Assessing high-order effects in feature importance via predictability decomposition

Daniele Marinazzo; Jesus M Cortes; Luca Faes; Marlis Ontivero-Ortega; Sebastiano Stramaglia

REVIEW 4 major objections 4 minor 22 references

Assessing high-order effects in feature importance via predictability decomposition

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

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Pith's one-line read This paper claims that a single Leave One Covariate Out importance score can be decomposed into unique, redundant, and synergistic components by comparing the smallest and largest LOCO values over feature subsets.

desk verdict Useful heuristic for decomposing LOCO, but the synergy/redundancy labels ride on an unproven analogy and need external benchmarking. read the letter →

arxiv 2412.09964 v2 pith:CI2DBBF3 submitted 2024-12-13 physics.data-an stat.ML

Marlis Ontivero-Ortega , Luca Faes , Jesus M Cortes , Daniele Marinazzo , Sebastiano Stramaglia This is my paper · ORCID

classification physics.data-anstat.ML

keywords featureimportanceLeaveOneCovariateOutpredictabilitydecompositionsynergyredundancyhigh-orderinteractionsexplainableartificialintelligencesuppressorvariables

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 usual single-number feature importance score can hide how features cooperate. It introduces Hi-Fi, a version of Leave One Covariate Out that searches for the subset of other features which most lowers the score and the subset which most raises it. The gap between those extremes is read as redundancy and synergy, and the leftover baseline is the feature's unique contribution. If right, a model's explanation can distinguish a feature that acts alone from one that matters only together with others, and can name which partners matter. The method is tested on a toy model, wine quality data, and proton/pion discrimination from simulated detector signals.

What carries the argument

The carrying identity is Eq. (2), $L_{z_{\max}}(X \to Y) = S + R + U$, with $U = L_{z_{\min}}$, $R = L_{\emptyset} - L_{z_{\min}}$, and $S = L_{z_{\max}} - L_{\emptyset}$, where $L_z$ is LOCO over subset $z$ and $L_{\emptyset}$ is the pairwise explained variance. The identity converts the definitions of redundancy as information retrievable from multiple sources and synergy as information available only from the whole into a search over feature subsets. The subsets are found exhaustively for small $n$ and, for larger $n$, by a greedy one-variable-at-a-time search with a permutation-based stopping rule.

What would settle it

For a small Gaussian system where a full information-theoretic partial information decomposition can be computed exactly, compare its redundancy and synergy values with Hi-Fi's $R$ and $S$; a qualitative disagreement across correlation strengths would show that the min/max LOCO labels are not tracking the quantities they are named after.

Watch

Extended reading notes

Core claim

On the paper's own terms, LOCO computed with the best subset $z_{\max}$ decomposes exactly as $L_{z_{\max}} = S + R + U$. Here $U$ is the LOCO value at the subset that minimizes it, the unique two-body contribution; $R$ is the gap between the pairwise explained variance and that minimum, the redundancy shared with other features; and $S$ is the gap from pairwise variance to the maximizing LOCO, the synergy supplied by partners. The authors therefore claim that the maximal predictive power of $X$ to $Y$ is a sum of a pure two-body effect and two cooperative effects, and that the optimizing subsets name which other features participate.

Load-bearing premise

The load-bearing premise is that the smallest and largest LOCO values over feature subsets can be read as redundancy and synergy; the paper states this mapping rather than proves it, and the variance-based measure deliberately gives up formal information-theoretic guarantees.

Editorial extensions

If this is right

Features that are redundant with others look less important under standard LOCO; Hi-Fi restores them by assigning the lost predictive power to $R$.
Features that matter only in combination, such as suppressors, are missed by pairwise explained variance; Hi-Fi assigns their effect to $S$ and names the partners.
In the particle discrimination example, $\beta$ and $p$ are synergistic because together they encode particle mass, while the detector responses are redundant with $\beta$ and $p$ because of Bethe-Bloch energy loss.
On the datasets analyzed, the greedy search reaches the same decomposition as exhaustive search, so the method can scale beyond the toy case.
The components depend on the chosen regression hypothesis space, so robustness can be checked by repeating the analysis with several models.

Reading between the lines

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

Beyond the paper's demonstrations, a direct check of the nomenclature would compute a true partial information decomposition on a Gaussian system and compare its $R$ and $S$ with Hi-Fi's components; the paper does not supply this validation.
The greedy search's stopping rule has only been compared with exhaustive search on small-$n$ datasets; in high-dimensional feature sets the global min or max could be unreachable, so the labels may depend on the search path.
Because the decomposition is built on prediction-error variance, the same recipe could be applied to other losses, making Hi-Fi a general way to expose cooperation in classification or survival models, not just regression.
The synergy term effectively detects suppressor variables, so Hi-Fi could double as a screening tool for suppression in regression diagnostics.

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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

Here's my take: the paper is a useful heuristic wrapped in an overclaim. The new move is replacing conditional mutual information with LOCO in the transfer entropy decomposition from [13], yielding a variance-based decomposition of feature importance into unique, redundant, and synergistic parts. That's a real and potentially practical extension for XAI, and the authors are transparent about sacrificing information-theoretic rigor. The toy example is well-constructed: linear suppressor effects for synergy via correlation, a nonlinear product term for structural synergy, and positive correlations for redundancy. Code is available, and the real-data applications (wine, GEANT particle discrimination) give plausible, interpretable results. The explicit comparison with DIP [22] is also a plus; the complementarity claim seems fair.

The soft spot is load-bearing, and it's the one the stress test identifies. The definitions U = Lzmin, R = L∅ - Lzmin, S = Lzmax - L∅ make Eq. (2) an algebraic identity. The semantic content rests entirely on the assertion that the subset minimizing Lz captures redundancy and the subset maximizing Lz captures synergy. That assertion is by analogy with [13], not derived. In PID, redundancy and synergy are axiomatic properties of how information from multiple sources combines; there is no reason the min and max of a single-variable LOCO over subsets must coincide with those quantities. The toy example is set up so the mapping works, so it doesn't independently validate the mapping. Given that the paper's novelty is precisely the semantic decomposition, the interpretations in Figures 3-6 go beyond what is established.

That said, I don't think this is fatal if framed differently. The method defines a well-specified, computable decomposition of LOCO: a baseline two-body component, a component obtained by removing the most redundant multiplet, and a component obtained by adding the most synergistic multiplet. That is a coherent exploratory tool even without the PID pedigree. What's missing is a benchmark: compare the Hi-Fi labels against an established PID measure on data where PID is computable (small n, discrete or Gaussian), to show that min/max LOCO actually tracks redundancy and synergy in the PID sense. The paper also lacks error bars and cross-validation. With n up to millions the in-sample estimates are probably fine, but for wine (n=6497) bootstrapping would have been easy. The greedy/exhaustive equivalence is stated without details; they say they verified it on these datasets, but no results are shown, and the greedy search procedure is only briefly described.

Who is this for? XAI practitioners who want a quick, variance-based way to see cooperative effects. They should treat U/R/S as descriptive of the regression model, not as information-theoretic quantities. The paper deserves peer review — the idea is worth airing, but it needs revision to either soften the semantic claims or provide external validation. I'd recommend sending it out, with a request for a PID benchmark and uncertainty quantification.

Referee Report

4 major / 4 minor

Summary. The paper proposes Hi-Fi, an adaptive version of Leave One Covariate Out (LOCO) in which, for a given driver variable X, one searches for the subset of covariates that minimizes and the subset that maximizes the LOCO value. Defining U = L_{zmin}, R = L_empty - L_{zmin}, and S = L_{zmax} - L_empty leads to the identity L_{zmax} = S + R + U (Eq. 2). The authors interpret U, R, and S as unique, redundant, and synergistic predictive contributions, respectively, and illustrate the method on a constructed Gaussian toy model, a wine-quality dataset, and a GEANT-simulated proton/pion discrimination problem. A greedy search with permutation-based stopping is used for scalability, and code is provided.

Significance. If the min/max LOCO decomposition can be rigorously connected to information-theoretic redundancy and synergy, Hi-Fi would be a simple and computationally efficient extension of one of the most widely used feature-importance measures. The algebraic identity is correct, the code is open, and the particle-physics interpretation involving the Bethe-Bloch formula is physically plausible. However, the central conceptual contribution currently rests on an analogy that is asserted rather than derived, and the empirical claims lack any form of uncertainty quantification. The paper is therefore potentially useful but needs substantial additional support before the labels 'unique', 'redundant', and 'synergistic' can be taken as factual descriptions of feature behavior.

major comments (4)

[Eq. (2) and surrounding definitions] The decomposition in Eq. (2) is an algebraic identity: for any subsets zmin and zmax, L_{zmax} = L_{zmin} + (L_empty - L_{zmin}) + (L_{zmax} - L_empty). The substantive claim is that the subset minimizing Lz 'captures the amount of redundancy R' and the subset maximizing Lz 'leads to the amount of synergy S,' but this mapping is asserted rather than derived. The paper's own concession that replacing entropy with variance 'sacrifices the formal rigor of information theory' (citing [11]) means that the labels unique, redundant, and synergistic are not automatically the PID quantities. I request either a formal derivation showing under which conditions min/max LOCO equals the corresponding PID components, or a benchmark against a recognized PID measure on small synthetic systems, or a clear reframing of the method as a heuristic whose names are only analogical. As written, the interpretations in Figs. 1, 3, and 5 are not supported as factual statements about information-theoretic redundancy and synergy.
[Toy example and Figs. 1-2] The toy example does not independently validate the mapping. Because U, R, and S are defined directly from Lzmin and Lzmax, a correctly implemented computation will always recover the labels used to construct the example. The example shows internal consistency, but it does not show that min/max LOCO tracks redundancy and synergy in the information-theoretic sense. A benchmark in which the true PID redundancy and synergy are known, such as a comparison against a recognized PID measure on the same Gaussian or categorical system, would be needed.
[Greedy search and final paragraph] The manuscript states that exhaustive evaluation gave the same results as the greedy search on the analyzed datasets, but it does not report the exhaustive-search procedure, the stopping thresholds, or any comparison. Because the greedy stopping rule involves a permutation-test significance threshold and multiple-comparison correction, the reader cannot assess whether the reported multiplets are stable or artifact-driven. Please provide the implementation details and either the exhaustive results or a sensitivity analysis across thresholds.
[Empirical results and Figs. 3-6] None of the empirical claims in Figs. 3-6 are accompanied by uncertainty quantification. The predictions are evaluated on the training data (per footnote 14), and the differences between, for example, the synergistic role assigned to residual sugar and the beta-p synergy are presented as point estimates only. The absence of confidence intervals or cross-validated errors makes it impossible to judge whether the reported differences are meaningful. This is load-bearing for the paper's concrete claims about wine quality and particle discrimination.

minor comments (4)

[Fig. 3 caption] The caption of Fig. 3 refers to the 'particle discrimination problem,' but the figure shows the Wine Quality dataset; the caption should be corrected.
[Text near reference [19]] The text says 'UCL Machine Learning Repository'; the correct name is the UCI Machine Learning Repository.
[Reproducibility details] The polynomial kernel degree is stated, but regularization or kernel scaling parameters, if any, are not specified, which makes exact reproduction difficult despite the provided code.
[Notation] The notation for the full set Z and the subset z is visually similar; please distinguish them clearly, for example by using bold or calligraphic symbols, to avoid ambiguity in Eqs. (1) and (2).

Circularity Check

2 steps flagged · score 5.0 of 10

The decomposition in Eq. (2) is an algebraic identity: U, R, and S are defined so that they sum to Lzmax, and the redundancy/synergy labels are imported by analogy from the authors' own Ref. [13] rather than derived from partial information decomposition.

self definitional [Methodology, page 5, Eq. (2) and preceding paragraph]
"it is intuitive that searching for zmin minimizing Lz(X → Y ) captures the amount of redundancy R ... we may define R = L∅(X → Y ) − Lzmin(X → Y ). The unique predictive power U ... will be given by U = Lzmin(X → Y ). On the other hand, searching for zmax maximizing Lz(X → Y ), leads to the amount of synergy S ... S = Lzmax(X → Y ) − L∅(X → Y ). It follows that: Lzmax(X → Y ) = S + R + U, (2)"

U, R, and S are defined as Lzmin, L∅−Lzmin, and Lzmax−L∅, so S+R+U = Lzmax is an algebraic identity that holds for any chosen subsets zmin and zmax. The decomposition therefore has no independent content; the three terms are constructed to sum to Lzmax. The semantic claims that U is a 'pure two-body effect', R is 'redundancy', and S is 'synergy' are asserted via 'it is intuitive' and 'we may define', not derived from partial information decomposition. All subsequent interpretations in Figs. 1–6 rest on this asserted labeling rather than on a demonstrated equivalence.
self citation load bearing [Introduction and methodology transition, pages 4–5]
"As discussed in [13], LZ(X → Y ) underestimates the importance of X when X exhibits redundancy with some variables in Z. Conversely, considering only the driver X ... overlooks synergies between X and Z. ... Adapting this framework by replacing conditional mutual information with LOCO enables a decomposition of LOCO into unique, redundant, and synergistic components"

The paper's central interpretive claim—that the subset minimizing Lz corresponds to redundancy and the subset maximizing Lz corresponds to synergy—is justified by reference to [13], which is authored by the same four authors (Stramaglia, Faes, Cortes, Marinazzo). The paper itself concedes that substituting variance for entropy 'sacrifices the formal rigor of information theory', so the min/max LOCO quantities are not shown to equal any established PID redundancy or synergy measure. Without an independent benchmark or derivation, the labels are an imported ansatz from the authors' prior work rather than a result established here.

full rationale

The paper is transparent about its definitions, but the core decomposition is definitional: Eq. (2) is a tautology given the definitions of U, R, and S. The main novel content is the semantic interpretation of the min/max LOCO terms as unique, redundant, and synergistic contributions, and that interpretation is inherited from the authors' own Ref. [13] by analogy. The paper's own admission that replacing entropy with variance 'sacrifices the formal rigor of information theory' reinforces that the mapping is an assumed analogy, not a proven equivalence. The toy example is a sanity check of the definitions rather than an independent validation, since it is constructed so that the heuristic yields the intended labels. The real-data applications do provide independent content in the form of computed LOCO values and greedy subset searches, and the physical interpretations (e.g., beta and p synergy in particle discrimination) are plausible; this independent anchoring prevents the paper from being fully circular. However, because the central decomposition and its labels reduce to definitions plus a load-bearing self-citation, a moderate circularity score is appropriate.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The method's central quantities U, R, S are defined from the extreme LOCO values rather than derived from an independent theory. The only free parameters are practical thresholds and model choices, but the interpretative axiom linking variance reductions to redundancy and synergy is the largest unverified assumption.

free parameters (3)

Permutation-test significance threshold for greedy stopping
The greedy search stops when the change in LOCO is compatible with a random effect below a threshold corrected for multiple comparisons; the threshold value is not given, yet it determines which variables enter the min/max multiplets and hence the U, R, S values.
Number of surrogates for permutation tests
Not stated in the paper; affects the reliability of the stopping rule and the reproducibility of the reported decompositions.
Polynomial kernel degree = 2
The toy and GEANT experiments use an inhomogeneous polynomial kernel of degree 2, chosen by hand. The paper acknowledges the decomposition depends on the hypothesis space, so this choice directly influences the reported importances.

assumptions (4)

standard math LOCO is non-negative for the considered regression hypothesis spaces (linear models, inhomogeneous polynomial and Gaussian kernels).
Invoked to define R and S as non-negative quantities; proof is referenced to [15] (Ancona and Stramaglia 2006), which is the authors' own prior work.
domain assumption In-sample mean squared error approximates generalization error when sample size is large.
Footnote [14] states training evaluation may be used when n is large; all numbers in the paper are in-sample, so the reported importances include potential overfitting effects.
ad hoc to paper For the analyzed datasets, greedy subset search yields the same result as exhaustive search.
Stated in the final discussion without showing the exhaustive results or the comparison procedure; this claim supports the practical scalability of the method.
ad hoc to paper The min/max LOCO search captures information-theoretic redundancy and synergy.
The core mapping from prediction-error reductions to PID-like concepts is asserted, not derived; the paper acknowledges it sacrifices formal information-theoretic rigor (ref [11]).

invented entities (1)

Hi-Fi decomposition components (U, R, S)
purpose: Provide a three-way split of LOCO feature importance into unique, redundant, and synergistic contributions.
These components are defined directly from the min/max LOCO search and have no external falsifiable handle; their interpretation as redundancy and synergy is assumed.

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Cite this review

Pith. "Pith review of Assessing high-order effects in feature importance via predictability decomposition." pith.science (2026). https://pith.science/paper/CI2DBBF3

@misc{pith2026241209964,
  author       = {Pith},
  title        = {Pith review of: Assessing high-order effects in feature importance via predictability decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CI2DBBF3}},
  note         = {Machine review of arXiv:2412.09964}
}

read the original abstract

Leveraging the large body of work devoted in recent years to describe redundancy and synergy in multivariate interactions among random variables, we propose a novel approach to quantify cooperative effects in feature importance, one of the most used techniques for explainable artificial intelligence. In particular, we propose an adaptive version of a well-known metric of feature importance, named Leave One Covariate Out (LOCO), to disentangle high-order effects involving a given input feature in regression problems. LOCO is the reduction of the prediction error when the feature under consideration is added to the set of all the features used for regression. Instead of calculating the LOCO using all the features at hand, as in its standard version, our method searches for the multiplet of features that maximize LOCO and for the one that minimize it. This provides a decomposition of the LOCO as the sum of a two-body component and higher-order components (redundant and synergistic), also highlighting the features that contribute to building these high-order effects alongside the driving feature. We report the application to proton/pion discrimination from simulated detector measures by GEANT.

Figures

Figures reproduced from arXiv: 2412.09964 by the authors.

**Figure 2.** FIG. 2: For each driving variable in the toy example (each row), the color value represents the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3: Left: The predictability decomposition by Hi-Fi, for the particle discrimination problem, [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: For each driving variable in the wine quality problem (each row), the color value represents [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5: Left: The predictability decomposition by Hi-Fi, for the particle discrimination problem, [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6: For each driving variable in the particle discrimination problem (each row), the color value [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

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Stramaglia, L

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G. Casalicchio, C. Molnar, and B. Bischl, Lecture Notes in Computer Science 1105, 655 (2019)

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