PCA of probability measures: Sparse and Dense sampling regimes

Elsa Cazelles; Gachon Erell; J\'er\'emie Bigot

REVIEW 3 major objections 6 minor 36 references

PCA of probability measures: Sparse and Dense sampling regimes

T0 review · 3 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper proves that PCA on n probability measures, each observed through m samples, has error n^{-1/2} + m^{-α}, with the two terms trading off in a sparse-to-dense transition.

desk verdict Solid double-asymptotic rates for PCA of embedded measures, with the main caveat that the LOT simulation uses an imported rate that does not actually cover the experiment. read the letter →

arxiv 2602.02190 v2 pith:CSRRTJ2Z submitted 2026-02-02 stat.ML cs.LG

Gachon Erell , Jérémie Bigot , Elsa Cazelles This is my paper

classification stat.MLcs.LG MSC 62H2562G20

keywords principalcomponentanalysisprobabilitymeasuresHilbertspaceembeddingskernelmeanembeddinglinearizedoptimaltransportslicedWassersteindoubleasymptoticregimesubsampling

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 studies PCA when the data are not points but n probability measures, each observed through m independent samples, embedded into a Hilbert space. Its central claim is that the estimation error of the empirical covariance operator and of the PCA projector splits into two additive parts: n^{-1/2}, from sampling only n measures, and r_m(Φ), the per-measure embedding error, which decays as m^{-α} with α > 0 depending on the embedding. This produces a sparse-to-dense transition: when m is small relative to n^{1/(2α)}, the per-measure error dominates; when m is large, the n^{-1/2} term dominates and further sampling within each measure no longer helps. The paper also proves the n^{-1/2} dense-regime rate is minimax optimal for covariance estimation, and argues in practice that subsampling each measure to the transition threshold preserves PCA accuracy while cutting the cost.

What carries the argument

The engine of the argument is the double decomposition of the error through an intermediate covariance operator Σ_n built from the true embeddings Φ(μ_i). The n-term E∥Σ-Σ_n∥_HS is controlled by the fourth-moment bound R^{1/2} n^{-1/2}; the m-term E∥Σ_n-Σ̂∥_HS is controlled by the per-measure embedding error r_m(Φ)=√E∥Φ(μ)-Φ(ˆμ)∥²_H, which the paper imports from earlier single-measure results. For PCA, the same split is obtained by writing the excess risk as ⟨Σ-Σ_n, P≤q-ˆP≤q⟩ + ⟨Σ_n-Σ̂, P≤q-ˆP≤q⟩; the first is bounded through operator concentration on projected subspaces, the second through a 2R√q-Lipschitz property of the map φ→⟨φ⊗φ, P⟩ and a Wasserstein distance between empirical distribut

What would settle it

Measure r_m(Φ) directly for one embedding and a fixed data class by computing E∥Φ(μ)-Φ(ˆμ)∥²_H across m; if its decay exponent differs from α, the predicted m^{-α} slope and threshold are off. For d=2 LOT, this measurement distinguishes the m^{-1/2} rate assumed in the paper's simulations from the m^{-1/4} rate implied by the cited d≤3 plug-in theorem.

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Extended reading notes

Core claim

Under a fourth-moment assumption on the embedding, Theorem 3.3 bounds the Hilbert–Schmidt error of the empirical covariance operator by R^{1/2} n^{-1/2} + 2 R^{1/4} r_m(Φ), where r_m(Φ) is the root mean-square distance between the embedding of a measure and the embedding of its m-sample estimate. Using known single-measure rates, this gives n^{-1/2} + m^{-α} with α = 1/2 for kernel mean and sliced-Wasserstein embeddings and α = 1/d for the linearized optimal transport embedding in the paper's Table 1. Theorem 3.7 transfers the same structure to PCA excess risk, replacing the n-term by a sum over the leading q eigenvalues and adding a 4R√q r_m(Φ) term; under polynomial eigenvalue decay this i

Load-bearing premise

The load-bearing premise is that the per-measure embedding error r_m(Φ) decays at the rate m^{-α} claimed in the cited single-measure literature; if that imported rate fails for the actual embedding and data, the sparse-regime rate and the transition threshold m ≍ n^{1/(2α)} collapse.

Editorial extensions

If this is right

For fixed n, the error cannot go below the n^{-1/2} floor; increasing m past the threshold m ≍ n^{1/(2α)} buys no statistical accuracy.
Because the sparse-regime rate is driven by m^{-α}, the choice of embedding changes the subsampling requirement: KME and SW need m ≳ n, while LOT needs m ≳ n^{d/2} in the paper's tabulated rates.
Subsampling each measure to the dense-regime threshold preserves the PCA representation; experiments on flow cytometry and 3D point clouds show stable low-dimensional projections at small m.
The dense-regime n^{-1/2} rate for covariance estimation is minimax, so no estimator using n measures and m samples can improve on it when m is large.
PCA excess risk inherits the same transition; with polynomial eigenvalue decay α > 3/2, excess risk is O(n^{-1/2} + √q m^{-α}) for q ≤ n.

Reading between the lines

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

Editorial extension: the theorems import r_m(Φ) from earlier single-measure results, so the sparse-regime rate and threshold m ≍ n^{1/(2α)} are only as reliable as those imported rates for the user's embedding and data class.
Editorial extension: the d=2 LOT simulation assumes r_m ≍ m^{-1/2}, while the cited d≤3 plug-in LOT result gives E∥T̂-T∥² ≍ m^{-1/2}, i.e. r_m ≍ m^{-1/4}; a direct measurement of r_m in d=2 would decide which rate governs the threshold.
Editorial extension: the Gaussian simulation violates the bounded-embedding assumption (Assumption 3.6) yet still shows the predicted m^{-1/2} behavior; this suggests the rates may hold beyond the stated assumptions, but it also means the experiments do not test the theorem's hypotheses.
Editorial extension: for fixed n, plotting E∥Σ̂-Σ∥_HS against m should show a plateau at the n^{-1/2} level; the crossover location gives an empirical estimate of the transition and a check on n^{1/(2α)}.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core of this paper is worth your time. The authors give a clean double-asymptotic decomposition for PCA on embedded probability measures: empirical covariance error bounded by n^{-1/2} + r_m(Phi), with a matching minimax lower bound for the dense regime, and an analogous excess-risk bound with a sqrt(q) factor on the intra-measure term. The proofs are detailed and internally consistent; Theorem 3.3 is a straightforward but correct bias-variance split, Theorem 3.4 is a standard two-point construction with explicit Gaussian models, and Theorem 3.7 adapts the Reiss-Wahl / Koltchinskii-Lounici machinery without obvious gaps. As far as I can tell, the double-asymptotic framework for PCA of random probability measures is genuinely new, and the minimax lower bound is a real addition. The paper earns a careful reading on the theory alone.

That said, the experimental validation is weaker than the theory, and the weak point is exactly where the reader's stress test lands. The rates for r_m(Phi) are all imported from single-measure results, which is fine in principle—the paper's contribution is the double-asymptotic aggregation, not the per-measure rate. But the LOT simulation uses r_m ~ m^{-1/2} for d=2 on the logic that m^{-1/d}=m^{-1/2}, while the cited results for the setting actually used (empirical reference measure, barycentric projection) give m^{-1/4} (up to logs) for d<=3. The paper's own Table 1 lists [26] as giving m^{-1/4} for barycentric projection, so there is an internal inconsistency. Additionally, Assumption 3.6, required for Theorem 3.7, is violated by the unbounded Gaussian measures and by LOT/SW embeddings of them. The simulations therefore do not actually validate the m^{-alpha} term for LOT, and the practical rule m ~ n^{d/2} rests on an unproved rate for the estimator used. This is fixable: either run LOT experiments in d>=5 where the compact-support rates apply, or prove Gaussian-specific LOT rates and weaken Assumption 3.6. Minor issues: no code, no error bars on the main simulation figures.

Overall: the theoretical core is sound and novel; the empirical claims need alignment with the stated assumptions. This deserves serious peer review—the referee should focus on the LOT rate mismatch and the Assumption 3.6 gap. I would cite the main theorems in my own work, and I'd bring it to a reading group interested in distribution-valued PCA or minimax covariance estimation.

Referee Report

3 major / 6 minor

Summary. The paper studies PCA for n i.i.d. random probability measures, each observed through m i.i.d. samples, after a Hilbert-space embedding Φ. It defines the empirical covariance operator of the embedded measures and bounds its expected Hilbert–Schmidt error by n^{-1/2} plus a per-measure sampling term r_m(Φ) (Theorem 3.3). It proves a minimax lower bound of order n^{-1/2} for covariance estimation (Theorem 3.4) and an upper bound for the PCA excess risk (Theorem 3.7) that again splits into an n-dependent term and a √q r_m(Φ) term. Corollary 3.8 specializes the PCA bound to polynomial and exponential eigenvalue decay. The paper also discusses the sparse–dense transition and the resulting recommendation for choosing the number of samples per measure, with simulations and real-data experiments using KME, LOT, and SW embeddings.

Significance. The paper's reduction of the double-asymptotic PCA problem to the single-measure sampling error r_m(Φ) is clean and potentially useful: if the imported rates are correct, it gives a transparent n^{-1/2}+m^{-α} decomposition and a corresponding subsampling rule. The proofs of the reduction (Appendix A and C) are detailed, and the minimax lower bound uses explicit Gaussian models rather than circular arguments. However, the advertised m^{-α} rates are not proved here; they are imported from prior work on single-measure embeddings. The numerical validation of the LOT embedding is not consistent with the manuscript's own cited rates in the d=2 setting used in Section 4.1, and the PCA excess-risk simulations use embeddings that violate Assumption 3.6. These issues are local to the interpretation, presentation, and validation, but they are load-bearing for the practical transition threshold and for the claim that the experiments validate the theorems.

major comments (3)

[§4.1, Table 1, Appendix F.2.1] The simulation claims that all three embeddings have sampling error r_m ≍ m^{-1/2} in d=2, and Figure 2 is said to confirm this. For LOT this is not supported by the manuscript's own cited rates. Table 1 explicitly targets d≥5, where r_m(Φ_LOT)≍m^{-1/d}; Section 4.1 chooses d=2 and writes m^{-1/d}=m^{-1/2}, but the cited one-sample result (Theorem F.5, [20, Cor. 7]) gives E∥T̂−T∥²≍m^{-1/2} for d≤3, i.e. r_m≍m^{-1/4}. The two-sample barycentric estimator used in Eq. (10) is even less covered by that result, and the [26] row of Table 1 gives r_m≍m^{-1/4} for the entropic barycentric projection. Thus no imported rate listed in the paper yields r_m≍m^{-1/2} for the LOT procedure actually implemented in §4.1. Consequently the statement that rates align in d=2 and the use of Figure 2 as validation of the LOT transition are not justified.
[§4.1 vs. Assumption 3.6] Theorem 3.7, the PCA excess-risk bound, is proved under Assumption 3.6, which requires the embedding to be uniformly bounded over all probability measures. The simulation in §4.1 uses Gaussian measures on R^d with: (i) KME with the linear kernel, whose embedding is x↦x^T b and is unbounded; (ii) LOT maps (σI−I)x+b on R^d; (iii) SW quantile functions on R, which are unbounded in t for varying σ. These violate Assumption 3.6. Therefore Figures 1(b)–2(b) do not fall under Theorem 3.7, and the claim that the experiments validate the PCA excess-risk rate is not supported. The authors should either restrict the simulations to compactly supported measures/bounded kernels or provide an excess-risk bound requiring only fourth moments.
[§3.3.1, Table 1, abstract] The main text states r_m(Φ_LOT)≍m^{-1/d} and derives the subsampling rule m≳n^{d/2} as a general statement, but the table caption restricts the displayed LOT rates to d≥5, and Appendix F.2.1 records different rates for d≤3 and d=4 from the same cited results. The sparse–dense transition advertised in the abstract depends on the correct value of α, so the dimension-dependent piecewise form of α for LOT should be stated in the main text. As written, the reader cannot tell that the transition threshold m≳n^{d/2} is only a high-dimensional LOT result and that in low dimensions the cited rates give a different threshold (e.g. m≍n^2 when r_m≍m^{-1/4}).

minor comments (6)

[Table 1] Typo in the caption: 'T able 1'. Also, the caption's '(d≥5)' qualification is not mentioned where Table 1 is used in §3.3.1; this should be cross-referenced explicitly.
[Figures 1 and 2] The metrics are min-max normalized to [0,1], so visual inspection alone is not a quantitative validation of the slopes. Please report fitted log-log slopes or residual-based checks for each embedding and regime.
[§4.1, Eq. (10)] The LOT experiments use an empirical reference measure ρ̂_{m0} with m0≠m in general, but the imported rates in Table 1 correspond to specific situations (true ρ vs. empirical ρ, m0=m or separate). Please clarify which row of Table 1 covers the exact procedure in Eq. (10) when m0 is fixed at 100 and m varies.
[Assumptions 3.1 and 3.6] Both assumptions use the symbol R for different bounds (fourth moment vs. uniform norm). Using R_4 and R_∞ would avoid confusion, especially because both appear in Theorem 3.7's proof.
[§2, §4.1] The paper says Φ(µ) is 'centered for simplicity'. In the SW simulation, with b∼N(0,τ_b²I) and σ∼N(1,τ_σ²), E[Φ_SW(µ)] is the nonzero function √2 erf^{-1}(2t−1). Please state whether the embeddings are centered before computing the closed-form Σ in Propositions D.1–D.3, and adjust the simulation description if so.
[Corollary 3.8] In the proof of the polynomial decay case, the text initially says α>1 and then specializes to α>3/2; state the assumption at the start of the corollary only.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper proves a decomposition in terms of r_m(Φ) and imports single-measure rates as external inputs; no prediction reduces to a fit or to the paper's own assumptions by construction.

full rationale

The central results Theorem 3.3 and Theorem 3.7 are upper bounds expressed in terms of r_m(Φ), the per-measure embedding error defined in Eq. (7). This is not circular: r_m(Φ) is an input quantity, not the target of the paper's derivation. The n^{-1/2} term in Theorem 3.3 is proved directly through Jensen's inequality, an expansion of the HS norm, and the fourth-moment Assumption 3.1 (Lemmas A.1-A.2). The m-dependent part is inherited from prior single-measure convergence results cited in Table 1 and Appendix F (e.g., [20], [1], [26], [8]); these are external results with stated assumptions, not results of the present paper. The minimax lower bound Theorem 3.4 is constructed from explicit Gaussian hypotheses with closed-form covariance operators, and the KL divergence is bounded directly; it does not reuse the upper bound or any fitted parameter. The Gaussian covariance formulas in Appendix D are derived analytically from the embeddings, not fitted to data. The numerical experiments compare observed errors to the theoretically predicted slopes using closed-form population covariances; no fitted constant from the experiments is used to obtain the theoretical rates. Some self-citations appear in the related-work section (e.g., [4], [32]) but they are not load-bearing for the main theorems. There is a legitimate correctness concern that the §4.1 LOT simulation uses an empirical reference measure and barycentric projection, whereas the m^{-1/d} rate in Table 1 for LOT assumes a true reference measure; likewise Assumption 3.6 (bounded Φ) is violated by Gaussian measures. However, these are assumption-applicability issues, not circular reductions: the paper's claimed rates do not reduce by construction to the data used to illustrate them. Therefore no circular step satisfying the quoted-evidence standard is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

Central claims rely on: (a) the random-measure i.i.d. model; (b) fourth-moment and bounded-embedding assumptions; (c) imported single-measure rates r_m(Φ); and, in the lower-bound proof, on a linear-kernel KME construction and explicit Gaussian models. No free parameters are fitted to data, and no new entities are introduced.

assumptions (7)

domain assumption Random measures μ_1,...,μ_n are i.i.d. copies of a random measure μ, and samples within each measure are i.i.d.
Sets up the double-asymptotic model; the n^{-1/2} term follows from independence across measures.
domain assumption Assumption 3.1: E||Φ(μ)||^4 ≤ R and E||Φ(μ̂)||^4 ≤ R for all m.
Ensures the covariance operator is trace-class and permits the fourth-moment bounds in Lemmas A.1 and A.2.
domain assumption Assumption 3.6: the embedding is uniformly bounded, i.e. ∃R, ∀μ, ||Φ(μ)||_H ≤ R.
Needed in Theorem 3.7 and Lemma C.6. It excludes LOT/SW on unbounded support; the paper's Gaussian simulations do not satisfy it.
domain assumption Known single-measure rates r_m(Φ) from Table 1 and Appendix F are correct and apply to the data at hand.
All m-dependence in the theorems is inherited from prior KME/LOT/SW embedding results; no new r_m rates are derived in this paper.
ad hoc to paper The minimax lower bound for KME is constructed with the linear kernel (unbounded).
Theorem 3.4 states the bound for 'the KME embedding' without qualification, but Lemma B.2 uses only the linear kernel; bounded-kernel KME is not covered.
ad hoc to paper Gaussian simulation supports r_m(Φ_LOT) ≍ m^{-1/2} in d=2.
Used to align rates in §4.1. Not established in the paper; the cited [20] d≤3 rate gives r_m ≍ m^{-1/4}.
standard math Spectral theorem, min-max principle, and trace/Hilbert–Schmidt facts for compact self-adjoint operators.
Used throughout the proofs (Appendix E, Theorems E.3 and E.4).

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Pith. "Pith review of PCA of probability measures: Sparse and Dense sampling regimes." pith.science (2026). https://pith.science/paper/CSRRTJ2Z

@misc{pith2026260202190,
  author       = {Pith},
  title        = {Pith review of: PCA of probability measures: Sparse and Dense sampling regimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CSRRTJ2Z}},
  note         = {Machine review of arXiv:2602.02190}
}

abstract

A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-\alpha}$ for the empirical covariance operator and the PCA excess risk, where $\alpha>0$ depends on the chosen embedding. This characterizes the relationship between the number $n$ of measures and the number $m$ of samples per measure, revealing a sparse (small $m$) to dense (large $m$) transition in the convergence behavior. Moreover, we prove that the dense-regime rate is minimax optimal for the empirical covariance error. Our numerical experiments validate these theoretical rates and demonstrate that appropriate subsampling preserves PCA accuracy while reducing computational cost.

Figures

Figures reproduced from arXiv: 2602.02190 by the authors.

**Figure 1.** Dense sampling regime (m = 1000 fixed, n varies from 10 to 1000). Flow cytometry datasets. We use publicly available flow cytometry datasets [7] from the T-cell panel of the Human Immunology Project Consortium (HIPC). Seven laboratories each stained three replicates of three cryo-preserved biological samples (denoted patient 1, 2, and 3), yielding a total of n = 7 × 3 × 3 = 63 datasets. Each dataset consists of meas… view at source ↗

**Figure 2.** Sparse sampling regime (n = 500 fixed, m varies from 10 to 500). 5 Conclusion Understanding the interplay between the number n of measures and the number m of samples per measure is essential for PCA of probability measures via Hilbert space embeddings. In this work, we characterized the convergence behavior of the empirical covariance operator and the PCA excess risk, establishing rates of the form n −1/2 + m−α, wh… view at source ↗

**Figure 3.** 2D PCA representation of the n = 63 HIPC measures for different subsample sizes m (per column) and three embeddings (KME, LOT, SW, per row respectively). Each plot shows the projection onto the first two principal components, with different markers indicating different laboratories. [2] Ryad Belhakem, Franck Picard, Vincent Rivoirard, and Angelina Roche. Minimax estimation of functional principal components from noi… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Evolution of mean and standard deviation of Procrustes disparity for different subsample sizes on two datasets. [8] Sinho Chewi, Jonathan Niles-Weed, and Philippe Rigollet. Statistical optimal transport. arXiv preprint arXiv:2407.18163, 3, 2024. [9] John B. Conway. A C…

**Figure 5.** Figure 5: Example of 3D shapes from the ModelNet10 dataset and their point cloud representation obtained by sampling m = 2000 points. [19] Vladimir Koltchinskii and Karim Lounici. Concentration inequalities and moment bounds for sample covariance operators. Bernoulli, pages 110–…

**Figure 6.** Figure 6: 2D PCA representation of the ModelNet10 dataset for different subsample sizes m and three embeddings (KME, LOT, SW). Each plot shows the projection onto the first two principal components. [29] Markus Reiss and Martin Wahl. Nonasymptotic upper bounds for the reconstruc…

**Figure 7.** Figure 7: 2D PCA representation of n = 20 gaussian measures for different subsamples sizes m and three embeddings (KME, LOT, SW). Each plot shows the projection onto the first two principal components. Notice that P θ (Y1), · · · , Pθ (Ym) are i.i.d samples from P θ #µ and P θ #…

**Figure 8.** Figure 8: Mean Procrustes disparity and standard deviation for different subsample sizes on the image dataset. the barycentric projection from the empirical reference measure ρˆm0 to each data as described in Section 4.1. Finally, the number of quantiles and the number of projec…

**Figure 9.** Figure 9: 2D PCA representation of the image dataset for different subsamples sizes m and three embeddings (KME, LOT, SW). Each plot shows the projection onto the first two principal components. 43 [PITH_FULL_IMAGE:figures/full_fig_p043_9.png]

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