NObSP: Functional Decomposition of Neural Networks via Oblique Subspace Projections
Alexander Caicedo, V\'ictor De La Hoz, Santiago Alf\'erez
cs.LG
Quick verdict
Decomposes NN predictions into per-feature functions via oblique projections; careful, honest paper, but functional-recovery evidence is synthetic-only.
Summary
The paper introduces NObSP, a post-hoc interpretability framework that decomposes a trained network's prediction into per-feature contribution functions plus an interaction residual. Exploiting the linearity of the final layer, it treats function evaluations as vectors in sample space and applies oblique subspace projections onto subspaces spanned by penultimate activations of isolated-feature inputs, reducing double counting when learned feature subspaces overlap. A partial-regression formulation (Theorem 4.1, Algorithm 1) cuts out-of-sample evaluation from O(N^2) to O(dz); NObSP-CAM produces backward-pass-free saliency after one-time calibration; contribution vectors double as retrieval embeddings. The authors connect the method to functional ANOVA and Kolmogorov-Arnold theory and prove uniqueness under subspace-intersection assumptions. Results: on a synthetic additive benchmark NObSP attains Function Reproduction Score 0.989 (KernelSHAP 0.966, Integrated Gradients 0.922); on tabular faithfulness correlation Integrated Gradients wins every dataset; on Region Deletion AUC across 24 vision configurations GradCAM is slightly but significantly ahead (difference -0.011, Holm-corrected p=0.0003); TinyImageNet kNN purity improves from 0.654 to 0.713 with halved neighbor distance. Main limitations: the interaction residual's magnitude is not quantified on real data, the only ground-truth functional-recovery test matches the method's additive assumption by construction, and vision calibration shares the evaluation split.
Review complete
Assessment at a glance
5
risk flags identified
● high 0
● medium 3
● low 2
5 strengths · 5 weaknesses · 5 author questions
Levels reflect verification complexity — not a quality score. Evolve never rates your paper.
Key claims under review
- For any network with a linear final layer, predictions decompose into per-feature contribution functions plus an interaction residual via oblique projections in sample space, uniquely when feature and reference subspaces intersect trivially (Theorems 3.1-3.3).
- A partial-regression formulation (Theorem 4.1, Algorithm 1) is equivalent to the oblique projector and reduces out-of-sample evaluation from O(N^2) to O(dz) with a single forward pass per feature.
- On a synthetic additive benchmark with known components, NObSP recovers the true contribution functions more accurately than KernelSHAP and Integrated Gradients (FRS 0.989 vs 0.966 vs 0.922).
- On standard faithfulness metrics NObSP is comparable but not superior: Integrated Gradients leads tabular faithfulness correlation on all nine datasets, and GradCAM leads Region Deletion AUC by a small statistically significant margin across 24 vision configurations.
- NObSP contribution vectors form retrieval embeddings that raise TinyImageNet nearest-neighbor class purity from 0.654 to 0.713 (K=8) and more than halve mean neighbor distance, without requiring ground-truth labels at inference.
Strengths
The central claim is tested with the right metric
Sections 5.1.4 and 6.1.2 evaluate function-level recovery against known ground-truth components (FRS, Table 8: 0.989 vs 0.966 KernelSHAP, 0.922 IG), including a null-feature false-positive check (0.009 vs 0.014/0.022) and a per-feature breakdown in Supplementary Table 3 (e.g., cubic: 0.986 vs 0.952/0.839) — exactly what a decomposition method should be judged on.
Unusually transparent negative results
The paper states plainly (Sections 6.1.1, 6.4, 7.3) that Integrated Gradients attains the best tabular faithfulness on all nine datasets (Table 7), GradCAM is significantly ahead on Region Deletion AUC (aggregate -0.011, CI [-0.012,-0.009], p=0.0003), and KernelSHAP wins local pairwise agreement (LCPDA) on 6 of 9 real datasets (Supplementary Table 4). This honesty is rare and strengthens trust.
Rigorous statistical reporting for the vision comparison
Practical algorithmic contribution with complete proofs
Nuanced treatment of the embedding's failure mode
Weaknesses
Interaction residual is never quantified on real data
Ground-truth evaluation matches the method's assumption by construction
Baseline sensitivity unexamined despite a PDP-style extrapolation concern
Vision calibration and metric evaluation share the same split
Marginal vision benefit over existing gradient-free options
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Frequently asked questions
What is NObSP and how does it explain a neural network?
NObSP (Nonlinear Oblique Subspace Projections) is a post-hoc interpretability method. For a trained network whose final layer is linear, it runs isolated inputs (only one feature active at a time) through the network, collects the penultimate activations, and uses oblique subspace projections to split each prediction into one contribution curve per feature (how that feature's value pushes the output up or down) plus a leftover interaction term. Unlike SHAP, which gives one importance number per feature, NObSP gives the full function of feature value versus contribution.
How is NObSP different from SHAP?
Both relate to functional ANOVA decompositions. SHAP allocates interaction effects among features and outputs a single scalar contribution per feature per prediction, answering 'how much does this feature matter?'. NObSP keeps the main effect of each feature as an explicit function of its value and leaves interactions in a separate residual, answering 'how does this feature's contribution behave across its range?'. The two are complementary rather than competitors; in the paper's own benchmarks SHAP-like KernelSHAP remains stronger on several real-data local-agreement tests.
Does NObSP outperform GradCAM or Integrated Gradients?
Not on the standard faithfulness metrics reported. Integrated Gradients achieves the best tabular faithfulness correlation on all nine datasets, and GradCAM is slightly but statistically significantly better than NObSP-CAM on Region Deletion AUC across 24 vision configurations (mean difference about 0.011). NObSP's advantage appears where ground-truth component functions are known: on a synthetic additive benchmark it reconstructs the true per-feature functions most accurately (score 0.989 versus 0.966 for KernelSHAP and 0.922 for Integrated Gradients), and it produces contribution functions rather than scalar scores.
Is NObSP code publicly available and is it expensive to run?
The paper links a reference implementation (github.com/santialferez/nobsp_lib). NObSP requires a one-time calibration on 1,000-5,000 samples to estimate its coefficient vectors; reported calibration times range from under a second (440-dimensional penultimate layers) to tens of minutes or hours for 1,280-dimensional layers on a single RTX 3090. After calibration, explaining a new sample costs one forward pass plus an inner product per feature, and NObSP-CAM needs no backward pass.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- The extracted text is complete through the supplementary material, but figures (Fig. 2 recovery curves, Figs. 3-5 CAM and retrieval panels) cannot be inspected; qualitative claims about DermaMNIST background suppression and retrieval panels rest on the authors' descriptions.
- Proofs were only available as sketched in the supplementary text; the underlying linear algebra (oblique projector identity, partial-regression equivalence) is standard and the derivations shown appear correct, but full verification was not performed against the linked code.
- Per-feature FRS and LCPDA supplement tables were included in the extraction; any additional supplement content (e.g., faithfulness correlation for the full 24 vision configurations) was summarized only via main-text references.
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