Combinatorial Penalties: Which structures are preserved by convex relaxations?

We consider the homogeneous and the non-homogeneous convex relaxations for combinatorial penalty functions defined on support sets. Our study identifies key differences in the tightness of the resulting relaxations through the notion of the lower combinatorial envelope of a set-function along with new necessary conditions for support identification. We then propose a general adaptive estimator for convex monotone regularizers, and derive new sufficient conditions for support recovery in the asymptotic setting

A totally unimodular view of structured sparsity

This paper describes a simple framework for structured sparse recovery based on convex optimization. We show that many structured sparsity models can be naturally represented by linear matrix inequalities on the support of the unknown parameters, where the constraint matrix has a totally unimodular (TU) structure. For such structured models, tight convex relaxations can be obtained in polynomial time via linear programming. Our modeling framework unifies the prevalent structured sparsity norms in the literature, introduces new interesting ones, and renders their tightness and tractability arguments transparent

Learning with Structured Sparsity: From Discrete to Convex and Back.

In modern-data analysis applications, the abundance of data makes extracting meaningful information from it challenging, in terms of computation, storage, and interpretability. In this setting, exploiting sparsity in data has been essential to the development of scalable methods to problems in machine learning, statistics and signal processing. However, in various applications, the input variables exhibit structure beyond simple sparsity. This motivated the introduction of structured sparsity models, which capture such sophisticated structures, leading to a significant performance gains and better interpretability. Structured sparse approaches have been successfully applied in a variety of domains including computer vision, text processing, medical imaging, and bioinformatics. The goal of this thesis is to improve on these methods and expand their success to a wider range of applications. We thus develop novel methods to incorporate general structure a priori in learning problems, which balance computational and statistical efficiency trade-offs. To achieve this, our results bring together tools from the rich areas of discrete and convex optimization. Applying structured sparsity approaches in general is challenging because structures encountered in practice are naturally combinatorial. An effective approach to circumvent this computational challenge is to employ continuous convex relaxations. We thus start by introducing a new class of structured sparsity models, able to capture a large range of structures, which admit tight convex relaxations amenable to efficient optimization. We then present an in-depth study of the geometric and statistical properties of convex relaxations of general combinatorial structures. In particular, we characterize which structure is lost by imposing convexity and which is preserved. We then focus on the optimization of the convex composite problems that result from the convex relaxations of structured sparsity models. We develop efficient algorithmic tools to solve these problems in a non-Euclidean setting, leading to faster convergence in some cases. Finally, to handle structures that do not admit meaningful convex relaxations, we propose to use, as a heuristic, a non-convex proximal gradient method, efficient for several classes of structured sparsity models. We further extend this method to address a probabilistic structured sparsity model, we introduce to model approximately sparse signals

An Efficient Streaming Algorithm for the Submodular Cover Problem

We initiate the study of the classical Submodular Cover (SC) problem in the data streaming model which we refer to as the Streaming Submodular Cover (SSC). We show that any single pass streaming algorithm using sublinear memory in the size of the stream will fail to provide any non-trivial approximation guarantees for SSC. Hence, we consider a relaxed version of SSC, where we only seek to find a partial cover. We design the first Efficient bicriteria Submodular Cover Streaming (ESC-Streaming) algorithm for this problem, and provide theoretical guarantees for its performance supported by numerical evidence. Our algorithm finds solutions that are competitive with the near-optimal offline greedy algorithm despite requiring only a single pass over the data stream. In our numerical experiments, we evaluate the performance of ESC-Streaming on active set selection and large-scale graph cover problems.Comment: To appear in NIPS'1

Combinatorial Penalties: Which structures are preserved by convex relaxations?

We consider the homogeneous and the non-homogeneous convex relaxations for combinatorial penalty functions defined on support sets. Our study identifies key differences in the tightness of the resulting relaxations through the notion of the lower combinatorial envelope of a set-function along with new necessary conditions for support identification. We then propose a general adaptive estimator for convex monotone regularizers, and derive new sufficient conditions for support recovery in the asymptotic setting

MAP Estimation for Bayesian Mixture Models with Submodular Priors

We propose a Bayesian approach where the signal structure can be represented by a mixture model with a submodular prior. We consider an observation model that leads to Lipschitz functions. Due to its combinatorial nature, computing the maximum a posteriori estimate for this model is NP-Hard, nonetheless our converging majorization-minimization scheme yields approximate estimates that, in practice, outperform state-of-the-art methods

Fairness in Streaming Submodular Maximization over a Matroid Constraint

Streaming submodular maximization is a natural model for the task of selecting a representative subset from a large-scale dataset. If datapoints have sensitive attributes such as gender or race, it becomes important to enforce fairness to avoid bias and discrimination. This has spurred significant interest in developing fair machine learning algorithms. Recently, such algorithms have been developed for monotone submodular maximization under a cardinality constraint. In this paper, we study the natural generalization of this problem to a matroid constraint. We give streaming algorithms as well as impossibility results that provide trade-offs between efficiency, quality and fairness. We validate our findings empirically on a range of well-known real-world applications: exemplar-based clustering, movie recommendation, and maximum coverage in social networks.Comment: Accepted to ICML 2

General Proximal Gradient Method: A Case for Non-Euclidean Norms

In this paper, we consider composite convex minimization problems. We advocate the merit of considering Generalized Proximal gradient Methods (GPM) where the norm employed is not Euclidean. To that end, we show the tractability of the general proximity operator for a broad class of structure priors by proposing a polynomial-time approach to approximately compute it. We also identify a special case of regularizers whose proximity operator admits an efficient greedy algorithm. We then introduce a proximity/projection-free accelerated variant of GPM. We illustrate numerically the benefit of non-Euclidean norms, on the estimation quality of the Lasso problem and on the time-complexity of the latent group Lasso problem

A scalable security protocol for wireless sensor networks

Since sensor nodes suffer from limited resources, in memory storage, computing power, energy capabilities, and transmission rates, available network security protocols are inadequate. Symmetric algorithms cannot provide the same degree of security as public-key algorithms, the fact of which has led us to devise a new algorithm, SHESP, which uses public keys within the limitations of sensor nodes. This paper aims to present a way to utilize existing public-key algorithms in the field of wireless sensor network security by dividing the network into clusters. Our algorithm supplies data confidentiality, node authentication and data integrity while remaining within acceptable memory, time and energy constraints. Also, an important feature we opted to establish, which was lacking in most security protocols, was enabling secure node-to-node communication, without the need to route through a distant base station. We provide theoretical as well as experimental evidence to validate our algorithm