This paper formalizes a latent variable inference problem we call {\em supervised pattern discovery}, the goal of which is to find sets of observations that belong to a single ``pattern.'' We discuss two versions of the problem and prove uniform risk bounds for both. In the first version, collections of patterns can be generated in an arbitrary manner and the data consist of multiple labeled collections. In the second version, the patterns are assumed to be generated independently by identically distributed processes. These processes are allowed to take an arbitrary form, so observations within a pattern are not in general independent of each other. The bounds for the second version of the problem are stated in terms of a new complexity measure, the quasi-Rademacher complexity.

In statistical learning theory, convex surrogates of the 0-1 loss are highly preferred because of the computational and theoretical virtues that convexity brings in. This is of more importance if we consider smooth surrogates as witnessed by the fact that the smoothness is further beneficial both computationally- by attaining an {\it optimal} convergence rate for optimization, and in a statistical sense- by providing an improved {\it optimistic} rate for generalization bound. In this paper we investigate the smoothness property from the viewpoint of statistical consistency and show how it affects the binary excess risk. We show that in contrast to optimization and generalization errors that favor the choice of smooth surrogate loss, the smoothness of loss function may degrade the binary excess risk. Motivated by this negative result, we provide a unified analysis that integrates optimization error, generalization bound, and the error in translating convex excess risk into a binary excess risk when examining the impact of smoothness on the binary excess risk. We show that under favorable conditions appropriate choice of smooth convex loss will result in a binary excess risk that is better than $O(1/\sqrt{n})$.

Mixed linear regression involves the recovery of two (or more) unknown vectors from unlabeled linear measurements; that is, where each sample comes from exactly one of the vectors, but we do not know which one. It is a classic problem, and the natural and empirically most popular approach to its solution has been the EM algorithm. As in other settings, this is prone to bad local minima; however, each iteration is very fast (alternating between guessing labels, and solving with those labels). In this paper we provide a new initialization procedure for EM, based on finding the leading two eigenvectors of an appropriate matrix. We then show that with this, a re-sampled version of the EM algorithm provably converges to the correct vectors, under natural assumptions on the sampling distribution, and with nearly optimal (unimprovable) sample complexity. This provides not only the first characterization of EM's performance, but also much lower sample complexity as compared to both standard (randomly initialized) EM, and other methods for this problem.

This work introduces a transformation-based learner model for classification forests. The weak learner at each split node plays a crucial role in a classification tree. We propose to optimize the splitting objective by learning a linear transformation on subspaces using nuclear norm as the optimization criteria. The learned linear transformation restores a low-rank structure for data from the same class, and, at the same time, maximizes the separation between different classes, thereby improving the performance of the split function. Theoretical and experimental results support the proposed framework.

Active learning refers to the learning protocol where the learner is allowed to choose a subset of instances for labeling. Previous studies have shown that, compared with passive learning, active learning is able to reduce the label complexity exponentially if the data are linearly separable or satisfy the Tsybakov noise condition with parameter $\kappa=1$. In this paper, we propose a novel active learning algorithm using a convex surrogate loss, with the goal to broaden the cases for which active learning achieves an exponential improvement. We make use of a convex loss not only because it reduces the computational cost, but more importantly because it leads to a tight bound for the empirical process (i.e., the difference between the empirical estimation and the expectation) when the current solution is close to the optimal one. Under the assumption that the norm of the optimal classifier that minimizes the convex risk is available, our analysis shows that the introduction of the convex surrogate loss yields an exponential reduction in the label complexity even when the parameter $\kappa$ of the Tsybakov noise is larger than $1$. To the best of our knowledge, this is the first work that improves the minimax rate of active learning by utilizing certain priori knowledge.

Joint matching over a collection of objects aims at aggregating information from a large collection of similar instances (e.g. images, graphs, shapes) to improve maps between pairs of them. Given multiple matches computed between a few object pairs in isolation, the goal is to recover an entire collection of maps that are (1) globally consistent, and (2) close to the provided maps --- and under certain conditions provably the ground-truth maps. Despite recent advances on this problem, the best-known recovery guarantees are limited to a small constant barrier --- none of the existing methods find theoretical support when more than $50\%$ of input correspondences are corrupted. Moreover, prior approaches focus mostly on fully similar objects, while it is practically more demanding to match instances that are only partially similar to each other. In this paper, we develop an algorithm to jointly match multiple objects that exhibit only partial similarities, given a few pairwise matches that are densely corrupted. Specifically, we propose to recover the ground-truth maps via a parameter-free convex program called MatchLift, following a spectral method that pre-estimates the total number of distinct elements to be matched. Encouragingly, MatchLift exhibits near-optimal error-correction ability, i.e. in the asymptotic regime it is guaranteed to work even when a dominant fraction $1-\Theta\left(\frac{\log^{2}n}{\sqrt{n}}\right)$ of the input maps behave like random outliers. Furthermore, MatchLift succeeds with minimal input complexity, namely, perfect matching can be achieved as soon as the provided maps form a connected map graph. We evaluate the proposed algorithm on various benchmark data sets including synthetic examples and real-world examples, all of which confirm the practical applicability of MatchLift.

In multiple instance learning, objects are sets (bags) of feature vectors (instances) rather than individual feature vectors. In this paper we address the problem of how these bags can best be represented. Two standard approaches are to use (dis)similarities between bags and prototype bags, or between bags and prototype instances. The first approach results in a relatively low-dimensional representation determined by the number of training bags, while the second approach results in a relatively high-dimensional representation, determined by the total number of instances in the training set. In this paper a third, intermediate approach is proposed, which links the two approaches and combines their strengths. Our classifier is inspired by a random subspace ensemble, and considers subspaces of the dissimilarity space, defined by subsets of instances, as prototypes. We provide guidelines for using such an ensemble, and show state-of-the-art performances on a range of multiple instance learning problems.

It is the main purpose of this paper to introduce a graph-valued stochastic process in order to model the spread of a communicable infectious disease. The major novelty of the SIR model we promote lies in the fact that the social network on which the epidemics is taking place is not specified in advance but evolves through time, accounting for the temporal evolution of the interactions involving infective individuals. Without assuming the existence of a fixed underlying network model, the stochastic process introduced describes, in a flexible and realistic manner, epidemic spread in non-uniformly mixing and possibly heterogeneous populations. It is shown how to fit such a (parametrised) model by means of Approximate Bayesian Computation methods based on graph-valued statistics. The concepts and statistical methods described in this paper are finally applied to a real epidemic dataset, related to the spread of HIV in Cuba in presence of a contact tracing system, which permits one to reconstruct partly the evolution of the graph of sexual partners diagnosed HIV positive between 1986 and 2006.

We introduce a new embarrassingly parallel parameter learning algorithm for Markov random fields with untied parameters which is efficient for a large class of practical models. Our algorithm parallelizes naturally over cliques and, for graphs of bounded degree, its complexity is linear in the number of cliques. Unlike its competitors, our algorithm is fully parallel and for log-linear models it is also data efficient, requiring only the local sufficient statistics of the data to estimate parameters.

We propose a novel method for clustering data which is grounded in information-theoretic principles and requires no parametric assumptions. Previous attempts to use information theory to define clusters in an assumption-free way are based on maximizing mutual information between data and cluster labels. We demonstrate that this intuition suffers from a fundamental conceptual flaw that causes clustering performance to deteriorate as the amount of data increases. Instead, we return to the axiomatic foundations of information theory to define a meaningful clustering measure based on the notion of consistency under coarse-graining for finite data.