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REVIEW 4 major objections 8 minor 108 references

Variable Selection Methods for Multivariate, Functional, and Complex Biomedical Data in the AI Age

T0 review · 4 major / 8 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A fast variable-selection solver for complex, metric-space biomedical responses.

desk verdict The paper's claimed equivalence to FRISO is false, so the metric-space consistency results collapse; the multivariate best-subset extension is plausible but the empirical and theoretical support is not there. read the letter →

arxiv 2501.06868 v1 pith:3JX7GRCP submitted 2025-01-12 stat.ML cs.LGstat.APstat.ME

Marcos Matabuena This is my paper · ORCID
classification stat.MLcs.LGstat.APstat.ME MSC 62R2062J0790C25
keywords variableselectionbestsubsetmetricspaceresponsesFréchetregressionsaddle-pointoptimizationdualsubgradientmethodfunctionaldatadigitalhealth
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 tries to establish that best-subset variable selection can be made practical for regression problems where the response is multivariate, functional, distributional, or graph-valued in a metric space. It formulates the selection problem as a cardinality-constrained ridge regression, relaxes the binary variables to a saddle-point problem, and solves it with a dual projected subgradient algorithm. The paper claims this solver matches the selection accuracy of the only existing general method for metric-space responses (FRISO) while running orders of magnitude faster, reaching sample sizes in the hundreds of thousands or millions. If true, this would let clinicians and data scientists select interpretable predictors for complex biomedical outcomes such as glucose distribution profiles and brain graphs at cohort scale.

What carries the argument

The engine is the dual saddle-point problem obtained by Fenchel-conjugating the losses and dualizing the ridge penalty, with binary support variables relaxed to the unit interval. The objective is linear in the support variables and concave in the dual variables, so Sion's minimax theorem allows the order of minimization and maximization to be interchanged; a tightness condition from Pilanci et al. guarantees that the relaxed solution is binary. The update loop in Algorithm 1 alternates a sorting step that selects the k variables with largest α⊤XjXj⊤α and a projected subgradient step on the dual variables. The perspective-function reformulation of the ridge penalty is the bridge used to claim equivalence with FRISO, the Fréchet Ridge Selection Operator, an individually penalized ridge regression for global Fréchet regression.

What would settle it

Construct a small counterexample with p = 2, n = 50, correlated predictors, and a distributional response; solve Equation (15) by exhaustive enumeration over the four possible subsets and solve Equation (10) with FRISO's optimizer. If the selected support sets differ for any matched tuning parameters (k, τ), Proposition 4 is false or only approximate. Alternatively, run Algorithm 1 on a problem where the saddle point is not unique and check whether the averaged iterate converges; divergence would falsify the convergence assumption.

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Extended reading notes

Core claim

The central claim is that a single optimization framework handles variable selection across many response types by treating the response space through a Hilbert-space embedding. For metric spaces of negative type, the squared distance embeds isometrically into a Hilbert space, so the global Fréchet regression model becomes a linear model in that transformed space, and the paper argues that the best-subset problem with a ridge penalty is equivalent to the FRISO individually penalized ridge regression problem (Proposition 4). On this equivalence it imports FRISO's consistency result. The computational discovery is that the Boolean relaxation of the saddle-point formulation is tight under a generic uniqueness condition, so a simple projected subgradient iteration, updating the support by sorting and the dual variable by one subgradient step, solves the relaxed problem and recovers the best subset.

Load-bearing premise

The load-bearing premise is Proposition 4: that the cardinality-constrained ridge problem in Equation (15) is exactly equivalent to FRISO's continuously penalized ridge problem in Equation (10); the paper states this equivalence without proof, and both the imported consistency theorem and the metric-space generalization stand on it.

Editorial extensions

If this is right

  • Selection of predictors for distributional and graph-valued responses becomes computationally feasible at cohort scale (n in hundreds of thousands), not just for datasets of a few hundred observations.
  • The same algorithm can fit different loss functions for each coordinate of a multivariate response, enabling mixed models that combine genomic, wearable, and electronic-health-record data.
  • Consistency of the selected variable set follows from FRISO's theory whenever the stated equivalence holds and the response space is of negative type.
  • Support for group and block constraints extends the method to one-hot encoded categorical predictors and additive spline bases.
  • On large multivariate linear problems the paper reports near-perfect support recovery with runtimes under tens of seconds.

Reading between the lines

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

  • If the equivalence with FRISO is proved and the constant-step-size subgradient convergence is established, the method would inherit a complete consistency theory and could replace lasso-type screening in personalized-medicine pipelines.
  • A concrete testable extension is to apply the solver to 2-Wasserstein distributional responses in clinical trial data, where current practice often collapses the distribution to mean and variance; the paper's own examples suggest that distributional outcomes can change which predictors are selected.
  • The claimed speedup rests on avoiding matrix inversions, which suggests the same dual subgradient strategy may accelerate other Fenchel-dual sparse regression problems beyond metric-space responses.
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Formalized claims in Lean

  1. Claim #1: The central claim is that a single optimization framework handles variable selection across many response types by treating the response space through a Hilbert-space embedding. For metric spaces of negative type, the squared distance embeds isometrically into a Hilbert space, so the global Fréchet regression model becomes a linear model in that transformed space, and the paper argues that the bes

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 8 minor

Summary. The paper proposes a best-subset variable selection framework for regression problems with multivariate, functional, and metric-space-valued responses. The method solves a cardinality-constrained empirical risk minimization with a ridge penalty (Eqs. 1/15), derives a saddle-point dual for convex losses (Theorem 1), relaxes the binary support variables to the unit interval, imports a tightness condition from Pilanci, Wainwright, and El Ghaoui (Theorem 2), and solves the relaxed problem with a projected dual subgradient algorithm (Algorithm 1). Section 2.3 claims, via Proposition 4, that the proposed problem is equivalent to the FRISO individual-ridge procedure of Tucker, Wu, and Müller, and on that basis imports FRISO's selection-consistency theorem (Theorem 5). The remainder applies the method to three diabetes datasets (multivariate biomarkers, bivariate longitudinal CGM outcomes, distributional glucose profiles) and reports simulations for multivariate Euclidean responses and 2-Wasserstein distribution responses, claiming several orders of magnitude speed improvement over competitors.

Significance. If established, the contribution would be genuinely valuable: a scalable ℓ₀-based solver for variable selection with complex responses, extending the Bertsimas–Pauphilet–Van Parys saddle-point approach to multivariate and metric-valued outcomes, with clinically motivated applications in diabetes research. The saddle-point dual for convex losses is plausible, the use of the Pilanci et al. Boolean-relaxation tightness result is appropriate, and the reported wall-clock times in Tables 4–5 (e.g., about 31 seconds for n = 100,000 with an m = 300 quantile grid) indicate real computational efficiency. However, the central theoretical bridge (Proposition 4) is unproven and appears false on its face (ℓ₀ + ℓ₂ versus an effective ℓ₁-type penalty), so the imported consistency theorem does not apply to the proposed method as stated; Algorithm 1 has no convergence analysis; and the headline speed claims are not backed by any competitor runtimes. These gaps undermine the paper's two central claims, so the manuscript cannot be accepted in its current form.

major comments (4)
  1. [2.3.5 (Proposition 4)] Proposition 4 asserts that problem (15) is equivalent to the FRISO individual-ridge problem (10), but no proof is given and the preceding perspective-function manipulation only rewrites (15); it does not relate (15) to (10). The equivalence is not a routine consequence of the definitions: for scalar squared-error loss, FRISO with constraints λ_j ≥ 0 and Σ_j λ_j ≤ τ induces the effective penalty βᵀ diag(1/λ)β whose minimum over λ is (Σ_j|β_j|)²/τ, an ℓ₁-type penalty, whereas (15) is an ℓ₀-cardinality-plus-ℓ₂ problem; ℓ₁-type and ℓ₀-type objectives are known to have different solution paths under correlated predictors, so the asserted equality fails in general. Because Theorem 5 from [50] is invoked solely through Proposition 4, the selection-consistency conclusion in Section 2.4 is not established for the estimator defined by (15) and Algorithm 1; Proposition 6, which is meant to supply the remaining conditions, is also stated without proof.
  2. [2.1.2 (Algorithm 1)] Algorithm 1 is presented as the paper's main computational contribution, but no convergence analysis is provided: the constant step size δ is unspecified, the projection operator P is not defined with respect to any norm or feasible set, and no theorem or proposition establishes even an asymptotic stationarity property or an O(δ) optimality gap for the averaged output (α̂_T, Ņ). The claim in Section 3.2.2 that "the algorithm obtained the optimal solution in just two seconds" therefore has no support; without a convergence guarantee (or at least a certified tolerance for solving the relaxed problem (3)), the speed and accuracy results in Sections 3–4 cannot be attributed to solving the stated optimization problem.
  3. [Abstract and Sections 4.1–4.2] The abstract claims "several orders of magnitude improvement over competitors," but the experiments do not provide the necessary baselines. Section 4.1 states explicitly that "we do not include a comparison with other approaches in this scenario," so the large-n multivariate timings in Table 4 have no competitor. In the distributional scenario (Section 4.2), the comparison with FRISO reports only that both methods select the true variable with proportion 1.00; Table 5 contains no FRISO runtimes, and the text limits FRISO to "datasets of less n = 300 observations" without quantifying its runtime even at n = 300. The speed advantage over FRISO or any other method is therefore never measured, and the abstract's accuracy claim is also unsupported: on the only comparative scenario the reported result is parity (1.00 versus 1.00), while the real-data sections (3.2–3.4) contain no competitor analyses and report in-sample R² only.
  4. [2.4 (Theorem 5, Remark 5)] Even if Proposition 4 were granted, the consistency statement does not connect to the proposed selector. Theorem 5 concerns the FRISO weights λ̂_j(τ_n), with selection defined by Î(τ) = {j : λ̂_j(τ) > 0}; the proposed method instead selects the k variables with the largest score from the saddle point of (3). The paper never shows that these two selection rules coincide, and Remark 5 concedes that Theorem 5 yields only λ̂_{j'} →_p 0, not exact support recovery. The opening sentence of Section 2.4, "we can select the correct variables as n grows to infinity," is therefore not justified for the estimator that is actually implemented; this is an internal-support gap rather than a disagreement with external consensus.
minor comments (8)
  1. [Throughout] The manuscript contains numerous typographical errors (e.g., "beetween", "contemporany", "ilustrate", "millons", "advanatgues", "uncertainity", "trejectories" in the Figure 2 caption, and "Weigth" and "Percentil" in Figure 6); a thorough proofread is needed.
  2. [Introduction and 2.3.2] The FRISO method is cited as reference [8] in the Introduction and in Section 2.3.2, but [8] is a review on wearable biosensors; the correct reference is the Tucker–Wu–Müller paper, which is unfortunately duplicated as both [50] and [73].
  3. [2.1.1] The tightness result is labeled "Theorem 1" a second time, with a stray "2" following the label; it should be renumbered as Theorem 2.
  4. [2.3.4 (Remark 4)] Remark 4 refers to "Equation 17" where equation (14) is meant.
  5. [3.4.1 versus Table 1] Section 3.4.1 describes a case-control study with 200 participants, while Table 1 reports "over 350 patients" for the same distributional application; the numbers should be reconciled.
  6. [2.4 and 4.1] The Conditions [A]–[D] referenced in Theorem 5 and the "complete results" for Section 4.1 are said to be in an Appendix that is not included in the manuscript.
  7. [4.1] The sample size is given as "n = 1000,000" in one place and "1, 000, 000" in another, while Table 4 reports n up to 100,000; the intended maximum sample size should be stated consistently.
  8. [Table 5] The units of "Average Time" in Table 5 are not specified, and the absence of any FRISO timing column makes the claimed speed comparison unverifiable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the consistency claim is imported from an external FRISO theorem; the unproven equivalence in Proposition 4 is a correctness gap, not a circular reduction.

full rationale

The paper's central statistical guarantee is inherited from the external FRISO consistency theorem (Theorem 5, cited to Tucker, Wu and Müller [50], none of whom are authors of this paper). Importing an external result is independent support rather than circularity, and the paper does not fit parameters and then relabel them as predictions. The main load-bearing step is Proposition 4, which asserts that Eq. (15), a cardinality-constrained ℓ0+ℓ2 ridge problem, is equivalent to FRISO's individually λ-penalized ridge problem. That assertion is made without proof and is mathematically non-obvious, since FRISO's penalty induced by the λ-constraint is ℓ1-type rather than ℓ0-type; however, an unproven or even false equivalence is a missing-support/correctness issue, not a circular one. The paper's equations are not identical by construction, and no same-author citation chain forces the conclusion. The unproven convergence of Algorithm 1 is likewise a missing proof. Thus no circular step can be exhibited, and the appropriate verdict is no significant circularity.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The method depends on a sparsity budget k and ridge strength gamma, neither of which is tuned by a data-driven rule. The metric-space theory rests on the negative-type embedding and on the Fréchet regression model, plus the unproven equivalence to FRISO. No new entities are invented.

free parameters (6)
  • k (sparsity budget) = 9 (NHANES), 3 (JDRF), 1/3/5/8 (simulations)
    User-specified cardinality bound in Problem (1); no data-driven selection rule or sensitivity analysis.
  • gamma (ridge strength) = not reported
    Controls ridge regularization in the objective; no tuning procedure or grid values given.
  • delta (subgradient step size) = not reported
    Constant step size in Algorithm 1; no formula, schedule, or convergence condition.
  • Stop criterion for Algorithm 1 = not specified
    The loop ends at an unspecified criterion; affects solution quality and runtime.
  • Kernel bandwidth h for density estimation = not reported
    Used in Eq. (24) to construct distributional glucose responses; not specified or varied.
  • Quantile grid size m = 300 or 500 grid points in applications; 50/150/300 in simulations
    Discretization level for functional/distributional responses; chosen by hand.
assumptions (6)
  • domain assumption Loss functions are convex with additive linear structure.
    Theorem 1 and the dual derivation require convexity; the paper cites convex losses in Table 2.
  • domain assumption Response space is a separable Hilbert space or embeds into one (negative type).
    Section 2.3.3 restricts to negative type spaces via Schoenberg embedding; not all metric spaces satisfy this.
  • domain assumption Global Fréchet regression model of Petersen and Müller is the correct regression model for metric responses.
    Section 2.3.1 adopts this model (Eq. 9) as the base estimator.
  • domain assumption Conditions [A-D] from the appendix hold.
    Theorem 5 cites conditions [A-D] and (U0)-(U2) of [78]; the paper does not verify them in the applications, and the appendix is not included in the preprint.
  • standard math Boolean relaxation tightness condition holds (distinct top-k components).
    Theorem 2 relies on [76, Proposition 1]; in correlated real data this is an assumption.
  • domain assumption Conditional Fréchet mean exists and is unique.
    Section 2.3 assumes uniqueness and existence of the minimizer in Eq. (8).

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Cite this review

Pith. "Pith review of Variable Selection Methods for Multivariate, Functional, and Complex Biomedical Data in the AI Age." pith.science (2026). https://pith.science/paper/3JX7GRCP

@misc{pith2026250106868,
  author       = {Pith},
  title        = {Pith review of: Variable Selection Methods for Multivariate, Functional, and Complex Biomedical Data in the AI Age},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3JX7GRCP}},
  note         = {Machine review of arXiv:2501.06868}
}
read the original abstract

Many problems within personalized medicine and digital health rely on the analysis of continuous-time functional biomarkers and other complex data structures emerging from high-resolution patient monitoring. In this context, this work proposes new optimization-based variable selection methods for multivariate, functional, and even more general outcomes in metrics spaces based on best-subset selection. Our framework applies to several types of regression models, including linear, quantile, or non parametric additive models, and to a broad range of random responses, such as univariate, multivariate Euclidean data, functional, and even random graphs. Our analysis demonstrates that our proposed methodology outperforms state-of-the-art methods in accuracy and, especially, in speed-achieving several orders of magnitude improvement over competitors across various type of statistical responses as the case of mathematical functions. While our framework is general and is not designed for a specific regression and scientific problem, the article is self-contained and focuses on biomedical applications. In the clinical areas, serves as a valuable resource for professionals in biostatistics, statistics, and artificial intelligence interested in variable selection problem in this new technological AI-era.

Figures

Figures reproduced from arXiv: 2501.06868 by the authors.

Figure 1
Figure 1. Variation in the mean and standard deviation of glucose values for a diabetic individual [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Average glucose trajectories (left) and standard deviation trejectories (right). [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Left: Raw CGM time series of two individuals. Center: The corresponding density functions. [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: P-values across the temporal domain of the statistical significance of each variable selected [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

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Reference graph

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Pith tools

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