OPTIMAL DETECTION OF SPARSE PRINCIPAL COMPONENTS IN HIGH DIMENSION

被引：151

作者：

Berthet, Quentin ^{[1
]}

Rigollet, Philippe ^{[1
]}

机构：

[1] Princeton Univ, Dept Operat Res & Financial Engn, Princeton, NJ 08544 USA

来源：

ANNALS OF STATISTICS | 2013年 / 41卷 / 04期

基金：

美国国家科学基金会;

关键词：

High-dimensional detection; sparse principal component analysis; spiked covariance model; semidefinite relaxation; minimax lower bounds; planted clique; LARGE HIDDEN CLIQUE; LARGEST EIGENVALUE; NORM; SIZE; PCA; SUBMATRICES; RELAXATIONS; CONSISTENCY; MATRICES; NOISY;

D O I：

10.1214/13-AOS1127

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

We perform a finite sample analysis of the detection levels for sparse principal components of a high-dimensional covariance matrix. Our minimax optimal test is based on a sparse eigenvalue statistic. Alas, computing this test is known to be NP-complete in general, and we describe a computationally efficient alternative test using convex relaxations. Our relaxation is also proved to detect sparse principal components at near optimal detection levels, and it performs well on simulated datasets. Moreover, using polynomial time reductions from theoretical computer science, we bring significant evidence that our results cannot be improved, thus revealing an inherent trade off between statistical and computational performance.

引用

页码：1780 / 1815

页数：36

共 76 条

[51] Hiding cliques for cryptographic security [J].

Juels, A ;

Peinado, M .

DESIGNS CODES AND CRYPTOGRAPHY, 2000, 20 (03) :269-280

[52]

Karp R.M., 1972, PLENUM PRESS SURV ST, P85, DOI 10.1007/978-1-4684-2001-2_9

[53]

KOIRAN P., 2012, PREPRINT

[54] Approximating the independence number and the chromatic number in expected polynomial time [J].

Krivelevich, M ;

Vu, VH .

JOURNAL OF COMBINATORIAL OPTIMIZATION, 2002, 6 (02) :143-155

[55] EXPECTED COMPLEXITY OF GRAPH PARTITIONING PROBLEMS [J].

KUCERA, L .

DISCRETE APPLIED MATHEMATICS, 1995, 57 (2-3) :193-212

[56]

Laurent B, 2000, ANN STAT, V28, P1302

[57] HIGH-DIMENSIONAL REGRESSION WITH NOISY AND MISSING DATA: PROVABLE GUARANTEES WITH NONCONVEXITY [J].

Loh, Po-Ling ;

Wainwright, Martin J. .

ANNALS OF STATISTICS, 2012, 40 (03) :1637-1664

[58] An augmented Lagrangian approach for sparse principal component analysis [J].

Lu, Zhaosong ;

Zhang, Yong .

MATHEMATICAL PROGRAMMING, 2012, 135 (1-2) :149-193

[59]

MA S., 2011, PREPRINT

[60] SPARSE PRINCIPAL COMPONENT ANALYSIS AND ITERATIVE THRESHOLDING [J].

Ma, Zongming .

ANNALS OF STATISTICS, 2013, 41 (02) :772-801

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