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Specific Algorithmic Interpretability of Neural Networks: A Case Study on Textures

Yanglin Zhang, Anneke von Seeger, Gilad Lerman, Ron Levie

arXiv:2610.04413v1cs.LGcs.AI

TL;DR

A trained network’s parameters may not retain a clear relation to the algorithm motivating its architecture, complicating interpretation of its computation. The paper initializes networks with an explicit algorithm for estimating class-distinguishing process statistics, then fine-tunes while controlling parameter movement. In texture classification, the fine-tuned scattering network outperforms all baselines when training data is extremely limited.

  • Problem

    A trained network’s computation can be difficult to understand, and fine-tuning an algorithmically motivated network can move its parameters away from the algorithmic initialization.

    Source: §1 · p. 1, §2.4 · p. 5

  • Method

    The framework initializes a network to estimate process statistics and fine-tunes it on real data while restricting its distance from that interpretable initialization.

    Source: §2.4 · p. 5, §3.2 · p. 7

  • Results

    The fine-tuned scattering network outperforms all baseline methods when training data is extremely limited.

    Source: §4 · p. 8

  • Takeaways & Limitations

    The framework offers an interpretable algorithm as an initialization for training over-parameterized networks in low-data regimes without large external-dataset pretraining.

    Source: §5 · pp. 9–10

    The framework is mainly suited to classification; extending it to generative models and other data modalities remains future work.

    Source: §5 · p. 10

Abstract

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We develop a principled framework for constructing neural networks whose specific parameter realizations admit an explicit algorithmic interpretation. Existing algorithm-inspired architectures can explain the computational structure of a network, yet after standard training the learned parameters need not retain a clear relation to the motivating algorithm. We address this gap as follows. First, we model each data point as a sample of a class-dependent stochastic process and assume that statistics of this process can be estimated from a single sample and these statistics are sufficient to distinguish the classes. We then construct a neural network whose initial parameters exactly implement an algorithm for estimating these statistics, making the network fully interpretable. To account for mismatch between the idealized model and real data, we fine-tune this network while controlling its deviation from the algorithmic initialization. The trained network hence roughly retains the interpretation of the initial network. A PAC-Bayesian analysis yields a uniform generalization bound whose complexity term scales with the fine-tuning radius, providing a statistical motivation for our approach. We instantiate the framework for texture classification using the scattering transform to estimate the discriminative statistics.

1 Introduction

The paper distinguishes an algorithm-inspired architecture from a particular trained network whose parameters retain an explicit algorithmic interpretation. It proposes constructing such networks from statistics of class-dependent stochastic processes and fine-tuning them while controlling parameter movement.

1 Introduction

  • Specific algorithmic interpretability links a particular parameter realization to an explicit algorithm, beyond explaining only a network’s architecture.The paper frames this as a way to understand the trained network’s decision process.Source: §1 · p. 2
  • A network initialized to estimate class-distinguishing statistics of an underlying process gives its parameters an explicit algorithmic interpretation.The data model treats each observation as a realization of a class-dependent stochastic process.Source: §1 · p. 2
  • Unconstrained training can erase an algorithmically motivated initialization’s relation to its algorithm; the paper addresses this gap with controlled fine-tuning and a PAC-Bayesian analysis.The introduction presents the bound as motivation for low-data training.Source: §1 · p. 2

1.1 Related Work

  • Algorithm-unfolding and related work explain computational structures shared across parameterizations, while the particular learned parameters may remain unexplained.The paper contrasts this generic interpretation with its parameter-specific target.Source: §1.1 · p. 2
  • Neural collapse describes class-concentrated, class-separated representations, and the paper uses it as a guiding objective for its initial algorithm.The cited work discusses neural collapse as a limiting behavior of training.Source: §1.1 · pp. 2–3
  • Wavelet scattering supplies multiscale statistics for stationary images, with windowed coefficients serving as estimates from a single realization.The paper uses these statistics to construct an explicit feature extractor for texture classification.Source: §1.1 · p. 3

2 Specific algorithmic interpretability via a model of perception

The framework starts from an explicit feature computation intended to organize classes, and models observations as samples from class-dependent processes whose distinguishing statistics can be estimated from one observation. It preserves the algorithmic reference during learning by restricting parameter movement, which also controls a generalization bound under a stability condition.

2.1 From neural collapse to data disentangling

  • The feature-extractor objective is to concentrate same-class representations and separate different classes, using an explicit computation rather than an opaque learned transformation.A data disentangler applies the same transformation independently to each observation, so class information must be recoverable from that observation.Source: §2.1 · pp. 3–4

2.2 A transparent stochastic data model

  • The stochastic model assumes class-separating process statistics that can be estimated from a single realization under ergodicity.The class-conditional processes and within-class variation are modeled explicitly.Source: §2.2 · p. 4
  • The stochastic pattern recognition hypothesis treats recognition as inferring, from one observation, statistics that distinguish its generating process.The paper conjectures that feature extractors may progressively estimate more discriminative process statistics.Source: §2.2 · p. 5

2.3 The algorithmic alignment gap

  • The algorithmic alignment gap is that a trained network’s architecture may retain an algorithmic template even when its learned parameters no longer implement that algorithm.Bridging the gap requires controlling parameter changes during learning.Source: §2.3 · p. 5

2.4 Bridging the algorithmic alignment gap: fine tuning

  • Fine-tuning starts from the interpretable realization and limits parameter movement so real-data adaptation can compensate for mismatch with the idealized model.The initialization need not itself perform competitively if a good predictor can be reached nearby.Source: §2.4 · p. 5
  • Under a local stability condition, Corollary 1 gives a uniform generalization bound for networks within a radius of a data-independent initialization.The bound’s complexity scales with the adaptation radius; restricting that distance retains much of the reference interpretation.Source: §2.4 · pp. 5–6
  • A smaller adaptation radius tightens the bound and can achieve a fixed generalization-gap guarantee with fewer training samples.The radius can be controlled with a regularizer or encouraged through conservative optimization.Source: §2.4 · p. 6
  • The interpretable initialization serves as a data-independent reference point, not as a requirement that the initial network already perform competitively.The framework permits substantial freedom in constructing that initial algorithm.Source: §2.4 · p. 6

3 Specific Algorithmic Interpretability In Texture classification

The texture-classification instantiation models images as finite observations of stationary processes and selects statistics that can be estimated from those observations while retaining class-relevant distinctions. Wavelet scattering supplies such statistics, and texture synthesis is used to inspect whether they preserve perceptually meaningful information.

3.1 Textures as stationary processes with subset statistics

  • Texture images are modeled as finite-window observations of stationary random fields, with the observed domain assumed large enough to approximate ergodic statistics.A stationary process can be decomposed as a mixture of ergodic processes.Source: §3.1 · pp. 6–7
  • Because only a finite subset of process statistics is practical, its selection must balance reliable estimation from finite images with retaining distinctions relevant to texture classes.Higher-order moments can be costly, motivating structured nonlinear statistics such as scattering.Source: §3.1 · p. 7

3.2 Wavelet scattering as a data disentangler

  • Wavelet scattering coefficients average path responses over space and, at sufficiently high scale, approximate process statistics from a finite image.Texture synthesis matches selected scattering statistics to inspect whether they retain perceptually meaningful texture information.Source: §3.2 · pp. 7–8, Figure 1 · p. 8

4 Experiments

The fine-tuned scattering network performs best when training data is extremely limited and is best or competitive on the remaining datasets as data increases, except that ScatNet B.A. surpasses it on MinC-2500 as data grows. The experiments also report benefits from anchoring to the scattering prior and a relationship between fine-tuning distance and generalization gap.

4 Experiments

  • Anchoring to the scattering prior improves accuracy over either no regularization or standard weight decay, and fine-tuned filters remain wavelet-like.The section reports test accuracy and generalization gap; the ablation and interpretability analyses are provided in Section C.3.Source: §4 · p. 8
  • With extremely limited training data, the fine-tuned scattering network outperforms all baselines; as data increases, ScatNet B.A. surpasses it on MinC-2500 while it is best or competitive elsewhere.Figure 2 plots training-data percentage against test accuracy; its caption identifies the proposed method as best when training data is small.Source: §4 · pp. 8–9, Figure 2 · p. 9
  • Smaller fine-tuning distance tends to accompany a lower generalization gap, with the strongest effect on MinC-2500, the least texture-like dataset.Fine-tuning distance is the normalized L2 distance between initial and final parameters; the randomly initialized network has a higher gap than the scattering-network trend suggests.Source: §4 · p. 9, Figure 3 · p. 9

5 Conclusion

The conclusion presents controlled fine-tuning from an algorithmically interpretable initialization as a way to retain a parameter-level algorithmic reference. The accompanying related-work and technical material distinguishes this goal from structural or post-hoc approaches and reviews the wavelet scattering construction.

5 Conclusion

  • The framework ties a particular parameter realization to an explicit algorithm and uses controlled fine-tuning to measure and limit deviation from that reference.The conclusion presents interpretable algorithm initializations as a way to train highly over-parameterized networks in low-data regimes without external large-dataset pretraining, combining domain knowledge with data-driven adaptation.Source: §5 · pp. 9–10

A.1 Additional related work on interpretability

  • Unlike structural alignment or post-hoc mechanistic analysis, the focus is whether a trained network's particular parameters retain explicit algorithmic meaning.Concept- and prototype-based approaches interpret intermediate representations or examples, whereas this work seeks an algorithmic account of network computation.Source: §A.1 · p. 15

A.2 Wavelet and Scattering Preliminaries

  • Scattering cascades predefined wavelet convolutions and modulus nonlinearities along paths, then spatially averages coefficients that estimate statistics under a stationary ergodic model.The reviewed construction treats the resulting coefficients as a finite family of statistics.Source: §A.2 · p. 15

A.2.1 Oriented wavelet transform

  • The oriented wavelet transform uses a finite discrete grid in practice; its frame lower bound preserves input information, while its upper bound and the modulus make scattering non-expansive.The modulus retains responses to structures at scale and orientation while varying slowly enough for stable averaging.Source: §A.2.1 · p. 16

A.2.2 Wavelet scattering transform

  • The windowed scattering transform collects pathwise wavelet-modulus outputs; frequency-decreasing paths are used, and the transform remains non-expansive under the Littlewood–Paley condition.In the discrete setting, the path collection is finite.Source: §A.2.2 · pp. 16–17

A.2.3 Scattering coefficients as statistics of an ergodic process.

For a stationary process, each scattering path defines a process statistic, and spatial averaging of a single observed texture provides an estimate of it. The scattering representation specifies both the measured functions and the estimation procedure, including nonlinear higher-order dependencies.

A.2.3 Scattering coefficients as statistics of an ergodic process

  • For a stationary process, the expected response along each scattering path is independent of spatial location and defines a statistic of the process.The path functions form a finite choice of the functions used to define statistics in the ergodic model.Source: §A.2.3 · p. 17
  • A scattering coefficient spatially averages a path function over translations of one observed image, approximating its process expectation when the process is ergodic and the averaging window is sufficiently wide.The resulting network estimates a prescribed collection of statistics from a single finite texture image.Source: §A.2.3 · p. 17
  • Scattering coefficients capture nonlinear higher-order dependencies, unlike the power spectrum's second-order information, while specifying both the functions measured and the algorithm estimating them.The expected scattering transform is described as having coefficients given by expectations of the pathwise modulus transforms.Source: §A.2.3 · p. 17

B.1 PAC-Bayesian Analysis Around a Data-Independent Initialization

The analysis uses an algorithmically defined, data-independent initialization as a reference for fine-tuning, then derives a uniform generalization bound under local parameter stability. It also checks scattering stability and uses texture synthesis to assess whether the selected scattering statistics retain perceptually meaningful information.

B.1 PAC-Bayesian Analysis Around a Data-Independent Initialization

  • The initialization is defined independently of the training sample and serves as the reference point for predictors obtained by fine-tuning within a neighborhood.The neighborhood is parameterized by radius R around Θ0.Source: §B.1 · p. 17

B.1.3 Uniform generalization bound

  • Under local perturbation stability, the PAC-Bayesian bound holds uniformly over a neighborhood of the data-independent initialization and depends on distance from it, not its predictive performance.The uniformity applies to all parameters in the specified neighborhood.Source: §B.1 · p. 17, §B.1.3 · p. 18

B.1.1 Margin risks and PAC-Bayesian margin bound

  • The analysis starts from a PAC-Bayesian margin bound and defines statistical and empirical margin risks, with zero-margin risk counting ties as errors.The bound uses a prior independent of the training data.Source: §B.1.1 · p. 18

B.1.2 Local perturbation stability

  • Local perturbation stability verifies the perturbation requirement uniformly for predictors near Θ0, enabling the neighborhood-wide bound.The condition controls bounded parameter perturbations around the initialization.Source: §B.1.1 · p. 18, §B.1.2 · p. 18

B.2 Local Perturbation Stability of Scattering

  • The predefined scattering feature extractor is locally stable under bounded parameter perturbations, and this stability extends to a predictor composed with a linear classifier.The scattering stability result assumes non-expansive predefined wavelet and final low-pass operators.Source: §B.2 · pp. 19–21

C.1 Texture Synthesis from Scattering Statistics

  • Texture synthesis matches spatially averaged scattering statistics; RGB synthesis additionally preserves correlations between centered, standardized color channels.The procedure keeps the scattering transform fixed while optimizing the image.Source: §C.1 · p. 22
  • Synthesized textures start from independent random images, so their resemblance to references is qualitative evidence that the selected statistics retain characteristic perceptual structures.Histogram matching is applied afterward for visualization and is not part of the optimization objective.Source: §C.1 · p. 22

C.2.1 Datasets and Sampling

The evaluation uses four texture datasets and class-stratified training subsets, with held-out test data and dataset-specific minimum training sizes.

C.2.1 Datasets and Sampling

  • Training subsets extend down to 0.2% for MinC-2500, 0.5% for EuroSAT, 10% for DTD, and 2% for SuroTex.At the smallest sizes, training uses 4, 8–9, 6, and 1 image per class, respectively.Source: §C.2.1 · p. 22
  • DTD evaluation uses five of its ten standard partitions; its official test split remains unchanged while training and validation are re-split class-stratified.The other datasets receive class-stratified test splits, followed by an 80/20 training-validation split.Source: §C.2.1 · p. 22
  • Subsampling affects only training data, leaving validation and test sets fixed across training sizes; validation accuracy selects checkpoints and triggers early stopping.All reported results use held-out test data.Source: §C.2.1 · p. 22

C.2.2 Model Architectures and Baselines

The proposed network realizes the scattering transform exactly at initialization, then permits wavelet-filter adaptation and restricted cross-channel mixing during fine-tuning.

C.2.2 Model Architectures and Baselines

  • The spatial implementation cascades small wavelet-filtering blocks with subsampling to realize multiscale scattering paths.Only frequency-decreasing paths are computed.Source: §C.2.2 · p. 23
  • Restricted channel mixing expands classical scattering, which does not mix feature channels, while limiting trainable parameters and retaining the scattering tree’s local organization.Off-diagonal kernels are zero at initialization and may change during fine-tuning.Source: §C.2.2 · pp. 23–24
  • At initialization, the convolutional architecture reproduces independent wavelet filtering and the predefined scattering transform; fine-tuning improves accuracy over fixed scattering and random initialization when data is extremely limited.The architecture permits wavelet filters and selected cross-channel interactions to adapt.Source: §C.2.2 · pp. 23–24, Figure 4 · p. 24
  • The fixed baseline uses the predefined scattering realization, whereas the random baseline uses the same trainable architecture with randomly initialized convolutional parameters.ScatNet B.A. keeps spatial wavelet filters fixed and learns transformations across scattering channels.Source: §C.2.2 · pp. 24–25
  • All models use batch normalization and a single-layer linear classifier, with global average pooling applied when needed to reduce spatial dimensions.The classifier is intended to operate on features that disentangle classes.Source: §C.2.2 · p. 25

C.2.3 Training, Evaluation, and Reproducibility

The experiments compare controlled fine-tuning with several training baselines and report performance across seeds. Fine-tuned scattering performs better than random initialization, while regularization and training-data size relate to the generalization gap.

C.2.3 Training, Evaluation, and Reproducibility

  • Fine-tuning trains the classifier first, then jointly trains the feature extractor and classifier while penalizing distance from the initial scattering parameters.The penalty strength controls the fine-tuning regularization.Source: §C.2.3 · p. 25
  • Architecture and learning-rate choices are tuned by dataset-specific sweeps, with depth, mixing width, pooling, and batch normalization varying across datasets.All four reported models use 5-pixel kernels, four wavelet orientations, complex modulus, and a final low-pass filter.Source: §C.2.3 · pp. 25–26
  • Each configuration used three seeds (42, 43, 44); Figures 2 and 9 show mean test accuracy with shaded ±1 SD bands, while Figures 3, 5–7, and 10 show seed means only.Figure 8 shows seed means with ±1 SD error bars on both axes; tabular results with standard deviation are reported in Section C.3.Source: §C.2.3 · p. 26

C.3.1 Test Accuracy versus Generalization Gap

  • Increasing regularization strength decreases generalization error on DTD, MinC-2500, and SuroTex; weight decay controls the gap at the cost of test accuracy on DTD, EuroSAT, and MinC-2500.Figure 5 plots test accuracy against generalization gap; Figure 6 shows the weight-decay trade-off.Source: Figure 5 · p. 27, Figure 6 · p. 28
  • The fine-tuned scattering network improves accuracy over fixed scattering and outperforms random initialization in accuracy and generalization gap, especially with limited training data.The original scattering transform has the smallest generalization gap across all four datasets.Source: §C.3.1 · p. 27

C.3.2 Regularization Ablation

The ablation contrasts regularization around different parameter anchors, showing that distance control from the scattering initialization can reduce generalization gap at less cost to test accuracy than weight decay. The neural-collapse analysis finds that fine-tuned scattering can improve on the original transform on two datasets, while stronger regularization keeps its normalized within-class variance closer to the original.

C.3.2 Regularization Ablation

  • Weight decay reduces the randomly initialized model’s generalization gap but can drive test accuracy near 0 on DTD; fine-tuning-distance regularization reduces the gap less while better preserving accuracy.The original scattering transform does not show the same accuracy collapse as the randomly initialized model with weight decay.Source: §C.3.2 · p. 28
  • Anchoring a randomly initialized model to the scattering prior improves test accuracy over standard weight decay, while fine-tuned scattering has a lower generalization gap on DTD and SuroTex.The fine-tuned model starts from and stays closer to its own scattering initialization; the randomly initialized model instead moves toward a separate scattering anchor.Source: §C.3.2 · p. 28, Figure 7 · p. 29

C.3.3 Neural Collapse

  • On DTD and MinC-2500, fine-tuned scattering outperforms the original scattering transform on normalized within-class variance and test accuracy, although ScatNet B.A. beats it on MinC-2500.Figure 8 compares classifier-subspace normalized within-class variance with test accuracy across datasets and regularization strengths.Source: Figure 8 · p. 30, §C.3.3 · p. 31
  • Normalized within-class variance is measured inside the classifier subspace, excluding feature variation orthogonal to that subspace because it does not change predictions.The measure is a ratio of within-class to between-class covariance traces projected into the classifier subspace.Source: §C.3.3 · p. 30
  • Stronger fine-tuning regularization keeps normalized within-class variance closer to the original scattering transform, especially on SuroTex and EuroSat.SuroTex is the only dataset where the original transform has significantly lower normalized within-class variance than all other methods.Source: §C.3.3 · p. 31

C.3.4 Model Interpretability Results

The interpretability analysis tests whether fine-tuned filters retain wavelet properties and whether the filter bank preserves its organization. The results indicate that distance regularization limits parameter shifts and helps maintain similarity to the initialized Morlet family.

C.3.4 Model Interpretability Results

  • The analysis checks interpretability both at the individual-filter level and at the level of the filter bank’s parameter organization.The scattering bank’s wavelets are distributed across scales and orientations, so wavelet-like individual filters alone are not sufficient evidence of an interpretable overall model.Source: §C.3.4 · pp. 31, 32, 34
  • At matched fine-tuning distance, fine-tuned filters remain substantially closer to the Morlet family than randomly initialized filters.Figure 10 plots fine-tuning distance against ρ, the normalized L2 distance from each filter to its best-fit Morlet wavelet.Source: Figure 10 · p. 32, §C.3.4 · p. 32
  • Fine-tuned filters retain a smaller zero-frequency component than randomly initialized filters, and stronger regularization shrinks it further; the worst filter’s degradation as training data grows remains bounded.Vanishing zero-frequency magnitude is necessary for a wavelet, but this metric alone cannot certify that a filter is one.Source: §C.3.4 · pp. 31–32
  • With λ=1, relative shifts in fitted Morlet parameters are close to zero across datasets, whereas λ=0 produces larger shifts on DTD, MinC-2500, and SuroTex.Parameter shifts compare each filter’s best-fit Morlet parameters with those of its matched filter in the initial bank.Source: Figure 11 · p. 33, §C.3.4 · p. 34

C.3.5 Full Numerical Results

The numerical-results tables report test accuracy, generalization gap, and classifier-subspace normalized within-class variance across architectures and training-set fractions. Results are summarized across random seeds, with additional data partitions for DTD.

C.3.5 Full Numerical Results

  • Reported results average three random seeds, and DTD additionally averages five data partitions; variability also includes training-subset selection, train-validation split, and model initialization.The scattering-transform and ScatNet B.A. filters are deterministic, but their classifiers and ScatNet B.A. analysis layers are randomly initialized.Source: §C.3.5 · p. 34
  • For DTD and EuroSat, the tables compare architectures on test accuracy, generalization gap, and normalized within-class variance across training-set fractions.Each cell reports a mean over seeds with standard deviation in parentheses; best test accuracy per column is bolded.Source: Table 4 · p. 34, Table 5 · p. 35, Table 6 · p. 35
  • For MinC-2500 and SuroTex, the tables report the same three metrics across architectures and training-set fractions, with results split across table parts.Each cell reports a mean over seeds with standard deviation in parentheses; best test accuracy per column is bolded.Source: Table 7 · p. 36, Table 8 · p. 37, Table 9 · p. 38
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