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Register a new userNon-Sparse PCA in High Dimensions via Cone Projected Power Iteration
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Added by
Yufei Yi
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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In this paper, we propose a cone projected power iteration algorithm to recover the first principal eigenvector from a noisy positive semidefinite matrix. When the true principal eigenvector is assumed to belong to a convex cone, the proposed algorithm is fast and has a tractable error. Specifically, the method achieves polynomial time complexity for certain convex cones equipped with fast projection such as the monotone cone. It attains a small error when the noisy matrix has a small cone-restricted operator norm. We supplement the above results with a minimax lower bound of the error under the spiked covariance model. Our numerical experiments on simulated and real data, show that our method achieves shorter run time and smaller error in comparison to the ordinary power iteration and some sparse principal component analysis algorithms if the principal eigenvector is in a convex cone.
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In this paper, we study the power iteration algorithm for the spiked tensor model, as introduced in [44]. We give necessary and sufficient conditions for the convergence of the power iteration algorithm. When the power iteration algorithm converges, for the rank one spiked tensor model, we show the estimators for the spike strength and linear functionals of the signal are asymptotically Gaussian; for the multi-rank spiked tensor model, we show the estimators are asymptotically mixtures of Gaussian. This new phenomenon is different from the spiked matrix model. Using these asymptotic results of our estimators, we construct valid and efficient confidence intervals for spike strengths and linear functionals of the signals.
In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension $ntimes p$ 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 $s_0le K_1sqrt{n/log p}$, and to fail with high probability if $s_0ge K_2 sqrt{n/log p}$ 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 $s_0le Ksqrt{n}$ (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.
An important challenge in statistical analysis lies in controlling the bias of estimators due to the ever-increasing data size and model complexity. Approximate numerical methods and data features like censoring and misclassification often result in analytical and/or computational challenges when implementing standard estimators. As a consequence, consistent estimators may be difficult to obtain, especially in complex and/or high dimensional settings. In this paper, we study the properties of a general simulation-based estimation framework that allows to construct bias corrected consistent estimators. We show that the considered approach leads, under more general conditions, to stronger bias correction properties compared to alternative methods. Besides its bias correction advantages, the considered method can be used as a simple strategy to construct consistent estimators in settings where alternative methods may be challenging to apply. Moreover, the considered framework can be easily implemented and is computationally efficient. These theoretical results are highlighted with simulation studies of various commonly used models, including the negative binomial regression (with and without censoring) and the logistic regression (with and without misclassification errors). Additional numerical illustrations are provided in the supplementary materials.
We study the statistical problem of estimating a rank-one sparse tensor corrupted by additive Gaussian noise, a model also known as sparse tensor PCA. We show that for Bernoulli and Bernoulli-Rademacher distributed signals and emph{for all} sparsity levels which are sublinear in the dimension of the signal, the sparse tensor PCA model exhibits a phase transition called the emph{all-or-nothing phenomenon}. This is the property that for some signal-to-noise ratio (SNR) $mathrm{SNR_c}$ and any fixed $epsilon>0$, if the SNR of the model is below $left(1-epsilonright)mathrm{SNR_c}$, then it is impossible to achieve any arbitrarily small constant correlation with the hidden signal, while if the SNR is above $left(1+epsilon right)mathrm{SNR_c}$, then it is possible to achieve almost perfect correlation with the hidden signal. The all-or-nothing phenomenon was initially established in the context of sparse linear regression, and over the last year also in the context of sparse 2-tensor (matrix) PCA, Bernoulli group testing, and generalized linear models. Our results follow from a more general result showing that for any Gaussian additive model with a discrete uniform prior, the all-or-nothing phenomenon follows as a direct outcome of an appropriately defined near-orthogonality property of the support of the prior distribution.
We consider the following multi-component sparse PCA problem: given a set of data points, we seek to extract a small number of sparse components with disjoint supports that jointly capture the maximum possible variance. These components can be computed one by one, repeatedly solving the single-component problem and deflating the input data matrix, but as we show this greedy procedure is suboptimal. We present a novel algorithm for sparse PCA that jointly optimizes multiple disjoint components. The extracted features capture variance that lies within a multiplicative factor arbitrarily close to 1 from the optimal. Our algorithm is combinatorial and computes the desired components by solving multiple instances of the bipartite maximum weight matching problem. Its complexity grows as a low order polynomial in the ambient dimension of the input data matrix, but exponentially in its rank. However, it can be effectively applied on a low-dimensional sketch of the data; this allows us to obtain polynomial-time approximation guarantees via spectral bounds. We evaluate our algorithm on real data-sets and empirically demonstrate that in many cases it outperforms existing, deflation-based approaches.
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