Structured sparse estimation has become an important technique in many areas of data analysis. Unfortunately, these estimators normally create computational difficulties that entail sophisticated algorithms. Our first contribution is to uncover a rich class of structured sparse regularizers whose polar operator can be evaluated efficiently. With such an operator, a simple conditional gradient method can then be developed that, when combined with smoothing and local optimization, significantly reduces training time vs. the state of the art. We also demonstrate a new reduction of polar to proximal maps that enables more efficient latent fused lasso.

We study the problem of online learning in finite episodic Markov decision processes (MDPs)where the loss function is allowed to change between episodes. The natural performance measure in this learning problem is the regret defined as the difference between the total loss of the best stationary policy and the total loss suffered by the learner. We assume that the learner is given access to a finite action space A and the state space X has a layered structure with L layers, so that state transitions are only possible between consecutive layers. We describe a variant of the recently proposed Relative Entropy Policy Search algorithm and show that its regret after T episodes is 2 L X A T log( X A /L) in the bandit setting and 2L T log( X A /L) in the full information setting, given that the learner has perfect knowledge of the transition probabilities of the underlying MDP. These guarantees largely improve previously known results under much milder assumptions andcannot be significantly improved under general assumptions.

In this work, we propose a general method for recovering low-rank three-order tensors, in which the data can be deformed by some unknown transformation and corrupted by arbitrary sparse errors. Since the unfolding matrices of a tensor are interdependent, we introduce auxiliary variables and relax the hard equality constraints by the augmented Lagrange multiplier method. To improve the computational efficiency, we introduce a proximal gradient step to the alternating direction minimization method. We have provided proof for the convergence of the linearized version of the problem which is the inner loop of the overall algorithm. Both simulations and experiments show that our methods are more efficient and effective than previous work. The proposed method can be easily applied to simultaneously rectify and align multiple images or videos frames. In this context, the state-of-the-art algorithms RASL'' and "TILT'' can be viewed as two special cases of our work, and yet each only performs part of the function of our method."

Recently, it has become evident that submodularity naturally captures widely occurring concepts in machine learning, signal processing and computer vision. In consequence, there is need for efficient optimization procedures for submodular functions, in particular for minimization problems. While general submodular minimization is challenging, we propose a new approach that exploits existing decomposability of submodular functions. In contrast to previous approaches, our method is neither approximate, nor impractical, nor does it need any cumbersome parameter tuning. Moreover, it is easy to implement and parallelize. A key component of our approach is a formulation of the discrete submodular minimization problem as a continuous best approximation problem. It is solved through a sequence of reflections and its solution can be automatically thresholded to obtain an optimal discrete solution. Our method solves both the continuous and discrete formulations of the problem, and therefore has applications in learning, inference, and reconstruction. In our experiments, we show the benefits of our new algorithms for two image segmentation tasks.

This paper addresses the problem of online learning in a dynamic setting. We consider a social network in which each individual observes a private signal about the underlying state of the world and communicates with her neighbors at each time period. Unlike many existing approaches, the underlying state is dynamic, and evolves according to a geometric random walk. We view the scenario as an optimization problem where agents aim to learn the true state while suffering the smallest possible loss. Based on the decomposition of the global loss function, we introduce two update mechanisms, each of which generates an estimate of the true state. We establish a tight bound on the rate of change of the underlying state, under which individuals can track the parameter with a bounded variance. Then, we characterize explicit expressions for the steady state mean-square deviation(MSD) of the estimates from the truth, per individual. We observe that only one of the estimators recovers the optimal MSD, which underscores the impact of the objective function decomposition on the learning quality. Finally, we provide an upper bound on the regret of the proposed methods, measured as an average of errors in estimating the parameter in a finite time.

This paper addresses the problem of unsupervised feature learning for text data. Our method is grounded in the principle of minimum description length and uses a dictionary-based compression scheme to extract a succinct feature set. Specifically, our method finds a set of word $k$-grams that minimizes the cost of reconstructing the text losslessly. We formulate document compression as a binary optimization task and show how to solve it approximately via a sequence of reweighted linear programs that are efficient to solve and parallelizable. As our method is unsupervised, features may be extracted once and subsequently used in a variety of tasks. We demonstrate the performance of these features over a range of scenarios including unsupervised exploratory analysis and supervised text categorization. Our compressed feature space is two orders of magnitude smaller than the full $k$-gram space and matches the text categorization accuracy achieved in the full feature space. This dimensionality reduction not only results in faster training times, but it can also help elucidate structure in unsupervised learning tasks and reduce the amount of training data necessary for supervised learning.

How does neural population process sensory information? Optimal coding theories assume that neural tuning curves are adapted to the prior distribution of the stimulus variable. Most of the previous work has discussed optimal solutions for only one-dimensional stimulus variables. Here, we expand some of these ideas and present new solutions that define optimal tuning curves for high-dimensional stimulus variables. We consider solutions for a minimal case where the number of neurons in the population is equal to the number of stimulus dimensions (diffeomorphic). In the case of two-dimensional stimulus variables, we analytically derive optimal solutions for different optimal criteria such as minimal L2 reconstruction error or maximal mutual information. For higher dimensional case, the learning rule to improve the population code is provided.

Seriation seeks to reconstruct a linear order between variables using unsorted similarity information. It has direct applications in archeology and shotgun gene sequencing for example. We prove the equivalence between the seriation and the combinatorial 2-sum problem (a quadratic minimization problem over permutations) over a class of similarity matrices. The seriation problem can be solved exactly by a spectral algorithm in the noiseless case and we produce a convex relaxation for the 2-sum problem to improve the robustness of solutions in a noisy setting. This relaxation also allows us to impose additional structural constraints on the solution, to solve semi-supervised seriation problems. We present numerical experiments on archeological data, Markov chains and gene sequences.

Nonnegative matrix factorization (NMF) is a popular data analysis method, the objective of which is to decompose a matrix with all nonnegative components into the product of two other nonnegative matrices. In this work, we describe a new simple and efficient algorithm for multi-factor nonnegative matrix factorization problem ({mfNMF}), which generalizes the original NMF problem to more than two factors. Furthermore, we extend the mfNMF algorithm to incorporate a regularizer based on Dirichlet distribution over normalized columns to encourage sparsity in the obtained factors. Our sparse NMF algorithm affords a closed form and an intuitive interpretation, and is more efficient in comparison with previous works that use fix point iterations. We demonstrate the effectiveness and efficiency of our algorithms on both synthetic and real data sets.

In this paper, we study the following new variant of prototype learning, called {\em $k$-prototype learning problem for 3D rigid structures}: Given a set of 3D rigid structures, find a set of $k$ rigid structures so that each of them is a prototype for a cluster of the given rigid structures and the total cost (or dissimilarity) is minimized. Prototype learning is a core problem in machine learning and has a wide range of applications in many areas. Existing results on this problem have mainly focused on the graph domain. In this paper, we present the first algorithm for learning multiple prototypes from 3D rigid structures. Our result is based on a number of new insights to rigid structures alignment, clustering, and prototype reconstruction, and is practically efficient with quality guarantee. We validate our approach using two type of data sets, random data and biological data of chromosome territories. Experiments suggest that our approach can effectively learn prototypes in both types of data.