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REVIEW 3 major objections 5 minor 31 references

Pathwise Explanation of ReLU Neural Networks

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper argues that a ReLU neural network is better explained by a connected subset of hidden units, called a path, than by all of its activated units, and that pathwise explanations are more consistent, decomposable, and…

desk verdict Solid path-sum math, but the path-selection heuristic doesn't measure path contribution, so the 'decision path' claim is unsubstantiated. read the letter →

arxiv 2506.18037 v1 pith:MPMJO4F6 submitted 2025-06-22 cs.LG cs.AI

Seongwoo Lim , Won Jo , Joohyung Lee , Jaesik Choi This is my paper · ORCID
classification cs.LGcs.AI
keywords ReLUnetworksexplainableAIpathwiseexplanationpiecewiselinearmodelsinputattributionunwrappinginsertiondeletionevaluationneuralnetworkinterpretability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that a ReLU neural network is better explained by a connected subset of hidden units, called a path, than by the whole set of activated units. For each such path the paper derives a piecewise linear model, so the explanation is literally the local linear function the network effectively uses along that path. Pathwise explanations are claimed to be more consistent with the input and can be decomposed, so a single decision can be split into components such as a fox's eyes and ears. On insertion and deletion tests over a curated ImageNet subset, pathwise attribution maps score at or above six standard attribution methods.

What carries the argument

A path is a connected set of hidden units in the network graph; a one-way complete path contains at most one unit per layer, and a multi-way complete path sums over all one-way complete paths contained in it. The object doing the work is the piecewise linear model $f_p(X) = (W_p X + b_p)\prod_{h\in p}\phi(h)$, whose weight $W_p$ is the product of weights along the path, or the sum of such products for a multi-way path. Because $f_p$ is a genuine term in the unfolded network expansion, a nonzero $f_p$ means the path is active. Algorithm 1 selects the active units by a softmax importance score, top-$k$ filtering per layer, and a threshold of $1/|\mathrm{classes}|$, with depth and width controlling the spatial scale of the explanation.

What would settle it

For a fixed test input, compute the path model $f_p(X)$ for the selected path and compare it with the network output $f(X)$; if $f_p$ is substantially different from $f(X)$ while all units in $p$ are active, or if ablating the units in $p$ leaves the prediction essentially unchanged, then the selected path is not the decision-making path and the method's central claim fails.

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Extended reading notes

Core claim

The paper shows that a ReLU network's output can be unfolded into a sum of path terms, one per one-way complete path, and that a path's term is nonzero exactly when the path is activated. For a multi-way complete path, the weight and bias of the path model are obtained by summing the weights and biases of all one-way complete paths inside it. The central proposal is to explain a prediction with only the path, or several paths, that Algorithm 1 selects, rather than with the local linear model derived from all activated hidden units. The authors argue this avoids mixing conflicting class evidence and enables explanations that can be decomposed either by high-level features or by separate objects in the input.

Load-bearing premise

The load-bearing premise is that Algorithm 1's heuristic, which ranks hidden units by softmax importance over the target class and keeps the top $k$ above the threshold $1/|\mathrm{classes}|$, picks exactly the units that form the decision-making path; a different ranking or threshold would select different units, and the claimed consistency and decomposability are not guaranteed.

Editorial extensions

If this is right

  • Explanations of a single input can be decomposed: distinct selected paths highlight separate features such as eyes and ears, or separate objects, so a user can see which input component each path supports.
  • Because each path's linear model is nonzero only when its units are activated, the attribution no longer mixes positive and negative class evidence, giving more consistent explanations than unwrapping with all units.
  • Varying depth and width in path construction continuously adjusts the explanation from whole-input attribution to local component attribution, offering a tunable scale not provided by fixed full-network methods.
  • On insertion and deletion metrics, pathwise attributions outperform six standard attribution methods in insertion and remain competitive in deletion across VGG-16, ResNet-18, and CIFAR-10 settings.
  • Misclassification diagnosis becomes explicit: explanations for the predicted and correct classes reveal the features, such as background details, that support the wrong answer.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not test this, but the unfolding theory suggests a purely path-based alternative to the softmax top-$k$ heuristic: greedily select the activated one-way complete paths with the largest $|W_pX + b_p|$, which would inherit the consistency guarantee directly from the expansion.
  • The observed common-path-per-class pattern suggests pathwise explanations could serve as a per-class probe for auditing which high-level units a model relies on, including detecting shortcut or spurious features, without additional training.
  • The framework should transfer to other piecewise-linear activations such as leaky ReLU, maxout, and quantized ReLU, because their indicator functions also admit a pathwise unfolding, though the paper does not test these cases.
  • A testable extension is counterfactual generation: since the selected path gives an explicit linear model, perturbing only the input directions it weights most should efficiently reveal minimal changes that flip the decision.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces a 'pathwise' explanation method for ReLU neural networks. Instead of using the activation states of all hidden units, it defines a path as a connected subset of hidden units and derives piecewise linear models for one-way and multi-way complete paths (Propositions 1 and 2). Algorithm 1 selects a subset of hidden units based on a softmax importance score, and the resulting linear model is used to produce input attribution maps. The paper claims that this pathwise explanation is clearer and more consistent than existing methods and that it outperforms them quantitatively and qualitatively. Experiments are reported on a curated 10-class ImageNet subset with VGG-16 and ResNet-18, and on CIFAR-10, using insertion and deletion metrics as well as qualitative visualizations.

Significance. The idea of explaining a ReLU network through a small, connected subset of hidden units is appealing, and the one-way complete path decomposition (Eqs. (2)-(4), Proposition 1) is mathematically straightforward and correctly grounded in the piecewise-linear structure of ReLU networks. The paper provides a public code repository and reports experiments across multiple architectures, which are strengths. However, the multi-way path theory is not correctly established, and the path-selection algorithm's importance score is not aligned with the path decomposition; consequently, the central claim that the selected path actually explains the decision is not yet supported. With substantial revision of the selection criterion and the theory, the approach could become a useful contribution to the interpretability literature.

major comments (3)
  1. [Section 3.3, Algorithm 1, lines 10-19] The importance score imp_j = softmax_c(W_{target,j} h_j^(n)) used for path selection measures the relative contrast of unit j's contribution to the target class versus other classes, not the magnitude of the path term f_p(X) from Eq. (2). A unit with a tiny value of |W_{target,j} h_j| can have imp_j close to 1 if all other class responses are even smaller, and it will pass the threshold alpha=1/|classes| even though every path through it contributes negligibly to the target logit. Conversely, a unit with a dominant path-term contribution but lower class-contrast can be discarded. Moreover, the algorithm does not enforce that selected units are activated (phi(h)=1); since h_j^(n) can be negative, a deactivated unit can have high imp_j, making f_p=0 even though W_p X is used as the attribution. The paper never reports the fraction of the target logit that the selected path carries. Because the central claim is that Algorithm 1 identifies the decision-making path, this lack of alignment between the selection objective and the path decomposition is a load-bearing gap. Please report the logit contribution of the selected path on test inputs, or change the selection criterion to one based on the actual path-term contribution, and explicitly filter to activated units.
  2. [Section 3.2 and Appendix A.2, Proposition 2] Proposition 2 states that for a multi-way complete path p, the piecewise linear model f_p(X) 'represents a summation of terms from Equation (2)'. As written, this is not correct. For a path containing two units h_i, h_j in the same hidden layer and a downstream unit h_k, the sum of the corresponding one-way terms from Eq. (2) is (W_{i,k} phi(h_i) + W_{j,k} phi(h_j)) phi(h_k), while f_p as defined in Eq. (5) is (W_{i,k}+W_{j,k}) phi(h_i) phi(h_j) phi(h_k). These coincide only when phi(h_i)=phi(h_j)=1. The proof in Appendix A.2 introduces a remainder alpha(h_i,h_j) but does not show that this remainder vanishes or otherwise complete the proof. This is not merely a cosmetic issue, because the multi-way path is used in all experiments with width>1, and the theoretical justification for the attribution map W_p X depends on the claim that f_p is the correct decomposition. The proposition should be restated with the appropriate activation-region qualification, and the proof should be completed, or the multi-way path explanation should be presented as a heuristic rather than as an exact decomposition.
  3. [Section 5.3, Table 1, and the Abstract] The quantitative support for the claim that the method 'outperforms others both quantitatively and qualitatively' is weak. In Table 1, the proposed method achieves the best insertion score (0.936) but is third-best among the compared methods on deletion (0.179, versus BlurIG 0.168 and IGs 0.176). No standard errors or significance tests are reported, and the experimental setup uses a curated 10-class subset of ImageNet without any description of the number of images per class or the train/test split. The Abstract and Conclusion overstate the results. Please report variance across multiple runs/samples, run appropriate significance tests (e.g., paired tests on the insertion/deletion AUC), and temper the claim to something like 'state-of-the-art or competitive on insertion, with competitive deletion performance' unless the additional analysis supports a stronger claim.
minor comments (5)
  1. [Section 3.1, Proposition 1] The main-text statement of Proposition 1 says f_p(X) is non-zero if and only if the path is activated, but the proof in Appendix A.1 adds the condition 'and (W_p X + b_p) is non-zero'. Please align the statement with the proof.
  2. [Section 3.3, Algorithm 1, lines 12-13] The softmax in lines 12-13 is applied over the class dimension of the matrix W diag(h^(n)), but the text says 'softmax(W^(N+1) h^(N))' without specifying the axis. Please clarify the axis and state the intended interpretation of the resulting importance score.
  3. [Section 5.3, Table 1] The hyperparameters depth and width used to produce the results in Table 1 (and the qualitative figures in Section 5.1-5.2) are not specified. Since the method's behavior depends strongly on these hyperparameters, please provide the exact values used.
  4. [Section 5, experimental setup] The curated 10-class subset of ImageNet is not described (which classes, how many images per class, how the subset was split). This makes the experiments difficult to reproduce; please include this information.
  5. [Abstract and Section 4] The term 'consistent' is used as a key advantage of the method, but it is never formally defined or quantified. Please provide a definition or a metric for consistency, or reframe the claim.

Circularity Check

1 steps flagged · score 4.0 of 10

Algorithm 1's target-contrast path selection makes the claimed 'consistent explanation' tautological, although the path decomposition itself is exact and the benchmarks are external.

  1. self definitional [Section 3.3 (Algorithm 1, Example 4) and Section 4 (Example 5)]
    "Then, we compute the importance of h(1)1 and h(1)2 : imp h(1)1 = e1×h(1)1 / (e1×h(1)1 + e−(1×h(1)1)) = 0.953, imp h(1)2 = e−1×h(1)2 / (e−1×h(1)2 + e−(−1×h(1)2)) = 0.269. We add h(1)1 to path(1). ... The top-k important units are those that increases the target class prediction value over predictions for other classes. ... our method offers a consistent explanation for predicting the white input as the positive class by using f p1 (x1,x2) instead of f p3 (x1,x2) even for the purple region."

    The 'consistency' exhibited for path p1 is not an independent finding: p1 is selected precisely because its unit-level softmax importance for the target class (0.953) exceeds the discarded unit's (0.269), i.e. it 'increases the target class prediction value over predictions for other classes.' The paper then cites f_p1's positive classification as evidence that the pathwise explanation is consistent. That conclusion is already contained in the selection criterion: any path chosen by this target-contrast rule will, by construction, point toward the target class.

full rationale

The mathematical path decomposition (Eqs. 2-7 and Propositions 1-2) is an exact unfolding of the ReLU network and is not circular. There is no load-bearing self-citation or imported uniqueness theorem. The circularity is localized to the passage from decomposition to explanation: Algorithm 1 defines importance as per-unit softmax target-contrast, and the paper's demonstration of a 'consistent explanation' in Example 5 is a restatement of that selection rule rather than an empirical or theoretical result. The insertion/deletion evaluations are external benchmarks and provide some independent content, as do the qualitative maps. However, because the central conceptual claim—that the method identifies the decision-making path and explains it consistently—is partially enforced by the selection objective, a moderate partial-circularity score is appropriate. Score 4 reflects partial circularity with substantial independent content remaining.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central mathematical identity (the path-sum expansion) is standard. The paper's own contributions are the path construction heuristic (Algorithm 1) and the choice of hyperparameters, which act as free parameters. No new physical or mathematical entities are introduced.

free parameters (3)
  • alpha (importance threshold) = 1/|classes|
    Set to the reciprocal of the number of classes in Algorithm 1 (Section 3.3). Appears without ablation; controls which hidden units are included in the path. This is a hand-chosen constant, not derived from the theory.
  • depth = varies from 1 to N (15 for VGG-16, 17 for ResNet-18, 4 for CIFAR10 toy model)
    User-selected number of layers included in the path. Experiments sweep this hyperparameter (Figures 8, E, G); the configuration used for the headline results in Table 1 is not stated, which acts as an implicit free parameter.
  • width = varies (e.g., 1, 2, 4, 8, 16, 32, 64, 128, 256, 512)
    Maximum number of hidden units per layer in the path, selected via top-k in Algorithm 1. As with depth, it is user-chosen and swept in the experiments.
assumptions (5)
  • standard math A ReLU feed-forward network can be unfolded as a sum over all one-way complete paths, as in Equation (2).
    This path-sum expansion is a known piecewise-linear representation of ReLU networks. The paper presents it as a starting point; it is not proven but is a standard algebraic identity.
  • standard math The property phi(x) + phi(-x) = 1 holds for the ReLU step function (Equation 8, used in the proof of Proposition 2).
    This is a direct property of the ReLU indicator function, used to decompose multi-way paths into one-way subsets.
  • standard math For a sub-network f1, the linearization W1 = d f1 / dX, b1 = f1(X) - W1 X is exact in the activation region of the input.
    ReLU networks are piecewise linear, so the local first-order Taylor expansion is exact within the same activation region. Invoked in Section 3.3 for incomplete paths.
  • ad hoc to paper The importance score softmax(W h^(n)) with top-k selection and threshold alpha identifies the hidden units that form the 'decision making path'.
    Algorithm 1 (Section 3.3) assumes this heuristic ranking captures the relevant decision process. No theoretical or empirical justification is given beyond the qualitative examples; the resulting explanation depends entirely on this choice.
  • domain assumption Neurons in higher layers of a trained CNN correspond to semantically meaningful features such as a fox's eye or ear.
    Section 5.1 states 'Assuming that neurons in the higher layers have high-level features like the fox's eye'; the explanation decomposition inherits this standard but not guaranteed property of deep features.

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Cite this review

Pith. "Pith review of Pathwise Explanation of ReLU Neural Networks." pith.science (2026). https://pith.science/paper/MPMJO4F6

@misc{pith2026250618037,
  author       = {Pith},
  title        = {Pith review of: Pathwise Explanation of ReLU Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MPMJO4F6}},
  note         = {Machine review of arXiv:2506.18037}
}
read the original abstract

Neural networks have demonstrated a wide range of successes, but their ``black box" nature raises concerns about transparency and reliability. Previous research on ReLU networks has sought to unwrap these networks into linear models based on activation states of all hidden units. In this paper, we introduce a novel approach that considers subsets of the hidden units involved in the decision making path. This pathwise explanation provides a clearer and more consistent understanding of the relationship between the input and the decision-making process. Our method also offers flexibility in adjusting the range of explanations within the input, i.e., from an overall attribution input to particular components within the input. Furthermore, it allows for the decomposition of explanations for a given input for more detailed explanations. Experiments demonstrate that our method outperforms others both quantitatively and qualitatively.

Figures

Figures reproduced from arXiv: 2506.18037 by the authors.

Figure 1
Figure 1. Comparison of the methods neural networks (Lapuschkin et al., 2019), and promoting consistent operation in our daily lives. ReLU is widely employed as an activation function due to its advantages, including mitigating the vanishing gradient problem and enabling efficient computation (Goodfellow et al., 2016). It is known that a Feed-Forward Neural Network with ReLU can be represented as a piecewise linear model (Sat… view at source ↗
Figure 2
Figure 2. Examples of paths with three hidden layers [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. An illustration of Proposition 1 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: Comparison of the proposed pathwise explanation and [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: An example of explanation decomposition through the [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: An example of the explanation for the incorrect predic [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: Quantitative explanation via insertion and deletion depending on the maximum number of units per layer (width) and the number of layers (depth) for path configuration. In our method, a depth of 15, which is the maximum, represents a complete path, while any lesser valu…
Figure 9
Figure 9. Figure 9: Qualitative explanation depending on the maximum number of units per layer ( [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

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