Sparse PCA via Covariance Thresholding
This provides a practical algorithm with provable support recovery guarantees for sparse PCA, addressing a bottleneck in high-dimensional data analysis where existing methods had limited theoretical backing.
The paper tackles the problem of sparse principal component analysis (PCA) in high-dimensional settings by proving that a covariance thresholding algorithm can correctly recover the support of sparse principal components with high probability for sparsity levels up to K√n, establishing a more general guarantee than previous conjectures. The result shows that no polynomial-time algorithm can significantly outperform this, based on recent lower bounds.
In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension $n\times p$ and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components $\mathbf{v}_1,\dots,\mathbf{v}_r$ has at most $s_0$ non-zero entries. We are particularly interested in the high dimensional regime wherein $p$ is comparable to, or even much larger than $n$. In an influential paper, \cite{johnstone2004sparse} introduced a simple algorithm that estimates the support of the principal vectors $\mathbf{v}_1,\dots,\mathbf{v}_r$ by the largest entries in the diagonal of the empirical covariance. This method can be shown to identify the correct support with high probability if $s_0\le K_1\sqrt{n/\log p}$, and to fail with high probability if $s_0\ge K_2 \sqrt{n/\log p}$ for two constants $0<K_1,K_2<\infty$. Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees. Here we analyze a covariance thresholding algorithm that was recently proposed by \cite{KrauthgamerSPCA}. On the basis of numerical simulations (for the rank-one case), these authors conjectured that covariance thresholding correctly recover the support with high probability for $s_0\le K\sqrt{n}$ (assuming $n$ of the same order as $p$). We prove this conjecture, and in fact establish a more general guarantee including higher-rank as well as $n$ much smaller than $p$. Recent lower bounds \cite{berthet2013computational, ma2015sum} suggest that no polynomial time algorithm can do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before.