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arxiv: 2506.11369 · v5 · submitted 2025-06-13 · 📊 stat.ME · stat.CO

Filtration-Based Learning of Multiscale Shared Structures for Multiple Functional Predictors

Shuhao Jiao , Hernando Ombao , Ian W. McKeague This is my paper

Pith reviewed 2026-05-19 10:16 UTC · model grok-4.3

classification 📊 stat.ME stat.CO
keywords functional predictorsshared structuresfiltrationmultiscale learningfunctional partial least squareshierarchical forestkinematicsaging patterns
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The pith

A filtration-based framework learns multiscale shared structures among multiple functional predictors by organizing them into a hierarchical forest that identifies common effects progressively from coarse to fine layers.

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

The paper aims to show that dependencies between a response and several functional predictors can be learned more effectively when those dependencies are allowed to appear at different scales, from effects shared across all predictors down to effects unique to one. It does so by building a hierarchical forest through successive filtration layers that separate shared components from specific ones in a coarse-to-fine sequence. If the claim holds, the resulting structure supports both better prediction and clearer interpretation of how predictors coordinate, as demonstrated on lower-limb angular kinematics where joint patterns linked to aging become visible. The authors combine a filtration-based pursuit step for discovering the structure with a filtrated functional partial least squares procedure for extracting the shared components and estimating coefficients. Simulations confirm that the layers recover the underlying organization while the full pipeline outperforms standard methods on held-out accuracy.

Core claim

The central claim is that response-predictor dependencies vary across representation dimensions and emerge at multiple resolutions ranging from globally shared effects to predictor-specific effects; therefore a hierarchical forest structure built through successive filtration layers can progressively identify them. Building on this structure, the authors develop a filtration-based pursuit pipeline for shared structure discovery together with a filtrated functional partial least squares method for shared component extraction and coefficient estimation. Simulation studies show that the framework recovers the dominant coarse-to-fine organization and yields improved prediction performance; the同じ

What carries the argument

hierarchical forest structure through successive filtration layers, which progressively identifies shared and predictor-specific components from coarse to fine scales

If this is right

  • Simulation studies recover the dominant coarse-to-fine organization of the underlying shared structures.
  • The framework yields improved prediction performance relative to competing methods.
  • Application to lower-limb angular kinematics improves evaluation accuracy and reveals interpretable joint coordination patterns associated with aging.

Where Pith is reading between the lines

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

  • The same filtration layering could be applied to other collections of curves or surfaces, such as multichannel time series in neuroscience or environmental monitoring, to test whether comparable coarse-to-fine shared patterns appear.
  • If the learned forest layers align with physically meaningful scales, they might serve as an automatic way to choose resolution-specific features before fitting downstream regression models.
  • The approach supplies a concrete way to represent how multiple objects interact at different levels of detail, which could be useful for problems that currently treat all dimensions as equally shared or equally distinct.

Load-bearing premise

Response-predictor dependencies vary across representation dimensions and emerge at multiple resolutions ranging from globally shared effects to predictor-specific effects, allowing successive filtration layers to separate them.

What would settle it

In simulated data constructed with known multiscale shared effects at distinct resolutions, the method should fail to recover the hierarchical organization or show no prediction gain over ordinary functional partial least squares; the same outcome on the lower-limb kinematics dataset would also falsify the utility claim.

Figures

Figures reproduced from arXiv: 2506.11369 by Hernando Ombao, Ian W. McKeague, Shuhao Jiao.

Figure 1
Figure 1. Figure 1: Angular kinematics (right-hand side) during treadmill walking averaged over [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Hierarchical forest structure of coefficient homogeneity. Here, [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Coefficient scores {bjd : d = 1, . . . , D} of the 10 coefficients. Note that the coefficient scores exhibit partial homogeneity. Specifically, the first two scores are set to 2 for all j = 1, . . . , p, reflecting the global homogeneity shared by 13 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The oracle forest structure underlying coefficient homogeneity. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The estimated fPLS basis functions for all the 6 layers. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Prediction MSEs of different methods. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Angular kinematic trajectories averaged across the left and right sides, with [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The identified forest structure under 100–115% normal walking speed (the [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The confidence intervals for coefficient scores and the PSS values. Different [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The partial least square basis functions in the first two layers. [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Box-plots of MSEs of different methods. The variance of the log-transformed [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
read the original abstract

It is crucial to learn the shared structures among functional predictors, as these structures characterize how predictor components exert common effects and, more generally, how predictors are homogeneously associated with the response. However, learning from multiple functional predictors is challenging because response-predictor dependencies may vary across representation dimensions and emerge at multiple resolutions, ranging from globally shared effects to predictor-specific effects. To address this issue, we propose a filtration-based shared structure learning framework for multiple functional predictors. The proposed framework organizes predictors through a hierarchical forest structure, in which shared and predictor-specific components are progressively identified from coarse to fine filtration layers. Building on this structure, we develop a filtration-based pursuit pipeline for shared structure discovery, together with a filtrated functional partial least squares method for shared component extraction and coefficient estimation under the learned shared structures. Simulation studies show that the proposed framework is able to recover the dominant coarse-to-fine organization of the underlying shared structures and yield improved prediction performance relative to competing methods. Applied to lower-limb angular kinematics, the proposed framework improves evaluation accuracy and reveals interpretable joint coordination patterns associated with aging. More broadly, it provides a new multiscale representation-learning perspective for complex data consisting of multiple multidimensional objects.

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 / 2 minor

Summary. The manuscript proposes a filtration-based framework for learning multiscale shared structures among multiple functional predictors. Predictors are organized into a hierarchical forest structure via successive filtration layers that progressively isolate globally shared effects down to predictor-specific components. A filtration-based pursuit pipeline discovers the structure, and a filtrated functional partial least squares procedure extracts shared components and estimates coefficients. Simulation studies are reported to recover the coarse-to-fine organization and improve prediction relative to competitors; an application to lower-limb angular kinematics data is said to yield higher accuracy and interpretable aging-related coordination patterns.

Significance. If the hierarchical filtration reliably isolates scale-specific dependencies without strong model assumptions, the work supplies a new multiscale representation-learning tool for functional data with multiple predictors. The emphasis on progressive identification from coarse to fine layers and the real-data interpretability in biomechanics could be useful for applications involving coordinated multidimensional objects.

major comments (2)
  1. [Simulation studies] Simulation studies section: the reported recovery of the 'dominant coarse-to-fine organization' is demonstrated only under data-generating processes that match the assumed hierarchical forest structure. When shared effects are instead non-hierarchical (e.g., overlapping groups at one scale or non-tree dependencies), the progressive filtration layers have no guaranteed mechanism to isolate the correct components, undermining the general claim that the framework recovers underlying shared structures.
  2. [Methods] Filtration-based pursuit pipeline subsection: the construction of the hierarchical forest and the definition of filtration layers rely on unspecified tuning parameters and stopping rules. Without explicit criteria or sensitivity analysis, it is unclear whether the progressive identification is robust or whether results depend on ad-hoc choices that could be tuned to favor the method.
minor comments (2)
  1. [Abstract] Abstract: quantitative details on prediction metrics, error bars, and the exact competing methods are omitted, making it difficult to gauge the magnitude of the reported improvements.
  2. [Methods] Notation: the distinction between 'filtration layers' and 'forest structure' is introduced without a clear diagram or pseudocode, complicating replication of the pipeline.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment point by point below, indicating planned revisions where appropriate to improve clarity and strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Simulation studies] Simulation studies section: the reported recovery of the 'dominant coarse-to-fine organization' is demonstrated only under data-generating processes that match the assumed hierarchical forest structure. When shared effects are instead non-hierarchical (e.g., overlapping groups at one scale or non-tree dependencies), the progressive filtration layers have no guaranteed mechanism to isolate the correct components, undermining the general claim that the framework recovers underlying shared structures.

    Authors: We agree that the reported simulations focus on data-generating processes consistent with the hierarchical forest structure for which the method is developed. The framework is intended to identify multiscale shared structures under this progressive coarse-to-fine organization, and the simulations demonstrate recovery and improved prediction in that setting. To address the concern about scope, we will add new simulation scenarios with non-hierarchical dependencies (such as overlapping groups or non-tree structures) in the revised manuscript. These will illustrate the method's behavior outside the primary assumption and help delineate the conditions under which the filtration approach is most appropriate. revision: yes

  2. Referee: [Methods] Filtration-based pursuit pipeline subsection: the construction of the hierarchical forest and the definition of filtration layers rely on unspecified tuning parameters and stopping rules. Without explicit criteria or sensitivity analysis, it is unclear whether the progressive identification is robust or whether results depend on ad-hoc choices that could be tuned to favor the method.

    Authors: We acknowledge that the current description of the filtration-based pursuit pipeline leaves the tuning parameters and stopping rules insufficiently detailed. In the revision, we will explicitly specify the criteria for constructing the hierarchical forest, the definition of filtration layers, and the stopping rules employed. We will also include a sensitivity analysis examining how variations in these parameters affect the recovered structure and downstream prediction performance, thereby demonstrating robustness and improving reproducibility. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation defines new filtration hierarchy independently of fitted outputs

full rationale

The paper defines a filtration-based framework that organizes multiple functional predictors into a hierarchical forest structure, with shared and predictor-specific components identified progressively across coarse-to-fine layers. It then specifies a pursuit pipeline and filtrated functional PLS for extraction and estimation under that structure. Simulation recovery and real-data kinematics results serve as external benchmarks rather than tautological re-derivations; no equation or claim reduces a prediction to a fitted parameter by construction, and no load-bearing step relies on self-citation chains or imported uniqueness theorems. The central claims remain self-contained against the stated model assumptions and validation experiments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that shared structures are hierarchically organizable via filtration layers; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Response-predictor dependencies vary across representation dimensions and emerge at multiple resolutions ranging from globally shared effects to predictor-specific effects.
    This premise directly motivates the construction of the hierarchical forest through successive filtration layers.

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Reference graph

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