We present a novel method for learning densities with bounded support which enables us to incorporate'hard' topological constraints. In particular, we show how emerging techniques from computational algebraic topology and the notion of persistent homology can be combined with kernel-based methods from machine learning for the purpose of density estimation. The proposed formalism facilitates learning of models with bounded support in a principled way, and - by incorporating persistent homology techniques in our approach - we are able to encode algebraic-topological constraints which are not addressed in current state of the art probabilistic models. We study the behaviour of our method on two synthetic examples for various sample sizes and exemplify the benefits of the proposed approach on a real-world dataset by learning a motion model for a race car. We show how to learn a model which respects the underlying topological structure of the racetrack, constraining the trajectories of the car.

We address the problem of general supervised learning when data can only be accessed through an (indefinite) similarity function between data points. Existing work on learning with indefinite kernels has concentrated solely on binary/multiclass classification problems. We propose a model that is generic enough to handle any supervised learning task and also subsumes the model previously proposed for classification. We give a "goodness" criterion for similarity functions w.r.t. a given supervised learning task and then adapt a well-known landmarking technique to provide efficient algorithms for supervised learning using "good" similarity functions. We demonstrate the effectiveness of our model on three important supervised learning problems: a) real-valued regression, b) ordinal regression and c) ranking where we show that our method guarantees bounded generalization error. Furthermore, for the case of real-valued regression, we give a natural goodness definition that, when used in conjunction with a recent result in sparse vector recovery, guarantees a sparse predictor with bounded generalization error. Finally, we report results of our learning algorithms on regression and ordinal regression tasks using non-PSD similarity functions and demonstrate the effectiveness of our algorithms, especially that of the sparse landmark selection algorithm that achieves significantly higher accuracies than the baseline methods while offering reduced computational costs.

We develop a new algorithm to cluster sparse unweighted graphs - i.e. partition the nodes into disjoint clusters so that there is higher density within clusters, and low across clusters. By sparsity we mean the setting where both the in-cluster and across cluster edge densities are very small, possibly vanishing in the size of the graph. Sparsity makes the problem noisier, and hence more difficult to solve. Any clustering involves a tradeoff between minimizing two kinds of errors: missing edges within clusters and present edges across clusters. Our insight is that in the sparse case, these must be penalized differently. We analyze our algorithm's performance on the natural, classical and widely studied "planted partition" model (also called the stochastic block model); we show that our algorithm can cluster sparser graphs, and with smaller clusters, than all previous methods. This is seen empirically as well.

In this paper we discuss a novel framework for multiclass learning, defined by a suitable coding/decoding strategy, namely the simplex coding, that allows us to generalize to multiple classes a relaxation approach commonly used in binary classification. In this framework, we develop a relaxation error analysis that avoids constraints on the considered hypotheses class. Moreover, using this setting we derive the first provably consistent regularized method with training/tuning complexity that is independent to the number of classes. We introduce tools from convex analysis that can be used beyond the scope of this paper.

Non-linear dynamical systems (DS) have been used extensively for building generative models of human behavior. Their applications range from modeling brain dynamics to encoding motor commands. Many schemes have been proposed for encoding robot motions using dynamical systems with a single attractor placed at a predefined target in state space. Although these enable the robots to react against sudden perturbations without any re-planning, the motions are always directed towards a single target. In this work, we focus on combining several such DS with distinct attractors, resulting in a multi-stable DS.

This paper studies a novel discriminative part-based model to represent and recognize object shapes with an "And-Or graph". We define this model consisting of three layers: the leaf-nodes with collaborative edges for localizing local parts, the or-nodes specifying the switch of leaf-nodes, and the root-node encoding the global verification. A discriminative learning algorithm, extended from the CCCP [23], is proposed to train the model in a dynamical manner: the model structure (e.g., the configuration of the leaf-nodes associated with the or-nodes) is automatically determined with optimizing the multi-layer parameters during the iteration. The advantages of our method are two-fold.

We theoretically analyze and compare the following five popular multiclass classification methods: One vs. All, All Pairs, Tree-based classifiers, Error Correcting Output Codes (ECOC) with randomly generated code matrices, and Multiclass SVM. In the first four methods, the classification is based on a reduction to binary classification.

In this paper we study sparsity-inducing nonconvex penalty functions using Lévy processes. We define such a penalty as the Laplace exponent of a subordinator. Accordingly, we propose a novel approach for the construction of sparsityinducing nonconvex penalties. Particularly, we show that the nonconvex logarithmic (LOG) and exponential (EXP) penalty functions are the Laplace exponents of Gamma and compound Poisson subordinators, respectively. Additionally, we explore the concave conjugate of nonconvex penalties. We find that the LOG and EXP penalties are the concave conjugates of negative Kullback-Leiber (KL) distance functions. Furthermore, the relationship between these two penalties is due to asymmetricity of the KL distance.

Recent work has shown this estimator to have strong statistical guarantees in recovering the true structure of the sparse inverse covariance matrix, or alternatively the underlying graph structure of the corresponding Gaussian Markov Random Field, even in very high-dimensional regimes with a limited number of samples. In this paper, we are concerned with the computational cost in solving the above optimization problem. Our proposed algorithm partitions the problem into smaller sub-problems, and uses the solutions of the sub-problems to build a good approximation for the original problem. Our key idea for the divide step to obtain a sub-problem partition is as follows: we first derive a tractable bound on the quality of the approximate solution obtained from solving the corresponding sub-divided problems. Based on this bound, we propose a clustering algorithm that attempts to minimize this bound, in order to find effective partitions of the variables. For the conquer step, we use the approximate solution, i.e., solution resulting from solving the sub-problems, as an initial point to solve the original problem, and thereby achieve a much faster computational procedure.

We offer a regularized, kernel extension of the multi-set, orthogonal Procrustes problem, or hyperalignment. Our new method, called Kernel Hyperalignment, expands the scope of hyperalignment to include nonlinear measures of similarity and enables the alignment of multiple datasets with a large number of base features. With direct application to fMRI data analysis, kernel hyperalignment is well-suited for multi-subject alignment of large ROIs, including the entire cortex. We report experiments using real-world, multi-subject fMRI data.