Classification via Incoherent Subspaces

This article presents a new classification framework that can extract individual features per class. The scheme is based on a model of incoherent subspaces, each one associated to one class, and a model on how the elements in a class are represented in this subspace. After the theoretical analysis an alternate projection algorithm to find such a collection is developed. The classification performance and speed of the proposed method is tested on the AR and YaleB databases and compared to that of Fisher's LDA and a recent approach based on on $\ell_1$ minimisation. Finally connections of the presented scheme to already existing work are discussed and possible ways of extensions are pointed out.Comment: 22 pages, 2 figures, 4 table

Factor Analysis of Moving Average Processes

The paper considers an extension of factor analysis to moving average processes. The problem is formulated as a rank minimization of a suitable spectral density. It is shown that it can be adequately approximated via a trace norm convex relaxation

Minimax rank estimation for subspace tracking

Rank estimation is a classical model order selection problem that arises in a variety of important statistical signal and array processing systems, yet is addressed relatively infrequently in the extant literature. Here we present sample covariance asymptotics stemming from random matrix theory, and bring them to bear on the problem of optimal rank estimation in the context of the standard array observation model with additive white Gaussian noise. The most significant of these results demonstrates the existence of a phase transition threshold, below which eigenvalues and associated eigenvectors of the sample covariance fail to provide any information on population eigenvalues. We then develop a decision-theoretic rank estimation framework that leads to a simple ordered selection rule based on thresholding; in contrast to competing approaches, however, it admits asymptotic minimax optimality and is free of tuning parameters. We analyze the asymptotic performance of our rank selection procedure and conclude with a brief simulation study demonstrating its practical efficacy in the context of subspace tracking.Comment: 10 pages, 4 figures; final versio

OBOE: Collaborative Filtering for AutoML Model Selection

Algorithm selection and hyperparameter tuning remain two of the most challenging tasks in machine learning. Automated machine learning (AutoML) seeks to automate these tasks to enable widespread use of machine learning by non-experts. This paper introduces OBOE, a collaborative filtering method for time-constrained model selection and hyperparameter tuning. OBOE forms a matrix of the cross-validated errors of a large number of supervised learning models (algorithms together with hyperparameters) on a large number of datasets, and fits a low rank model to learn the low-dimensional feature vectors for the models and datasets that best predict the cross-validated errors. To find promising models for a new dataset, OBOE runs a set of fast but informative algorithms on the new dataset and uses their cross-validated errors to infer the feature vector for the new dataset. OBOE can find good models under constraints on the number of models fit or the total time budget. To this end, this paper develops a new heuristic for active learning in time-constrained matrix completion based on optimal experiment design. Our experiments demonstrate that OBOE delivers state-of-the-art performance faster than competing approaches on a test bed of supervised learning problems. Moreover, the success of the bilinear model used by OBOE suggests that AutoML may be simpler than was previously understood

Optimal Principal Component Analysis in Distributed and Streaming Models

We study the Principal Component Analysis (PCA) problem in the distributed and streaming models of computation. Given a matrix $A \in R^{m \times n},$ a rank parameter $k < rank(A)$, and an accuracy parameter $0 < \epsilon < 1$, we want to output an $m \times k$ orthonormal matrix $U$ for which $$|| A - U U^T A ||_F^2 \le \left(1 + \epsilon \right) \cdot || A - A_k||_F^2,$$ where $A_k \in R^{m \times n}$ is the best rank-$k$ approximation to $A$. This paper provides improved algorithms for distributed PCA and streaming PCA.Comment: STOC2016 full versio