High-dimensional analysis of semidefinite relaxations for sparse principal components (0803.4026v2)

Abstract: Principal component analysis (PCA) is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix. However, PCA is well known to behave poorly in the ``large $p$, small $n$'' setting, in which the problem dimension $p$ is comparable to or larger than the sample size $n$. This paper studies PCA in this high-dimensional regime, but under the additional assumption that the maximal eigenvector is sparse, say, with at most $k$ nonzero components. We consider a spiked covariance model in which a base matrix is perturbed by adding a $k$-sparse maximal eigenvector, and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector, as follows: (a) a simple diagonal thresholding method, which transitions from success to failure as a function of the rescaled sample size $\theta_{\mathrm{dia}}(n,p,k)=n/[k^2\log(p-k)]$; and (b) a more sophisticated semidefinite programming (SDP) relaxation, which succeeds once the rescaled sample size $\theta_{\mathrm{sdp}}(n,p,k)=n/[k\log(p-k)]$ is larger than a critical threshold. In addition, we prove that no method, including the best method which has exponential-time complexity, can succeed in recovering the support if the order parameter $\theta_{\mathrm{sdp}}(n,p,k)$ is below a threshold. Our results thus highlight an interesting trade-off between computational and statistical efficiency in high-dimensional inference.

• The paper establishes conditions under which SDP accurately recovers sparse eigenvectors in high-dimensional spiked covariance models.
• The paper shows that diagonal thresholding, despite its computational efficiency, demands a larger sample size for reliable support recovery.
• The paper highlights a trade-off between statistical precision and computational cost, offering insights for future high-dimensional PCA algorithm design.

High-Dimensional Analysis of Semidefinite Relaxations for Sparse Principal Components

Principal Component Analysis (PCA) is a classical method widely used for dimensionality reduction by identifying dominant eigenvectors of a covariance matrix. The traditional implementation of PCA assumes that the problem dimension $p$ is fixed and largely outnumbered by the sample size $n$. This assumption falters in high-dimensional settings where $p$ approaches or even exceeds $n$. The paper "High-Dimensional Analysis of Semidefinite Relaxations for Sparse Principal Components" by Amini and Wainwright addresses this issue by analyzing conditions under which sparse principal components can be accurately recovered from high-dimensional data.

The authors focus on sparse principal component analysis (SPCA), a variant where the principal components (i.e., the leading eigenvectors) are assumed to be sparse. They evaluate two methods to recover sparse eigenvectors in a spiked covariance model: diagonal thresholding and semidefinite programming (SDP) relaxation.

Key Contributions and Results

1. Sparse Eigenvector Recovery: The paper investigates a spiked covariance model, specifically oriented towards the recovery of sparse eigenvectors. This model adds a sparse perturbation to a base covariance matrix. The goal is to determine the precise conditions under which the support set of the maximal eigenvector can be recovered. The analysis details two procedures: diagonal thresholding and SDP relaxation.
2. Diagonal Thresholding: This method is acknowledged for its computational efficiency. However, it suffers from requiring a large number of observations for accurate recovery. Amini and Wainwright characterize the performance of diagonal thresholding via a critical rescaled sample size defined as $\theta_{\text{dia}}(n,p,k) = n/[k\log(p-k)]$. They show that diagonal thresholding succeeds with high probability if this parameter exceeds a predisposed threshold, otherwise, it almost surely fails.
3. Semidefinite Programming Relaxation: Compared to diagonal thresholding, the SDP relaxation method offers significantly improved statistical efficiency. The authors establish that SDP can accurately recover the support set of the sparse eigenvector when the rescaled sample size $\theta_{\text{sdp}}(n,p,k) = n/[k\log(p-k)]$ surpasses a certain threshold. The paper emphasizes that this condition is not only sufficient but also close to necessary, supported by a subsequent information-theoretic analysis.
4. Trade-off Between Statistical and Computational Efficiencies: A major theme in the paper is the trade-off between statistical efficiency and computational cost. While SDP requires fewer observations to recover the eigenvector accurately, it involves higher computational complexity (as compared to diagonal thresholding). The complexity of the SDP is approximately $O(np^2 + p^3 \log p)$.
5. Information-Theoretic Bounds: Utilizing information-theoretic methods, the paper establishes fundamental limits on the recovery probability. It indicates that no method can recover the support of the eigenvectors if the rescaled sample size $\theta_{\text{sdp}}$ lies below the established threshold.

Implications and Future Work

The implications of this research are significant for the fields of high-dimensional statistics and machine learning, particularly in applications where dimensionality reduction is paramount, such as genomics and finance. The insights regarding computational versus statistical efficiency could guide the design of algorithms that balance these aspects according to the specificities of practical datasets.

Future developments could address the relaxation of Gaussian assumptions in sampling methodologies, exploring the performance of these methods under more general distributions. Additionally, exploring the scalability of semidefinite programming to even larger datasets is likely of interest, possibly leveraging advances in optimization techniques to mitigate computational burdens.

In conclusion, the work provides a rigorous statistical and computational analysis of recovering sparse principal components from high-dimensional data, bridging a gap between theoretical promise and practical utility in PCAinduced dimensionality reduction.