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REVIEW 3 major objections 4 minor 60 references

OMENN: One Matrix to Explain Neural Networks

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that a neural network can be collapsed into a single matrix per input whose entries sum exactly to the model's output logit.

desk verdict OMENN's exact completeness for smooth activations is real and worth knowing, but the empirical paper as written overclaims and hides a VGG failure in the supplementary. read the letter →

arxiv 2412.02399 v1 pith:KR2XH2QL submitted 2024-12-03 cs.LG cs.AIcs.CV

Adam Wróbel , Miko{l}aj Janusz , Bartosz Zieliński , Dawid Rymarczyk This is my paper · ORCID
classification cs.LGcs.AIcs.CV
keywords explainableAIattributionmethodsdynamiclinearityneuralnetworkinterpretabilityVisionTransformerconvolutionalcompletenesspropertypost-hocexplanation
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 introduces OMENN (One Matrix to Explain Neural Networks), a post-hoc method that represents a trained neural network as a single matrix for each input. The authors show that every layer of a Vision Transformer or CNN—convolutions, attention, normalization, residual connections, and GELU or SWISH activations—can be rewritten as an input-dependent affine transformation. Composing these transformations collapses the whole network into one matrix, and the paper's central claim is that the sum of all entries in the explanation matrix equals the network's output logit for the explained class. This exact completeness, if it holds, would make OMENN's pixel attributions faithful by construction rather than approximate, and it would apply to already-trained models without retraining or architectural changes. The authors also report benchmark results on FunnyBirds and a faithfulness metric where OMENN is competitive with or exceeds existing attribution methods.

What carries the argument

The key machinery is dynamic linearity: each layer $l_i(x)=W_i(x)x + b_i(x)$ acts as an affine transformation whose parameters depend on the input. The paper establishes this form for fully-connected layers, convolutions via doubly block-Toeplitz matrices and im2col, normalization, residual connections, multi-head self-attention, and dynamic activations such as GELU and SWISH, which factor as $\phi(x)=x\,\xi(x)$ and become diagonal linear maps. To absorb additive biases, the input is augmented with an extra channel of ones and layer parameters are merged into augmented matrices, so that a whole network collapses into a single matrix $\tilde{\Omega}_w$. The explanation matrix $C$ is the Hadamard-product decomposition of that collapse, and equation (27) states that its entries sum exactly to the network output, which is the completeness property that carries the method's claims of exactness.

What would settle it

Evaluate OMENN on the reported benchmarks using the raw matrix before post-processing; if the raw version scores no better than the gradient baseline, the completeness property does not carry the empirical results.

Watch

Extended reading notes

Core claim

The central discovery is that a broad class of modern networks can be exactly re-expressed as one input-dependent affine map, yielding an attribution matrix whose entries sum to the output. The construction augments the input with a constant channel of ones so that biases become part of a combined weight matrix, then multiplies the per-layer linear maps together. For the explanation matrix $C$ defined as $C = \sum_d (C_w \odot X)[:,d] + C_b$, with $C_w$ the input-multiplying part and $C_b$ the bias part, the identity $\sum C = f(x)$ holds. The paper contrasts this with gradient-based saliency, which assigns zero contribution to a GELU input at $x=-0.75$ even though the function value is $-0.17$, while OMENN's local linear coefficient captures that contribution exactly.

Load-bearing premise

The paper's reported benchmarks treat the post-processed explanation maps as OMENN, even though the post-processing steps (dropping negative entries, quantile clipping, smoothing) break the exact sum-to-logit equality that the method's completeness guarantee depends on.

Editorial extensions

If this is right

  • OMENN can be applied to pre-trained ViTs and CNNs without any retraining or architecture change, unlike B-Cos or QSENN, which require modifying the model.
  • For piecewise-linear activations such as ReLU, OMENN's explanation coincides with image times gradient; for GELU and SWISH it departs from the gradient, correcting cases where the gradient is zero despite a nonzero contribution.
  • The exact sum-to-logit property provides a built-in consistency check: any explanation that does not sum to the logit can be identified immediately, and the raw OMENN matrix can serve as ground truth for evaluating other attribution methods.
  • On the FunnyBirds benchmark, OMENN achieves the highest mean score on ViT-B/16 and VGG-16 among the compared methods, and on the Quantus faithfulness metric it roughly doubles the score of the second-best method for ViT-B/16.

Reading between the lines

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

  • Because the raw matrix preserves exact completeness while the post-processed maps do not, a natural extension is to evaluate whether the reported benchmark gains survive when using the raw matrix; this would separate the value of the exact decomposition from the value of the smoothing step.
  • The same dynamic-linearity collapse should extend to other architectures built from GELU/SWISH and attention, such as transformer decoders or hybrid models; testing OMENN on a text or audio model would check whether the proof's assumptions travel beyond vision.
  • The paper's completeness identity suggests a new diagnostic: the difference between the raw OMENN sum and a post-processed map's sum measures how much information the visualization pipeline discards, which could guide the design of explanation displays that preserve faithfulness.
  • Another testable extension is to use the OMENN matrix directly in perturbation-based faithfulness evaluations, since the raw matrix gives exact per-pixel contributions; comparing raw versus smoothed maps on existing benchmarks would quantify the cost of interpretability post-processing.
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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 / 4 minor

Summary. The paper introduces OMENN, a post-hoc attribution method that represents a neural network as a single input-dependent matrix for a given input, based on the 'dynamic linearity' property that each layer can be expressed as an affine or input-dependent affine transformation. The authors derive how fully-connected, convolutional, normalization, attention, and activation layers (including GELU and SWISH) can be reformulated, and show that the resulting explanation matrix C satisfies the completeness property sum(C) = f(x) (Eq. 27). The method is evaluated on the FunnyBirds and Quantus faithfulness benchmarks with ViT and CNN backbones, where the authors claim competitive or superior performance. The theoretical derivations in Sections 3.2-3.5 are mathematically sound for the listed layers, and the completeness identity is a clean contribution. However, the paper suffers from a major ambiguity about whether the reported experiments use the raw matrix or the post-processed version described in Section 4, and the main text's claim of consistent outperformance is contradicted by the supplementary VGG-16 faithfulness table.

Significance. If the exactness and completeness properties are taken as the core contribution, OMENN offers a principled way to obtain pixel-level attributions that exactly reconstruct the logit for modern architectures with smooth activations. The derivation is clear and the completeness property is a rigorous, parameter-free guarantee for the raw matrix. This distinguishes OMENN from heuristic gradient-based methods and provides a potential tool for exact decomposition in ViTs. The authors also correctly identify the limitations of gradient-based attributions at near-stationary points. However, the empirical evaluation does not currently establish that the proposed method, as actually used, retains these guarantees: the post-processing steps in Section 4 break the completeness identity, and the benchmark numbers may reflect a different pipeline than the theoretical object. The missing comparison with FullGrad, the closest related exact method, is a significant gap. Thus the theoretical contribution is promising but the empirical evidence, as presented, is not yet convincing.

major comments (3)
  1. [Section 4, Efficient Implementation; Eq. (27)] The paper claims a completeness property sum(C) = f(x) for the explanation matrix C, but Section 4 states that, after computing C, the authors 'disregard all elements with negative contributions' and apply 'quantile-based outlier removal and smoothing using a mean filter.' These operations change the matrix, so the post-processed map no longer satisfies Eq. (27). The manuscript never states whether the reported FunnyBirds results (Table 3, Figure 4) and faithfulness scores (Tables 1-2) use the raw matrix or the post-processed one, nor does it report the quantile threshold and filter kernel. If the evaluations use the post-processed maps, then the empirical validation is of 'OMENN + clipping + smoothing', not of the exact decomposition that is the paper's central theoretical contribution. The authors must report raw and post-processed results separately and provide the post-processing hyperparameters, or modify the method to preserve completeness under post-processing (e.g., by redistributing discarded contributions). Without this, the connection between theory and experiments is unsubstantiated.
  2. [Section 5, Comparison of Faithfulness; Supplementary Table 2] The main text states that OMENN 'consistently outperforms all competing approaches' on the faithfulness benchmark, based on the ViT-B/16 results in Table 1. However, the supplementary Table 2 for VGG-16 reports an OMENN faithfulness score of 0.023 ± 0.003, which is lower than GradCAM (0.065), LRP (0.057), and Integrated Gradients (0.045). The main text only remarks that 'results for convolutional model are provided in the Supplementary Materials,' without acknowledging that the reported numbers are not superior for VGG-16. This is a direct contradiction of the 'consistently outperforms' claim. The authors should either revise the claim to make it backbone-specific, or provide a discussion explaining the VGG-16 result and what it implies for the applicability of OMENN.
  3. [Sections 2 and 5; related work [53]] The paper motivates OMENN partly by contrast with FullGrad (Gradient NN Representation, cited as [53]), which provides an exact gradient+bias explanation but is restricted to ReLU/LeakyReLU networks. FullGrad is the most closely related method to OMENN, and it is directly applicable to the VGG-16 and ResNet-50 backbones used in the FunnyBirds and faithfulness experiments. Yet FullGrad is not included in any benchmark comparison (Tables 1-3, Figure 4). Without comparing to FullGrad, the claimed advantage of OMENN — extension to smooth activations while maintaining exactness — is not empirically tested. The authors should either include FullGrad in the evaluations or explicitly justify its absence.
minor comments (4)
  1. [Section 4] The post-processing hyperparameters (quantile cutoff, mean filter size) are not reported anywhere in the manuscript or supplementary material. Please specify these values, as the benchmark results depend on them.
  2. [Supplementary, Figure 9 caption] The word 'tremnedous' in the caption should be 'tremendous'.
  3. [Section 3.2] There is a typo: 'SWISH (Self-Gated Activation Funstion)' should be 'Self-Gated Activation Function'.
  4. [Section 3.6] The paper uses 'exact' and 'precise' to describe OMENN, but the post-processing in Section 4 breaks the exactness of the raw matrix. It would be clearer to explicitly state that the theoretical exactness applies only to the raw decomposition, and that the reported qualitative and quantitative results may use a modified map.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the completeness identity is derived from the network's own weights by an explicit algebraic construction, with no fitted parameters or load-bearing self-citation.

full rationale

The derivation chain in OMENN is an explicit algebraic construction: each layer is rewritten as an input-dependent affine map (Eqs. 4, 11, 14, 16), the composition is collapsed to a single matrix product (Eqs. 17-19), biases are folded into an augmented input with a channel of ones (Eqs. 20-23), and the final completeness statement sum(C) = f(x) (Eq. 27) follows by direct substitution from Eq. 26 and Eq. 24. No parameter is fitted to any benchmark; the equality is exact by construction and does not import any result from the authors' prior work. The related-work citations to prototype papers by the same group (Refs. 43, 48, 49) are contextual and not load-bearing for the OMENN derivation. The only notable gap is that Section 4's post-processing (negative-value suppression, quantile outlier removal, mean filtering) breaks Eq. 27, and the paper does not state whether benchmark tables report raw or processed OMENN maps; this is an evaluation-transparency issue rather than circular reasoning, so it does not raise the circularity score.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The method has no learned parameters, but it relies on a domain assumption about layer expressibility and an ad hoc choice for smooth activations. The post-processing hyperparameters are unquantified and could be tuned to benchmarks.

free parameters (3)
  • post-processing quantile threshold = not specified
    Section 4: quantile-based outlier removal; threshold value not given and may be tuned to benchmarks.
  • mean filter size = not specified
    Section 4: smoothing using a mean filter; kernel size not given.
  • negative contribution policy = all negative contributions set to zero
    Section 4: 'we disregard all elements with negative contributions'; this is an ad hoc binary filter that breaks the completeness sum.
assumptions (4)
  • domain assumption Every layer in the target networks can be expressed as an affine or input-dependent affine transformation (Eq. 4).
    Section 3.1 assumes this and gives examples for fully-connected, attention, activations, normalization, convolution, residual connections. It is not proved for all possible layers, e.g., max-pooling or custom ops.
  • ad hoc to paper For GELU/SWISH, the identity phi(x)=x*xi(x) with xi(x)=Phi(x) or sigma(x) is the appropriate contribution decomposition.
    Section 3.2 defines dynamic linear activation functions with this multiplier; there are infinitely many such decompositions, and the paper gives no criterion selecting this one.
  • standard math The Kronecker product identities (Eqs. 6-9) hold for the used dimensions.
    Used in fully-connected and attention vectorization; standard linear algebra.
  • domain assumption The augmented channel of ones propagates through all layers, including convolution with zero padding, without changing the output (Eqs. 20-22).
    Supplementary section 'Combining parameters' defines how the ones channel is carried through convolution (with the K_bi kernel and normalization by s_i), attention, and activations. This requires careful bookkeeping and is not automatically true for all padding/stride choices.

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Cite this review

Pith. "Pith review of OMENN: One Matrix to Explain Neural Networks." pith.science (2026). https://pith.science/paper/KR2XH2QL

@misc{pith2026241202399,
  author       = {Pith},
  title        = {Pith review of: OMENN: One Matrix to Explain Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KR2XH2QL}},
  note         = {Machine review of arXiv:2412.02399}
}
read the original abstract

Deep Learning (DL) models are often black boxes, making their decision-making processes difficult to interpret. This lack of transparency has driven advancements in eXplainable Artificial Intelligence (XAI), a field dedicated to clarifying the reasoning behind DL model predictions. Among these, attribution-based methods such as LRP and GradCAM are widely used, though they rely on approximations that can be imprecise. To address these limitations, we introduce One Matrix to Explain Neural Networks (OMENN), a novel post-hoc method that represents a neural network as a single, interpretable matrix for each specific input. This matrix is constructed through a series of linear transformations that represent the processing of the input by each successive layer in the neural network. As a result, OMENN provides locally precise, attribution-based explanations of the input across various modern models, including ViTs and CNNs. We present a theoretical analysis of OMENN based on dynamic linearity property and validate its effectiveness with extensive tests on two XAI benchmarks, demonstrating that OMENN is competitive with state-of-the-art methods.

Figures

Figures reproduced from arXiv: 2412.02399 by the authors.

Figure 1
Figure 1. This figure illustrates OMENN, a novel XAI method [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. OMENN method provides an explanation in the form of a single matrix, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. An example highlighting the difference between our [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Results on the FunnyBirds benchmark for the ViT-B/16 backbone show that OMENN achieves the highest score among various explainability methods. Notably, Chefer LRP ranks as the second-best approach, with only a marginal difference from OMENN, indicating that these metho…
Figure 5
Figure 5. Figure 5: Examples of explanations generated by various XAI [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 8
Figure 8. Figure 8: Examples of explanations generated by various XAI [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 6
Figure 6. Figure 6: Examples of explanations generated by various XAI [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Examples of explanations generated by various XAI [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 9
Figure 9. Figure 9: Results of OMENN on FunnyBirds framework using CNN backbones, VGG16 and ResNet50. OMENN surpasses all other XAI [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.