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Estimation of the number of principal components in high-dimensional multivariate extremes

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper builds AIC and BIC estimators for the number of significant principal components of the angular measure in multivariate extremes, and proves conditions under which they recover the true count.

desk verdict First real attempt at information criteria for PCA dimension in extremes; fixed-dim results are solid, but the high-dimensional consistency proofs are one-paragraph delegations with a uniformity gap that needs to be closed. read the letter →

arxiv 2505.22437 v1 pith:TEAZ6T7B submitted 2025-05-28 stat.ME

Lucas Butsch , Vicky Fasen-Hartmann This is my paper
classification stat.ME MSC 62G3262H2560G7062G20
keywords AICBICprincipalcomponentanalysismultivariateextremesangularmeasurespikedcovariancemodelhigh-dimensionalconsistencyregularvariation
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

The dependence structure of multivariate extremes can be summarized by the angular measure, a distribution on the unit sphere whose covariance has a few dominant directions when the data are concentrated near a lower-dimensional subspace. This paper proposes AIC and BIC information criteria to estimate how many of those directions are significant, and proves when the estimates are consistent. In fixed dimension the BIC is weakly consistent while the AIC is not; in the high-dimensional setting, where the dimension and the number of extreme observations grow at the same rate, the AIC is consistent under a gap condition on the spiked eigenvalue and the BIC is consistent when that eigenvalue grows faster than the logarithm of the dimension. The result replaces the subjective 'elbow' choice in a scree plot with an automated rule with proven large-sample behavior.

What carries the argument

The central object is the ordered empirical spectrum $\hat\lambda_{n,1} \ge \cdots \ge \hat\lambda_{n,d}$ of $\hat\Sigma_n$, built from the $k_n$ observations with largest norm; both information criteria are functionals of these eigenvalues, so their consistency is inherited from the asymptotic behavior of the spectrum. In the high-dimensional case that behavior is governed by the Marchenko-Pastur law, with the map $\phi_c(x) = x(1 + c/(x-1))$ giving the limit of a distant spike's empirical eigenvalue and $(1\pm\sqrt{c})^2$ marking the bulk edges. The directional model $\Theta^{(n)} = \Gamma^{(n)1/2}V^{(n)}/\|\Gamma^{(n)1/2}V^{(n)}\|$, with i.i.d. symmetric entries of $V^{(n)}$ and finite fourth moment, is what makes the trailing eigenvalues of $\Sigma^{(n)}$ exactly equal, keeping the spike location $p^*$ well-defined.

What would settle it

Simulate the directional model with $p^* = 1$, $c = 0.5$, and a distant spike whose size makes $\xi_{n,p^*}/\log(d_n)$ tend to 0; if the BIC$^\circ$ selects $p^*$ with probability tending to 1, then the 'not weakly consistent' claim in Theorem 4.4(a) is wrong. Alternatively, take a fixed spike satisfying $\xi_{p^*} > 1+\sqrt{c}$ but violating gap condition (4.1); if AIC$^\circ$ still selects $p^*$ almost surely, Theorem 4.2(b) is false.

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Extended reading notes

Core claim

Under the spiked covariance model $\lambda_1 \ge \cdots \ge \lambda_{p^*} > \lambda_{p^*+1} = \cdots = \lambda_{d-1}$ for the covariance $\Sigma$ of the angular measure, the paper defines information criteria from the empirical eigenvalues $\hat\lambda_{n,i}$ of $\hat\Sigma_n$, the covariance of the $k_n$ observations with largest norm. Its central results are consistency statements: for fixed $d$, $\mathbb{P}(\mathrm{BIC}_{k_n}(p) > \mathrm{BIC}_{k_n}(p^*)) \to 1$ for every $p \neq p^*$ (Theorem 3.6), while the AIC can overestimate with positive asymptotic probability (Theorem 3.3). In the high-dimensional directional model with $d_n/k_n \to c > 0$, the AIC variants are weakly consistent when a distant spike $\xi_{p^*} > 1+\sqrt{c}$ satisfies the gap condition (4.1) or (4.2), or when $\xi_{n,p^*} \to \infty$ with $\xi_{n,1} = o(\sqrt{d_n})$ (Theorems 4.2 and 4.7); the BIC variants are weakly consistent when $\xi_{n,p^*}/\log d_n \to \infty$ and fail when this ratio tends to 0 (Theorems 4.4 and 4.8).

Load-bearing premise

The load-bearing premise is that the trailing eigenvalues of the angular-measure covariance are exactly equal after the $p^*$-th one, so that 'the number of significant components' is a single well-defined location; the paper's precipitation analysis shows real spectra can keep decreasing, and then that target is less clear.

Editorial extensions

If this is right

  • In fixed dimension, use the BIC to select the number of significant components; it recovers the true $p^*$ with probability tending to 1, whereas the AIC tends to overestimate.
  • In the high-dimensional regime with a distant spike, the AIC variants are the right tool when the gap condition (4.1) or (4.2) holds, and the BIC variants are right when the spike grows faster than $\log d_n$.
  • When $\xi_{n,p^*}/\log d_n \to 0$, the BIC cannot be trusted to find the true dimension; it will underestimate.
  • The new eigenvalue limits for the angular-measure covariance can be reused beyond information criteria, for example in testing or threshold choice for extreme-value PCA.
  • On 500-station precipitation data, the criteria cut the dimension of extreme dependence to about 25 (AIC*) or 5–9 (BIC*), showing that the penalty choice strongly controls the resulting model size.

Reading between the lines

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

  • A natural next step, hinted at in the paper's conclusion, is to relax the exactly-equal-trailing-eigenvalues assumption to a band of eigenvalues near the bulk edge; the same Marchenko-Pastur techniques should still give approximate consistency conditions.
  • The BIC condition $\xi_{n,p^*}/\log d_n \to \infty$ suggests a practical diagnostic that the authors do not spell out: estimate the leading eigenvalue's separation from the bulk and compare it with $\log d_n$ before trusting BIC in applications.
  • Because the directional model is one of several tail models, the same information-criterion analysis could be extended to other regularly varying constructions, such as hidden regular variation or kernel-based angular measures, though the paper does not do that.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes AIC- and BIC-type information criteria for estimating the number p* of significant principal components in the covariance matrix of the angular measure of a multivariate regularly varying random vector, under a spiked covariance assumption. Two settings are treated. In Model A, the dimension d is fixed and the number of extreme observations k_n tends to infinity with k_n/n → 0; the paper proves that the AIC is not consistent (Theorem 3.3 with a counterexample in Example 3.5) and that the BIC is weakly consistent (Theorem 3.6). In Model B, the dimension d_n increases with d_n/k_n → c ∈ (0, ∞), and the analysis is restricted to a directional model whose angular component is Γ(n)^{1/2}V / ||Γ(n)^{1/2}V||. The paper derives asymptotic results for the empirical eigenvalues (Theorems 2.7 and 2.9) and then states sufficient conditions, under 0 < c < 1 and c > 1, for AIC- and BIC-type estimators to be weakly consistent (Theorems 4.2, 4.4, 4.7, 4.8), with gap conditions (4.1) and (4.2) for the AIC and a requirement ξ_{n,p*}/log(d_n) → ∞ for the BIC. The performance of the criteria is illustrated in simulations and in an application to German precipitation data.

Significance. If the high-dimensional consistency theorems are fully established, the paper would provide the first principled, automatic method for choosing the number of principal components in multivariate extreme-value PCA, a problem that so far has been handled mainly by scree plots and risk plots. The fixed-dimensional results are clear and in line with classical information-criterion theory; the random-matrix derivations in Appendix A, in particular Theorem A.1 connecting the empirical eigenvalues of the angular measure to those of the underlying Gaussian-like matrix, are a substantive contribution. The paper also supplies code and gives a candid discussion of the restrictiveness of the spiked assumption in the precipitation application. However, the advertised high-dimensional consistency theorems are not fully proved as written, because the Appendix C proofs delegate to an external result without verifying the required uniformity over the growing candidate set. This is a load-bearing gap that must be addressed before the main claims can be accepted.

major comments (2)
  1. [Appendix C, proofs of Theorems 4.2, 4.4, 4.7, 4.8] The proofs of the four high-dimensional consistency theorems consist of one-paragraph statements that the proofs of Bai, Choi and Fujikoshi (2018) for bξ_{n,i} 'can be carried out step by step' for d_n λhat_{n,i}. This transfer is not automatic. The information-criterion differences involve averages of the trailing eigenvalues with lower summation index p+1 ranging over all candidate dimensions p = 1, ..., q_n, where q_n = o(d_n). Theorems 2.7(c)-(d) and 2.9(c)-(d) establish convergence of such averages only for one fixed truncation q_n = o(d_n), not uniformly over all p ≤ q_n. Controlling the argmin over the growing set {1, ..., q_n} requires simultaneous control of these statistics; pointwise convergence for each fixed p does not suffice when q_n grows. In addition, the BCF proofs are almost-sure arguments, whereas the paper only provides convergence in probability for fixed truncation points. The manuscript therefore needs either a full proof of the required sup-over-p uniformity or an explicit uniform convergence lemma showing that the replacement d_n λhat_{n,i} for bξ_{n,i} preserves the BCF arguments.
  2. [Section 2.1, Proposition 2.1 and Remark 2.2] Proposition 2.1, the asymptotic normality of √k_n (Σhat_n − Σ), is stated without proof; Remark 2.2 says only that the techniques of Larsson and Resnick (2012) can be generalized under the technical assumption (A4). This proposition is load-bearing: Theorem 2.3 and hence the fixed-dimensional consistency results in Section 3 (Theorems 3.3 and 3.6) rely on it. Because (A4) is a nonstandard uniform condition on truncated moments, the statement is not a routine citation, and the paper should either provide the proof or give a precise reference with the exact result covering the vectorized covariance estimator.
minor comments (5)
  1. [Section 5.2, noisy directional model] The model X(n) = Γ(n)^{1/2} V / ||Γ(n)^{1/2} V|| · Z + ε, with ε being entrywise absolute Gaussian noise, is used in simulations but is not shown to satisfy the directional Model B or to be multivariate regularly varying; the simulation results for this model are illustrative rather than a direct validation of the theorems.
  2. [Definitions 4.1 and 4.6] The definitions state p = 1, ..., d_n − 2 (or k_n − 2) but the estimator is defined as argmin over 1 ≤ p ≤ q_n; the paper should explicitly require p* ≤ q_n eventually and state how q_n is chosen in the simulations and application.
  3. [Section 6, Figure 8 and Section 7] The text says the scaled eigenvalue increments 'are nearly constant' after some point, but no quantitative criterion is given; the authors themselves acknowledge in Section 7 that the empirical eigenvalues do not stabilize, so the application should be framed even more explicitly as an exploratory illustration outside the spiked model.
  4. [Introduction, line 'Principle Component Analysis'] The phrase 'Principle Component Analysis' should be 'Principal Component Analysis'.
  5. [Equation (1.1) and Model B] In (1.1) the equal trailing eigenvalues are listed as λ_{p*+1} = ... = λ_{d−1}, while in the high-dimensional setting the statement in Lemma 2.5 includes λ_{dn}; the notation should be made uniform.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: consistency proofs are built on external random-matrix theorems; the self-citation [9] is contextual and not load-bearing.

full rationale

The paper's central claims are the consistency theorems for information criteria in fixed and growing dimension. The AIC/BIC definitions are taken from Fujikoshi and Sakurai [22] and Bai, Choi and Fujikoshi [4] (external), and the asymptotic eigenvalue results in Section 2 are derived from external random matrix theory (Bai-Yin, Bai-Yao, Bai-Silverstein, Silverstein) plus the paper's own Theorem A.1, which is proven using those results. The directional model is an explicitly stated model class, not a fitted output; no parameter is calibrated to data and then presented as a prediction. In the fixed-dimensional case, Theorems 3.3 and 3.6 are proved directly from the empirical eigenvalue expansions. In the high-dimensional case, Appendix C delegates the final step to the external BCF proofs via 'step by step' replacement; however this is an appeal to an independent, prior theorem, not a self-citation, and any unverified uniformity transfer would be a proof gap rather than circularity. The only places the authors cite their own earlier work [9] are the introduction and the concluding discussion on the choice of kn; that citation is contextual and does not carry any of the consistency arguments. Simulations generate data with known p*, and the precipitation application makes no claim of recovering a fitted value. Thus there is no circular step; the derivation chain is self-contained with respect to external benchmarks.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim depends on the spiked covariance assumption and, in the high-dimensional regime, on the directional model, plus standard random matrix theory. The tuning parameters kn and qn are free inputs. The technical condition (A4) is specific to this paper's proof of the covariance CLT. No new entities are introduced.

free parameters (2)
  • kn (number of extreme observations) = varies by user; 1% to 15% of n in the precipitation application
    The criteria are computed on the kn observations with the largest norms. The theory requires kn → ∞ and kn/n → 0, but the finite-sample estimate changes strongly with kn (Table 6.1).
  • qn (number of candidate dimensions) = d/2 in the precipitation analysis
    The high-dimensional criteria minimize over p = 1, ..., qn, and the consistency theorem requires qn = o(d_n). The choice of qn affects the selected p* in finite samples.
assumptions (6)
  • domain assumption Multivariate regular variation of index α with spectral vector Θ (Model A1 and Model B1).
    This is the standard EVT setup that defines the angular measure; it is assumed rather than tested in the application.
  • domain assumption Spiked covariance structure: λ1 ≥ ... ≥ λ_{p*} > λ_{p*+1} = ... = λ_{d-1} (Eq. 1.1).
    Defines the target p* as the number of significant components. The precipitation data example shows eigenvalues that decrease without a plateau, which the authors concede contradicts this assumption (Section 7).
  • domain assumption Directional model with i.i.d. symmetric V entries, finite fourth moment, and independent Fréchet radial part (Section 2.2).
    The high-dimensional consistency theorems (Section 4) are proved only under this parametric family.
  • ad hoc to paper Technical condition (A4) on the uniform convergence of truncated moments.
    Introduced to support the CLT for the empirical covariance matrix (Proposition 2.1), whose proof is omitted (Remark 2.2).
  • domain assumption Eigenvalue conditions: ξ_{n,p*} > 1 + √c (distant spike) or ξ_{n,p*} → ∞ with ξ_{n,1} = o(d_n^{1/2}).
    Required for the eigenvalue limits in Theorems 2.7 and 2.9, which drive the consistency results.
  • domain assumption Gaussian likelihood functional form used to define AIC and BIC.
    The criteria are motivated by a Gaussian log-likelihood even though Θ is not Gaussian; consistency is proved for the resulting functionals under the stated models.

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Pith. "Pith review of Estimation of the number of principal components in high-dimensional multivariate extremes." pith.science (2026). https://pith.science/paper/TEAZ6T7B

@misc{pith2026250522437,
  author       = {Pith},
  title        = {Pith review of: Estimation of the number of principal components in high-dimensional multivariate extremes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TEAZ6T7B}},
  note         = {Machine review of arXiv:2505.22437}
}
abstract

For multivariate regularly random vectors of dimension $d$, the dependence structure of the extremes is modeled by the so-called angular measure. When the dimension $d$ is high, estimating the angular measure is challenging because of its complexity. In this paper, we use Principal Component Analysis (PCA) as a method for dimension reduction and estimate the number of significant principal components of the empirical covariance matrix of the angular measure under the assumption of a spiked covariance structure. Therefore, we develop Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) to estimate the location of the spiked eigenvalue of the covariance matrix, reflecting the number of significant components, and explore these information criteria on consistency. On the one hand, we investigate the case where the dimension $d$ is fixed, and on the other hand, where the dimension $d$ converges to $\infty$ under different high-dimensional scenarios. When the dimension $d$ is fixed, we establish that the AIC is not consistent, whereas the BIC is weakly consistent. In the high-dimensional setting, with techniques from random matrix theory, we derive sufficient conditions for the AIC and the BIC to be consistent. Finally, the performance of the different AIC and BIC versions is compared in a simulation study and applied to high-dimensional precipitation data.

Figures

Figures reproduced from arXiv: 2505.22437 by the authors.

Figure 1
Figure 1. Simulations for the directional model with [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Simulations for directional factor data with [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Simulations for noisy directional factor data with [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Simulations for noisy directional factor data with [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Simulations for spiked angular Gaussian data with [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Left figure: Map of Germany with all stations highlighted by black dots. Right figure: Map [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: The estimated number pbn of significant eigenvalues determined by AIC∗ and BIC∗ plotted against kn. 0.00 0.25 0.50 0.75 1.00 0 20 40 60 0.0 0.1 0.2 0.3 0 20 40 60 AIC* BIC* [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: For kn = 76, on the left hand side the scaled ordered empirical eigenvalues λb n,i/λb n,1, i = 1, . . ., 75 and on the right hand side the differences of the ordered empirical eigenvalues divided by the value of the largest eigenvalue (λb n,i −λb n,i+1)/λb n,1, i = 1, …

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