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arxiv: 2605.20604 · v1 · pith:BWNG6CK4new · submitted 2026-05-20 · 📊 stat.ME

Conditional regularized halfspace depth for sparse functional data and its applications

Hyemin Yeon , Xiongtao Dai , Sara Lopez-Pintado This is my paper

Pith reviewed 2026-05-21 03:22 UTC · model grok-4.3

classification 📊 stat.ME
keywords sparse functional datahalfspace depthconditional depthfunctional data analysisdata depthranking methodsgrowth curves
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The pith

Conditional regularized halfspace depth ranks sparse functional data directly from observations without reconstructing trajectories.

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

The paper introduces conditional regularized halfspace depth to order functional data observed only at scattered irregular points. It sets this depth equal to the lowest conditional halfspace probability for the full underlying trajectory given the sparse points that were actually recorded. The construction works straight from the limited measurements and skips the usual step of estimating the unseen segments of each curve. The resulting orderings support rank-based tests and are demonstrated on infant growth data collected at irregular intervals.

Core claim

CRHD is defined as the infimum of conditional halfspace probabilities of the underlying trajectory given the observed sparse measurements, thereby enabling depth evaluation directly at sparse observations without requiring trajectory reconstruction. The paper establishes several basic theoretical properties that confirm its behavior as a depth measure and shows that the approach remains applicable even when observations are extremely sparse.

What carries the argument

The conditional regularized halfspace depth, defined as the infimum of conditional halfspace probabilities for the trajectory given the sparse measurements.

If this is right

  • CRHD applies to functional data observed at only a handful of irregular times.
  • It produces rankings that can be used directly in rank-based statistical procedures.
  • The method avoids bias that can arise from preliminary curve reconstruction steps.
  • It is illustrated on an infant growth dataset with irregular measurement times.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same conditional-probability construction could be examined for other forms of incomplete high-dimensional data where full reconstruction is unreliable.
  • It may offer a route to robust outlier detection in longitudinal studies that collect only a few observations per subject.
  • Performance under varying patterns of missingness, such as clustered observation times, remains a natural next check.

Load-bearing premise

Conditional halfspace probabilities for the infinite-dimensional trajectory can be defined, regularized, and estimated from sparse measurements without introducing substantial bias or instability.

What would settle it

Apply CRHD to simulated data generated from known full trajectories, subsample each trajectory to sparse points, compute the induced ordering, and compare it with the ordering obtained from full-data depth; systematic reversal of expected ranks would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.20604 by Hyemin Yeon, Sara Lopez-Pintado, Xiongtao Dai.

Figure 1
Figure 1. Figure 1: Empirical rank correlations of ACRHD, PCRHD, and TwoStageRHD values at [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical power of the KW tests based on both averaged and plug-in CRHDs [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Filtered height trajectories from the infant growth data, obtained by retaining [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
read the original abstract

Many functional datasets are observed sparsely and irregularly. Ordering such data is challenging because only limited information is available from each observation, while the underlying trajectories remain infinite-dimensional. This paper develops a novel depth notion for sparse functional data, called the conditional regularized halfspace depth (CRHD). CRHD is defined as the infimum of conditional halfspace probabilities of the underlying trajectory given the observed sparse measurements, thereby enabling depth evaluation directly at sparse observations without requiring trajectory reconstruction. We study several basic theoretical properties of CRHD that clarify its behavior as a depth measure. The proposed depth is applicable even to extremely sparsely observed functional data, overcoming key limitations of existing sparse functional depths that often rely on reconstructed curves. In addition, CRHD induces meaningful rankings for complex functional data. Its numerical performance is demonstrated through rank-based tests, and its practical utility is illustrated using an infant growth dataset.

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

3 major / 2 minor

Summary. The manuscript proposes the conditional regularized halfspace depth (CRHD) for sparsely and irregularly observed functional data. CRHD is defined as the infimum over directions u of the conditional probability that the underlying infinite-dimensional trajectory satisfies a halfspace condition given the finite sparse point observations. The authors state that this construction permits direct depth evaluation on the observed data points without explicit trajectory reconstruction. They examine basic theoretical properties of CRHD as a depth measure, claim applicability to extremely sparse regimes, demonstrate that it produces meaningful rankings, and illustrate performance via rank-based tests together with an application to an infant growth dataset.

Significance. If the central construction can be shown to be well-defined and estimable without implicitly reintroducing reconstruction bias, the work would address a genuine methodological gap in functional data analysis. Existing sparse functional depths frequently rely on preliminary smoothing or basis expansion, which can distort ordering; a depth that operates directly on the conditional law would be a useful addition. The reported theoretical properties and the real-data illustration on growth curves provide a foundation for further development, though the magnitude of the advance hinges on whether the regularization step delivers the claimed separation from reconstruction-based approaches.

major comments (3)
  1. [Section 2] Definition of CRHD (Section 2): the claim that the depth is evaluated 'directly at sparse observations without requiring trajectory reconstruction' is not yet supported by an explicit argument showing that the conditional probability P(X in halfspace | observations at t1..tk) remains identifiable and stable under the chosen regularization without effectively constraining the trajectory to a finite-dimensional subspace. In nonparametric settings the conditional law of an infinite-dimensional process given finitely many point evaluations is typically non-identifiable; any practical estimator must therefore impose structure (basis truncation, kernel smoothing, or parametric covariance) whose effect on the resulting depth values needs to be quantified.
  2. [Section 3] Theoretical properties (Section 3): the manuscript states that several basic properties of CRHD are established, yet the provided text does not contain the derivations or the precise regularity conditions under which the infimum over directions is attained and the depth is monotone or convex. Without these details it is difficult to assess whether the properties survive the regularization step or whether they reduce to known properties of unconditional halfspace depth after the conditional law is approximated.
  3. [Section 4] Estimation procedure (Section 4): the numerical implementation must rely on an estimator of the conditional halfspace probability. The manuscript should clarify whether this estimator is constructed from a finite sample of sparsely observed curves and whether consistency or rates are proved; if the estimator implicitly reconstructs trajectories via the same regularization used in the definition, the advantage over existing reconstruction-based depths requires explicit comparison on the same sparse regimes.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a concise statement of the precise form of regularization (e.g., basis dimension, penalty parameter, or kernel bandwidth) employed in the definition and estimation of CRHD.
  2. [Section 2] Notation for the sparse observation times and the conditioning sigma-field should be introduced once and used consistently throughout the theoretical sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We address each major comment point by point below, indicating revisions where the manuscript will be strengthened to provide additional clarity and rigor.

read point-by-point responses

  1. Referee: [Section 2] Definition of CRHD (Section 2): the claim that the depth is evaluated 'directly at sparse observations without requiring trajectory reconstruction' is not yet supported by an explicit argument showing that the conditional probability P(X in halfspace | observations at t1..tk) remains identifiable and stable under the chosen regularization without effectively constraining the trajectory to a finite-dimensional subspace. In nonparametric settings the conditional law of an infinite-dimensional process given finitely many point evaluations is typically non-identifiable; any practical estimator must therefore impose structure (basis truncation, kernel smoothing, or parametric covariance) whose effect on the resulting depth values needs to be quantified.

    Authors: We appreciate the referee's emphasis on identifiability. The regularization in the definition of CRHD is introduced precisely to ensure the conditional probability is well-defined and depends only on the sparse observations and the regularization parameter, without performing explicit trajectory reconstruction or projecting onto a finite basis. We will revise Section 2 to include a detailed argument establishing identifiability under the chosen regularization, along with an analysis of how the regularization parameter affects depth values and preserves the infinite-dimensional character of the underlying process. revision: yes

  2. Referee: [Section 3] Theoretical properties (Section 3): the manuscript states that several basic properties of CRHD are established, yet the provided text does not contain the derivations or the precise regularity conditions under which the infimum over directions is attained and the depth is monotone or convex. Without these details it is difficult to assess whether the properties survive the regularization step or whether they reduce to known properties of unconditional halfspace depth after the conditional law is approximated.

    Authors: We agree that the derivations and regularity conditions should be presented more explicitly. In the revised manuscript we will supply complete proofs for the properties of CRHD, including conditions ensuring the infimum is attained and establishing monotonicity and convexity. These proofs will explicitly address the effect of regularization and demonstrate that the properties do not reduce to those of the unconditional halfspace depth. revision: yes

  3. Referee: [Section 4] Estimation procedure (Section 4): the numerical implementation must rely on an estimator of the conditional halfspace probability. The manuscript should clarify whether this estimator is constructed from a finite sample of sparsely observed curves and whether consistency or rates are proved; if the estimator implicitly reconstructs trajectories via the same regularization used in the definition, the advantage over existing reconstruction-based depths requires explicit comparison on the same sparse regimes.

    Authors: The estimator is constructed directly from finite samples of sparsely observed curves via empirical conditional probabilities under the regularization. We will clarify this construction in the revised Section 4, add consistency results with convergence rates, and include explicit numerical comparisons against reconstruction-based depths in the same sparse observation regimes to quantify the practical advantage. revision: yes

Circularity Check

0 steps flagged

No significant circularity; CRHD definition is a direct construction from conditional probabilities without reduction to fitted inputs or self-citations

full rationale

The paper defines CRHD explicitly as the infimum of conditional halfspace probabilities of the underlying trajectory given sparse measurements. This is presented as a novel depth notion enabling direct evaluation without reconstruction. No equations or sections in the provided abstract or description show the definition reducing by construction to its own inputs, fitted parameters renamed as predictions, or load-bearing self-citations. The regularization is invoked to make the conditional law tractable, but the derivation chain remains independent of the target result itself. This is the common honest non-finding for papers introducing new depth measures via explicit probabilistic definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5680 in / 935 out tokens · 35996 ms · 2026-05-21T03:22:08.733560+00:00 · methodology

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