We present the first efficient and provably consistent estimator for the robust regression problem. The area of robust learning and optimization has generated a significant amount of interest in the learning and statistics communities in recent years owing to its applicability in scenarios with corrupted data, as well as in handling model mis-specifications. In particular, special interest has been devoted to the fundamental problem of robust linear regression where estimators that can tolerate corruption in up to a constant fraction of the response variables are widely studied. Surprisingly however, to this date, we are not aware of a polynomial time estimator that offers a consistent estimate in the presence of dense, unbounded corruptions. In this work we present such an estimator, called CRR. This solves an open problem put forward in the work of [3]. Our consistency analysis requires a novel two-stage proof technique involving a careful analysis of the stability of ordered lists which may be of independent interest.

Neural Information Processing Systems http://nips.cc/

We consider the problem of estimating multiple related functions computed by weighted automata (WFA). We first present a natural notion of relatedness between WFAs by considering to which extent several WFAs can share a common underlying representation. We then introduce the novel model of vector-valued WFA which conveniently helps us formalize this notion of relatedness. Finally, we propose a spectral learning algorithm for vector-valued WFAs to tackle the multitask learning problem. By jointly learning multiple tasks in the form of a vector-valued WFA, our algorithm enforces the discovery of a representation space shared between tasks. The benefits of the proposed multitask approach are theoretically motivated and showcased through experiments on both synthetic and real world datasets.

We present a novel deep neural network architecture for unsupervised subspace clustering. This architecture is built upon deep auto-encoders, which non-linearly map the input data into a latent space. Our key idea is to introduce a novel self-expressive layer between the encoder and the decoder to mimic the "selfexpressiveness" property that has proven effective in traditional subspace clustering. Being differentiable, our new self-expressive layer provides a simple but effective way to learn pairwise affinities between all data points through a standard backpropagation procedure. Being nonlinear, our neural-network based method is able to cluster data points having complex (often nonlinear) structures. We further propose pre-training and fine-tuning strategies that let us effectively learn the parameters of our subspace clustering networks. Our experiments show that our method significantly outperforms the state-of-the-art unsupervised subspace clustering techniques.

Linear Dynamical Systems (LDSs) are fundamental tools for modeling spatiotemporal data in various disciplines. Though rich in modeling, analyzing LDSs is not free of difficulty, mainly because LDSs do not comply with Euclidean geometry and hence conventional learning techniques can not be applied directly. In this paper, we propose an efficient projected gradient descent method to minimize a general form of a loss function and demonstrate how clustering and sparse coding with LDSs can be solved by the proposed method efficiently. To this end, we first derive a novel canonical form for representing the parameters of an LDS, and then show how gradient-descent updates through the projection on the space of LDSs can be achieved dexterously. In contrast to previous studies, our solution avoids any approximation in LDS modeling or during the optimization process. Extensive experiments reveal the superior performance of the proposed method in terms of the convergence and classification accuracy over state-of-the-art techniques.

Neural Information Processing Systems http://nips.cc/

We study this problem in the high-dimensional regime where the number of observations are fewer than the dimension of the weight vector. We assume that the weight vector belongs to some closed set (convex or nonconvex) which captures known side-information about its structure. We focus on the realizable model where the inputs are chosen i.i.d.

Many machine learning tools for regression are based on recursive partitioning of the covariate space into smaller regions, where the regression function can be estimated locally. Among these, regression trees and their ensembles have demonstrated impressive empirical performance. In this work, we shed light on the machinery behind Bayesian variants of these methods. In particular, we study Bayesian regression histograms, such as Bayesian dyadic trees, in the simple regression case with just one predictor. We focus on the reconstruction of regression surfaces that are piecewise constant, where the number of jumps is unknown.

In this paper, we study convergence properties of the gradient variant of Expectation-Maximization algorithm [11] for Gaussian Mixture Models for arbitrary number of clusters and mixing coefficients. We derive the convergence rate depending on the mixing coefficients, minimum and maximum pairwise distances between the true centers, dimensionality and number of components; and obtain a near-optimal local contraction radius. While there have been some recent notable works that derive local convergence rates for EM in the two symmetric mixture of Gaussians, in the more general case, the derivations need structurally different and non-trivial arguments. We use recent tools from learning theory and empirical processes to achieve our theoretical results.