The problem of minimizing a continuously differentiable convex function over an intersection of closed convex sets is ubiquitous in applied mathematics. It is particularly interesting when it is easy to project onto each separate set, but nontrivial to project onto their intersection. Algorithms based on Newton's method such as the interior point method are viable for small to medium-scale problems. However, modern applications in statistics, engineering, and machine learning are posing problems with potentially tens of thousands of parameters or more. We revisit this convex programming problem and propose an algorithm that scales well with dimensionality. Our proposal is an instance of a sequential unconstrained minimization technique and revolves around three ideas: the majorization-minimization (MM) principle, the classical penalty method for constrained optimization, and quasi-Newton acceleration of fixed-point algorithms. The performance of our distance majorization algorithms is illustrated in several applications.

The performance of a modulation classifier is highly sensitive to channel signal-to-noise ratio (SNR). In this paper, we focus on amplitude-phase modulations and propose a modulation classification framework based on centralized data fusion using multiple radios and the hybrid maximum likelihood (ML) approach. In order to alleviate the computational complexity associated with ML estimation, we adopt the Expectation Maximization (EM) algorithm. Due to SNR diversity, the proposed multi-radio framework provides robustness to channel SNR. Numerical results show the superiority of the proposed approach with respect to single radio approaches as well as to modulation classifiers using moments based estimators.

The problem of adaptive noisy clustering is investigated. Given a set of noisy observations $Z_i=X_i+\epsilon_i$, $i=1,...,n$, the goal is to design clusters associated with the law of $X_i$'s, with unknown density $f$ with respect to the Lebesgue measure. Since we observe a corrupted sample, a direct approach as the popular {\it $k$-means} is not suitable in this case. In this paper, we propose a noisy $k$-means minimization, which is based on the $k$-means loss function and a deconvolution estimator of the density $f$. In particular, this approach suffers from the dependence on a bandwidth involved in the deconvolution kernel. Fast rates of convergence for the excess risk are proposed for a particular choice of the bandwidth, which depends on the smoothness of the density $f$. Then, we turn out into the main issue of the paper: the data-driven choice of the bandwidth. We state an adaptive upper bound for a new selection rule, called ERC (Empirical Risk Comparison). This selection rule is based on the Lepski's principle, where empirical risks associated with different bandwidths are compared. Finally, we illustrate that this adaptive rule can be used in many statistical problems of $M$-estimation where the empirical risk depends on a nuisance parameter.

We introduce kernel nonparametric tests for Lancaster three-variable interaction and for total independence, using embeddings of signed measures into a reproducing kernel Hilbert space. The resulting test statistics are straightforward to compute, and are used in powerful interaction tests, which are consistent against all alternatives for a large family of reproducing kernels. We show the Lancaster test to be sensitive to cases where two independent causes individually have weak influence on a third dependent variable, but their combined effect has a strong influence. This makes the Lancaster test especially suited to finding structure in directed graphical models, where it outperforms competing nonparametric tests in detecting such V-structures.

We consider the stochastic approximation problem where a convex function has to be minimized, given only the knowledge of unbiased estimates of its gradients at certain points, a framework which includes machine learning methods based on the minimization of the empirical risk. We focus on problems without strong convexity, for which all previously known algorithms achieve a convergence rate for function values of O(1/n^{1/2}). We consider and analyze two algorithms that achieve a rate of O(1/n) for classical supervised learning problems. For least-squares regression, we show that averaged stochastic gradient descent with constant step-size achieves the desired rate. For logistic regression, this is achieved by a simple novel stochastic gradient algorithm that (a) constructs successive local quadratic approximations of the loss functions, while (b) preserving the same running time complexity as stochastic gradient descent. For these algorithms, we provide a non-asymptotic analysis of the generalization error (in expectation, and also in high probability for least-squares), and run extensive experiments on standard machine learning benchmarks showing that they often outperform existing approaches.

Many natural signals exhibit a sparse representation, whenever a suitable describing model is given. Here, a linear generative model is considered, where many sparsity-based signal processing techniques rely on such a simplified model. As this model is often unknown for many classes of the signals, we need to select such a model based on the domain knowledge or using some exemplar signals. This paper presents a new exemplar based approach for the linear model (called the dictionary) selection, for such sparse inverse problems. The problem of dictionary selection, which has also been called the dictionary learning in this setting, is first reformulated as a joint sparsity model. The joint sparsity model here differs from the standard joint sparsity model as it considers an overcompleteness in the representation of each signal, within the range of selected subspaces. The new dictionary selection paradigm is examined with some synthetic and realistic simulations.

Tensor factorizations have become increasingly popular approaches for various learning tasks on structured data. In this work, we extend the Rescal tensor factorization, which has shown state-of-the-art results for multi-relational learning, to account for the binary nature of adjacency tensors. We study the improvements that can be gained via this approach on various benchmark datasets and show that the logistic extension can improve the prediction results significantly.

While several papers have investigated computationally and statistically efficient methods for learning Gaussian mixtures, precise minimax bounds for their statistical performance as well as fundamental limits in high-dimensional settings are not well-understood. In this paper, we provide precise information theoretic bounds on the clustering accuracy and sample complexity of learning a mixture of two isotropic Gaussians in high dimensions under small mean separation. If there is a sparse subset of relevant dimensions that determine the mean separation, then the sample complexity only depends on the number of relevant dimensions and mean separation, and can be achieved by a simple computationally efficient procedure. Our results provide the first step of a theoretical basis for recent methods that combine feature selection and clustering.

A prototypical blind signal separation problem is the so-called cocktail party problem, with n people talking simultaneously and n different microphones within a room. The goal is to recover each speech signal from the microphone inputs. Mathematically this can be modeled by assuming that we are given samples from an n-dimensional random variable X=AS, where S is a vector whose coordinates are independent random variables corresponding to each speaker. The objective is to recover the matrix A^{-1} given random samples from X. A range of techniques collectively known as Independent Component Analysis (ICA) have been proposed to address this problem in the signal processing and machine learning literature. Many of these techniques are based on using the kurtosis or other cumulants to recover the components. In this paper we propose a new algorithm for solving the blind signal separation problem in the presence of additive Gaussian noise, when we are given samples from X=AS+\eta, where \eta is drawn from an unknown, not necessarily spherical n-dimensional Gaussian distribution. Our approach is based on a method for decorrelating a sample with additive Gaussian noise under the assumption that the underlying distribution is a linear transformation of a distribution with independent components. Our decorrelation routine is based on the properties of cumulant tensors and can be combined with any standard cumulant-based method for ICA to get an algorithm that is provably robust in the presence of Gaussian noise. We derive polynomial bounds for the sample complexity and error propagation of our method.

We give an overview of our journal paper on applying iterative-deepening A* to the traveling tournament problem, a combinatorial optimization problem from the sports scheduling literature. This approach involved combining past ideas and creating new ideas to help reduce node expansion. This resulted in a state-of-the-art approach for optimally solving instances of the traveling tournament problem. It was the first approach to solve the classic NL10 and CIRC10 instances, which had not been solved since the problem’s introduction.