arxiv: 2605.00889 · v1 · submitted 2026-04-27 · 💻 cs.CV · cs.LG

Recognition: unknown

On the explainability of max-plus neural networks

Ikhlas Enaieh (S2A , LTCI) , Olivier Fercoq (S2A , Garc\'ia \'Angel (DATSI , UPM)

Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:55 UTC · model grok-4.3

classification 💻 cs.CV cs.LG

keywords max-plus neural networksexplainabilityneural network interpretabilitypixel fragilityPneumoniaMnistSHAPintegrated gradients

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The pith

Max-plus neural networks determine their output value from a single most-activated neuron, which directly supports a pixel fragility measure for explanations.

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

The paper shows that linear-min-max neural networks, interpretable at start as k-medoids clustering under the infinity norm and trained by subgradient descent, remain universal approximators while preserving traceability. Because the final output is always set by one dominant neuron, the authors construct a pixel fragility score that checks whether altering an individual pixel could flip the classification. On the PneumoniaMnist chest X-ray dataset this score produces explanations that compare favorably with SHAP and Integrated Gradients. The approach therefore supplies model-native attributions without post-hoc approximation, which matters for safety-critical vision tasks where users need to know exactly which image regions drive a decision.

Core claim

In max-plus neural networks a single most-activated neuron governs the output value. This property lets the authors define a pixel fragility measure that quantifies whether a change to one pixel can alter the classification decision. Experiments on PneumoniaMnist show the resulting explanations match or exceed the fidelity of SHAP and Integrated Gradients.

What carries the argument

The single most-activated neuron property, which fixes the network output and thereby allows direct computation of each pixel's influence on that output.

If this is right

Explanations can be read directly from the network activations without training auxiliary models.
The fragility score identifies individual pixels whose modification is likely to change the predicted class.
The same traceability holds after training, not only at initialization.
The method applies at least as well as standard attribution techniques on medical-image classification.

Where Pith is reading between the lines

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

The single-neuron rule could be used to prune or regularize networks so that only the dominant path remains active, potentially improving both speed and interpretability.
Because the property originates from the max-plus algebra, similar traceability may appear in other tropical or max-min architectures.
If the fragility measure correlates with human-annotated lesion locations on chest X-rays, it could serve as an automated quality check for model decisions in clinical workflows.

Load-bearing premise

The decision process can always be reduced to one most-activated neuron whose value determines the output, and that this reduction yields a faithful pixel fragility score.

What would settle it

An observation that the classification output changes even though the most-activated neuron remains unaffected, or that perturbing the highest-fragility pixel leaves the output unchanged while a low-fragility pixel alters it, on the same PneumoniaMnist images.

Figures

Figures reproduced from arXiv: 2605.00889 by Garc\'ia \'Angel (DATSI, Ikhlas Enaieh (S2A, LTCI), Olivier Fercoq (S2A, UPM).

read the original abstract

We investigate the explanability properties of the recently proposed linear-min-max neural networks. At initialization, they can be interpreted as k-medoids with the infinity norm as a distance. Then, they are trained using subgradient descent to better fit the data. The model has been shown to be a universal approximator. Yet, we can trace the decision process because a single most activated neuron is responsible for the value of the output. Using this property, we designed a pixel fragility measure that determines whether changes to a single pixel may be responsible to a change in the classification output. Experiments on the PneumoniaMnist dataset show that this explanation for the output of the neural network compares favorably to SHAP and Integrated Gradient.

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.

Desk Editor's Note private letter to a colleague

The paper's main contribution is a pixel fragility measure that exploits the single most-activated neuron property in max-plus networks, showing better alignment with human intuition than SHAP or Integrated Gradients on PneumoniaMnist, though the multi-layer causal link looks shaky.

read the letter

The punchline is that max-plus networks let you trace the output to one dominant neuron by design, and the authors convert that into a pixel fragility score for image explanations. On PneumoniaMnist it lines up better with expected behavior than the usual post-hoc methods, which is the concrete result worth noting first. They also recall that these networks start as k-medoids under the infinity norm and train via subgradient descent while remaining universal approximators, but the explainability angle is the new piece. The single-neuron responsibility is a structural feature they leverage directly instead of approximating after the fact, and that is genuinely cleaner than many black-box explainers. The fragility measure itself is defined from the activation property and tested head-to-head on the medical dataset, which gives it a clear empirical hook. The soft spot is the jump from neuron responsibility to pixel responsibility once you have more than one layer. The final output is a composition of max and min-linear operations, so the argmax at the top can depend on sets of intermediate paths rather than a unique pixel. Nothing in the abstract shows they invert those paths rigorously or test whether the fragility score remains faithful under deeper stacking. The comparison is also on a single dataset with no reported significance tests or controls for network depth, so the favorable result is suggestive but not yet robust. This is for people already working with tropical or max-plus architectures who want an explanation method that stays inside the model rather than bolted on. A reader who cares about medical imaging trust or inherently interpretable networks could get something useful from the fragility idea if the multi-layer part checks out. I would send it to peer review so the methods section and the exact fragility formula get proper scrutiny.

Referee Report

2 major / 2 minor

Summary. The paper examines the explainability of linear-min-max neural networks, which at initialization act as k-medoids clustering under the infinity norm and are trained via subgradient descent while remaining universal approximators. The central claim is that the output is always determined by a single most-activated neuron, enabling a pixel fragility measure that attributes potential classification changes to individual input pixels; experiments on PneumoniaMnist show this measure compares favorably to SHAP and Integrated Gradients.

Significance. If the single-neuron responsibility property propagates faithfully through layers and the derived fragility measure yields causally accurate pixel attributions, the work would strengthen the case for architecture-specific interpretability in max-plus networks, combining universal approximation with an efficient, non-perturbation-based explanation tool that could outperform generic methods like SHAP in targeted domains such as medical imaging.

major comments (2)

[Section on the pixel fragility measure and network architecture] The single most-activated neuron property is invoked to justify the pixel fragility measure, but in a multi-layer setting the final argmax depends on intermediate max selections whose pre-images are sets of activations rather than unique input pixels. This makes it unclear whether the fragility score correctly isolates responsibility to specific pixels after composition of max and linear-min operations (see the section defining the pixel fragility measure and the multi-layer architecture description).
[Experiments section] The experimental claim of favorable comparison on PneumoniaMnist lacks the exact definition or formula for the pixel fragility measure, any statistical tests, sample sizes, hyperparameter controls, or baseline implementation details for SHAP and Integrated Gradients. Without these, the empirical support for the central explainability claim cannot be verified (see the Experiments section).

minor comments (2)

[Abstract] The abstract contains the typographical error 'explanability' which should read 'explainability'.
[Abstract] The abstract phrase 'responsible to a change in the classification output' should be revised to 'responsible for a change in the classification output' for grammatical precision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our work regarding the explainability of max-plus neural networks. We address each major comment in detail below and will revise the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

Referee: The single most-activated neuron property is invoked to justify the pixel fragility measure, but in a multi-layer setting the final argmax depends on intermediate max selections whose pre-images are sets of activations rather than unique input pixels. This makes it unclear whether the fragility score correctly isolates responsibility to specific pixels after composition of max and linear-min operations (see the section defining the pixel fragility measure and the multi-layer architecture description).

Authors: We appreciate this observation. The single most-activated neuron property is defined recursively across layers: at each max layer, one dominant activation is selected, and the linear-min operations map this back to previous layer activations. The pixel fragility measure is computed by tracing this dominant path from the output neuron back to the input pixels, assigning fragility scores based on the sensitivity along this path. While pre-images under max are indeed sets, the measure focuses on the selected dominant branch. To make this explicit, we will expand the definition section with a formal recursive definition and a worked example for multi-layer networks. revision: yes
Referee: The experimental claim of favorable comparison on PneumoniaMnist lacks the exact definition or formula for the pixel fragility measure, any statistical tests, sample sizes, hyperparameter controls, or baseline implementation details for SHAP and Integrated Gradients. Without these, the empirical support for the central explainability claim cannot be verified (see the Experiments section).

Authors: We agree that additional details are necessary for reproducibility and verification. In the revised manuscript, we will provide the precise mathematical formula for the pixel fragility measure, specify the number of samples used from PneumoniaMnist (e.g., the test set size or subset evaluated), include statistical significance tests comparing the measures, detail the hyperparameters for training the network and for the baseline methods, and describe the implementation of SHAP and Integrated Gradients (including any libraries or custom code used). revision: yes

Circularity Check

0 steps flagged

No circularity: pixel fragility measure follows from architectural max property without reduction to fits or self-citations

full rationale

The derivation begins from the max-plus network structure itself, where the output is defined as the maximum over neuron activations; the claim that a single most-activated neuron determines the output value is therefore a direct consequence of the max operation rather than a fitted parameter, renamed result, or self-citation. The pixel fragility measure is constructed by tracing responsibility through this architectural selection, and the PneumoniaMnist experiments serve only as external empirical comparison to SHAP and Integrated Gradients. No load-bearing step equates a prediction to its own input by construction, and the universal-approximator reference is external to the explainability argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work relies on the previously established universal approximation property of the networks and their initialization as k-medoids; the new contribution is the fragility measure derived from the activation property.

axioms (2)

domain assumption Linear-min-max neural networks are universal approximators
Stated as previously shown; invoked to support the model's capability before introducing explainability.
domain assumption At initialization the networks correspond to k-medoids with infinity norm
Used to ground the initial interpretability claim.

invented entities (1)

pixel fragility measure no independent evidence
purpose: Quantify whether a single pixel change can alter the classification output via the dominant neuron
Newly proposed in the paper to leverage the single-neuron responsibility property.

pith-pipeline@v0.9.0 · 5435 in / 1546 out tokens · 34101 ms · 2026-05-09T20:55:41.181338+00:00 · methodology

discussion (0)

Reference graph

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