Signal-plus-noise matrix models: eigenvector deviations and fluctuations

Cape, Joshua; Tang, Minh; Priebe, Carey E.

doi:10.1093/biomet/asy070

Cited by 42 publications

(43 citation statements)

References 23 publications

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“…This paper focuses on decomposing the matrixÛ −U W U and on establishing the two-to-infinity norm as a useful tool for matrix perturbation analysis. In the time since this work was first made publicly available, there has been a flurry of activity within the statistics, computer science, and mathematics communities devoted to obtaining refined entrywise singular vector and eigenvector perturbation bounds [1,10,18,38,53]. Among the observations made earlier in this paper, it is useful to keep in mind that…”

Section: In Contrast Spectral Norm Analysis Implies the Weaker Two-tmentioning

confidence: 99%

The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics

Cape¹,

Tang²,

Priebe³

2019

Ann. Statist.

106

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The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors.This paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two-to-infinity norm. Motivated by preliminary deterministic Procrustes analysis, we consider a general matrix perturbation setting in which we derive a new Procrustean matrix decomposition. Together with flexible machinery developed for the two-to-infinity norm, this allows us to conduct a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity. Our analysis yields singular vector entrywise perturbation bounds for a range of popular matrix noise models, each of which has a meaningful associated statistical inference task. In addition, we demonstrate how the two-to-infinity norm is the preferred norm in certain statistical settings. Specific applications discussed in this paper include covariance estimation, singular subspace recovery, and multiple graph inference.Both our Procrustean matrix decomposition and the technical machinery developed for the two-to-infinity norm may be of independent interest.

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Section: In Contrast Spectral Norm Analysis Implies the Weaker Two-tmentioning

confidence: 99%

The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics

Cape¹,

Tang²,

Priebe³

2019

Ann. Statist.

106

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“…Theorem 4.4 (below) relates the quantity U − V W 2→∞ to both the sin Θ distance and to the row structure of V and V ⊥ . We remark that in various (statistical) settings involving (stochastic) matrix perturbations, the term implicitly bounded by s U,V V ⊥ 2→∞ can be analyzed more delicately to yield an improved leading-order term, as pursued in [5,6].…”

Section: Orthogonal Procrustes and Norm-dependent Optimalitymentioning

confidence: 99%

“…As such, the results discussed here hold in broad generality. By way of contrast, stronger results can be obtained under additional structural assumptions, particularly in the classical matrix perturbation setting A := A + E. In statistics, for example, the papers [3,5,6] consider such settings where U (derived from some matrix A) is viewed as a (stochastic) perturbation of V (derived from some matrix A), making it possible to considerably improve (probabilistically) upon certain bounds (e.g., Theorem 4.4).…”

mentioning

confidence: 99%

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Orthogonal Procrustes and norm-dependent optimality

Cape

2020

ELA

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This note revisits the classical orthogonal Procrustes problem and investigates the norm-dependent geometric behavior underlying Procrustes alignment for subspaces. It presents generic, deterministic bounds quantifying the performance of a specified Procrustes-based choice of subspace alignment. Numerical examples illustrate the theoretical observations and offer additional, empirical findings which are discussed in detail. This note complements recent advances in statistics involving Procrustean matrix perturbation decompositions and eigenvector estimation.

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“…This approach is similar to the proposal of Fan et al, 24 who derived the robust properties of the covariance matrix estimator. Cape et al 25 discussed the asymptotic properties of low‐rank approximations showing the unbiasedness and normality of the limiting distribution. Thus, the matrix

VAR false[\hat{ϑ} false]

is computed by using the spectral decomposition of the positive semidefinite Hessian matrix

H (\hat{ϑ}) = V Λ V^{⊤}

, such that

VAR false[\hat{ϑ} false] \approx V_{*} Λ_{*}^{- 1} V_{*}^{⊤},

where V is the matrix of the eigenvectors of Hessian, and the matrix V ∗ denotes a sub‐matrix of eigenvectors associated with the positive eigenvalues of

H (\hat{ϑ})

.…”

Section: Estimation Proceduresmentioning

confidence: 99%

Modeling swine population dynamics at a finer temporal resolution

Sartore

Wei

Abayomi

et al. 2020

Appl Stoch Models Bus & Ind

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The United States Department of Agriculture's National Agricultural Statistics Service (NASS) uses probability surveys of hog owners to estimate quarterly hog inventories in the United States at the national and state levels. NASS also receives data from external sources. A panel of commodity experts forms the Agricultural Statistics Board (ASB). The ASB establishes the NASS official estimates for each quarter by taking into account survey estimates and other relevant sources of information that are available in numerical and non‐numerical form. The aim of this article is to propose an estimation method of hog inventories by combining the NASS proprietary survey results, the hog transaction data, the past ASB panel expert analyses, biological dynamics, and the inter‐inventory relationship constraints. This approach downscales the official estimates to provide monthly estimates according to well‐defined biological growth patterns. The model developed in this study provides national estimates that may inform the quarterly reports.

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Signal-plus-noise matrix models: eigenvector deviations and fluctuations

Cited by 42 publications

References 23 publications

The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics

The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics

Orthogonal Procrustes and norm-dependent optimality

Modeling swine population dynamics at a finer temporal resolution

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