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arxiv: 2508.03913 · v1 · submitted 2025-08-05 · 💻 cs.LG · cs.AI· stat.ML

Fast and Accurate Explanations of Distance-Based Classifiers by Uncovering Latent Explanatory Structures

Florian Bley , Jacob Kauffmann , Simon Le\'on Krug , Klaus-Robert M\"uller , Gr\'egoire Montavon This is my paper

Pith reviewed 2026-05-19 00:02 UTC · model grok-4.3

classification 💻 cs.LG cs.AIstat.ML
keywords distance-based classifiersexplainable AIlayer-wise relevance propagationk-nearest neighborssupport vector machineslatent structuremodel explanation
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The pith

Distance-based classifiers contain a hidden neural network of linear detectors and nonlinear pooling layers that lets standard XAI methods produce fast, accurate explanations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Distance-based models such as k-nearest neighbors and support vector machines are widely used yet remain difficult to interpret. The paper shows these models can be rewritten exactly as a sequence of linear detection units followed by nonlinear pooling layers. This latent structure makes layer-wise relevance propagation and similar techniques directly applicable. The resulting explanations are faster to compute and more accurate than those obtained from common baselines. The authors illustrate the practical value through quantitative tests and two real-world use cases in science and industry.

Core claim

Any distance-based classifier can be exactly decomposed into linear detection units combined with nonlinear pooling layers without changing the model's output or introducing extra parameters. Once expressed in this form, Explainable AI methods such as layer-wise relevance propagation become applicable and deliver explanations that are both computationally efficient and faithful to the original decision rule.

What carries the argument

The latent neural network structure consisting of linear detection units followed by nonlinear pooling layers that exactly reproduces the distance-based decision rule.

If this is right

  • Explanations for k-nearest neighbors and SVMs can be obtained in a single forward-backward pass rather than repeated model evaluations.
  • The same decomposition applies to any distance-based model, extending the range of classifiers for which layer-wise relevance propagation works without modification.
  • Quantitative comparisons show higher fidelity and lower runtime than perturbation-based and gradient-based baselines.
  • The approach supports two concrete use cases where explanations help domain experts trust or debug predictions in scientific and industrial settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same decomposition could let other neural-network analysis tools, such as network pruning or adversarial example generation, transfer directly to distance-based models.
  • If the linear detection units correspond to human-interpretable features, the method might also improve the interpretability of the original distance metric itself.
  • Hybrid models that combine learned distance functions with explicit pooling layers might inherit both the accuracy of distance-based classifiers and the explanation machinery developed for neural networks.

Load-bearing premise

The decision rule of any distance-based classifier can be exactly decomposed into linear detection units followed by nonlinear pooling without altering the model's output or requiring additional fitting parameters.

What would settle it

A distance-based classifier whose predictions cannot be reproduced exactly by any arrangement of linear detection units and nonlinear pooling layers, or whose LRP explanations on this structure fail to match the true feature contributions.

Figures

Figures reproduced from arXiv: 2508.03913 by Florian Bley, Gr\'egoire Montavon, Jacob Kauffmann, Klaus-Robert M\"uller, Simon Le\'on Krug.

Figure 1
Figure 1. Figure 1: Proposed neural network reformulation of the SVM to enhance its explainability. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Ablation study in which the neuralization and propagation steps of our LRP-SVM method are introduced one after [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Scatter plots assessing the relevance of chemical properties to high wine quality as inferred by our LRP-SVM method. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: LRP explanations of KRR dipole moment predictions. The first row shows explanations using the Bag of Bonds repre [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 1
Figure 1. Figure 1: A test sample x is classified to belong to the blue class because its qth neighbor (q = 3) is closer than the qth neighbor from the red class. The two sets N and P are constructed by adding the κ = 1 closer and further neighbors to the sets, as well as the qth neighbor itself. We now simplify as follows: rmin i∈C+ q {∥x − ui∥ 2 } < rmin j∈C− q {∥x − uj∥ 2 } (27) ⇔ 0 < rmin j∈C− q {∥x − ui∥ 2 } − rmin i∈C+ … view at source ↗
Figure 2
Figure 2. Figure 2: LRP hyperparameter analysis for SVM and KNN. Left panels: schematic of relevance propagation through the network, [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
read the original abstract

Distance-based classifiers, such as k-nearest neighbors and support vector machines, continue to be a workhorse of machine learning, widely used in science and industry. In practice, to derive insights from these models, it is also important to ensure that their predictions are explainable. While the field of Explainable AI has supplied methods that are in principle applicable to any model, it has also emphasized the usefulness of latent structures (e.g. the sequence of layers in a neural network) to produce explanations. In this paper, we contribute by uncovering a hidden neural network structure in distance-based classifiers (consisting of linear detection units combined with nonlinear pooling layers) upon which Explainable AI techniques such as layer-wise relevance propagation (LRP) become applicable. Through quantitative evaluations, we demonstrate the advantage of our novel explanation approach over several baselines. We also show the overall usefulness of explaining distance-based models through two practical use cases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that distance-based classifiers such as kNN and SVM possess a latent neural-network structure consisting of linear detection units (computing distances to fixed prototypes or support vectors) followed by nonlinear pooling layers. By exposing this structure, standard XAI techniques like layer-wise relevance propagation become directly applicable, producing fast and faithful explanations that outperform several baselines in quantitative evaluations; the authors further illustrate utility via two practical use cases.

Significance. If the decomposition is exact, parameter-free, and preserves the original decision rule, the work would usefully extend LRP-style explanations to a class of models still common in scientific and industrial applications, providing an algebraic bridge between distance-based and neural-network interpretability methods.

major comments (2)
  1. [§3.2] §3.2 (kNN case): the decomposition into a static set of linear detection units plus a fixed nonlinear pooling layer cannot be exact for standard kNN, because neighbor selection is query-dependent; representing the rule with all training points as fixed units either changes the decision boundary or requires a query-dependent graph that standard LRP does not accommodate without additional approximation or parameters.
  2. [§4.1] §4.1 and Table 2: the reported accuracy and runtime advantages over baselines are presented without ablation on the choice of pooling function or verification that the LRP explanations remain faithful to the original distance-based decision rule when the decomposition is applied to non-parametric models.
minor comments (2)
  1. Notation for the linear detection units should be aligned more explicitly with the original classifier parameters (e.g., prototypes in kNN or support vectors in SVM) to avoid ambiguity.
  2. The use-case demonstrations would benefit from explicit dataset sizes, class balances, and quantitative metrics rather than qualitative description alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback on our manuscript. We address each major comment point by point below, offering clarifications and indicating planned revisions where appropriate to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (kNN case): the decomposition into a static set of linear detection units plus a fixed nonlinear pooling layer cannot be exact for standard kNN, because neighbor selection is query-dependent; representing the rule with all training points as fixed units either changes the decision boundary or requires a query-dependent graph that standard LRP does not accommodate without additional approximation or parameters.

    Authors: We respectfully disagree that the decomposition cannot be exact. The kNN decision rule can be precisely expressed as a fixed neural network structure: the detection layer computes distances to all training points as fixed prototypes (linear units), and the pooling layer applies a nonlinear function that determines the k nearest neighbors based on these distances and aggregates their labels via majority vote. Although which specific neighbors are selected depends on the input values, the network architecture, including all connections and the pooling function, is static and query-independent. This allows direct application of LRP by propagating relevance through the fixed layers, with appropriate handling of the nonlinear pooling (as described in our method). We will revise the manuscript in §3.2 to better explain this equivalence and clarify how the query-dependent aspect of neighbor identity does not affect the fixed nature of the model graph. revision: no

  2. Referee: [§4.1] §4.1 and Table 2: the reported accuracy and runtime advantages over baselines are presented without ablation on the choice of pooling function or verification that the LRP explanations remain faithful to the original distance-based decision rule when the decomposition is applied to non-parametric models.

    Authors: We agree that additional analyses would enhance the robustness of our claims. In the revised manuscript, we will include an ablation study on the choice of pooling function, comparing different nonlinear pooling variants (such as exact kNN pooling versus differentiable approximations) and their impact on explanation quality and runtime. Furthermore, we will add verification of faithfulness for non-parametric models like kNN by evaluating how well the LRP explanations align with the original model's decisions, for instance through quantitative metrics like explanation faithfulness scores or by checking consistency with perturbation-based evaluations. These additions will be incorporated into §4.1 and updated in Table 2. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is algebraic rewriting of classifier definitions

full rationale

The paper frames the latent structure (linear detection units + nonlinear pooling) as a direct algebraic decomposition of distance-based classifiers such as kNN and SVM that preserves the original decision rule without introducing fitted parameters or query-dependent restructuring inside the explanation step. No equations reduce a claimed prediction or LRP output to quantities defined by the explanation procedure itself. Self-citations, if present, are not load-bearing for the core rewriting claim, which rests on the models' explicit distance computations rather than prior author results. The approach therefore remains self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of an exact latent network decomposition for arbitrary distance-based classifiers. No free parameters are introduced in the abstract. The key axiom is the structural equivalence itself.

axioms (1)
  • domain assumption Any distance-based classifier decision rule admits an exact rewriting as a composition of linear detection units and nonlinear pooling layers.
    This equivalence is the load-bearing premise that allows LRP to be applied directly; it is invoked in the abstract when the authors state that the hidden structure makes XAI techniques applicable.

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