Estimating a vector from noisy quadratic observations is a task that arises naturally in many contexts, from dimensionality reduction, to synchronization and phase retrieval problems. It is often the case that additional information is available about the unknown vector (for instance, sparsity, sign or magnitude of its entries). Many authors propose non-convex quadratic optimization problems that aim at exploiting optimally this information. However, solving these problems is typically NP-hard. We consider a simple model for noisy quadratic observation of an unknown vector $\bvz$. The unknown vector is constrained to belong to a cone $\Cone \ni \bvz$. While optimal estimation appears to be intractable for the general problems in this class, we provide evidence that it is tractable when $\Cone$ is a convex cone with an efficient projection. This is surprising, since the corresponding optimization problem is non-convex and --from a worst case perspective-- often NP hard. We characterize the resulting minimax risk in terms of the statistical dimension of the cone $\delta(\Cone)$. This quantity is already known to control the risk of estimation from gaussian observations and random linear measurements. It is rather surprising that the same quantity plays a role in the estimation risk from quadratic measurements.

We study the distributed computing setting in which there are multiple servers, each holding a set of points, who wish to compute functions on the union of their point sets. A key task in this setting is Principal Component Analysis (PCA), in which the servers would like to compute a low dimensional subspace capturing as much of the variance of the union of their point sets as possible. Given a procedure for approximate PCA, one can use it to approximately solve problems such as $k$-means clustering and low rank approximation. The essential properties of an approximate distributed PCA algorithm are its communication cost and computational efficiency for a given desired accuracy in downstream applications. We give new algorithms and analyses for distributed PCA which lead to improved communication and computational costs for $k$-means clustering and related problems. Our empirical study on real world data shows a speedup of orders of magnitude, preserving communication with only a negligible degradation in solution quality. Some of these techniques we develop, such as input-sparsity subspace embeddings with high correctness probability with a dimension and sparsity independent of the error probability, may be of independent interest.

We provide a new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that we call noisy power method. Our result characterizes the convergence behavior of the algorithm when a large amount noise is introduced after each matrix-vector multiplication. The noisy power method can be seen as a meta-algorithm that has recently found a number of important applications in a broad range of machine learning problems including alternating minimization for matrix completion, streaming principal component analysis (PCA), and privacy-preserving spectral analysis. Our general analysis subsumes several existing ad-hoc convergence bounds and resolves a number of open problems in multiple applications. A recent work of Mitliagkas et al.~(NIPS 2013) gives a space-efficient algorithm for PCA in a streaming model where samples are drawn from a spiked covariance model. We give a simpler and more general analysis that applies to arbitrary distributions. Moreover, even in the spiked covariance model our result gives quantitative improvements in a natural parameter regime. As a second application, we provide an algorithm for differentially private principal component analysis that runs in nearly linear time in the input sparsity and achieves nearly tight worst-case error bounds. Complementing our worst-case bounds, we show that the error dependence of our algorithm on the matrix dimension can be replaced by an essentially tight dependence on the coherence of the matrix. This result resolves the main problem left open by Hardt and Roth (STOC 2013) and leads to strong average-case improvements over the optimal worst-case bound.

In this on-going work, I explore certain theoretical and empirical implications of data transformations under the PCA. In particular, I state and prove three theorems about PCA, which I paraphrase as follows: 1). PCA without discarding eigenvector rows is injective, but looses this injectivity when eigenvector rows are discarded 2). PCA without discarding eigen- vector rows preserves pair-wise distances, but tends to cause pair-wise distances to shrink when eigenvector rows are discarded. 3). For any pair of points, the shrinkage in pair-wise distance is bounded above by an L1 norm reconstruction error associated with the points. Clearly, 3). suggests that there might exist some correlation between shrinkages in pair-wise distances and mean square reconstruction error which is defined as the sum of those eigenvalues associated with the discarded eigenvectors. I therefore decided to perform numerical experiments to obtain the corre- lation between the sum of those eigenvalues and shrinkages in pair-wise distances. In addition, I have also performed some experiments to check respectively the effect of the sum of those eigenvalues and the effect of the shrinkages on classification accuracies under the PCA map. So far, I have obtained the following results on some publicly available data from the UCI Machine Learning Repository: 1). There seems to be a strong cor- relation between the sum of those eigenvalues associated with discarded eigenvectors and shrinkages in pair-wise distances. 2). Neither the sum of those eigenvalues nor pair-wise distances have any strong correlations with classification accuracies. 1

Principal Component Analysis (PCA) has wide applications in machine learning, text mining and computer vision. Classical PCA based on a Gaussian noise model is fragile to noise of large magnitude. Laplace noise assumption based PCA methods cannot deal with dense noise effectively. In this paper, we propose Cauchy Principal Component Analysis (Cauchy PCA), a very simple yet effective PCA method which is robust to various types of noise. We utilize Cauchy distribution to model noise and derive Cauchy PCA under the maximum likelihood estimation (MLE) framework with low rank constraint. Our method can robustly estimate the low rank matrix regardless of whether noise is large or small, dense or sparse. We analyze the robustness of Cauchy PCA from a robust statistics view and present an efficient singular value projection optimization method. Experimental results on both simulated data and real applications demonstrate the robustness of Cauchy PCA to various noise patterns.

Neurons in higher cortical areas, such as the prefrontal cortex, are known to be tuned to a variety of sensory and motor variables. The resulting diversity of neural tuning often obscures the represented information. Here we introduce a novel dimensionality reduction technique, demixed principal component analysis (dPCA), which automatically discovers and highlights the essential features in complex population activities. We reanalyze population data from the prefrontal areas of rats and monkeys performing a variety of working memory and decision-making tasks. In each case, dPCA summarizes the relevant features of the population response in a single figure. The population activity is decomposed into a few demixed components that capture most of the variance in the data and that highlight dynamic tuning of the population to various task parameters, such as stimuli, decisions, rewards, etc. Moreover, dPCA reveals strong, condition-independent components of the population activity that remain unnoticed with conventional approaches.

Sequential or online dimensional reduction is of interests due to the explosion of streaming data based applications and the requirement of adaptive statistical modeling, in many emerging fields, such as the modeling of energy end-use profile. Principal Component Analysis (PCA), is the classical way of dimensional reduction. However, traditional Singular Value Decomposition (SVD) based PCA fails to model data which largely deviates from Gaussian distribution. The Bregman Divergence was recently introduced to achieve a generalized PCA framework. If the random variable under dimensional reduction follows Bernoulli distribution, which occurs in many emerging fields, the generalized PCA is called Logistic PCA (LPCA). In this paper, we extend the batch LPCA to a sequential version (i.e. SLPCA), based on the sequential convex optimization theory. The convergence property of this algorithm is discussed compared to the batch version of LPCA (i.e. BLPCA), as well as its performance in reducing the dimension for multivariate binary-state systems. Its application in building energy end-use profile modeling is also investigated.

Sparse principal component analysis (sparse PCA) aims at finding a sparse basis to improve the interpretability over the dense basis of PCA, meanwhile the sparse basis should cover the data subspace as much as possible. In contrast to most of existing work which deal with the problem by adding some sparsity penalties on various objectives of PCA, in this paper, we propose a new method SPCArt, whose motivation is to find a rotation matrix and a sparse basis such that the sparse basis approximates the basis of PCA after the rotation. The algorithm of SPCArt consists of three alternating steps: rotate PCA basis, truncate small entries, and update the rotation matrix. Its performance bounds are also given. SPCArt is efficient, with each iteration scaling linearly with the data dimension. It is easy to choose parameters in SPCArt, due to its explicit physical explanations. Besides, we give a unified view to several existing sparse PCA methods and discuss the connection with SPCArt. Some ideas in SPCArt are extended to GPower, a popular sparse PCA algorithm, to overcome its drawback. Experimental results demonstrate that SPCArt achieves the state-of-the-art performance. It also achieves a good tradeoff among various criteria, including sparsity, explained variance, orthogonality, balance of sparsity among loadings, and computational speed.

Principal component analysis (PCA) is a standard tool in modern data analysis - in diverse fields from neuroscience to computer graphics - because it is a simple, nonparametric method for extracting relevant information from confusing data sets. With minimal effort PCA provides a roadmap for how to reduce a complex data set to a lower dimension to reveal the sometimes hidden, simplified structures that often underlie it. The goal of this tutorial is to provide both an intuitive feel for PCA, and a thorough discussion of this topic. We will begin with a simple example and provide an intuitive explanation of the goal of PCA. We will continue by adding mathematical rigor to place it within the framework of linear algebra to provide an explicit solution.

We propose a new high dimensional semiparametric principal component analysis (PCA) method, named Copula Component Analysis (COCA). The semiparametric model assumes that, after unspecified marginally monotone transformations, the distributions are multivariate Gaussian. COCA improves upon PCA and sparse PCA in three aspects: (i) It is robust to modeling assumptions; (ii) It is robust to outliers and data contamination; (iii) It is scale-invariant and yields more interpretable results. We prove that the COCA estimators obtain fast estimation rates and are feature selection consistent when the dimension is nearly exponentially large relative to the sample size. Careful experiments confirm that COCA outperforms sparse PCA on both synthetic and real-world datasets.