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Beyond $L_2$: Generalizing Abductive Latent Explanations to Diverse Prototype-Based Architectures

Jules Soria, Alban Grastien, Romain Xu-Darme, Julien Girard-Satabin, Zakaria Chihani, Daniela Cancila

arXiv:2608.16773v1cs.LG

TL;DR

Existing ALE methods are confined to Euclidean latent spaces, limiting formal explanations for non-Euclidean prototype architectures. This paper generalizes ALE across geometric variants and validates the framework on trained image classifiers, enabling quantitative cross-architecture interpretability comparisons.

  • Problem

    ALEs are designed around Euclidean latent-space distances, leaving formal explanations for prototype architectures using non-Euclidean representations insufficiently covered.

  • Method

    The paper extends ALE reasoning by mapping architectures to existing bounds or deriving architecture-specific algorithms, including spherical-cap intersection bounds.

  • Results

    PIP-Net yielded the smallest absolute explanation sizes across all datasets, averaging 18–103 prototypes, while Simplex yielded the smallest relative sizes overall at 0.1%–0.2%.

  • Takeaways & Limitations

    The unified framework enables quantitative cross-architecture comparisons and highlights design principles associated with better formal interpretability.

  • Takeaways & Limitations

    Absolute explanation sizes are confounded by latent-space dimensions and prototype counts, so comparisons should emphasize relative size.

Abstract

from arXiv · show

Prototype-based neural networks are hailed as interpretable-by-design architectures. Recently, Abductive Latent Explanations (ALE) were introduced to provide formal, mathematically guaranteed explanations that leverage the intrinsic structure of these networks to ensure both predictive safety and human readability. ALEs rely on computing tight bounds on latent space distances to produce formal explanations. However, existing ALE formulations are rigidly confined to Euclidean latent spaces. This leaves a critical gap: modern state-of-the-art architectures increasingly rely on non-Euclidean representations - such as spherical metrics, Gaussian densities, and dimensional projections - rendering current formal explanation methods incompatible. In this work, we generalize the ALE framework to support non-Euclidean prototype architectures. For each geometric variant, we systematically derive how to either map the architecture to existing bounds or construct novel, architecture-specific bounding algorithms. We validate our theoretical constructions by computing subset-minimal formal explanations on fully trained image classifiers. By unifying these diverse models under a single formal framework, we enable the first rigorous, cross-architecture comparison of their interpretability.

1 Introduction

The paper situates explainability and interpretability as central to trustworthy ML, especially for critical systems, and builds on ALEs to formalize explanations for prototype-based neural networks. It extends ALEs across prototype modalities, similarity functions, and pooling layers, then applies geometric, solver-free reasoning to compare architectures experimentally.

  • Motivation: Explainability and interpretability support trustworthy ML, with transparency especially important in critical energy, transportation, and medical applications.The paper connects this need to standards and regulations including the AI Act, GDPR, and ISO/IEC JTC1 42.
  • Formal explanations: Prototype-based neural networks justify predictions through similar training examples, while ALEs extract subset-minimal prototype sets that guarantee predictions.ALEs bridge formal explainable AI and prototype-based neural networks by building on abductive explanations.
  • Experimental validation: Experiments apply ALEs’ solver-free geometric reasoning and compare different architectures.The paper presents extensive experimentation to showcase the validity of the method.
  • Contributions: The framework adapts ALEs to other prototype modalities, similarity functions, and pooling layers.These adaptations demonstrate the framework’s flexibility and usefulness across prototype-based architectures.

2 Preliminaries

Prototypical part networks map images to latent patches, compare those patches with class-relevant prototypes, and classify via prototype-weighted linear scores. ALEs formalize interpretable similarity statements whose induced bounds tighten prototype activations and can guarantee the predicted class.

  • Prototypical part networks: The backbone maps an image X to a latent representation z composed of H1×W1 patches zl ∈ R^D.The input image lies in F = R^(H0×W0×C0), while z lies in Z = R^(H1×W1×D).
  • Prototypical part networks: Prototypes are training-image patches deemed representative of classes, and activation measures patch–prototype similarity.Different architectures implement activation differently, motivating the framework’s focus on diverse activation measures.
  • Prototypical part networks: Each prototype’s activation is the maximum similarity over all image patches: aj = maxl∈L sim(zl, pj).This aggregates local patch comparisons into one activation value per prototype.
  • Prototypical part networks: A linear layer W assigns prototype contributions to classes, and prediction selects the class with the highest score, argmax W^T a.Weights are traditionally positive for prototypes typical of a class and lower than zero otherwise.
  • Abductive Latent Explanations: ALEs collect interpretable statements about latent patches and prototypes, including similarity information that implicitly constrains other prototype relationships.For Euclidean distances, distances to two prototypes restrict a patch to intersecting hyperspheres, enabling bounds on its distance to another prototype.
  • Abductive Latent Explanations: Adding prototype or patch–prototype facts tightens similarity and activation intervals, ultimately bounding class-score differences until one class is proved greater than all others.This proof establishes the network’s classification as guaranteed by the explanation.

3 Related Works

The related-work review focuses on Formal XAI and prototype-based interpretable models, whose intersection motivates principled transparency. It situates the paper amid increasingly diverse prototype geometries while restricting analysis to atomic-prototype architectures.

  • Formal XAI and Prototype-Based Models: The review covers Formal XAI and prototype-based interpretable machine learning as the two subfields motivating this work.Formal XAI seeks mathematically rigorous explanations, while prototype models provide an interpretable architectural foundation.
  • Formal XAI: Abductive Explanations are prime implicants of decision functions, while traditional Formal XAI operates on input features and is computationally NP-hard.Formal XAI has also recently been applied to computer vision, including distance-restricted AXp methods such as VeriX.
  • Prototype-Based Architectures: ProtoPNet introduced latent-space prototypes classified by squared L2 distance, inspiring variants using Grassmannian representations, cosine similarity, dimensional projection, and probabilistic prototypes.Probabilistic approaches represent prototypes as distributions, including von-Mises-Fischer, singular Gaussian, or Gaussian-mixture forms.
  • Prototype-Based Architectures: Table 1 summarizes prototype-network module evolution and the architectural components requiring corresponding changes to formal explanations.The table is described as a non-exhaustive list of recent state-of-the-art prototype-based networks.
  • Scope: Networks with over-prototypes are excluded because component-activation bounds can be combined as a weighted sum to bound prototype activations.The paper therefore focuses on architectures with atomic prototypes.

4 Theoretical Insights for new ALEs … Spherical Cap Intersection Approximation

The paper extends ALEs beyond Euclidean geometry by formalizing top-k explanations and deriving sound, progressively tighter similarity bounds for spherical prototype architectures. For cosine-based models, spherical-cap intersections support activation bounds that can certify predictions.

  • 4.1 Top-k Explanations: Top-k ALEs include the k most activated prototypes and guarantee every excluded prototype has lower activation than each included prototype.The number of prototypes required for a formal explanation is not known before inference.
  • 4.1 Top-k Explanations: An iterative, activation-agnostic algorithm adds the highest-activated excluded prototype until the explanation guarantees the model’s prediction.Because it does not depend on the activation function, the procedure applies across prototype architectures.
  • 4.2 Cosine Similarity: TesNet and related models use dot-product similarity with unit-norm prototypes, giving their latent representations spherical geometry.Dot products evaluate vectors with identical angles differently when their amplitudes differ.
  • 4.2 Cosine Similarity: Cosine similarity and cosine distance cannot directly provide the required metric bounds because cosine distance violates the triangle inequality.The paper therefore uses angular distance, which yields tighter cosine-similarity bounds.
  • Triangle Inequality for Cosine Similarity: Using angular-distance inequalities, known latent–prototype similarities refine bounds on excluded prototypes as explanation pairs are added incrementally.These refined bounds allow the ALE algorithm to continue until the prediction is formally guaranteed.
  • Spherical Cap Intersection Approximation: The refined similarity bounds yield activation bounds for excluded prototypes and can certify, or fail to certify, the model’s prediction.This extends the Euclidean hypersphere-intersection idea to spherical-cap intersections for cosine-based prototype networks.
  • Spherical Cap Intersection Approximation: The spherical-cap method constructs a minimal-radius cap containing the intersection of two geodesically convex prototype caps under the stated triangle-feasibility conditions.The cap center lies on the great circle through the two prototype centers and is obtained by geodesic orthogonal projection.
  • Spherical Cap Intersection Approximation: Adding prototype caps monotonically decreases the bounding radius, thereby tightening the angular-distance and cosine-similarity bounds for unknown prototypes.The construction remains entirely spherical and provides closed-form, provably sound bounds.

4.3 Dimensional Projection

Dimensional projection in PIP-Net maps patch activations to a probability simplex using softmax over prototypes, with each vector dimension representing similarity to an indexed prototype.

  • 4.3 Dimensional Projection: PIP-Net applies softmax over prototypes for each patch, mapping activations to the probability simplex Δm−1.For each location l and prototype j, the resulting representation is defined through sim(z_l, p_j).
  • 4.3 Dimensional Projection: Each dimension of the projected vector corresponds to similarity with one indexed prototype, so prototypes define the representation’s dimensions.The prototype index set is P = {1, . . . , D}.

Simplex Explanation · Sparse-Weight Explanation

The framework adapts explanations to simplex-valued activations by tracking conserved probability mass and residual bounds. For sparse-weight heads, explanations begin with prototypes directly weighted for the predicted class and expand only when competing-class bounds remain unresolved.

  • Simplex Explanation: Softmax makes the spatial explanation a conservation of probability mass rather than a geometric relation.Similarity values satisfy sim(z_l, p_j) ∈ [0, 1], with their sum over prototypes equal to 1.
  • Simplex Explanation: Prototype activations are bounded by a_j = max_l sim(z_l, p_j) ∈ [0, 1], while the negative baseline is a− = 0.These bounds are used for Top-k explanations.
  • Simplex Explanation: Adding patch–prototype pairs to an explanation consumes softmax mass and strictly limits the maximum residual contribution of unexplained prototypes.The remaining global upper bound is determined by the maximum residual mass.
  • Sparse-Weight Explanation: PIP-Net uses a sparse, non-negative decision head, unlike traditional ProtoPNets whose dense heads let every prototype influence every class.The weights satisfy w_j,c ≥ 0.
  • Sparse-Weight Explanation: For the predicted class, the initial explanation contains every prototype with non-zero class weight, and this set gives its exact score.Any superset E ⊇ E0 preserves the exact predicted-class score because all omitted prototypes have zero weight for that class.
  • Sparse-Weight Explanation: Because softmax activations are globally bounded by 1.0, competing-class outputs receive explicit upper bounds.This bound applies to any competing class c′ ≠ ˆc.
  • Sparse-Weight Explanation: If every competing bound remains below the predicted score, the prediction is formally guaranteed; otherwise, the explanation adds the activation most reducing the strongest competitor.The strongest competitor is c∗ = argmax_c′≠ˆc o_c′.

4.4 Isotropic Gaussian Similarity

For isotropic Gaussian prototypes, ALEs recover Euclidean latent distances from Gaussian similarities, apply hypersphere-intersection bounds, and map the resulting bounds back to model activations. This decoupled pipeline supports rapid subset-minimal explanations across prototype architectures.

  • Gaussian-to-Euclidean Mapping: Isotropic Gaussian prototypes replace latent distance with negative log-likelihood while recovering the true Euclidean distance under isotropic covariance.Each prototype has covariance Σ_j = σ^2 I.
  • Similarity-Agnostic Bounding: Monotonic similarity transformations make the bounding logic agnostic to whether the model uses Gaussian or heavy-tailed similarity functions.The method maps either similarity form into a universal Euclidean space before bounding.
  • Bounding Pipeline: The pipeline applies inverse similarity mapping, Euclidean hypersphere intersection, target-metric projection, and forward activation mapping.HIA constructs a minimal enclosing hypersphere around the exact intersection of observed hyperspheres.
  • Explanation Computation: Decoupling geometric intersection from the model forward pass enables rapid computation of subset-minimal explanations across prototype architectures.Similarity bounds are inverted so maximum distance gives the lower similarity bound and minimum distance gives the upper bound.

4.5 Focal Similarity

Focal Similarity extends prototype pooling by subtracting average prototype activation from the maximum patch similarity, suppressing background activations and encouraging localization. Within ALE, existing patch-wise similarity bounds are aggregated to produce formally guaranteed focal-activation bounds without changing the geometric solver.

  • Focal Similarity: ProtoPool replaces max pooling with relative focal pooling, subtracting the average latent-representation activation from the maximum prototype similarity.Focal Similarity can accompany any predefined or novel similarity function because it changes only pooling before final prototype activations.
  • Focal Similarity: Spatial ALE bounds for every latent patch provide the basis for bounding the maximum and expected patch similarities required by focal pooling.The method derives strict bounds for these aggregate statistics for each prototype.
  • Focal Similarity: Because focal activation subtracts the expected similarity from the maximum, its guaranteed extrema are obtained by cross-subtracting the opposing bounds.This produces the final formally guaranteed activation bounds.
  • Focal Similarity: Aggregating patch-wise similarity bounds extends spatial ALEs to ProtoPool while preserving the underlying geometric solver.The extension handles focal-similarity penalties without modifying the solver.

5 Experiments

The experiments evaluate generalized ALE methods on four prototype-based architectures across three fine-grained image-classification datasets. Results show that spatial bounding techniques compress the relative hypothesis space, often with additional computational overhead.

  • Experimental setup: Four architectures were trained on three fine-grained image-classification datasets to extract subset-minimal explanations with rigid mathematical guarantees.The architectures were ProtoPNet, PIP-Net, TesNet, and Isotropic Gaussian; the datasets were Oxford Flowers 102, Oxford IIIT Pet, and CUB200.
  • Evaluation metrics: Explanation methods were evaluated using absolute size, relative size, and computing time.These metrics capture explanation compactness, comparability across paradigms, and the operational cost of explanation generation.
  • Evaluation metrics: Relative Size normalizes explanation size by each ALE paradigm’s maximum possible theoretical size for fair cross-paradigm comparison.Global paradigms are bounded by the number of prototypes, while spatial paradigms scale from |L| to |L| × |P|.
  • Results: The generalized spatial bounding techniques compressed the relative hypothesis space, often at the cost of computational overhead.The paper further analyzes the architectural inductive biases underlying these trade-offs in Section 6.

6 Discussion

The generalized ALE framework enables quantitative comparison across diverse prototype-based networks, revealing architecture-dependent trade-offs between explanation compactness and computation time. Results also show that latent dimensionality and post-training prototype reduction affect how raw explanation sizes should be interpreted.

  • The generalized ALE framework provides the first quantitative comparison of formal interpretability across diverse prototype-based networks.
  • PIP-Net has the smallest absolute explanation sizes across datasets, averaging 18 to 103 prototypes, and the fastest computation at ∼0.07s.On CUB-200, its Top-k ALE reaches 18 ± 38, the smallest absolute size across architectures and datasets.
  • Simplex explanations achieve extreme compression of 0.1%–0.2%, suggesting favorable scaling with latent-space dimensions.
  • Scaled HIA in Gaussian models produces the smallest relative explanation sizes on Oxford datasets at 4.6%–6.2%, but computation takes up to 142s.On CUB-200, Scaled TI and Scaled HIA exceed the time budget entirely.
  • Cosine TI and Spherical HIA achieve the smallest Oxford Flowers absolute size of 187 and a competitive Oxford Pets relative size of 3.2%, but time out on CUB-200.The passage attributes this scalability failure to cosine similarity complicating spatial reasoning.
  • Raw explanation sizes require caution because latent-space dimensions confound architectural comparisons, while prototype pruning or merging would mechanically reduce all absolute ALE sizes.

7 Conclusion

The work extends ALE definitions across modern prototype-based networks. Experiments enable quantitative model comparisons and identify design principles associated with stronger formal interpretability.

  • ALE definitions are extended to cover the diverse landscape of modern prototype-based networks.
  • Experiments support quantitative comparisons among prototype-based models.
  • The study highlights design principles leading to better formal interpretability.
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