Tractable Shapley Values and Interactions via Tensor Networks

Chao Li; Farzaneh Heidari; Guillaume Rabusseau

REVIEW 4 major objections 4 minor 12 references

Tractable Shapley Values and Interactions via Tensor Networks

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Exact Shapley values can be computed in polynomial time for any model well approximated by a multilinear tensor-network surrogate, replacing the exponential coalition enumeration with a handful of structured evaluations.

desk verdict TN-SHAP is a real algorithmic contribution — polynomial-time Shapley and SII reads off a tensor-network surrogate — but it silently swaps the distributional coalition game for a zero-baseline game and never measures the epsilon its main theorem needs. read the letter →

arxiv 2510.22138 v3 pith:AHSFNLW3 submitted 2025-10-25 cs.LG

Farzaneh Heidari , Chao Li , Guillaume Rabusseau This is my paper · ORCID

classification cs.LG

keywords Shapleyvaluesinteractionindicestensornetworksmultilinearextensionpolynomialinterpolationmodelexplanationscoalitionalgameslow-rankapproximation

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 claims that the exponential cost of Shapley values and Shapley interaction indices — normally O(2^n) coalition queries — can be replaced by a few structured evaluations. The device is a multilinear tensor-network surrogate: when each feature is lifted to [φ_i(x_i),1] and multiplied by a diagonal selector Diag(t,1), every coalition of size s contributes a term proportional to t^s, so the whole sweep over coalitions collapses into interpolation of a single polynomial. An n×n solve recovers the size-aggregated marginal contributions, and Shapley weights turn them into attributions; the surrogate makes each evaluation cheap. The paper proves exactness on the surrogate plus a 2ε error bound to the true model if coalition values are uniformly ε-accurate, and reports millisecond attributions that are orders of magnitude faster than sampling baselines at comparable accuracy.

What carries the argument

The engine is the diagonal selector S_r(t)=Diag(t,1) applied to lifted feature vectors x̃_r=[φ_r(x_r),1]: it scales the data channel by t for included features and keeps a constant bias channel, so setting t=1 includes a feature, t=0 excludes it by zeroing its data channel. Contracting all selectors at once makes the output a univariate polynomial in t whose coefficients are the size-aggregated marginal contributions; a single Vandermonde solve extracts them. The multiplier is the tensor network itself: with bond dimension χ, each probe evaluation costs O(poly(χ)), cutting total cost from O(2^n) to O(n·poly(χ)+n^2).

What would settle it

On a dataset with n ≤ 12, fit the surrogate exactly as described, then exhaustively enumerate all 2^n coalitions and compute the worst-case coalition-value error between surrogate and true model under both the observed and interventional coalition definitions. If that supremum is not small compared with the output scale, Theorem 4.1's 2ε bound is vacuous and the reported accuracy is attributable to the surrogate's local fit rather than to the lemma.

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Extended reading notes

Core claim

TN-SHAP's central claim is that coalitional quantities — Shapley values and order-k Shapley interaction indices — are linear probes of the coefficient tensor of a multilinear map, so they can be extracted without visiting 2^n coalitions. For the probe G_i(t;x) that difference-includes feature i while scaling every other feature by t, multilinearity forces G_i to be a degree-(n−1) polynomial whose t^s coefficient aggregates all marginal contributions over coalitions of size s; evaluating at n distinct points and solving a Vandermonde system recovers these exactly, and the Shapley value is their size-weighted sum. The same construction, with inclusion–exclusion over a size-k subset and a signe

Load-bearing premise

The load-bearing premise is that a low-rank multilinear surrogate reproduces the true model's coalition values uniformly within a small ε — with switching a feature off implemented by zeroing its data channel while keeping a constant bias — so that the Shapley values computed on the surrogate are those of a game close to the original.

Editorial extensions

If this is right

All n single-feature Shapley values require only 2n² tensor-network forwards plus an n² linear solve, so attribution for models with hundreds of locally multilinear features becomes routine.
Pairwise interactions cost 4(n−1) forwards and order-k interactions cost 2^k(n−k+1) forwards (or one per probe via the signed-toggle identity), with the error bound scaling as 2^k ε.
Any surrogate whose coalition values are uniformly within ε of the true model yields Shapley values within 2ε, independent of the number of features.
Feature maps that are nonlinear per feature (polynomial, Fourier, or learned embeddings) can be inserted without breaking multilinearity, so surrogate fidelity and tractability are independently tunable.

Reading between the lines

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

The same probe-and-interpolate structure applies to any size-weighted coalitional index, not just Shapley and SII, so the method opens a route to exact semivalues of cooperative games of large n.
The decisive quantity for the guarantee — sup_C |v_g(C)−v_f(C)| — is never measured in the experiments; enumerating all coalitions for a small dataset would test how often the 2ε bound actually binds.
Since one surrogate is fitted per local cohort and reused for every point in it, the amortized training cost is modest, making the method attractive for explaining whole test sets rather than single instances.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: TN-SHAP does something useful. It gives a clean way to compute Shapley values and order-k Shapley interactions on a multilinear tensor-network surrogate with O(n·poly(χ)+n²) cost per index, replacing exponential coalition enumeration with diagonal probes and a Vandermonde solve. The algebraic core is Owen's multilinear-extension observation, but the packaging — selectors as trainable feature maps, TN contraction, local cohort amortization — is genuinely new and practically motivated. The central derivation in Section 3.2 checks out: a multilinear map evaluated at t-scaled inputs produces size-aggregated coefficients, and the Shapley weights recover the index. Theorem 4.1 is a correct triangle-inequality bound, and the experiments include rank ablations and a code link, which is more than many papers in this area ship.

The soft spot matters. Section 2.1 defines coalition values as conditional/marginal expectations over the data distribution, v_obs and v_int. The algorithm, however, probes the surrogate with selectors S_i(t)=Diag(t,1); at t=0 the feature channel is zeroed and the bias channel is set to 1. So excluded features are replaced by zero, not by a conditional mean or a draw from D. The paper never proves these two games coincide, and for standardized but correlated UCI features they generally do not. The stress-test note has this right: TN-SHAP computes Shapley values of the zero-baseline game v_g(C)=g({x_i}_{i∈C},{0}_{i∉C}), not necessarily of v_obs or v_int. Theorem 4.1's epsilon is then an assumption about the wrong comparison — or at best an unmeasured one. The experiments report accuracy against "exact" SII but don't say which game the exhaustive enumeration used. If it used zero-imputation on the teacher, the strong cosine numbers validate the surrogate fit, not the Section 2.1 quantity.

A smaller inconsistency: Table 1 and Algorithm 3 list 2^k forwards per probe for k-way interactions, while Section 10.3's signed-toggle identity claims one forward per probe. These can't both be the implemented complexity. The paper should say which is real and how Table 1 relates to it.

These are fixable. The right response is not rejection but a substantive revision: state the zero-baseline game explicitly, prove or disprove equivalence under concrete conditions (independence plus zero-mean standardization would do), and measure epsilon on held-out coalitions. I'd send it to a serious referee. The method is worth engaging with, and the flaw is in the framing, not in the core algebra.

For peer review: accept. This deserves referee time, with the semantic mismatch and the complexity inconsistency as the main review asks.

Referee Report

4 major / 4 minor

Summary. TN-SHAP proposes to compute Shapley values and order-k Shapley interaction indices by fitting a tensor-network (TN) surrogate that is multilinear in lifted feature vectors [phi_i(x_i),1]^T, then probing it with diagonal selectors S_i(t)=Diag(t,1). The core algebraic claim is that for a multilinear surrogate the size-aggregated marginal contributions are the coefficients of a degree-(n-1) polynomial, recoverable from n evaluations and one Vandermonde solve, reducing the exponential coalition enumeration to O(n*poly(chi)+n^2) per index. The paper provides a theoretical approximation bound (Theorem 4.1), a tractability theorem (Theorem 4.2), and experiments on synthetic and UCI regression tasks showing large wall-clock speedups over KernelSHAP-IQ. The central derivation on the surrogate is sound, but the paper does not establish that the zero-baseline game induced by S_i(0) matches the coalition-value definitions in Section 2.1, and the key approximation error epsilon is never measured.

Significance. The main algebraic insight is correct and useful: for a multilinear game v(C)=sum_{T subseteq C} c_T, Shapley values depend only on size-aggregated marginals, and those marginals can be recovered from O(n) diagonal probes. This is a clean, parameter-free identity that makes the exact computation on the surrogate genuinely tractable. The TN contraction machinery is a novel computational vehicle for this old observation and has the potential to be useful for low-rank or locally multilinear models. The paper also ships an implementation, which aids reproducibility. However, the significance for explaining the original model is contingent on two unverified premises: (i) that zero-imputation coalition values agree with the declared target v_obs/v_int, and (ii) that the surrogate achieves small uniform coalition-value error, epsilon. Theorem 4.1 is a mathematically valid conditional bound, but as a guarantee about the original model it is not connected to the experiments.

major comments (4)

[Sec. 2.1 vs. Sec. 3.2, Eq. (3.3)] The target game is not the game being computed. Section 2.1 defines v_obs and v_int as expectations over the data distribution D (Eqs. 2.1-2.2), but the selectors S_i(t)=Diag(t,1) replace an excluded feature by [0,1]^T (Eq. 3.3), giving the zero-baseline game v_g(C)=g({x_i}_{i in C},{0}_{i notin C}). No argument is given that these coincide for standardized but correlated UCI features. Theorem 4.1's assumption sup_C |v_g-v_f|<=epsilon therefore compares v_g with the wrong v_f unless v_f is also defined by zero-imputation. The paper should explicitly adopt zero-imputation as the intended Explainability semantics and discuss its implications, or change the probe to implement the declared baseline.
[Sec. 4, Theorem 4.1; Sec. 5] The uniform coalition-value error epsilon is never measured. The experiments report cosine similarity and MSE against exact teacher SII values, but not sup_C |v_g(C)-v_f(C)|. Thus the theoretical guarantee is not empirically validated, and the phrase 'comparable accuracy' is not tied to the theorem's hypothesis. Please report the uniform error (e.g., over a representative subset of coalitions) or explicitly state that the experiments validate only the surrogate-fidelity heuristic, not Theorem 4.1.
[Table 1, Sec. 4] The stated interaction error '2k epsilon' is incorrect for k>=3. For order-k SII, each inclusion-exclusion marginal Delta_C v(T) is a signed sum of 2^k coalition values, each with error at most epsilon, so the marginal error is at most 2^k epsilon; with SII weights summing to 1, the index error is at most 2^k epsilon. The k=3 entry should be 8 epsilon, not 6 epsilon. The proof sketch of Theorem 4.1 covers only single-feature indices and does not support the interaction entries. Please correct the bound and provide the derivation.
[Sec. 5.2 and Appendix 11.1] The TN surrogate fitting procedure is underspecified. The text describes a 'binary tensor-tree surrogate (rank chi=16)', an MLP feature map, a Gaussian neighborhood, and '2n^2 selector-weighted interpolation configurations', but never defines the loss function or the tensor-tree parameterization being optimized. Without the exact training objective (e.g., least-squares over teacher outputs at these configurations, with any regularization), the empirical results cannot be reproduced and it is unclear whether the selector probes are part of the training data or only of the inference. Please provide the full training protocol.

minor comments (4)

[Abstract and Table 1] The complexity statement is per index or per pair, not total for all interactions. The paper should state explicitly that computing all n Shapley values costs O(n^2*poly(chi)+n^3), and computing all pairs costs O(n^3*poly(chi)+n^4) under the per-pair algorithm, to avoid the impression that all order-2 interactions are obtained in O(n^2) total.
[Sec. 2.1 and Sec. 3.1] The initial exposition says 'setting x_j=1 for j in C and x_j=0 otherwise yields f(x)=sum_{T subseteq C} c_T' (Sec. 2.1, Eq. 2.3). This holds for binary inputs, but the method later uses actual values x_i and a separate bias channel. The text should clarify that this is a simplified motivation and that the actual probe uses the lifted features with x_i values retained.
[Sec. 5.1] The synthetic multilinear validation reports only runtime, not recovery error. Since this is the only setting where exactness on the surrogate is analytically clear, please report the actual Shapley/interaction recovery error against enumeration on the surrogate, not just wall-clock times.
[References] Several references are duplicated (Grabisch and Roubens 1999a/b, Muschalik et al. 2024a/b, Tsai et al. 2023a/b). Consolidate to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: TN-SHAP's exactness on the surrogate is a proven algebraic identity, and the error bound is a conditional Lipschitz argument; cited prior work is independent.

full rationale

The derivation chain is self-contained. Section 3.2 defines the diagonal probe G_i(t) and proves (via multilinearity) that its coefficients are exactly the size-aggregated marginal contributions of the surrogate; recovering them by interpolation and weighting by Shapley weights is an algebraic identity, not an input re-used as output. Theorem 4.1 is a conditional Lipschitz bound: if the surrogate's coalition values are uniformly epsilon-close to the target's, the index error is at most 2epsilon; the proof is a triangle-inequality calculation. No fitted Shapley values are used as training targets, and no load-bearing self-citation appears (references to Owen 1972, Roth 1988, and TN literature are external). The main caveats are validation gaps, not circularity: the paper never measures epsilon, and the selector-based game S_i(0)x_tilde_i=[0,1]^T is not shown to equal the expectation-based v_obs/v_int of Sec. 2.1 for correlated features, so the surrogate-fidelity premise is unverified; the Limitations section itself acknowledges the multilinearity assumption. These are correctness/fidelity concerns, not definitional reductions.

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

The method introduces no new physical or empirical entities. Its free parameters are surrogate capacity and local-training hyperparameters. The dominant axioms are the local-multilinearity assumption and the implicit zero-imputation coalition semantics, both of which are load-bearing for the claim that TN-SHAP computes the Shapley values of the original model.

free parameters (4)

TN bond dimension chi = 16 (15-20 in practice)
Chosen per experiment; controls surrogate capacity and the poly(chi) complexity factor. Not derived from first principles.
Feature-map MLP width = 64 ReLU units, one hidden layer
Used to smooth local behavior and improve multilinearity; no tuning procedure or sensitivity analysis is reported.
Local cohort size and Gaussian neighborhood = 100 k-NN neighbors; M≈50-100 with sigma=0.1×std
Hyperparameters for local surrogate training that affect fidelity, amortization, and the claimed 25-1000x speedups.
Number of structured probe configurations = 2n^2
Determined by the algorithm for all n features, but the choice of Chebyshev-Gauss nodes and their count affect interpolation conditioning.

assumptions (4)

domain assumption The local model behavior is sufficiently multilinear that a low-rank TN surrogate approximates all coalition values with small uniform error epsilon.
Required by Theorem 4.1; experiments report local R^2 and cosine similarity but never measure sup_C |v_g - v_f|.
ad hoc to paper Excluding a feature means replacing its data channel with 0 (bias-only), and this matches the desired coalition-value definition.
Implicit in Eqs. (3.2)-(3.4) and never justified against the observational/interventional definitions v_obs and v_int in Eqs. (2.1)-(2.2) for correlated features.
standard math Ten-sor-network forward contraction costs O(poly(chi)) and Vandermonde systems can be solved stably in O(n^2).
Standard results cited from Oseledets 2011 and Bjorck-Pereyra 1970; used throughout the complexity analysis.
standard math The signed-toggle identity (10.5) holds because g is multilinear in every lifted input channel.
A direct consequence of Eq. (2.4), but the paper never states whether experiments use the 2^k-forward inclusion-exclusion or the signed-toggle single-forward variant.

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Cite this review

Pith. "Pith review of Tractable Shapley Values and Interactions via Tensor Networks." pith.science (2026). https://pith.science/paper/AHSFNLW3

@misc{pith2026251022138,
  author       = {Pith},
  title        = {Pith review of: Tractable Shapley Values and Interactions via Tensor Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHSFNLW3}},
  note         = {Machine review of arXiv:2510.22138}
}

read the original abstract

We show how to replace the O(2^n) coalition enumeration over n features behind Shapley values and Shapley-style interaction indices with a few-evaluation scheme on a tensor-network (TN) surrogate: TN-SHAP. The key idea is to represent a predictor's local behavior as a factorized multilinear map, so that coalitional quantities become linear probes of a coefficient tensor. TN-SHAP replaces exhaustive coalition sweeps with just a small number of targeted evaluations to extract order-k Shapley interactions. In particular, both order-1 (single-feature) and order-2 (pairwise) computations have cost O(n*poly(chi) + n^2), where chi is the TN's maximal cut rank. We provide theoretical guarantees on the approximation error and tractability of TN-SHAP. On UCI datasets, our method matches enumeration on the fitted surrogate while reducing evaluation by orders of magnitude and achieves 25-1000x wall-clock speedups over KernelSHAP-IQ at comparable accuracy, while amortizing training across local cohorts.

Figures

Figures reproduced from arXiv: 2510.22138 by the authors.

**Figure 2.** Two TNs with selector matrices Sk(t): tensor train (left) and balanced binary tree (right). This applies selectors Sr(t) to all features r ̸= i (the "complement features"), while evaluating feature i in two configurations: • Included: Si(1)˜xi = [xi , 1]⊤ (feature present) • Off (bias): Si(0)˜xi = [0, 1]⊤ (feature absent) The difference isolates the marginal contribution of feature i across all coalitions of comple… view at source ↗

**Figure 3.** Concrete: runtime (log y) vs. cosine (x); k=2 (left), k=3 (right). TN-SHAP achieves higher similarity at millisecond scale. Dua and Graff, 2019; Tsanas and Xifara, 2012) MLP (Goodfellow et al., 2016) teachers) comparing TN surrogates to sampling baselines; and (3) Rank ablations & training dynamics on synthetic teachers to quantify capacity requirements and convergence behavior. 5.1 Validation on Synthetic Multili… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Training dynamics: R2 vs. epoch for the student fit (Train R2 ) and SII (k=1, 2, 3). thetic teachers where ground truth is exactly computable. Setup. We train student TNs of varying rank to approximate teachers with known structure: (1) a TNtree teacher with rank 16…

**Figure 5.** Figure 5: Rank ablation heatmap: safe-R2 for student ranks under a ground-truth tensor network of rank 14. Scaling Experiments. For dimensions d ∈ {10, 20, 30, 40, 50}, we generated CP-decomposed multilinear functions and fitted rank-16 tensor tree students. Each configuration …

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Reference graph

Works this paper leans on

12 extracted references · 5 linked inside Pith

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Adebayo, J., Gilmer, J., Muelly, M., Goodfellow, I., Hardt, M., and Kim, B. (2018). Sanity checks for saliency maps. InAdvances in Neural Information Processing Systems, pages 9505–9515. Ballester-Ripoll, R. (2022). Tensor approximation of cooperative games and their semivalues.Int. J. Ap- prox. Reasoning, 142(C):94–108. Ballester-Ripoll, R., Paredes, E. ...

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while maintaining controlled approximation error. The proofs rely on the multilinear structure of the surrogate and the size-aggregation property of Shapley weights, showing how diagonal selectors enable efficient extraction of all coalition values through polynomial interpolation. Proof of Theorem 4.1.Fix a center pointx. For any modelh, we abbreviate th...

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of each featureiin C. The signs implement inclusion–exclusion overC, while the selectorsSr(t)modulate the contribution of the F arzaneh Heidari, Chao Li, Guillaume Rabusseau remainingn−kfeatures. By multilinearity,Q C(t;x)is a univariate polynomial intof degree at mostn−k: QC(t;x) = n−kX s=0 c(C) s (x)t s, where each coefficientc(C) s (x)aggregates the co...

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Each configuration used 10 test points with exact ground truth for runtime comparison against KernelSHAP-IQ

Scaling Experiments.For dimensionsd∈ {10,20,30,40,50}, we generated CP-decomposed multilinear func- tions and fitted rank-16 tensor tree students. Each configuration used 10 test points with exact ground truth for runtime comparison against KernelSHAP-IQ. TN-SHAP achieved consistent millisecond-scale attribution across all dimensions, with runtime scaling...

2024
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For reproducibility, we fixed random seeds (seed=42 for synthetic experiments, seed=2711 for UCI benchmarks) and recorded complete hardware specifications for each run

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Adebayo, J., Gilmer, J., Muelly, M., Goodfellow, I., Hardt, M., and Kim, B. (2018). Sanity checks for saliency maps. InAdvances in Neural Information Processing Systems, pages 9505–9515. Ballester-Ripoll, R. (2022). Tensor approximation of cooperative games and their semivalues.Int. J. Ap- prox. Reasoning, 142(C):94–108. Ballester-Ripoll, R., Paredes, E. ...

arXiv 2018

[2] [2]

Grabisch, M

MIT Press. Grabisch, M. and Roubens, M. (1999a). An axiomatic approach to the concept of interaction among play- ers in cooperative games.International Journal of game theory, 28(4):547–565. Grabisch, M. and Roubens, M. (1999b). An axiomatic approach to the concept of interaction among play- ers in cooperative games.International Journal of Game Theory, 2...

arXiv 2010

[3] [7]

while maintaining controlled approximation error. The proofs rely on the multilinear structure of the surrogate and the size-aggregation property of Shapley weights, showing how diagonal selectors enable efficient extraction of all coalition values through polynomial interpolation. Proof of Theorem 4.1.Fix a center pointx. For any modelh, we abbreviate th...

1976

[4] [8]

The signs implement inclusion–exclusion overC, while the selectorsSr(t)modulate the contribution of the F arzaneh Heidari, Chao Li, Guillaume Rabusseau remainingn−kfeatures

of each featureiin C. The signs implement inclusion–exclusion overC, while the selectorsSr(t)modulate the contribution of the F arzaneh Heidari, Chao Li, Guillaume Rabusseau remainingn−kfeatures. By multilinearity,Q C(t;x)is a univariate polynomial intof degree at mostn−k: QC(t;x) = n−kX s=0 c(C) s (x)t s, where each coefficientc(C) s (x)aggregates the co...

2020

[5] [14]

Each configuration used 10 test points with exact ground truth for runtime comparison against KernelSHAP-IQ

Scaling Experiments.For dimensionsd∈ {10,20,30,40,50}, we generated CP-decomposed multilinear func- tions and fitted rank-16 tensor tree students. Each configuration used 10 test points with exact ground truth for runtime comparison against KernelSHAP-IQ. TN-SHAP achieved consistent millisecond-scale attribution across all dimensions, with runtime scaling...

2024

[6] [32]

E., editor (1988).The Shapley Value: Essays in Honor of Lloyd S

Roth, A. E., editor (1988).The Shapley Value: Essays in Honor of Lloyd S. Shapley. Studies in Game The- ory. Cambridge University Press, Cambridge, UK and New York, NY, USA. Rudin, W. (1976).Principles of Mathematical Analy- sis. McGraw–Hill, 3 edition. Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). Learning representations by back- propagat...

arXiv 1988

[7] [2010]

units, producing scalar outputs that are concatenated with a bias term. For every test instance, the training data consist of two parts: a Gaussian neighborhood—using eitherM= 100or the number of test samples, whichever is smaller, with standard deviationσ= 0.1×std(Xtrain)—and2n 2 structured selector- weighted probes placed at Chebyshev–Gauss nodes (Trefe...

2013

[8] [2012]

Allfeaturesandtargetswerestandardizedwithscikit-learn’sStandardScaler (Pedregosa et al., 2011)

regres- siontasksusedinourexperiments. Allfeaturesandtargetswerestandardizedwithscikit-learn’sStandardScaler (Pedregosa et al., 2011). For local explanations, we selected cohorts of 100 neighbors via k-NN in standardized feature space around each test instance. Table 5: Dataset characteristics used in our experiments. Dataset Task # Samples # Features Tar...

2011