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Consistency of Sparse PCA in High Dimension, Low Sample Size Contexts

Dan Shen, Haipeng Shen, J. S. Marron

arXiv:1104.4289v1math.STstat.TH

TL;DR

Dense PCA loadings can be hard to interpret, motivating sparse PCs with few nonzero loadings; the paper studies their consistency in HDLSS settings. It analyzes thresholding-based sparse PCA methods and finds a region where sparse PCA is consistent even though conventional PCA is strongly inconsistent, with an oracle result clarifying the boundary.

  • Problem

    Conventional PCA typically uses all variables, making its loadings challenging to interpret, while sparse PCA consistency in HDLSS settings is the paper’s central question.

    Source: §1 · pp. 1–2

  • Method

    The paper studies thresholding sparse PC directions, including thresholding through the dual covariance matrix and a penalized RSPCA method.

    Source: §2 · p. 6, §3 · p. 10

  • Results

    Sparse PCA is consistent when 0 ≤ β < α ≤ 1, although conventional PCA is consistent only for α > 1; an oracle sparse procedure is strongly inconsistent when α < β.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The illustrated consistency region shows sparse PCA can remain consistent under small-spike conditions where conventional PCA is strongly inconsistent.

    Source: §1 · pp. 2–3

    Under non-Gaussian variations, the first empirical eigenvector’s largest-magnitude entry need not be correct, so simple thresholding can fail; similar considerations apply to other sparse PCA methods.

    Source: §6 · p. 20

Abstract

from arXiv · show

Sparse Principal Component Analysis (PCA) methods are efficient tools to reduce the dimension (or the number of variables) of complex data. Sparse principal components (PCs) are easier to interpret than conventional PCs, because most loadings are zero. We study the asymptotic properties of these sparse PC directions for scenarios with fixed sample size and increasing dimension (i.e. High Dimension, Low Sample Size (HDLSS)). Under the previously studied spike covariance assumption, we show that Sparse PCA remains consistent under the same large spike condition that was previously established for conventional PCA. Under a broad range of small spike conditions, we find a large set of sparsity assumptions where Sparse PCA is consistent, but PCA is strongly inconsistent. The boundaries of the consistent region are clarified using an oracle result.

1. Introduction.

The paper studies sparse PCA in HDLSS settings, where conventional PCA loadings can be difficult to interpret, and asks when sparse directions remain consistent despite weak spikes. It develops consistency conditions and an oracle boundary for a single-component spiked model, then outlines the paper’s analyses.

1. Introduction.

  • The paper’s contribution is identifying conditions where sparse PCA is consistent while conventional PCA is strongly inconsistent, and bounding that region with an oracle result.The analysis focuses on a single-component spiked covariance model; α measures first-eigenvalue dominance and β measures the number of nonzero entries in the first population eigenvector.Source: §1 · pp. 2–3
  • Sparse PCA is consistent for 0 ≤ β < α ≤ 1, where conventional PCA is consistent only when α > 1.An oracle sparse PCA procedure is strongly inconsistent when 0 ≤ α < β ≤ 1; β cannot exceed 1 in the illustrated model.Source: §1 · p. 3, Figure 1 · p. 4

1.1. Notation and Assumptions.

  • The illustrative model has λ1 = d^α, all other eigenvalues equal to 1, and a first population eigenvector with ⌊d^β⌋ nonzero entries.Those nonzero entries need not occupy the first coordinates or be equal.Source: §1.1 · p. 5

1.2. Roadmap of the paper.

  • The roadmap moves from thresholding and RSPCA consistency results to oracle strong inconsistency, simulations, and discussion of extending results beyond Gaussian distributions.The paper notes that obtaining HDLSS consistency for sparse PCA beyond Gaussianity is challenging.Source: §1.2 · pp. 5–6

2. Consistency of a simple thresholding method for sparse PCA in HDLSS.

Simple thresholding can produce a consistent sparse PC direction in HDLSS settings where the conventional sample eigenvector is strongly inconsistent. The section gives conditions for this consistency and explains why thresholding is applied through the dual covariance matrix.

2. Consistency of a simple thresholding method for sparse PCA in HDLSS.

  • Thresholding the sample eigenvector directly is challenging as dimension grows, so the method is formulated using the dual covariance matrix.The sample eigenvector’s entries shrink because it is a unit vector with a rapidly growing number of entries.Source: §2 · p. 6
  • Thresholding must avoid both extremes: a threshold that is too large zeros all entries, while one that is too small retains noise.The section gives sufficient lower and upper bounds on the threshold sequence.Source: §2 · p. 7

2.1. Consistency of the simple thresholding method.

  • The general result allows the first population eigenvector to have ⌊d^β⌋ nonzero entries, but requires its nonzero entries to stay a certain distance from zero.Entries near zero could resemble noise in the empirical eigenvector.Source: §2.1 · p. 8
  • A mixing condition makes the remaining components’ effect negligible, allowing the signal component to dominate and yield consistency.The passage links this condition to rapid convergence of the maximum noise term.Source: §2.1 · pp. 8–9
  • Under the theorem’s stated conditions, the simple-thresholding estimator is consistent with the first population eigenvector.The result covers a first population eigenvector with ⌊d^β⌋ nonzero entries.Source: §2.1 · pp. 7, 9

3. Asymptotic properties of RSPCA.

The section analyzes regularized sparse PCA (RSPCA) through a penalized sum-of-squares criterion. It describes how adding a sparsity-inducing penalty changes the rank-one approximation and leads to an iterative thresholding procedure.

3. Asymptotic properties of RSPCA.

  • RSPCA minimizes a penalized sum-of-squares criterion over a data-loading vector and a unit sample-space vector.The penalty is a sum of componentwise penalties and may use a sparsity-inducing function.Source: §3 · p. 10
  • Without the penalty, singular value decomposition yields the best rank-one approximation, whose normalized loading vector is the first empirical PC.With the penalty, the normalized minimizer defines the sparse PC loading vector.Source: §3 · p. 10
  • With a penalty, minimization is iterative, and the penalty-specific thresholding function is applied componentwise to the data projected onto the sample-space vector.The penalty parameter controls the thresholding function.Source: §3 · p. 10

3. Repeat Step 2 setting ˜uold

The section establishes consistency results for RSPCA’s first-step and updated sparse loadings under suitable thresholding choices and assumptions. It also relates the first RSPCA iteration with hard thresholding to simple thresholding.

3. Repeat Step 2 setting ˜uold

  • With hard thresholding, the first RSPCA iteration produces the same sparse loading vector as simple thresholding.The thresholding parameter in the two procedures corresponds in this case.Source: §3 · p. 11
  • The first-step sparse loading vector is consistent with the population eigenvector under the assumptions of Theorem 2.2.Different thresholding parameters yield different convergence rates.Source: §3 · p. 11
  • Theorem 3.3 analyzes the updated dual vector when the previous sparse loading is consistent, with an ε-condition for its stated result.The preceding loading’s consistency rate is also part of the theorem’s setup.Source: §3 · p. 12
  • With a proper thresholding parameter and the stated assumptions, the updated RSPCA sparse loading vector remains consistent with the population eigenvector.The section also gives a condition under which consistency holds at every updating step.Source: §3 · p. 12

4. Strong Inconsistency.

The section constructs an oracle sparse PCA estimator using the known zero locations of the population eigenvector. Even with that information, the estimator is strongly inconsistent when the spike index is smaller than the sparsity index.

4. Strong Inconsistency.

  • The oracle procedure uses the known nonzero coordinates to form a lower-dimensional subspace PCA estimate, then restores zeros in the remaining coordinates.For Example 1.1, the nonzero coordinates are the first ⌊d^β⌋ entries.Source: §4 · pp. 13–14
  • The oracle estimator is defined to have the same sparsity as the population eigenvector.The setup assumes the first ⌊d^β⌋ coordinates are nonzero and the rest are zero.Source: §4 · p. 13
  • Even the oracle estimator is strongly inconsistent with the population eigenvector when α < β.The result bounds what thresholding methods can achieve because they must also estimate the zero locations.Source: §4 · p. 13

5. Simulations for sparse PCA.

The simulations compare PCA, simple thresholding (ST), and RSPCA across spike and sparsity settings. They show better sparse-method performance in many cases when α > β, but poor performance when α < β.

5. Simulations for sparse PCA.

  • The simulations use n = 25, d = 10,000, and 100 realizations for each of 20 spike-and-sparsity pairs.The reported pairs combine α values 0.2, 0.4, 0.6, and 0.8 with β values 0, 0.1, 0.3, 0.5, and 0.7.Source: §5 · pp. 14–15
  • BIC selects the RSPCA threshold parameter at each iterative step using a penalized-regression criterion whose degrees of freedom count nonzero loading entries.In the α = 0.6, β = 0.1 setting, the selected threshold values are very close across replications.Source: §5 · pp. 16–17
  • For α = 0.6 and β = 0.1, RSPCA performs well within the indicated consistency range, with small Type I and Type II errors there.BIC-selected threshold values work well in this α > β setting.Source: Figure 2 · p. 16, §5 · p. 17
  • Across almost all tested cases, ST and RSPCA outperform PCA; at α = 0.4 and β = 0.3, RSPCA performs better than ST, while their Type II Errors are almost the same.The Figure 3 box plots compare angles and error rates for PCA, ST, and RSPCA.Source: §5 · pp. 17–18, Figure 3 · p. 18
  • For α = 0.2 and β = 0.7, all RSPCA angle curves exceed 80 degrees and BIC-selected angles are close to 90 degrees; Type I Error is also close to one.The results illustrate poor performance when α < β.Source: §5 · p. 19, Figure 4 · p. 19

6. Non-Gaussian Variations.

The section considers HDLSS data under a high-dimensional normal distribution and discusses the difficulty of extending the results to more general distributions. A discrete example with sample size one illustrates a failure mode for simple thresholding.

6. Non-Gaussian Variations.

  • Extending the theorems to more general distributions is left as future work, and the authors note sparse PCA may fail in extreme cases.The text introduces a discrete example to illustrate this concern.Source: §6 · pp. 19–20
  • With sample size n = 1, the first empirical eigenvector is determined by the observed data vector, and the example examines whether its largest absolute entry is the first coordinate.The stated event is that |x1,1| exceeds all of |x2,1|,...,|xd,1|.Source: §6 · p. 20
  • Simple thresholding can fail because the first empirical eigenvector's first entry is not guaranteed to have the largest absolute value.The passage says this prevents reliably thresholding out the right entries; similar considerations apply to other sparse PCA methods.Source: §6 · p. 20

7. Proofs.

The proofs use results on dependent extremes under a mixing condition to establish the stated theorem results. They also show strong inconsistency for an oracle estimator in the setting of Theorem 4.1.

7.1. Proofs of Theorem 2.2 and Theorem 2.3.

  • Theorem 2.2 and Theorem 2.3 proofs use dependent extreme-value results for a standard normal sequence satisfying a mixing condition.A proposition from those results is used to control the right side of (2.6) through a lemma.Source: §7 · p. 21, §7.1 · p. 21
  • The supplied excerpt reports that 1 is consistent with u1 and gives convergence rate d; it also states that the proof of Theorem 2.3 finishes.Source: §7.1 · p. 26

7.3. Proof of Theorem 4.1.

  • Under the stated eigenvalue conditions, the oracle estimator is strongly inconsistent with u1, establishing the boundary result for Theorem 4.1.The proof invokes strong inconsistency of the oracle estimator in the reduced problem and transfers it to the oracle estimator.Source: §7.3 · p. 27

Supplement B: Additional Simulation Results

The supplement points to additional simulation results for twenty spike-index and sparsity-index pairs indicated in Figure 1.

Supplement B: Additional Simulation Results

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