Sparse Autoencoders Can Induce Local Geometry for Mitigating Catastrophic Forgetting

Kamil Faber; Katarzyna Filus; Łukasz Pawela

cs.LGOct 15, 2026

Sparse Autoencoders Can Induce Local Geometry for Mitigating Catastrophic Forgetting

Authors: Kamil Faber, Katarzyna Filus, Łukasz Pawela

Organizations: Faculty of Computer Science, AGH University of Krakow, Krakow, Poland · Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Gliwice, Poland

Abstract

Catastrophic forgetting remains a challenge in supervised continual learning, as adapting to new tasks causes representation drift and loss of previously learned information. This raises the question of which representational changes should be constrained while preserving plasticity. We therefore introduce Sparse Autoencoder Geometric Anchoring (SAGA), which converts a Sparse Autoencoder (SAE) trained at each task boundary into local, low-rank constraints on representation drift. Replay examples specify where the constraints apply, while leading right singular directions of the encoder Jacobian specify which changes are penalized. The resulting geometry is input dependent and can be retained without keeping the SAE for subsequent tasks' training. We establish its local relation to latent code matching, define its spectral truncation error, and derive a conditional bound on frozen-readout drift. With replay and snapshot history, the method achieves the highest mean Class-IL accuracy and lowest mean forgetting among the evaluated ER, DER++, ER-ACE, and SGD baselines on three pretrained ViT-B/16 benchmarks, with accuracy gains of 12.15, 4.79, and 3.22 percentage points over the strongest conventional baseline on CIFAR-100, CUB-200, and ImageNet-R, respectively. From scratch results are more mixed, while rank ablations and geometry controls show that SAE-induced low-rank geometry offers useful stability-plasticity trade-offs.

Figures & tables

	CIFAR-100		CUB-200		ImageNet-R
Configuration	Acc.	Forget.	Acc.	Forget.	Acc.	Forget.
Vanilla Latent	$31.58\pm 3.19$	$72.95\pm 3.64$	$37.89\pm 0.79$	$51.36\pm 0.70$	$33.76\pm 0.36$	$64.69\pm 0.25$
Vanilla Jacobian	$27.57\pm 0.95$	$78.69\pm 1.13$	$33.03\pm 0.80$	$60.94\pm 0.46$	$28.15\pm 1.09$	$71.86\pm 1.28$
Latent + History	$56.65\pm 0.46$	$45.77\pm 0.54$	$61.21\pm 1.22$	$27.77\pm 1.26$	$48.61\pm 0.29$	$49.33\pm 0.24$
Jacobian + History	$51.37\pm 1.89$	$52.12\pm 2.09$	$59.53\pm 0.37$	$31.79\pm 0.38$	$45.06\pm 1.47$	$53.00\pm 1.45$
Latent + Replay	$70.96\pm 0.21$	$25.82\pm 0.84$	$64.45\pm 1.64$	$20.96\pm 1.53$	$53.78\pm 0.19$	$42.32\pm 0.36$

Table 1: SAE anchoring ablations for Class-IL scenario with pretrained ViT-B/16. Results are mean

\pm

std over three seeds. Higher accuracy (

\uparrow

) and lower forgetting (

\downarrow

) are better.

	CIFAR-100		TinyImageNet
Configuration	Acc.	Forget.	Acc.	Forget.
Vanilla Latent	$8.03\pm 0.74$	$59.32\pm 3.53$	$6.64\pm 0.19$	$57.79\pm 3.37$
Vanilla Jacobian	$8.16\pm 1.20$	$69.50\pm 5.43$	$7.32\pm 0.45$	$61.78\pm 4.50$
Latent + History	$7.90\pm 0.34$	$58.14\pm 4.18$	$6.44\pm 0.08$	$58.85\pm 3.26$
Jacobian + History	$8.20\pm 0.52$	$66.50\pm 7.42$	$5.89\pm 0.62$	$56.40\pm 5.09$
Latent + Replay	$15.78\pm 0.29$	$55.26\pm 2.07$	$17.79\pm 0.83$	$38.55\pm 4.37$

Table 2: SAE anchoring ablations for Class-IL scenario with ResNet-18 trained from scratch . Results are mean

\pm

std over three seeds. Higher accuracy (

\uparrow

) and lower forgetting (

\downarrow

) are better.

	CIFAR-100		CUB-200		ImageNet-R
Method	Acc.	Forget.	Acc.	Forget.	Acc.	Forget.
SAGA (Jacobian + Replay + History)	$\mathbf{76.01\pm 0.46}$	$\mathbf{24.24\pm 0.43}$	$\mathbf{73.68\pm 0.79}$	$\mathbf{15.38\pm 0.92}$	$\mathbf{61.58\pm 0.51}$	$\mathbf{34.48\pm 0.40}$
DER++	$57.18\pm 2.52$	$46.04\pm 2.88$	$68.89\pm 3.17$	$19.57\pm 3.60$	$58.36\pm 2.09$	$38.42\pm 2.36$
ER-ACE	$63.86\pm 3.34$	$38.01\pm 3.78$	$56.93\pm 1.79$	$29.34\pm 2.04$	$57.09\pm 1.50$	$38.79\pm 1.62$
ER	$45.61\pm 2.54$	$58.93\pm 2.86$	$64.22\pm 3.17$	$26.10\pm 3.43$	$52.85\pm 1.22$	$44.40\pm 1.33$
SGD	$9.78\pm 0.04$	$93.77\pm 0.15$	$9.02\pm 0.48$	$77.48\pm 0.89$	$9.31\pm 0.23$	$81.94\pm 0.18$

Table 3: Class-IL accuracy and forgetting against continual learning baselines on three pretrained ViT-B/16 settings. Results are mean

\pm

std over three seeds. Higher accuracy (

\uparrow

) and lower forgetting (

\downarrow

) are better.

	CIFAR-100		TinyImageNet
Method	Acc.	Forget.	Acc.	Forget.
SAGA (Jacobian + Replay + History)	$28.53\pm 0.52$	$38.46\pm 2.64$	$28.27\pm 4.43$	$\mathbf{12.84\pm 2.24}$
DER++	$\mathbf{36.28\pm 0.24}$	$37.71\pm 1.20$	$29.90\pm 5.00$	$37.61\pm 3.96$
ER-ACE	$32.16\pm 0.99$	$\mathbf{37.41\pm 3.88}$	$\mathbf{32.37\pm 0.49}$	$29.98\pm 2.04$
ER	$18.28\pm 0.39$	$71.89\pm 0.64$	$25.13\pm 0.49$	$52.09\pm 0.34$
SGD	$8.37\pm 0.71$	$68.55\pm 3.87$	$6.84\pm 0.72$	$53.83\pm 6.67$

Table 4: Class-IL accuracy and forgetting against continual learning baselines with a ResNet-18 trained from scratch . Results are mean

\pm

std over three seeds. Higher accuracy (

\uparrow

) and lower forgetting (

\downarrow

) are better.

	CIFAR-100		ImageNet-R		CUB-200
Configuration	Class-IL	Task-IL	Class-IL	Task-IL	Class-IL	Task-IL
Jacobian ( $r=4$ ) + Replay + History	$75.55\pm 0.56$	$97.45\pm 0.09$	$59.41\pm 0.35$	$88.44\pm 0.36$	$70.25\pm 1.35$	$85.76\pm 0.52$
Jacobian ( $r=8$ ) + Replay + History	$75.71\pm 0.33$	$97.43\pm 0.17$	$60.63\pm 0.76$	$88.55\pm 0.48$	$71.87\pm 0.49$	$85.92\pm 0.67$
Jacobian ( $r=16$ ) + Replay + History	$76.48\pm 0.28$	$97.41\pm 0.04$	$61.64\pm 0.63$	$88.45\pm 0.42$	$74.06\pm 0.82$	$86.23\pm 0.34$
Jacobian ( $r=32$ ) + Replay + History	$78.08\pm 0.36$	$97.50\pm 0.02$	$63.94\pm 0.16$	$88.87\pm 0.24$	$74.52\pm 0.61$	$86.30\pm 0.27$
Jacobian ( $r=64$ ) + Replay + History	$79.45\pm 0.11$	$97.43\pm 0.13$	$64.34\pm 0.56$	$89.02\pm 0.10$	$74.77\pm 0.34$	$86.01\pm 0.74$

Table 5: Effect of the retained Jacobian rank

r

for a TopK SAE with

K=64

on accuracy, with normalized anchor weights (constant nominal penalty strength across

r

) and pretrained ViT-B/16. Results are mean

\pm

std over three seeds. Higher results are better (

\uparrow

	CUB-200 ( $r=16$ )		ImageNet-R ( $r=16$ )		CIFAR-100 ( $r=8$ )
Reference	Acc.	Forget.	Acc.	Forget.	Acc.	Forget.
Random	$70.68\pm 0.52$	$19.05\pm 0.84$	$61.11\pm 0.80$	$34.78\pm 1.03$	$71.69\pm 1.07$	$29.11\pm 1.12$
PCA	$71.64\pm 0.92$	$17.71\pm 1.12$	$61.37\pm 0.25$	$34.95\pm 0.30$	$74.98\pm 0.55$	$25.45\pm 0.62$
Classifier SVD	$67.27\pm 1.54$	$22.66\pm 1.67$	$56.69\pm 0.25$	$39.81\pm 0.52$	$\mathbf{77.34\pm 1.01}$	$\mathbf{22.73\pm 0.97}$
SAE Jacobian	$\mathbf{72.55\pm 1.19}$	$\mathbf{16.64\pm 1.40}$	$\mathbf{61.44\pm 0.30}$	$\mathbf{34.67\pm 0.21}$	$76.08\pm 0.64$	$24.16\pm 0.75$

Table 6: Class-IL accuracy and forgetting for different reference geometries on pretrained ViT-B/16. All variants use normalized anchor weighting. Results are mean

\pm

std over three seeds. Higher accuracy (

\uparrow

) and lower forgetting (

\downarrow

) are better.

	Random		PCA		Classifier SVD		SAE Jacobian
Rank $r$	Acc.	Forget.	Acc.	Forget.	Acc.	Forget.	Acc.	Forget.
4	$20.83\pm 0.39$	$51.76\pm 2.74$	$19.81\pm 2.30$	$\mathbf{32.57\pm 7.77}$	$21.99\pm 1.55$	$42.86\pm 6.33$	$\mathbf{23.63\pm 0.71}$	$46.65\pm 3.06$
8	$20.69\pm 3.96$	$46.23\pm 5.20$	$22.48\pm 0.95$	$\mathbf{36.46\pm 7.52}$	$22.52\pm 1.26$	$43.63\pm 8.14$	$\mathbf{24.88\pm 1.85}$	$37.74\pm 2.69$
16	$23.39\pm 1.07$	$45.26\pm 2.44$	$\mathbf{27.71\pm 2.21}$	$\mathbf{27.62\pm 7.20}$	$26.29\pm 1.33$	$41.92\pm 1.19$	$27.65\pm 1.44$	$35.44\pm 4.94$
64	$23.22\pm 1.72$	$45.73\pm 1.71$	$19.07\pm 8.06$	$\mathbf{28.81\pm 6.75}$	$24.39\pm 5.52$	$40.52\pm 5.01$	$\mathbf{26.19\pm 4.58}$	$33.03\pm 4.27$

Table 7: Class-IL accuracy and forgetting for different reference geometries and ranks on CIFAR-100 with ResNet-18 trained from scratch (

r=K

). All variants use normalized anchor weighting. Results are mean

\pm

std over three seeds. Higher accuracy (

\uparrow

) and lower forgetting (

\downarrow

) are better.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

	CIFAR-100		CUB-200		ImageNet-R
Configuration	Acc.	Forget.	Acc.	Forget.	Acc.	Forget.
Vanilla Latent	$89.34\pm 0.30$	$9.07\pm 0.42$	$74.97\pm 0.29$	$10.30\pm 0.32$	$84.29\pm 0.62$	$8.65\pm 0.45$
Vanilla Jacobian	$93.10\pm 0.59$	$5.96\pm 0.67$	$79.23\pm 1.49$	$9.62\pm 1.38$	$84.00\pm 0.53$	$9.83\pm 0.65$
Latent + History	$95.66\pm 0.12$	$2.79\pm 0.15$	$82.28\pm 0.81$	$4.61\pm 0.83$	$87.33\pm 0.60$	$6.41\pm 0.45$
Jacobian + History	$96.57\pm 0.13$	$2.00\pm 0.12$	$84.21\pm 0.22$	$4.49\pm 0.26$	$87.30\pm 0.22$	$6.15\pm 0.53$
Latent + Replay	$94.59\pm 0.37$	$3.24\pm 0.37$	$81.01\pm 0.43$	$4.35\pm 0.53$	$86.66\pm 0.28$	$6.27\pm 0.14$

Appendix

Table 8: Task-IL accuracy and forgetting for SAE anchoring ablations with a pretrained ViT-B/16. Results are mean

\pm

std over three seeds.

	CIFAR-100		TinyImageNet
Configuration	Acc.	Forget.	Acc.	Forget.
Vanilla Latent	$59.03\pm 4.49$	$9.12\pm 2.16$	$60.86\pm 1.28$	$1.78\pm 0.24$
Vanilla Jacobian	$54.51\pm 8.59$	$21.82\pm 3.10$	$47.29\pm 5.19$	$22.41\pm 2.19$
Latent + History	$65.88\pm 2.20$	$2.89\pm 0.63$	$61.34\pm 0.66$	$2.01\pm 0.19$
Jacobian + History	$68.50\pm 5.34$	$5.79\pm 0.92$	$59.19\pm 3.94$	$3.47\pm 0.44$
Latent + Replay	$67.19\pm 1.47$	$9.21\pm 2.46$	$56.24\pm 2.06$	$7.94\pm 1.43$

Appendix

Table 9: Task-IL accuracy and forgetting for SAE anchoring ablations with a ResNet-18 trained from scratch.

	CIFAR-100		CUB-200		ImageNet-R
Method	Acc.	Forget.	Acc.	Forget.	Acc.	Forget.
Ours	$\mathbf{97.54\pm 0.10}$	$\mathbf{1.03\pm 0.18}$	$\mathbf{86.38\pm 0.34}$	$\mathbf{2.42\pm 0.47}$	$\mathbf{88.76\pm 0.39}$	$\mathbf{4.66\pm 0.23}$
DER++	$91.64\pm 0.91$	$7.90\pm 0.87$	$83.14\pm 0.98$	$5.30\pm 1.03$	$85.77\pm 0.49$	$8.42\pm 0.53$
ER-ACE	$93.90\pm 1.10$	$5.11\pm 1.24$	$77.92\pm 1.46$	$7.83\pm 1.65$	$84.10\pm 0.38$	$9.97\pm 0.40$
ER	$95.16\pm 0.33$	$3.94\pm 0.39$	$83.01\pm 0.75$	$5.76\pm 0.79$	$86.00\pm 0.58$	$7.82\pm 0.68$
SGD	$74.31\pm 1.13$	$22.07\pm 1.37$	$39.48\pm 1.03$	$43.63\pm 1.21$	$53.57\pm 1.96$	$32.78\pm 2.16$

Appendix

Table 10: Task-IL accuracy and forgetting against continual learning baselines on pretrained ViT-B/16.

	CIFAR-100		TinyImageNet
Method	Acc.	Forget.	Acc.	Forget.
Ours	$74.35\pm 0.43$	$\mathbf{3.93\pm 0.90}$	$61.84\pm 3.38$	$\mathbf{3.04\pm 1.34}$
DER++	$\mathbf{75.15\pm 0.24}$	$8.03\pm 1.14$	$66.08\pm 4.69$	$8.81\pm 3.47$
ER-ACE	$70.00\pm 0.56$	$12.34\pm 3.40$	$61.10\pm 0.55$	$12.88\pm 1.09$
ER	$68.13\pm 0.86$	$17.28\pm 0.71$	$\mathbf{66.09\pm 0.31}$	$9.65\pm 0.44$
SGD	$32.49\pm 3.01$	$41.89\pm 2.35$	$17.13\pm 2.25$	$42.96\pm 5.99$

Appendix

Table 11: Task-IL accuracy and forgetting against continual learning baselines with a ResNet-18 trained from scratch.

Dictionary size	Expansion	Class-IL	Task-IL
256	$0.5\times$	$\mathbf{28.53\pm 0.52}$	$\mathbf{74.35\pm 0.43}$
512	$1\times$	$26.67\pm 0.87$	$71.29\pm 0.63$
1024	$2\times$	$24.66\pm 5.33$	$69.49\pm 4.39$
2048	$4\times$	$28.07\pm 1.41$	$71.80\pm 1.09$
4096	$8\times$	$28.10\pm 2.15$	$71.51\pm 1.22$

Appendix

Table 12: Effect of SAE dictionary size on accuracy for CIFAR-100 with a ResNet-18 trained from scratch, at fixed

K=r=16

. Expansion factor is relative to the backbone’s

512

-dimensional feature space. Results are mean

\pm

std over three seeds. Higher results are better (

\uparrow

Dataset	Classes	Tasks	Train / Test images	Input	Backbone
CIFAR-100	100	$10\times 10$	50,000 / 10,000	$32\times 32$	ResNet-18 (scratch)
TinyImageNet	200	$10\times 20$	100,000 / 10,000	$64\times 64$	ResNet-18 (scratch)
CIFAR-100	100	$10\times 10$	50,000 / 10,000	$224\times 224$	ViT-B/16 (pretrained)
ImageNet-R	200	$10\times 20$	24,000 / 6,000	$224\times 224$	ViT-B/16 (pretrained)
CUB-200	200	$10\times 20$	5,994 / 5,794	$224\times 224$	ViT-B/16 (pretrained)

Appendix

Table 13: Datasets. “Tasks” is the number of tasks

\times

classes per task. Input: image size seen by the network. Train-time augmentation is listed in the text; test images are only resized (and, where stated, center-cropped) and normalized.

Setting	Backbone	lr	Anchor weight	Epochs/task	Buffer	Optimizer	Batch
CIFAR-100	ViT-B/16 (pretrained)	$3\times 10^{-5}$	$3.0$	$10$	$500$	Adam	$64$
ImageNet-R	ViT-B/16 (pretrained)	$3\times 10^{-5}$	$0.3$	$10$	$500$	Adam	$64$
CUB-200	ViT-B/16 (pretrained)	$3\times 10^{-5}$	$0.3$	$10$	$500$	Adam	$64$
CIFAR-100	ResNet-18 (scratch)	$0.1$	$0.3$	$20$	$500$	SGD	$64$
TinyImageNet	ResNet-18 (scratch)	$0.03$	$1.0$	$20$	$5120$	SGD	$64$

Appendix

Table 14: Tuned hyperparameters and fixed training settings for our method SAGA (Jacobian + Replay + History), by setting.

Backbone; datasets	$D$	$m$	$K$	$r$	$N$	SAE params
ViT-B/16; CIFAR-100, CUB-200, ImageNet-R	768	256	16	16	500	394k
ResNet-18; CIFAR-100	512	256	16	16	500	263k
ResNet-18; TinyImageNet	512	256	16	16	5120	263k

Appendix

Table 15: SAE hyperparameters for the main results.

D

: feature dimension;

m

: dictionary size;

K

: TopK sparsity;

r

: retained Jacobian rank;

N

: buffer samples per snapshot (SAE training set and anchors); SAE parameters:

2Dm+m+D

. The training hyperparameters (Adam, learning rate

10^{-3}

20

epochs, batch size

64

) are identical in all settings.

Setting	Ours (s/task)	SGD (s/task)	Ratio
CUB-200 (ViT-B/16, pretrained)	$114.3$	$16.6$	$6.9\times$
ImageNet-R (ViT-B/16, pretrained)	$429.8$	$67.0$	$6.4\times$
CIFAR-100 (ViT-B/16, pretrained)	$904.2$	$138.6$	$6.5\times$
CIFAR-100 (ResNet-18, scratch)	$82.6$	$12.9$	$6.4\times$
TinyImageNet (ResNet-18, scratch)	$302.2$	$42.8$	$7.1\times$

Appendix

Table 16: Mean wall-clock time per task (seconds), our method vs. SGD, across the five settings with timing instrumentation. Ratio = Ours / SGD.

Catastrophic Forgetting

Catastrophic forgetting is the loss of previously learned behavior when a model adapts to new tasks or changing data.

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Aug 16, 2026

Geometry of Forgetting: Representation Flux in Continual Learning

Catastrophic forgetting remains a fundamental obstacle to continual learning, where neural networks lose previously acquired knowledge while learning new tasks. Existing methods primarily mitigate forgetting through parameter regularization or experience replay, while the representation-space dynamics associated with forgetting remain less understood. We investigate latent representation evolution during sequential learning and introduce representation flux, a geometric measure of sample-level representation displacement across training. We show that representation flux is strongly associated with catastrophic forgetting across multiple benchmarks, with temporal analyses indicating that elevated flux can precede subsequent performance degradation. Representation displacement is also associated with confidence degradation, while complementary geometric properties provide additional information about sample-level forgetting. Motivated by these observations, we propose FlowLess-R, a representation-space regularization method that constrains replay representations relative to stored references while allowing continued learning. FlowLess-R is architecture-agnostic and integrates into replay-based methods through a representation-matching term. Experiments on SplitMNIST, SplitFashionMNIST, SplitCIFAR10, and SplitTinyImageNet show improved final average accuracy and reduced forgetting with ER, DER++, and ER-ACE. Our results identify representation flux as an informative geometric marker of forgetting and show that stabilizing latent representations provides a simple strategy for mitigating catastrophic forgetting.

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Continual learning studies how models can adapt to new tasks while retaining previously acquired knowledge. Although a broad spectrum of methods has been proposed to mitigate catastrophic forgetting, the field remains predominantly performance-driven, with limited insight into what forgetting actually corresponds to within the vision model's representation space. Prior work has primarily analyzed forgetting through task-level performance or coarse measures of representational drift, without disentangling output-level accessibility from changes in finer-grained internal structure. To this end, we propose a diagnostic framework that leverages Sparse Autoencoders (SAEs) to define a task-anchored latent feature space, enabling analysis of how task-specific information evolves at a finer granularity, where individual SAE latents are treated as concept proxies for recurring and relatively disentangled visual patterns in the model's internal computations. Within this framework, we decompose forgetting into apparent concept deletion, recoverability, and decodability. We show that a large portion of seemingly lost concept-level information can often be recovered under linearity assumption, with concept decodability degrading as more tasks are introduced. Overall, our findings suggest that a significant part of concept-level forgetting can be attributed to changes in the representational accessibility rather than complete information erasure.

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The Stable Recovery Manifold: Geometric Principles Governing Recoverability in Continual Learning

Catastrophic forgetting is often viewed as the destruction of previously learned knowledge during sequential learning. Building on the Accessibility Collapse framework, we investigate the geometric structure of recoverability in continual learning. Using Split CIFAR-100 and a sequentially trained ResNet-18, we analyze recoverability, representational drift, and recovery complexity across ten tasks. We introduce Recovery Subspace Dimensionality (k_t), a measure of the minimum number of singular directions required to preserve 90 percent of full probe performance. Contrary to our Recoverability Diffusion hypothesis, recovery dimensionality remains stable throughout training (mean k_t = 8.0) despite substantial representational drift. Principal-angle drift strongly predicts recoverability (r = -0.862), and a simple geometric model explains 82.2 percent of recoverability variance. These findings support the Stable Recovery Manifold hypothesis, suggesting that forgotten knowledge remains compactly decodable despite representational reorganization. The results indicate that catastrophic forgetting is primarily an accessibility and manifold-alignment problem rather than information destruction.

Ayushman Trivedi, Bhavika Melwani

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Abstract

Catastrophic Forgetting

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