We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order-$3$ tensor $T$ of the form $T = \tau \cdot v_0^{\otimes 3} + A$, where $\tau \geq 0$ is a signal-to-noise ratio, $v_0$ is a unit vector, and $A$ is a random noise tensor, the goal is to recover the planted vector $v_0$. For the case that $A$ has iid standard Gaussian entries, we give an efficient algorithm to recover $v_0$ whenever $\tau \geq \omega(n^{3/4} \log(n)^{1/4})$, and certify that the recovered vector is close to a maximum likelihood estimator, all with high probability over the random choice of $A$. The previous best algorithms with provable guarantees required $\tau \geq \Omega(n)$. In the regime $\tau \leq o(n)$, natural tensor-unfolding-based spectral relaxations for the underlying optimization problem break down (in the sense that their integrality gap is large). To go beyond this barrier, we use convex relaxations based on the sum-of-squares method. Our recovery algorithm proceeds by rounding a degree-$4$ sum-of-squares relaxations of the maximum-likelihood-estimation problem for the statistical model. To complement our algorithmic results, we show that degree-$4$ sum-of-squares relaxations break down for $\tau \leq O(n^{3/4}/\log(n)^{1/4})$, which demonstrates that improving our current guarantees (by more than logarithmic factors) would require new techniques or might even be intractable. Finally, we show how to exploit additional problem structure in order to solve our sum-of-squares relaxations, up to some approximation, very efficiently. Our fastest algorithm runs in nearly-linear time using shifted (matrix) power iteration and has similar guarantees as above. The analysis of this algorithm also confirms a variant of a conjecture of Montanari and Richard about singular vectors of tensor unfoldings.

Linear regression models depend directly on the design matrix and its properties. Techniques that efficiently estimate model coefficients by partitioning rows of the design matrix are increasingly popular for large-scale problems because they fit well with modern parallel computing architectures. We propose a simple measure of {\em concordance} between a design matrix and a subset of its rows that estimates how well a subset captures the variance-covariance structure of a larger data set. We illustrate the use of this measure in a heuristic method for selecting row partition sizes that balance statistical and computational efficiency goals in real-world problems.

In support vector machine (SVM) applications with unreliable data that contains a portion of outliers, non-robustness of SVMs often causes considerable performance deterioration. Although many approaches for improving the robustness of SVMs have been studied, two major challenges remain in robust SVM learning. First, robust learning algorithms are essentially formulated as non-convex optimization problems. It is thus important to develop a non-convex optimization method for robust SVM that can find a good local optimal solution. The second practical issue is how one can tune the hyperparameter that controls the balance between robustness and efficiency. Unfortunately, due to the non-convexity, robust SVM solutions with slightly different hyper-parameter values can be significantly different, which makes model selection highly unstable. In this paper, we address these two issues simultaneously by introducing a novel homotopy approach to non-convex robust SVM learning. Our basic idea is to introduce parametrized formulations of robust SVM which bridge the standard SVM and fully robust SVM via the parameter that represents the influence of outliers. We characterize the necessary and sufficient conditions of the local optimal solutions of robust SVM, and develop an algorithm that can trace a path of local optimal solutions when the influence of outliers is gradually decreased. An advantage of our homotopy approach is that it can be interpreted as simulated annealing, a common approach for finding a good local optimal solution in non-convex optimization problems. In addition, our homotopy method allows stable and efficient model selection based on the path of local optimal solutions. Empirical performances of the proposed approach are demonstrated through intensive numerical experiments both on robust classification and regression problems.

In spite of the recent surge of interest in quantile regression, joint estimation of linear quantile planes remains a great challenge in statistics and econometrics. We propose a novel parametrization that characterizes any collection of non-crossing quantile planes over arbitrarily shaped convex predictor domains in any dimension by means of unconstrained scalar, vector and function valued parameters. Statistical models based on this parametrization inherit a fast computation of the likelihood function, enabling penalized likelihood or Bayesian approaches to model fitting. We introduce a complete Bayesian methodology by using Gaussian process prior distributions on the function valued parameters and develop a robust and efficient Markov chain Monte Carlo parameter estimation. The resulting method is shown to offer posterior consistency under mild tail and regularity conditions. We present several illustrative examples where the new method is compared against existing approaches and is found to offer better accuracy, coverage and model fit.

In the last twenty-five years (1990-2014), algorithmic advances in integer optimization combined with hardware improvements have resulted in an astonishing 200 billion factor speedup in solving Mixed Integer Optimization (MIO) problems. We present a MIO approach for solving the classical best subset selection problem of choosing $k$ out of $p$ features in linear regression given $n$ observations. We develop a discrete extension of modern first order continuous optimization methods to find high quality feasible solutions that we use as warm starts to a MIO solver that finds provably optimal solutions. The resulting algorithm (a) provides a solution with a guarantee on its suboptimality even if we terminate the algorithm early, (b) can accommodate side constraints on the coefficients of the linear regression and (c) extends to finding best subset solutions for the least absolute deviation loss function. Using a wide variety of synthetic and real datasets, we demonstrate that our approach solves problems with $n$ in the 1000s and $p$ in the 100s in minutes to provable optimality, and finds near optimal solutions for $n$ in the 100s and $p$ in the 1000s in minutes. We also establish via numerical experiments that the MIO approach performs better than {\texttt {Lasso}} and other popularly used sparse learning procedures, in terms of achieving sparse solutions with good predictive power.

Stochastic gradient algorithms estimate the gradient based on only one or a few samples and enjoy low computational cost per iteration. They have been widely used in large-scale optimization problems. However, stochastic gradient algorithms are usually slow to converge and achieve sub-linear convergence rates, due to the inherent variance in the gradient computation. To accelerate the convergence, some variance-reduced stochastic gradient algorithms, e.g., proximal stochastic variance-reduced gradient (Prox-SVRG) algorithm, have recently been proposed to solve strongly convex problems. Under the strongly convex condition, these variance-reduced stochastic gradient algorithms achieve a linear convergence rate. However, many machine learning problems are convex but not strongly convex. In this paper, we introduce Prox-SVRG and its projected variant called Variance-Reduced Projected Stochastic Gradient (VRPSG) to solve a class of non-strongly convex optimization problems widely used in machine learning. As the main technical contribution of this paper, we show that both VRPSG and Prox-SVRG achieve a linear convergence rate without strong convexity. A key ingredient in our proof is a Semi-Strongly Convex (SSC) inequality which is the first to be rigorously proved for a class of non-strongly convex problems in both constrained and regularized settings. Moreover, the SSC inequality is independent of algorithms and may be applied to analyze other stochastic gradient algorithms besides VRPSG and Prox-SVRG, which may be of independent interest. To the best of our knowledge, this is the first work that establishes the linear convergence rate for the variance-reduced stochastic gradient algorithms on solving both constrained and regularized problems without strong convexity.

Microarray is one of the essential technologies used by the biologist to measure genome-wide expression levels of genes in a particular organism under some particular conditions or stimuli. As microarrays technologies have become more prevalent, the challenges of analyzing these data for getting better insight about biological processes have essentially increased. Due to availability of artificial intelligence based sophisticated computational techniques, such as artificial neural networks, fuzzy logic, genetic algorithms, and many other nature-inspired algorithms, it is possible to analyse microarray gene expression data in more better way. Here, we reviewed artificial intelligence based techniques for the analysis of microarray gene expression data. Further, challenges in the field and future work direction have also been suggested.

The objective in extreme multi-label learning is to train a classifier that can automatically tag a novel data point with the most relevant subset of labels from an extremely large label set. Embedding based approaches make training and prediction tractable by assuming that the training label matrix is low-rank and hence the effective number of labels can be reduced by projecting the high dimensional label vectors onto a low dimensional linear subspace. Still, leading embedding approaches have been unable to deliver high prediction accuracies or scale to large problems as the low rank assumption is violated in most real world applications. This paper develops the X-One classifier to address both limitations. The main technical contribution in X-One is a formulation for learning a small ensemble of local distance preserving embeddings which can accurately predict infrequently occurring (tail) labels. This allows X-One to break free of the traditional low-rank assumption and boost classification accuracy by learning embeddings which preserve pairwise distances between only the nearest label vectors. We conducted extensive experiments on several real-world as well as benchmark data sets and compared our method against state-of-the-art methods for extreme multi-label classification. Experiments reveal that X-One can make significantly more accurate predictions then the state-of-the-art methods including both embeddings (by as much as 35%) as well as trees (by as much as 6%). X-One can also scale efficiently to data sets with a million labels which are beyond the pale of leading embedding methods.

We consider the problem of identifying underlying community-like structures in graphs. Towards this end we study the Stochastic Block Model (SBM) on $k$-clusters: a random model on $n=km$ vertices, partitioned in $k$ equal sized clusters, with edges sampled independently across clusters with probability $q$ and within clusters with probability $p$, $p>q$. The goal is to recover the initial "hidden" partition of $[n]$. We study semidefinite programming (SDP) based algorithms in this context. In the regime $p = \frac{\alpha \log(m)}{m}$ and $q = \frac{\beta \log(m)}{m}$ we show that a certain natural SDP based algorithm solves the problem of {\em exact recovery} in the $k$-community SBM, with high probability, whenever $\sqrt{\alpha} - \sqrt{\beta} > \sqrt{1}$, as long as $k=o(\log n)$. This threshold is known to be the information theoretically optimal. We also study the case when $k=\theta(\log(n))$. In this case however we achieve recovery guarantees that no longer match the optimal condition $\sqrt{\alpha} - \sqrt{\beta} > \sqrt{1}$, thus leaving achieving optimality for this range an open question.

In this paper, we propose AutoCompete, a highly automated machine learning framework for tackling machine learning competitions. This framework has been learned by us, validated and improved over a period of more than two years by participating in online machine learning competitions. It aims at minimizing human interference required to build a first useful predictive model and to assess the practical difficulty of a given machine learning challenge. The proposed system helps in identifying data types, choosing a machine learn- ing model, tuning hyper-parameters, avoiding over-fitting and optimization for a provided evaluation metric. We also observe that the proposed system produces better (or comparable) results with less runtime as compared to other approaches.