}

of the spike trains and fraction of disagreeing spikes at initialisation and after training, and relative logit deviation and per-sample prediction agreement after training. n.d.: undefined, because the reference output contains no spikes.

Group	$T$	Clean	Full-orbit	$\Delta_{\mathrm{eq}}$	$\bar{r}$	Energy ( $\mu$ J)
$C_{4}$	4	93.42 $\pm$ 1.72	93.41 $\pm$ 1.70	$7.8\times 10^{-8}$	0.253	21.70
$C_{4}$	6	94.87 $\pm$ 1.55	94.87 $\pm$ 1.52	$7.8\times 10^{-8}$	0.245	29.09
$C_{4}$	8	95.58 $\pm$ 1.46	95.58 $\pm$ 1.46	$7.8\times 10^{-8}$	0.237	36.50
$C_{4}$	12	96.02 $\pm$ 1.38	96.01 $\pm$ 1.27	$7.8\times 10^{-8}$	0.221	51.38
$C_{4}$	16	96.35 $\pm$ 1.31	96.34 $\pm$ 1.29	$7.8\times 10^{-8}$	0.205	66.22
$C_{8}$	4	93.42 $\pm$ 1.72	87.21 $\pm$ 1.85	$1.6\times 10^{-1}$	0.253	21.70

Table 5: Sensitivity of SGE-Net to the number of time steps

T

on MNIST: clean and full-orbit test accuracy (%, mean and

95\%

confidence interval over four seeds), mean firing rate

\bar{r}

and estimated energy (Equation 22 ) of the trained networks.

\Delta_{\mathrm{eq}}

is the membrane error of the three-block stack of Figure 2 at depth

3

, at initialisation, run for

T

steps.

Layer	CNN	SCNN	GCNN	SGE-Net
conv/lift- $1$	$0.113$	$0.113$	$0.226$	$0.226$
conv/gconv- $2$	$1.806$	$1.806$	$7.225$	$7.225$
conv/gconv- $3$	$0.903$	$0.903$	$3.613$	$3.613$
conv/gconv- $4$	$1.806$	$1.806$	$7.225$	$7.225$
classifier	$3\times 10^{-4}$	$3\times 10^{-4}$	$2\times 10^{-4}$	$2\times 10^{-4}$
Total	$4.63$	$4.63$	$18.29$	$18.29$

Table 6: Millions of multiply-accumulates per inference, by layer, at the benchmarked widths. Hidden group convolutions cost

G^{2}=16

times a plain convolution of the same per-slot width.

Model	Params	$T=6$					$T=8$
Model	Params	$\bar{r}$	Syn.	Neuron	Total ( $\mu$ J)	Ratio	$\bar{r}$	Syn.	Neuron	Total ( $\mu$ J)	Ratio
CNN	16,890	N/A	21.29	0.00	21.53	1.00 $\times$	N/A	21.29	0.00	21.53	1.00 $\times$
SCNN	16,890	0.274	6.26	1.24	8.95	0.42 $\times$	0.264	8.01	1.66	11.59	0.54 $\times$
GCNN	16,514	N/A	84.13	0.00	84.61	3.93 $\times$	N/A	84.13	0.00	84.61	3.93 $\times$
SGE-Net	16,514	0.245	23.72	2.48	29.09	1.35 $\times$	0.237	29.33	3.31	36.50	1.69 $\times$

Table 7: Estimated per-inference arithmetic energy (

\mu

J) at 45 nm CMOS with 32-bit floating point (Equation 22 ), on MNIST with

C_{4}

augmentation at

p4

(

|G|=4

). Syn.: synaptic term; Neuron: neuron-update term; the total also includes normalisation and pooling. Firing rates

\bar{r}

are measured at each

T

. Ratio: total relative to CNN. Memory traffic is excluded.

Figure 6: Analytical energy estimate at 45 nm CMOS with 32-bit floating point, excluding memory traffic. (a) Estimated energy of SGE-Net relative to the plain CNN against the number of time steps

T

, with the

C_{4}

full-orbit accuracy at each point (Table 5 ); the dotted line marks parity. (b) Estimated per-inference energy of the four models at

T=6

and

T=8

, split into synaptic, neuron-update and auxiliary terms (Table 7 ). CNN and GCNN do not depend on

T

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Setting	Model	MNIST			FMNIST			KMNIST			CIFAR10
Setting	Model	Prelim.	Final	$\Delta$	Prelim.	Final	$\Delta$	Prelim.	Final	$\Delta$	Prelim.	Final	$\Delta$
Clean	CNN	99.21	99.20	$-0.01$	91.95	91.13	$-0.82$	94.67	93.88	$-0.79$	80.88	79.93	$-0.95$
	SCNN	98.73	98.68	$-0.05$	90.84	89.84	$-1.00$	89.49	89.20	$-0.29$	76.67	75.77	$-0.90$
	GCNN	98.16	97.86	$-0.30$	91.82	91.37	$-0.45$	91.93	91.61	$-0.32$	80.84	79.81	$-1.03$
	SGE-Net	95.75	95.58	$-0.17$	88.64	87.78	$-0.86$	89.57	88.30	$-1.27$	73.61	73.58	$-0.03$
$C_{4}$ , ✗ ✗	CNN	46.78	43.17	$-3.61$	37.58	34.53	$-3.05$	36.89	35.28	$-1.61$	48.17	46.73	$-1.44$

Appendix

Table 8: Mean test accuracy (%) under the preliminary and the final pipeline (Appendix I ) for the clean setting and the exact groups, and the change

\Delta

= final

-

preliminary. Final values are those of Tables 1 to 3 ; confidence intervals are given there.

Setting	Model	MNIST			FMNIST			KMNIST			CIFAR10
Setting	Model	Prelim.	Final	$\Delta$	Prelim.	Final	$\Delta$	Prelim.	Final	$\Delta$	Prelim.	Final	$\Delta$
$C_{8}$ , ✗ ✗	CNN	40.44	41.02	$+0.58$	29.45	25.79	$-3.66$	25.51	24.05	$-1.46$	39.20	38.96	$-0.24$
	SCNN	40.69	41.35	$+0.66$	29.55	26.46	$-3.09$	25.42	23.65	$-1.77$	37.91	37.91	$+0.00$
	GCNN	69.50	87.88	$+18.38$	61.89	59.31	$-2.58$	56.64	56.12	$-0.52$	64.51	64.92	$+0.41$
	SGE-Net	77.75	85.62	$+7.87$	56.43	58.42	$+1.99$	51.75	53.31	$+1.56$	58.02	59.75	$+1.73$
$C_{8}$ , ✓ ✓	CNN	95.95	95.74	$-0.21$	85.21	83.26	$-1.95$	73.49	72.86	$-0.63$	72.14	67.51	$-4.63$

Appendix

Table 9: Mean test accuracy (%) under the preliminary and the final pipeline for group order

8

, and the change

\Delta

= final

-

preliminary. ✗ ✗ : trained on clean images; ✓ ✓ : trained with random group transformations.

Data	Model	$C_{8}$		$D_{8}$
Data	Model	no disc	disc	no disc	disc
MNIST	GCNN (ReLU)	57.74 $\pm$ 18.77	87.46 $\pm$ 0.86	71.92 $\pm$ 43.17	87.03 $\pm$ 8.16
MNIST	GCNN ( $T$ -level quantiser)	61.04 $\pm$ 5.54	84.84 $\pm$ 25.45	66.62 $\pm$ 6.03	84.32 $\pm$ 3.97
MNIST	GCNN (noisy ReLU)	64.69 $\pm$ 24.04	87.76 $\pm$ 5.04	68.21 $\pm$ 2.00	87.43 $\pm$ 0.29
MNIST	SGE-Net (spiking)	66.82 $\pm$ 23.07	85.41 $\pm$ 18.30	63.32 $\pm$ 22.60	84.75 $\pm$ 7.98
KMNIST	GCNN (ReLU)	55.02 $\pm$ 10.62	56.18 $\pm$ 10.98	54.60 $\pm$ 7.43	56.99 $\pm$ 4.28

Appendix

Table 10: Full-orbit accuracy (%) at group order

8

without training augmentation, without and with the disc of Section 4.1.3 , for GCNN with three activations and for SGE-Net . Mean and

95\%

confidence interval over four seeds. These are training runs separate from those of Tables 2 and 3 , so the entries with the disc differ from those tables by less than

0.7

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