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arxiv: 2605.24113 · v1 · pith:RKNJJWJQnew · submitted 2026-05-22 · 💻 cs.LG · math.DG· math.OC· math.ST· stat.TH

Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions

Willem Diepeveen , Deanna Needell This is my paper

Pith reviewed 2026-06-30 16:30 UTC · model grok-4.3

classification 💻 cs.LG math.DGmath.OCmath.STstat.TH
keywords Riemannian archetypal analysispullback geometrydeformed star distributionsarchetypal analysismanifold learninggeodesic interpolationinterpretable machine learningdata projection
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The pith

Riemannian archetypal analysis projects data onto manifolds of geodesically convex archetype combinations via pullback geometry.

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

The paper develops a Riemannian version of archetypal analysis to handle non-linear data structures while retaining the interpretability of classical archetypes. It introduces a class of deformed star distributions that induce a data-driven pullback Riemannian geometry, supplying a statistical interpretation for the resulting manifold mappings. The Riemannian archetypal mapping is defined as the projection onto the manifold of geodesically convex combinations of archetypes and is computed via convex relaxation followed by non-convex refinement. Experiments on synthetic examples and MNIST data show that the approach yields meaningful geodesics, denoising projections, and geometry-aware classifications.

Core claim

The Riemannian archetypal mapping is defined as a projection onto the manifold of geodesically convex combinations of archetypes under the pullback Riemannian geometry induced by deformed star distributions, providing a statistically interpretable non-linear extension of classical archetypal analysis that combines interpretability with expressive power through a practical optimization scheme of convex relaxation and non-convex refinement.

What carries the argument

The Riemannian archetypal mapping (RAM) as the projection onto the manifold of geodesically convex combinations of archetypes, enabled by pullback Riemannian geometry on deformed star distributions.

If this is right

  • The method produces meaningful geodesics between archetypes on the learned manifold.
  • It yields useful denoising projections of input data points.
  • It enables geometry-aware classifications of data.
  • It clarifies remaining limitations in the convex-plus-non-convex optimization scheme.

Where Pith is reading between the lines

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

  • The framework could be tested on additional high-dimensional datasets to assess scalability beyond MNIST.
  • More effective schemes for learning the deformed star distributions from data might reduce the gap to optimality.
  • Similar pullback constructions could be explored for other linear interpretable methods to create Riemannian variants.

Load-bearing premise

Data-driven pullback geometry defined on deformed star distributions supplies a valid statistical interpretation of the manifold mappings for real-valued data.

What would settle it

An experiment where the learned geodesics fail to produce better denoising or classification performance than classical linear archetypal analysis on the MNIST dataset would falsify the claimed practical advantage of the Riemannian extension.

Figures

Figures reproduced from arXiv: 2605.24113 by Deanna Needell, Willem Diepeveen.

Figure 1
Figure 1. Figure 1: Data from the deformed star distribution (blue) are projected in the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: geodesics under a pullback geometry that is possibly modified by a diffeomorphism [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Using the default pullback geometry for a data distribution [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Using either the default pullback geometry or its modified variant for a data distribution [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The Riemannian archetypal mapping (RAM) – initialized from relaxed RAM weights – projects data points [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The pullback logarithmic mappings (pink) do not carry information on how far away the archetypes are [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Instead of fitting a single radial function to the entire dataset, we first decompose the latent data (left) using [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The proposed construction (38) combined with (47) yields geodesics between archetypes that remain close [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: After training the normalizing flow with loss (37) on the digit 3 from MNIST, the latent representations [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The archetypes v (j) θ ∗ (40) for j = 1, . . . , 10 obtained from Riemannian Archetypal analysis (39) – under the pullback geometry generated by ϕθ ∗ – sketch a reasonable picture of the different types of threes in the data set. γ ℓ 2 x,y (t) γ ϕθ∗ x,y (t) γ φ x,y (t) γ φ,iso x,y (t) [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: When interpolating between two validation images of the digit three (left- and right-most columns), the [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Validation images (top row) are first mapped using the relaxed RAM (middle row), which then serves as an [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: After training the normalizing flow with loss (37) on MNIST, the latent representations [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The archetypes v (j) θ ∗ (40) for j = 1, . . . , 100 obtained from Riemannian Archetypal analysis (39) – under the pullback geometry generated by ϕθ ∗ – on each of the labeled subsets of the data set sketch a reasonable picture of the different types of digits in the data set. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: When interpolating between different digits (left- and right-most columns), the linear path (first rows) [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Validation data points (top row) are first mapped using the relaxed RAM (middle row), which then serves [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: For each of the validation points in Figure 15, we look at how its mass is distributed over the archetypes, [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
read the original abstract

Classical archetypal analysis is appealing for its interpretability, but its linear geometry can limit performance on data with strongly non-linear structure; at the same time, existing neural extensions improve flexibility while often weakening the geometric meaning of archetypes and interpolations. In this work, we develop a Riemannian version of archetypal analysis based on data-driven pullback geometry for real-valued data, with the goal of combining the interpretability of classical archetypal analysis with the expressive power of modern non-linear models. We introduce a class of deformed star distributions together with associated pullback Riemannian geometry to provide a statistical interpretation of the resulting manifold mappings, define the Riemannian archetypal mapping (RAM) as a projection onto the manifold of geodesically convex combinations of archetypes, and propose a practical optimization scheme based on convex relaxation followed by non-convex refinement. We further propose a learning scheme that yields reasonable, albeit generally suboptimal, deformed star distributions from data. Experiments on synthetic examples and MNIST show that the resulting framework produces meaningful geodesics, useful denoising projections, and geometry-aware classifications, while also clarifying where current optimization limitations remain.

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 paper develops Riemannian Archetypal Analysis (RAM) for non-linear interpretable data analysis. It introduces a class of deformed star distributions to induce data-driven pullback Riemannian geometry, defines RAM as the projection onto the manifold of geodesically convex combinations of archetypes, proposes an optimization scheme via convex relaxation followed by non-convex refinement, and includes a learning scheme to obtain the distributions from data. Experiments on synthetic examples and MNIST are reported to produce meaningful geodesics, denoising projections, and geometry-aware classifications.

Significance. If the central construction is valid, the framework could usefully combine the interpretability of classical archetypal analysis with non-linear flexibility while retaining geometric meaning for archetypes and interpolations. The reported experiments indicate practical utility on both synthetic and image data, but the significance hinges on whether the pullback geometry supplies a coherent statistical model rather than an arbitrary construction.

major comments (2)
  1. [Abstract] Abstract: the assertion that deformed star distributions together with their pullback Riemannian geometry 'provide a statistical interpretation' of the manifold mappings is load-bearing for the central claim, yet the abstract (and the described construction) supplies no independent derivation linking the pullback metric to a standard statistical object such as a density or likelihood; the geometry is data-driven by construction, leaving open whether geodesics and convex combinations correspond to any coherent probabilistic model for real-valued data.
  2. [Learning scheme description] The learning scheme that produces deformed star distributions from data is presented as yielding 'reasonable, albeit generally suboptimal' results, but without external benchmarks or verification that the induced manifold satisfies the statistical-interpretation requirement, the scheme risks circularity: the distributions are fitted to enable the geometry that is then claimed to interpret the mappings.
minor comments (2)
  1. [Optimization] The optimization scheme is described at a high level (convex relaxation then non-convex refinement) but lacks explicit convergence analysis or guarantees that the refined solution remains on the geodesically convex hull; this should be clarified with pseudocode or a dedicated subsection.
  2. [Experiments] MNIST experiments report qualitative benefits for denoising and classification but provide no quantitative metrics (e.g., reconstruction error, classification accuracy) or comparisons against linear archetypal analysis or other non-linear baselines.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment below and indicate the revisions we plan to make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that deformed star distributions together with their pullback Riemannian geometry 'provide a statistical interpretation' of the manifold mappings is load-bearing for the central claim, yet the abstract (and the described construction) supplies no independent derivation linking the pullback metric to a standard statistical object such as a density or likelihood; the geometry is data-driven by construction, leaving open whether geodesics and convex combinations correspond to any coherent probabilistic model for real-valued data.

    Authors: The deformed star distributions are explicitly constructed as a parametric family of probability distributions on the data space. The pullback Riemannian geometry is then induced by considering the mapping from the simplex of archetype weights to the parameters of these distributions. This provides a statistical interpretation because the archetypes correspond to specific distributions in the family, and geodesically convex combinations correspond to paths within this parametric family, allowing for probabilistic interpretations of interpolations and projections. We acknowledge that this does not constitute a full likelihood model for general real-valued data beyond the induced geometry. To address the concern, we will revise the abstract to clarify that the statistical interpretation is in terms of the parametric family of distributions rather than a direct density estimation framework. revision: yes

  2. Referee: [Learning scheme description] The learning scheme that produces deformed star distributions from data is presented as yielding 'reasonable, albeit generally suboptimal' results, but without external benchmarks or verification that the induced manifold satisfies the statistical-interpretation requirement, the scheme risks circularity: the distributions are fitted to enable the geometry that is then claimed to interpret the mappings.

    Authors: We agree that the learning scheme is heuristic, as explicitly stated in the manuscript. The scheme fits the distributions to the data to capture its structure, after which the geometry is derived. While this could appear circular, the experiments on synthetic data and MNIST demonstrate that the resulting manifolds yield meaningful geodesics and classifications, providing empirical support. We do not provide external benchmarks comparing to other distribution learning methods, which is a limitation. In the revision, we will expand the discussion of the learning scheme to include its limitations and potential for future improvements with more rigorous validation. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces deformed star distributions and associated pullback geometry as a new construction to support the Riemannian archetypal mapping, along with a proposed learning scheme and optimization procedure. No equations or self-citations are available in the provided text that reduce any central claim (such as the statistical interpretation or manifold mappings) to a fitted input, self-definition, or prior author result by construction. The framework is presented as a novel combination of interpretability and non-linear modeling, with experiments on synthetic data and MNIST serving as external validation points. The derivation chain is therefore self-contained against the stated assumptions and does not exhibit load-bearing reductions of the enumerated kinds.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The central claim rests on the newly introduced deformed star distributions and the assumption that pullback geometry on them yields a statistically meaningful manifold; no free parameters or standard axioms are specified in the abstract.

invented entities (1)
  • deformed star distributions no independent evidence
    purpose: to provide a statistical interpretation of manifold mappings under pullback Riemannian geometry
    Introduced in the abstract as a new class of distributions for the method

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