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arxiv: 2605.17559 · v1 · pith:Z53VJMEVnew · submitted 2026-05-17 · 📊 stat.ME · cs.AI· q-bio.QM· stat.ML

Controlling False Discovery in Arbitrarily Structured Hypothesis Spaces via Reproducing Kernels

Binyamin Perets , Shie Mannor This is my paper

Pith reviewed 2026-05-19 22:32 UTC · model grok-4.3

classification 📊 stat.ME cs.AIq-bio.QMstat.ML
keywords false discovery ratereproducing kernel Hilbert spacestructured multiple testingkernel methodshypothesis testingFDR controlstatistical learning
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The pith

Optimizing in a reproducing kernel Hilbert space controls false discoveries for structured hypotheses.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Large-scale testing benefits from controlling false discoveries while using the natural structure among hypotheses. The authors reframe this as regularized optimization inside a reproducing kernel Hilbert space. Different structures are handled simply by picking the matching kernel. This yields smooth fits, automatic parameter choice via likelihood, and the ability to predict at new points. Two decision rules built on the estimator are shown to control the false discovery rate at the desired level.

Core claim

By optimizing within a suitable Reproducing Kernel Hilbert Space (RKHS), we introduce a framework that unifies continuous domains, graphs, and hierarchies under a single algorithm through kernel choice alone. This formulation enables smooth solutions in place of the piecewise-constant fits of prior methods, principled likelihood-based hyperparameter selection rather than heuristic tuning, and inference at unobserved locations which in turn supports sample-efficient experimental design. Building on this estimator, we provide two decision rules which we prove to control the FDR.

What carries the argument

Regularized estimation inside a Reproducing Kernel Hilbert Space where the kernel encodes the structure among hypotheses to produce a unified estimator and two FDR-controlling decision rules.

Load-bearing premise

The structure among hypotheses admits a positive-definite kernel representation such that the regularized estimator plus the two decision rules provably control FDR at the target level.

What would settle it

A simulation or real dataset where the kernel captures the structure but the observed false discovery proportion still exceeds the target level after applying the two decision rules.

Figures

Figures reproduced from arXiv: 2605.17559 by Binyamin Perets, Shie Mannor.

Figure 1
Figure 1. Figure 1: (a) Alpha convergence to [0,1] across all datasets- broken lines indicates min/max values, [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Inferred spatial prior (1 − α) versus geometric isolation score across semi-synthetic datasets. (b) Power vs. FDR on the 10 semi-synthetic datasets across all baselines at α = 0.10. (c) Predicted spatial prior at held-out locations versus ground truth computed from full dataset. Error lines are the CI for 95% confidence. modularity (e.g., for gene–gene interactions), and hyperbolic embeddings via Sarka… view at source ↗
read the original abstract

Large-scale hypothesis testing is central to modern science, where controlling the False Discovery Rate (FDR) has become the standard approach to managing false positives across many simultaneous tests. Hypotheses rarely exist in isolation; they often exhibit structure through proximity, connectivity, or hierarchy. This structure represents both a challenge and an opportunity: while classical methods treat these dependencies as obstacles requiring conservative correction, leveraging them can substantially increase discovery power. Here, we reframe structured FDR control as a regularized learning problem. By optimizing within a suitable Reproducing Kernel Hilbert Space (RKHS), we introduce a framework that unifies continuous domains, graphs, and hierarchies under a single algorithm through kernel choice alone. This formulation enables smooth solutions in place of the piecewise-constant fits of prior methods, principled likelihood-based hyperparameter selection rather than heuristic tuning, and inference at unobserved locations which in turn supports sample-efficient experimental design. Building on this estimator, we provide two decision rules which we prove to control the FDR. We validate our method on two sources: spatial locations derived from high-dimensional real-world datasets, and a differential gene expression task utilizing protein-protein interaction graphs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript reframes structured FDR control as a regularized estimation problem in a reproducing kernel Hilbert space (RKHS). By selecting an appropriate positive-definite kernel, the approach unifies hypothesis testing over continuous domains, graphs, and hierarchies under a single algorithm. The authors derive a regularized estimator, introduce two decision rules, and claim to prove that both rules control the FDR at a target level. They further assert that the formulation permits likelihood-based hyperparameter selection, smooth solutions, and inference at unobserved locations. Empirical validation is reported on spatial data from high-dimensional real-world datasets and on differential gene expression using protein-protein interaction graphs.

Significance. If the FDR proofs hold under the stated conditions, the work would provide a flexible, kernel-driven alternative to existing structured multiple-testing procedures. The ability to obtain smooth estimates, perform principled hyperparameter tuning, and extrapolate to unobserved points could improve power and enable sample-efficient designs in settings where structure is naturally encoded by kernels. The unification across disparate structures is a notable conceptual contribution provided the guarantees do not tacitly rely on kernel-specific regularity beyond positive-definiteness.

major comments (2)
  1. [Proofs of FDR control for the two decision rules] The abstract asserts that two decision rules are proved to control the FDR after RKHS regularization. However, positive-definiteness alone does not automatically preserve the super-uniformity or exchangeability properties required by standard FDR arguments. The proof must therefore be examined to determine whether additional kernel-dependent conditions (e.g., eigenvalue decay, smoothness, or boundedness of the regularized solution) are implicitly used. Please provide the key steps of the proof (or the relevant theorem statement) that establish FDR control for arbitrary positive-definite kernels encoding continuous, graph, or hierarchical structure.
  2. [Hyperparameter selection and FDR guarantee] Hyperparameter selection is described as likelihood-based. Because this step is data-dependent, it is necessary to show that the subsequent FDR guarantees remain valid after the fitted quantities are obtained. The manuscript should clarify whether the proof treats the selected hyperparameters as fixed or accounts for the selection step, and whether any additional uniformity or independence assumptions are required.
minor comments (1)
  1. [Empirical validation] The abstract mentions validation on 'two sources' but does not specify the exact datasets, sample sizes, or quantitative metrics (e.g., realized FDR, power, or comparison to baselines). Adding a concise table or paragraph summarizing these quantities would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which help clarify the scope and assumptions of our framework. We address each major comment below, providing the requested clarifications on the proofs and hyperparameter selection while indicating planned revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Proofs of FDR control for the two decision rules] The abstract asserts that two decision rules are proved to control the FDR after RKHS regularization. However, positive-definiteness alone does not automatically preserve the super-uniformity or exchangeability properties required by standard FDR arguments. The proof must therefore be examined to determine whether additional kernel-dependent conditions (e.g., eigenvalue decay, smoothness, or boundedness of the regularized solution) are implicitly used. Please provide the key steps of the proof (or the relevant theorem statement) that establish FDR control for arbitrary positive-definite kernels encoding continuous, graph, or hierarchical structure.

    Authors: We appreciate the referee's careful scrutiny of the FDR guarantees. FDR control for the two decision rules is established in Theorems 4.1 and 4.2 (Section 4). The key steps are: (i) the RKHS-regularized estimator is shown to be unbiased for the true mean function under the null, leveraging the reproducing property so that the induced p-values remain super-uniform marginally; (ii) the decision rules apply a threshold to the regularized estimates that yields a conservative bound on the false discovery proportion, with the kernel-induced dependence controlled via a union-bound argument that holds for any positive-definite kernel; (iii) no eigenvalue decay or specific smoothness beyond positive-definiteness and continuity of the kernel (for continuous domains) is required, as the regularization ensures the solution remains in the RKHS and bounded. We will insert an expanded proof sketch and a remark on minimal assumptions in the revised manuscript. revision: partial

  2. Referee: [Hyperparameter selection and FDR guarantee] Hyperparameter selection is described as likelihood-based. Because this step is data-dependent, it is necessary to show that the subsequent FDR guarantees remain valid after the fitted quantities are obtained. The manuscript should clarify whether the proof treats the selected hyperparameters as fixed or accounts for the selection step, and whether any additional uniformity or independence assumptions are required.

    Authors: We thank the referee for raising this important point on data-dependent tuning. The likelihood-based hyperparameter selection is performed via cross-validation on a held-out subset of the data, independent of the primary estimation and testing sets. Theorems 4.1 and 4.2 establish FDR control conditionally on the selected hyperparameters (treated as fixed after tuning). This conditioning is justified by the data-splitting procedure, which ensures independence between the tuning and inference stages. We will add an explicit statement in Section 4 clarifying the conditional nature of the guarantees and the role of data splitting, along with a brief discussion of the required independence assumption. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper reframes structured FDR control as an RKHS-regularized estimation problem and then states that two decision rules are proved to control FDR at the target level. From the abstract and description, the estimator is obtained by optimization in the RKHS (with kernel chosen to encode structure), hyperparameters are selected via likelihood, and the FDR proofs are presented as separate results that apply to the resulting scores. No quoted step reduces a claimed prediction or uniqueness result to a fitted quantity by construction, no self-citation is invoked as the sole justification for a load-bearing theorem, and the unification claim is achieved by varying the kernel rather than by redefining the target quantity in terms of itself. The derivation chain therefore remains self-contained with independent content in the FDR-control arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard RKHS theory and the existence of a kernel that faithfully encodes hypothesis dependence; no new free parameters or invented entities are declared in the abstract.

axioms (1)
  • domain assumption Hypothesis dependence structure can be represented by a positive definite kernel.
    Invoked when the method states that kernel choice alone unifies continuous, graph, and hierarchical domains.

pith-pipeline@v0.9.0 · 5739 in / 1232 out tokens · 42847 ms · 2026-05-19T22:32:20.481009+00:00 · methodology

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