Over the last years, many variations of the quadratic k-means clustering procedure have been proposed, all aiming to robustify the performance of the algorithm in the presence of outliers. In general terms, two main approaches have been developed: one based on penalized regularization methods, and one based on trimming functions. In this work, we present a theoretical analysis of the robustness and consistency properties of a variant of the classical quadratic k-means algorithm, the robust k-means, which borrows ideas from outlier detection in regression. We show that two outliers in a dataset are enough to breakdown this clustering procedure. However, if we focus on "well-structured" datasets, then robust k-means can recover the underlying cluster structure in spite of the outliers. Finally, we show that, with slight modifications, the most general non-asymptotic results for consistency of quadratic k-means remain valid for this robust variant.

In recent years, structured matrix recovery problems have gained considerable attention for its real world applications, such as recommender systems and computer vision. Much of the existing work has focused on matrices with low-rank structure, and limited progress has been made on matrices with other types of structure. In this paper we present non-asymptotic analysis for estimation of generally structured matrices via the generalized Dantzig selector based on sub-Gaussian measurements. We show that the estimation error can always be succinctly expressed in terms of a few geometric measures such as Gaussian widths of suitable sets associated with the structure of the underlying true matrix. Further, we derive general bounds on these geometric measures for structures characterized by unitarily invariant norms, a large family covering most matrix norms of practical interest. Examples are provided to illustrate the utility of our theoretical development.

Neuroscience experiments often require training animals to perform tasks designed to elicit various sensory, cognitive, and motor behaviors. Training typically involves a series of gradual adjustments of stimulus conditions and rewards in order to bring about learning. However, training protocols are usually hand-designed, relying on a combination of intuition, guesswork, and trial-and-error, and often require weeks or months to achieve a desired level of task performance. Here we combine ideas from reinforcement learning and adaptive optimal experimental design to formulate methods for adaptive optimal training of animal behavior. Our work addresses two intriguing problems at once: first, it seeks to infer the learning rules underlying an animal's behavioral changes during training; second, it seeks to exploit these rules to select stimuli that will maximize the rate of learning toward a desired objective.

We propose and study a new class of submodular functions called deep submodular functions (DSFs). We define DSFs and situate them within the broader context of classes of submodular functions in relationship both to various matroid ranks and sums of concave composed with modular functions (SCMs). Notably, we find that DSFs constitute a strictly broader class than SCMs, thus motivating their use, but that they do not comprise all submodular functions. Interestingly, some DSFs can be seen as special cases of certain deep neural networks (DNNs), hence the name. Finally, we provide a method to learn DSFs in a max-margin framework, and offer preliminary results applying this both to synthetic and real-world data instances.

Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent progress in proving various non-convex algorithms converge from a good initial point, it remains unclear why random or arbitrary initialization suffices in practice. We prove that the commonly used non-convex objective function for matrix completion has no spurious local minima --- all local minima must also be global. Therefore, many popular optimization algorithms such as (stochastic) gradient descent can provably solve matrix completion with \textit{arbitrary} initialization in polynomial time.

We consider a transfer-learning problem by using the parameter transfer approach, where a suitable parameter of feature mapping is learned through one task and applied to another objective task. Then, we introduce the notion of the local stability of parametric feature mapping and parameter transfer learnability, and thereby derive a learning bound for parameter transfer algorithms. As an application of parameter transfer learning, we discuss the performance of sparse coding in self-taught learning. Although self-taught learning algorithms with plentiful unlabeled data often show excellent empirical performance, their theoretical analysis has not been studied. In this paper, we also provide the first theoretical learning bound for self-taught learning.

Feature selection is one of the most fundamental problems in machine learning. An extensive body of work on information-theoretic feature selection exists which is based on maximizing mutual information between subsets of features and class labels. Practical methods are forced to rely on approximations due to the difficulty of estimating mutual information. We demonstrate that approximations made by existing methods are based on unrealistic assumptions. We formulate a more flexible and general class of assumptions based on variational distributions and use them to tractably generate lower bounds for mutual information. These bounds define a novel information-theoretic framework for feature selection, which we prove to be optimal under tree graphical models with proper choice of variational distributions. Our experiments demonstrate that the proposed method strongly outperforms existing information-theoretic feature selection approaches.

Tensor decomposition is positioned to be a pervasive tool in the era of big data. In this paper, we resolve many of the key algorithmic questions regarding robustness, memory efficiency, and differential privacy of tensor decomposition. We propose simple variants of the tensor power method which enjoy these strong properties. We propose the first streaming method with a linear memory requirement. Moreover, we present a noise calibrated tensor power method with efficient privacy guarantees. At the heart of all these guarantees lies a careful perturbation analysis derived in this paper which improves up on the existing results significantly.

Predicting the behavior of human participants in strategic settings is an important problem in many domains. Most existing work either assumes that participants are perfectly rational, or attempts to directly model each participant's cognitive processes based on insights from cognitive psychology and experimental economics. In this work, we present an alternative, a deep learning approach that automatically performs cognitive modeling without relying on such expert knowledge. We introduce a novel architecture that allows a single network to generalize across different input and output dimensions by using matrix units rather than scalar units, and show that its performance significantly outperforms that of the previous state of the art, which relies on expert-constructed features.

Many applications of machine learning involve structured output with large domain, where learning of structured predictor is prohibitive due to repetitive calls to expensive inference oracle. In this work, we show that, by decomposing training of Structural Support Vector Machine (SVM) into a series of multiclass SVM problems connected through messages, one can replace expensive structured oracle with Factorwise Maximization Oracle (FMO) that allows efficient implementation of complexity sublinear to the factor domain. A Greedy Direction Method of Multiplier (GDMM) algorithm is proposed to exploit sparsity of messages which guarantees $\epsilon$ sub-optimality after $O(log(1/\epsilon))$ passes of FMO calls. We conduct experiments on chain-structured problems and fully-connected problems of large output domains. The proposed approach is orders-of-magnitude faster than the state-of-the-art training algorithms for Structural SVM.