This work is concerned with the non-negative robust principal component analysis (PCA), where the goal is to recover the dominant non-negative principal component of a data matrix precisely, where a number of measurements could be grossly corrupted with sparse and arbitrary large noise. Most of the known techniques for solving the robust PCA rely on convex relaxation methods by lifting the problem to a higher dimension, which significantly increase the number of variables. As an alternative, the well-known Burer-Monteiro approach can be used to cast the robust PCA as a non-convex and non-smooth $\ell_1$ optimization problem with a significantly smaller number of variables. In this work, we show that the low-dimensional formulation of the symmetric and asymmetric positive robust PCA based on the Burer-Monteiro approach has benign landscape, i.e., 1) it does not have any spurious local solution, 2) has a unique global solution, and 3) its unique global solution coincides with the true components. An implication of this result is that simple local search algorithms are guaranteed to achieve a zero global optimality gap when directly applied to the low-dimensional formulation. Furthermore, we provide strong deterministic and statistical guarantees for the exact recovery of the true principal component. In particular, it is shown that a constant fraction of the measurements could be grossly corrupted and yet they would not create any spurious local solution.

Your data is the life-giving fuel to your Machine Learning model. There are always many ML techniques to choose from and apply to a particular problem, but without a lot of good data you won't get very far. Data is often the driver behind most of your performance gains in a Machine Learning application. Sometimes that data can be complicated. You have so much of it that it may be challenging to understand what it all means and which parts are actually important.

Anomalies and outliers are common in real-world data, and they can arise from many sources, such as sensor faults. Accordingly, anomaly detection is important both for analyzing the anomalies themselves and for cleaning the data for further analysis of its ambient structure. Nonetheless, a precise definition of anomalies is important for automated detection and herein we approach such problems from the perspective of detecting sparse latent effects embedded in large collections of noisy data. Standard Graphical Lasso-based techniques can identify the conditional dependency structure of a collection of random variables based on their sample covariance matrix. However, classic Graphical Lasso is sensitive to outliers in the sample covariance matrix. In particular, several outliers in a sample covariance matrix can destroy the sparsity of its inverse. Accordingly, we propose a novel optimization problem that is similar in spirit to Robust Principal Component Analysis (RPCA) and splits the sample covariance matrix $M$ into two parts, $M=F+S$, where $F$ is the cleaned sample covariance whose inverse is sparse and computable by Graphical Lasso, and $S$ contains the outliers in $M$. We accomplish this decomposition by adding an additional $ \ell_1$ penalty to classic Graphical Lasso, and name it "Robust Graphical Lasso (Rglasso)". Moreover, we propose an Alternating Direction Method of Multipliers (ADMM) solution to the optimization problem which scales to large numbers of unknowns. We evaluate our algorithm on both real and synthetic datasets, obtaining interpretable results and outperforming the standard robust Minimum Covariance Determinant (MCD) method and Robust Principal Component Analysis (RPCA) regarding both accuracy and speed.

Designing and modifying complex hull forms for optimal vessel performances have been a major challenge for naval architects. In the present study, Principal Component Analysis (PCA) is introduced to compress the geometric representation of a group of existing vessels, and the resulting principal scores are manipulated to generate a large number of derived hull forms, which are evaluated computationally for their calm-water performances. The results are subsequently used to train a Deep Neural Network (DNN) to accurately establish the relation between different hull forms and their associated performances. Then, based on the fast, parallel DNN-based hull-form evaluation, the large-scale search for optimal hull forms is performed.

We propose robust sparse reduced rank regression and robust sparse principal component analysis for analyzing large and complex high-dimensional data with heavy-tailed random noise. The proposed methods are based on convex relaxations of rank-and sparsity-constrained non-convex optimization problems, which are solved using the alternating direction method of multipliers (ADMM) algorithm. For robust sparse reduced rank regression, we establish non-asymptotic estimation error bounds under both Frobenius and nuclear norms, while existing results focus mostly on rank-selection and prediction consistency. Our theoretical results quantify the tradeoff between heavy-tailedness of the random noise and statistical bias. For random noise with bounded $(1+\delta)$th moment with $\delta \in (0,1)$, the rate of convergence is a function of $\delta$, and is slower than the sub-Gaussian-type deviation bounds; for random noise with bounded second moment, we recover the results obtained under sub-Gaussian noise. Furthermore, the transition between the two regimes is smooth. For robust sparse principal component analysis, we propose to truncate the observed data, and show that this truncation will lead to consistent estimation of the eigenvectors. We then establish theoretical results similar to those of robust sparse reduced rank regression. We illustrate the performance of these methods via extensive numerical studies and two real data applications.

A novel approach is put forth that utilizes data similarity, quantified on a graph, to improve upon the reconstruction performance of principal component analysis. The tasks of data dimensionality reduction and reconstruction are formulated as graph filtering operations, that enable the exploitation of data node connectivity in a graph via the adjacency matrix. The unknown reducing and reconstruction filters are determined by optimizing a mean-square error cost that entails the data, as well as their graph adjacency matrix. Working in the graph spectral domain enables the derivation of simple gradient descent recursions used to update the matrix filter taps. Numerical tests in real image datasets demonstrate the better reconstruction performance of the novel method over standard principal component analysis.

In this paper, we propose to adopt the diffusion approximation tools to study the dynamics of Oja's iteration which is an online stochastic gradient descent method for the principal component analysis. Oja's iteration maintains a running estimate of the true principal component from streaming data and enjoys less temporal and spatial complexities. We show that the Oja's iteration for the top eigenvector generates a continuous-state discrete-time Markov chain over the unit sphere. We characterize the Oja's iteration in three phases using diffusion approximation and weak convergence tools. Our three-phase analysis further provides a finite-sample error bound for the running estimate, which matches the minimax information lower bound for principal component analysis under the additional assumption of bounded samples.

Principal component analysis (PCA) is arguably the most popular tool in multivariate exploratory data analysis. In this paper, we consider the question of how to handle heterogeneous variables that include continuous, binary, and ordinal. In the probabilistic interpretation of low-rank PCA, the data has a normal multivariate distribution and, therefore, normal marginal distributions for each column. If some marginals are continuous but not normal, the semiparametric copula-based principal component analysis (COCA) method is an alternative to PCA that combines a Gaussian copula with nonparametric marginals. If some marginals are discrete or semi-continuous, we propose a new extended PCA (XPCA) method that also uses a Gaussian copula and nonparametric marginals and accounts for discrete variables in the likelihood calculation by integrating over appropriate intervals. Like PCA, the factors produced by XPCA can be used to find latent structure in data, build predictive models, and perform dimensionality reduction. We present the new model, its induced likelihood function, and a fitting algorithm which can be applied in the presence of missing data. We demonstrate how to use XPCA to produce an estimated full conditional distribution for each data point, and use this to produce to provide estimates for missing data that are automatically range respecting. We compare the methods as applied to simulated and real-world data sets that have a mixture of discrete and continuous variables.

Robust PCA has drawn significant attention in the last decade due to its success in numerous application domains, ranging from bio-informatics, statistics, and machine learning to image and video processing in computer vision. Robust PCA and its variants such as sparse PCA and stable PCA can be formulated as optimization problems with exploitable special structures. Many specialized efficient optimization methods have been proposed to solve robust PCA and related problems. In this paper we review existing optimization methods for solving convex and nonconvex relaxations/variants of robust PCA, discuss their advantages and disadvantages, and elaborate on their convergence behaviors. We also provide some insights for possible future research directions including new algorithmic frameworks that might be suitable for implementing on multi-processor setting to handle large-scale problems.

In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product) [13]. Induced by the t-product, we first rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor average rank, and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm. These definitions, their relationships and properties are consistent with matrix cases. Equipped with the new tensor nuclear norm, we then solve the TRPCA problem by solving a convex program and provide the theoretical guarantee for the exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA as a special case. Numerical experiments verify our results, and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method.