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Signal-Noise Factorization Isolates Nuisance Variation into Removable Subspaces

Sakin Kirti, Joel Zylberberg

arXiv:2610.00751v1cs.LGcs.CVstat.ML

TL;DR

The paper investigates how hidden-representation geometry affects generalization, targeting signal-signal and signal-noise factorization with training regularizers. Across its classification evaluations, SNF improves performance under out-of-distribution distortion while SSF has little or no benefit. The authors link the SNF result to separating nuisance variation from signal-carrying directions.

  • Problem

    The paper asks how specific geometric properties of hidden representations shape downstream task performance, focusing on signal-signal and signal-noise factorization.

    Source: §1 · p. 1

  • Method

    The authors apply separate SSF and SNF regularizers during training and examine their effects on representation geometry and classification performance.

    Source: §1 · p. 2

  • Results

    SNF regularization improves categorization on out-of-distribution distortions while SSF has little impact; on BloodMNIST corruptions, L2+SNF reaches 61.03±0.83% versus 57.93±0.96% for L2 alone.

    Source: §1 · p. 2, §5.2 · p. 7

  • Takeaways & Limitations

    Explicit signal-noise factorization can be targeted to improve out-of-distribution robustness, with nuisance variation separated from signal-carrying directions.

    Source: §6 · pp. 8–9

    Experiments cover only two datasets and one architecture, ResNet-18, so the authors cannot say whether the effects hold for substantially different architectures.

    Source: §6.2 · p. 9

Abstract

from arXiv · show

Recent theoretical work identified fundamental properties of representation geometry that shape inference ability of deep neural networks. These include signal-noise factorization (SNF), the ability to segregate signal from noise, and signal-signal factorization (SSF), the ability to segregate task-specific and task-irrelevant signals. Here, we built regularizers that reinforce these two properties during training. We compared networks trained with these regularizers to $L_2$-regularized baseline networks on the CIFAR-100 classification task to understand how our regularizers shape representation geometry and impact performance on a well-known computer vision baseline. Enhancing SNF via regularization improved model performance but enhancing SSF did not. Motivated by biomedical applications, we investigated how our regularizers affected performance on the BloodMNIST dataset treated with MedMNIST-C corruptions at five severity levels, and found even larger performance gains using the SNF regularizer. To understand the mechanism by which SNF-regularization produces improved performance, we analyzed the nuisance subspaces across regularization regimes, finding that the SNF-regularized models represent noise in distinct subspaces, separate from class-relevant signal. Because this geometry is explicit, the dominant corruption-induced directions can be estimated on held-out data and projected out of the representations. This manipulation led to a substantial gain in accuracy. These results show that regularizers that enforce signal-noise factorization can produce substantial improvements on computer vision tasks that contain out-of-distribution image distortions at inference time. They also highlight how shaping representations affects model performance: isolating nuisance variables from categorical ones is more important than maintaining factorized representations of categorical variables.

1 INTRODUCTION

The paper asks how specific properties of hidden-representation geometry shape task performance, targeting signal-signal and signal-noise factorization with training regularizers. It contrasts this intervention with prior diagnostic or observational approaches and reports that SNF, unlike SSF, improves categorization under out-of-distribution distortions.

1 INTRODUCTION

  • The study asks how hidden-representation structure shapes downstream performance, focusing on SSF, which segregates task-specific from task-irrelevant signals, and SNF, which separates signal from noise.These geometric properties were identified by prior theoretical work as critical to generalization.Source: §1 · p. 1
  • SNF regularization improves categorization on out-of-distribution distortions by segregating distortion effects from class-relevant signal, while SSF has little impact on categorization.The authors compare both targeted regularizers on CIFAR-100 and BloodMNIST against unregularized and L2-regularized baselines.Source: §1 · p. 2

1.1 RELATED WORK

  • Unlike diagnostic methods that describe representation contents, targeted training regularizers manipulate structural properties to examine their effects on readout accuracy.The paper presents its approach as complementary to probing classifiers and other post-training representation analyses.Source: §1.1 · p. 2
  • Prior activation-statistics and domain-adaptation methods reshape representations for downstream objectives, whereas these regularizers target interpretable geometry to study its link to performance.The paper also distinguishes its intervention from observational studies that correlate weight-decay-induced geometry with performance.Source: §1.1 · p. 2

2 FACTORIZATION-BASED REGULARIZERS

The paper targets two representation properties with separate regularizers: SSF shapes class-to-class geometry, while SNF suppresses residual variance in signal-coding directions. The factorization measures relate signal power to class structure or nuisance variance.

2 FACTORIZATION-BASED REGULARIZERS

  • The factorization framework focuses on f (SSF) and s (SNF), motivated by a generalization bound whose other terms become asymptotically unimportant as training-set size grows.The schematic contrasts entangled unregularized representations, orthogonal class means under SSF, and nuisance variation displaced from class-signal directions under SNF.Source: §2 · p. 3, Figure 1 · p. 3

2.1 PRELIMINARIES

  • The s measure compares total signal power with residual noise after the class-predictive component is projected out, while f measures how signal patterns across classes align with class structure.Higher s indicates that class-signal information is well-separated from nuisance variance; the column space of Φ is the signal subspace.Source: §2.1 · p. 3

2.2 SIGNAL-SIGNAL FACTORIZATION (SSF)

  • SSF penalizes deviation of the class-to-class covariance Φ^TΦ from a scaled identity, encouraging class representations to be orthogonal and equal-norm.This structure is intended to represent each class independently.Source: §2.2 · p. 3

2.3 SIGNAL-NOISE FACTORIZATION (SNF)

  • SNF minimizes residual activation variance not explained by the signal subspace, suppressing noise-subspace variance within signal-coding directions.The inverse label-covariance term is computed per batch rather than assuming perfectly balanced sampling.Source: §2.3 · p. 4

3 MODEL TRAINING AND ARCHITECTURE

The experiments use CIFAR-100 and BloodMNIST to evaluate geometric regularizers across natural-image and biomedical classification settings. Both use ResNet-18 models with regularization strengths selected by validation performance.

3 MODEL TRAINING AND ARCHITECTURE

  • The evaluation spans CIFAR-100, a standard computer-vision benchmark, and BloodMNIST, a biomedical classification task with substantial inter- and intra-class nuisance variation.Source: §3 · p. 4
  • Both datasets use ResNet-18 with cross-entropy loss and geometric regularizers applied to the penultimate layer.Source: §3 · p. 4
  • Regularization strengths are selected by joint validation sweeps, with unregularized and L2-only baselines included for comparison.The baseline L2 strength is selected by the same procedure.Source: §3 · p. 4

4 MODEL AND REGULARIZER EVALUATION

The evaluation measures representation geometry, test performance, and corruption robustness, then probes whether corruption-induced variation occupies removable representation subspaces. The analysis uses CIFAR-100 and BloodMNIST evaluations described in the supplied methods.

4 MODEL AND REGULARIZER EVALUATION

  • The evaluation examines how regularizers affect model geometry and performance, including whether nuisance variation can be isolated in a subspace.Source: §4 · p. 4

4.1 MODEL GEOMETRY

  • The geometry analysis computes Wakhloo’s s and f metrics on penultimate activations to assess whether CIFAR-100 models achieve the regularizers’ targeted properties.Source: §4.1 · p. 4

4.2 MODEL PERFORMANCE

  • Performance testing measures clean test accuracy, opposite-strength augmentation transfer on CIFAR-100, and BloodMNIST performance across corruption types and severity levels.The figure describes panels for f, s, CIFAR-100 accuracy, and cross-strength augmentation evaluation.Source: §4.2 · p. 4, Figure 2 · p. 5

4.3 MECHANISTIC SUBSPACE ANALYSIS

  • The mechanistic analysis extracts clean and corrupted BloodMNIST representations, uses top-50 SVD directions of corruption perturbations, and tests projection and cross-corruption transfer.Projection dimensions are swept from 0 to 50; the transfer matrix applies each source-corruption subspace to the other corruption types.Source: §4.3 · pp. 4–5

5 RESULTS

SNF regularization improved performance on corrupted BloodMNIST and CIFAR-100, while SSF did not improve generalization. Mechanistic analyses found that SNF separates corruption-induced variation from the signal subspace, allowing a validation-estimated subspace projection to improve test accuracy.

5.1 REPRESENTATION GEOMETRY AND CIFAR-100 PERFORMANCE

  • L2+SNF improved accuracy over no regularization on CIFAR-100 and achieved the best overall accuracy with L2, whereas SSF lowered accuracy versus its unregularized and L2-paired baselines.SSF produced the largest increase in the geometric measures f and s, while SNF produced an intermediate increase.Source: §5.1 · pp. 5–6

5.2 REGULARIZERS APPLIED TO CELL TYPE CLASSIFICATION

  • 61.03±0.83% accuracy with L2+SNF versus 57.93±0.96% with L2 and 57.27±0.91% with L2+SSF across 11 corruptions and 5 severities, with no measurable clean-accuracy cost.The comparison covers BloodMNIST test images under MedMNIST-C-style corruptions.Source: Figure 3 · p. 6, §5.2 · p. 7

5.1 REPRESENTATION GEOMETRY AND CIFAR-100 PERFORMANCE

  • Under weak-to-strong augmentation transfer, L2+SNF models degraded least; strong-to-weak transfer produced similar 1–2% accuracy drops across regularizers.The transfer evaluation used separate weak- and strong-augmentation models, with n = 3 models per condition.Source: §5.1 · p. 6

5.3 MECHANISTIC SUBSPACE ANALYSIS

  • 69.3±0.8% test accuracy after projecting out 10 validation-estimated directions, up from 60.9±0.6% without projection in the L2+SNF model.The projection used a subspace estimated on an independent validation split; analogous projections degraded accuracy for L2 and L2+SSF models.Source: Figure 4 · p. 7, §5.3 · p. 8

5.2 REGULARIZERS APPLIED TO CELL TYPE CLASSIFICATION

  • L2+SNF gains reached 5.7% on defocus blur, with largest gains concentrated on blur and photometric corruptions; L2+SSF reduced performance across corruption types.SNF was also associated with modest sensitivity improvements across classes and variable specificity changes.Source: Figure 3 · p. 6, §5.2 · p. 7

5.3 MECHANISTIC SUBSPACE ANALYSIS

  • SNF shifted corruption-induced variance out of the signal subspace: 16.3% remained within it and 83.7% fell in its orthogonal complement, where the variance occupied a higher-rank subspace.L2 and L2+SSF retained majorities of corruption-induced variance within the signal subspace; the higher rank under SNF reflects separation, not global noise suppression.Source: §5.3 · p. 8

6 CONCLUSION

SNF regularization produced meaningful gains, particularly under distribution shift, while SSF did not yield consistent benefits. The conclusion connects separating nuisance variation from signal to potential robustness in nuisance-heavy settings, while noting the limited evaluation scope and open research directions.

6 CONCLUSION

  • SNF regularization produced meaningful gains, particularly under distribution shift, while SSF did not yield consistent benefits; orthogonalizing nuisance variation enabled recovery on out-of-distribution samples.The authors attribute the recovery to nuisance variation being separated from signal-carrying directions.Source: §6 · pp. 8–9

6.1 BROADER IMPACTS

  • Separating nuisance variation from diagnostic signal may help address hospital-specific scanner and staining artifacts that can impair cross-hospital generalization.The authors frame this as a potential path toward diagnostic tools that generalize more reliably across clinical sites.Source: §6.1 · p. 9

6.2 LIMITATIONS AND FUTURE WORK

  • Evaluation is limited to a small set of regularizers, ResNet-18, and two datasets, so broader validation across datasets, modalities, and shift types remains needed.The authors note that CIFAR-100 and BloodMNIST represent different domains, providing initial evidence of generality.Source: §6.2 · p. 9
  • Future work could test joint SSF and SNF regularization, extend SNF to self-supervised or contrastive pretraining, and analyze L2’s distinct mechanisms.The authors observed coupling between SSF and SNF penalties even when applied independently.Source: §6.2 · p. 9

AI USE STATEMENT

The authors used generative AI tools to assist with code and manuscript editing, but not draft writing or ideation; they reviewed the assisted work and checked AI-generated code.

AI USE STATEMENT

  • Claude was used to generate and refine code and to edit manuscript text and figures; literature search was aided by undermind.ai, and AI-assisted code was code-reviewed before functional validation.The authors report no AI assistance with draft writing or ideation and take responsibility for the final content.Source: AI USE STATEMENT · p. 9

ETHICS STATEMENT

The authors report no human-subject research, sensitive-domain applications, or identified concerns related to bias, fairness, privacy, or security.

ETHICS STATEMENT

  • The work involved no human subjects, user studies, dataset releases, or sensitive-domain applications, and the authors identify no significant ethical concerns beyond standard machine learning research.They also report no relevant conflicts of interest or external sponsorship.Source: ETHICS STATEMENT · p. 9

REPRODUCIBILITY STATEMENT

The paper describes its experimental materials and methods and states that code will be made available.

REPRODUCIBILITY STATEMENT

  • The paper points to appendices for architecture, training, statistical reporting, and corruption-protocol details, and states that experiment and figure reproduction code will be linked upon acceptance or rejection.Experiments used three seeds; the statistical reporting approach is described in Appendix A.2.2.Source: REPRODUCIBILITY STATEMENT · p. 10

A.1 EXTENDED RESULTS AND FIGURES

The appendix provides extended figure guides and implementation details for CIFAR-100 and BloodMNIST, alongside protocols for corruption and subspace analyses.

A.1 EXTENDED RESULTS AND FIGURES

  • Figures A.1–A.3 cover regularizer-loss terms across regimes, classwise BloodMNIST sensitivity and specificity, and performance after projecting corruption subspaces across source and target types.In Figure A.3, rows identify the corruption projected out and columns the corruption tested.Source: Figure A.3 · p. 13, Figure A.1 · p. 13, Figure A.2 · p. 13

A.2.1 ARCHITECTURE DETAILS

  • The experiments use randomly initialized ResNet-18s, apply SNF at the penultimate layer with a shared BatchNorm layer, and report three-seed means without formal statistical tests.CIFAR-100 training uses a stratified batch sampler because SNF requires every class in each batch.Source: §A.2.1 · p. 14, §A.2.2 · p. 14, §A.2.3 · p. 14

A.2.4 CIFAR-100 AUGMENTATION TEST

  • CIFAR-100 augmentation-transfer models are trained at mutually exclusive weak or strong augmentation strengths and evaluated at the opposite strength; CIFAR-100-C supplies corruption data for geometry measurements.The standard CIFAR-100 augmentations include reflect padding, random cropping, and horizontal flipping.Source: §A.2.4 · p. 14, §A.2.7 · p. 15

A.2.6 HYPERPARAMETER SEARCH

  • Hyperparameters are selected by validation accuracy through separate L2 and factorization-regularizer sweeps, with test data reserved for final evaluation.CIFAR-100 additionally evaluates SNF and SSF without L2; the appendix tables list search ranges and selected values for both datasets.Source: §A.2.6 · p. 15, Table A.1 · p. 15, Table A.2 · p. 15

A.2.8 BLOODMNIST AND MEDMNIST-C CORRUPTIONS

  • BloodMNIST contains eight pre-cropped and preprocessed blood-cell types, and its test set receives the standard MedMNIST-C corruptions at five severity levels.BloodMNIST training uses 64×64 images and a batch size of 112; the dataset has class imbalance.Source: §A.2.5 · pp. 14–15, §A.2.8 · p. 15

A.2.9 MECHANISTIC SUBSPACE ANALYSIS

  • The subspace analysis extracts 512-dimensional penultimate-layer representations across clean data and 55 corruption variants, then uses a centered perturbation matrix and truncated SVD to estimate leading noise directions.The reported effective rank and captured variance describe only the retained top-50 perturbation components.Source: §A.2.9 · pp. 15–16
  • The analysis also decomposes corruption-induced variance into components inside and outside the signal subspace.Individual-sample perturbations are used for the decomposition.Source: §A.2.9 · p. 16
  • For recovery, top-k singular-vector directions are projected out before classification with a frozen head; a separate 11 × 11 matrix tests each corruption’s subspace on every target corruption.Matched-subspace entries measure within-corruption recovery, while off-diagonal entries measure cross-corruption recovery.Source: §A.2.9 · pp. 16–17
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