We study the behavior of block (cid:96)1/(cid:96)2 regularization for multivariate regression, where a K-dimensional response vector is regressed upon a fixed set of p co- variates. The problem of support union recovery is to recover the subset of covariates that are active in at least one of the regression problems. Study- ing this problem under high-dimensional scaling (where the problem parame- ters as well as sample size n tend to infinity simultaneously), our main result is to show that exact recovery is possible once the order parameter given by θ(cid:96)1/(cid:96)2(n, p, s): n/[2ψ(B) log(p s)] exceeds a critical threshold. Here n is the sample size, p is the ambient dimension of the regression model, s is the size of the union of supports, and ψ(B) is a sparsity-overlap function that measures a combination of the sparsities and overlaps of the K-regression coefficient vectors that constitute the model. This sparsity-overlap function reveals that block (cid:96)1/(cid:96)2 regularization for multivariate regression never harms performance relative to a naive (cid:96)1-approach, and can yield substantial improvements in sample complexity (up to a factor of K) when the regression vectors are suitably orthogonal rela- tive to the design.

By attempting to simultaneously partition both the rows (examples) and columns (features) of a data matrix, Co-clustering algorithms often demonstrate surpris- ingly impressive performance improvements over traditional one-sided (row) clustering techniques. A good clustering of features may be seen as a combinatorial transformation of the data matrix, effectively enforcing a form of regularization that may lead to a better clustering of examples (and vice-versa). In many applications, partial supervision in the form of a few row labels as well as column labels may be available to potentially assist co-clustering. In this paper, we develop two novel semi-supervised multi-class classification algorithms motivated respectively by spectral bipartite graph partitioning and matrix approximation (e.g., non-negative matrix factorization) formulations for co-clustering. These algorithms (i) support dual supervision in the form of labels for both examples and/or features, (ii) provide principled predictive capability on out-of-sample test data, and (iii) arise naturally from the classical Representer theorem applied to regularization problems posed on a collection of Reproducing Kernel Hilbert Spaces.

This paper introduces a new approach to constructing meaningful lower dimensional representations of sets of data points. We argue that constraining the mapping between the high and low dimensional spaces to be a diffeomorphism is a natural way of ensuring that pairwise distances are approximately preserved. Accordingly we develop an algorithm which diffeomorphically maps the data near to a lower dimensional subspace and then projects onto that subspace. The problem of solving for the mapping is transformed into one of solving for an Eulerian flow field which we compute using ideas from kernel methods. We demonstrate the efficacy of our approach on various real world data sets.

Principal Components Analysis (PCA) has become established as one of the key tools for dimensionality reduction when dealing with real valued data. Approaches such as exponential family PCA and non-negative matrix factorisation have successfully extended PCA to non-Gaussian data types, but these techniques fail to take advantage of Bayesian inference and can suffer from problems of overfitting and poor generalisation. This paper presents a fully probabilistic approach to PCA, which is generalised to the exponential family, based on Hybrid Monte Carlo sampling. We describe the model which is based on a factorisation of the observed data matrix, and show performance of the model on both synthetic and real data.

We consider the problem of binary classification where the classifier may abstain instead of classifying each observation. The Bayes decision rule for this setup, known as Chow's rule, is defined by two thresholds on posterior probabilities. From simple desiderata, namely the consistency and the sparsity of the classifier, we derive the double hinge loss function that focuses on estimating conditional probabilities only in the vicinity of the threshold points of the optimal decision rule. We show that, for suitable kernel machines, our approach is universally consistent. We cast the problem of minimizing the double hinge loss as a quadratic program akin to the standard SVM optimization problem and propose an active set method to solve it efficiently.

In this paper we focus on training deep neural networks for visual recognition tasks. One challenge is the lack of an informative regularization on the network parameters, to imply a meaningful control on the computed function. We propose a training strategy that takes advantage of kernel methods, where an existing kernel function represents useful prior knowledge about the learning task of interest. We derive an efficient algorithm using stochastic gradient descent, and demonstrate very positive results in a wide range of visual recognition tasks.

We propose a nonparametric Bayesian factor regression model that accounts for uncertainty in the number of factors, and the relationship between factors. To accomplish this, we propose a sparse variant of the Indian Buffet Process and couple this with a hierarchical model over factors, based on Kingman's coalescent. We apply this model to two problems (factor analysis and factor regression) in gene-expression data analysis.

A novel center-based clustering algorithm is proposed in this paper. We first formulate clustering as an NP-hard linear integer program and we then use linear programming and the duality theory to derive the solution of this optimization problem. This leads to an efficient and very general algorithm, which works in the dual domain, and can cluster data based on an arbitrary set of distances. Despite its generality, it is independent of initialization (unlike EM-like methods such as K-means), has guaranteed convergence, and can also provide online optimality bounds about the quality of the estimated clustering solutions. To deal with the most critical issue in a center-based clustering algorithm (selection of cluster centers), we also introduce the notion of stability of a cluster center, which is a well defined LP-based quantity that plays a key role to our algorithm's success.

Stochastic relational models provide a rich family of choices for learning and predicting dyadic data between two sets of entities. Previously empirical Bayesian inference was applied, which is however not scalable when the size of either object sets becomes tens of thousands. In this paper, we introduce a Markov chain Monte Carlo (MCMC) algorithm to scale the model to very large-scale dyadic data. Both superior scalability and predictive accuracy are demonstrated on a collaborative filtering problem, which involves tens of thousands users and a half million items.

Recently, supervised dimensionality reduction has been gaining attention, owing to the realization that data labels are often available and strongly suggest important underlying structures in the data. In this paper, we present a novel convex supervised dimensionality reduction approach based on exponential family PCA and provide a simple but novel form to project new testing data into the embedded space. This convex approach successfully avoids the local optima of the EM learning. Moreover, by introducing a sample-based multinomial approximation to exponential family models, it avoids the limitation of the prevailing Gaussian assumptions of standard PCA, and produces a kernelized formulation for nonlinear supervised dimensionality reduction. A training algorithm is then devised based on a subgradient bundle method, whose scalability can be gained through a coordinate descent procedure.