We consider the high-dimensional sparse linear regression problem of accurately estimating a sparse vector using a small number of linear measurements that are contaminated by noise. It is well known that the standard cadre of computationally tractable sparse regression algorithms---such as the Lasso, Orthogonal Matching Pursuit (OMP), and their extensions---perform poorly when the measurement matrix contains highly correlated columns. To address this shortcoming, we develop a simple greedy algorithm, called SWAP, that iteratively swaps variables until convergence. SWAP is surprisingly effective in handling measurement matrices with high correlations. In fact, we prove that SWAP outputs the true support, the locations of the non-zero entries in the sparse vector, under a relatively mild condition on the measurement matrix. Furthermore, we show that SWAP can be used to boost the performance of any sparse regression algorithm. We empirically demonstrate the advantages of SWAP by comparing it with several state-of-the-art sparse regression algorithms.

Metric learning is a key problem for many data mining and machine learning applications, and has long been dominated by Mahalanobis methods. Recent advances in nonlinear metric learning have demonstrated the potential power of non-Mahalanobis distance functions, particularly tree-based functions. We propose a novel nonlinear metric learning method that uses an iterative, hierarchical variant of semi-supervised max-margin clustering to construct a forest of cluster hierarchies, where each individual hierarchy can be interpreted as a weak metric over the data. By introducing randomness during hierarchy training and combining the output of many of the resulting semi-random weak hierarchy metrics, we can obtain a powerful and robust nonlinear metric model. This method has two primary contributions: first, it is semi-supervised, incorporating information from both constrained and unconstrained points. Second, we take a relaxed approach to constraint satisfaction, allowing the method to satisfy different subsets of the constraints at different levels of the hierarchy rather than attempting to simultaneously satisfy all of them. This leads to a more robust learning algorithm. We compare our method to a number of state-of-the-art benchmarks on $k$-nearest neighbor classification, large-scale image retrieval and semi-supervised clustering problems, and find that our algorithm yields results comparable or superior to the state-of-the-art, and is significantly more robust to noise.

In this paper, a new descriptor selection method for selecting an optimal combination of important descriptors of sulfonamide derivatives data, named self tuned reweighted sampling (STRS), is developed. descriptors are defined as the descriptors with large absolute coefficients in a multivariate linear regression model such as partial least squares(PLS). In this study, the absolute values of regression coefficients of PLS model are used as an index for evaluating the importance of each descriptor Then, based on the importance level of each descriptor, STRS sequentially selects N subsets of descriptors from N Monte Carlo (MC) sampling runs in an iterative and competitive manner. In each sampling run, a fixed ratio (e.g. 80%) of samples is first randomly selected to establish a regresson model. Next, based on the regression coefficients, a two-step procedure including rapidly decreasing function (RDF) based enforced descriptor selection and self tuned sampling (STS) based competitive descriptor selection is adopted to select the important descriptorss. After running the loops, a number of subsets of descriptors are obtained and root mean squared error of cross validation (RMSECV) of PLS models established with subsets of descriptors is computed. The subset of descriptors with the lowest RMSECV is considered as the optimal descriptor subset. The performance of the proposed algorithm is evaluated by sulfanomide derivative dataset. The results reveal an good characteristic of STRS that it can usually locate an optimal combination of some important descriptors which are interpretable to the biologically of interest. Additionally, our study shows that better prediction is obtained by STRS when compared to full descriptor set PLS modeling, Monte Carlo uninformative variable elimination (MC-UVE).

We describe an adaptation of the simulated annealing algorithm to nonparametric clustering and related probabilistic models. This new algorithm learns nonparametric latent structure over a growing and constantly churning subsample of training data, where the portion of data subsampled can be interpreted as the inverse temperature beta(t) in an annealing schedule. Gibbs sampling at high temperature (i.e., with a very small subsample) can more quickly explore sketches of the final latent state by (a) making longer jumps around latent space (as in block Gibbs) and (b) lowering energy barriers (as in simulated annealing). We prove subsample annealing speeds up mixing time N^2 -> N in a simple clustering model and exp(N) -> N in another class of models, where N is data size. Empirically subsample-annealing outperforms naive Gibbs sampling in accuracy-per-wallclock time, and can scale to larger datasets and deeper hierarchical models. We demonstrate improved inference on million-row subsamples of US Census data and network log data and a 307-row hospital rating dataset, using a Pitman-Yor generalization of the Cross Categorization model.

We apply the OSCAR (octagonal selection and clustering algorithms for regression) in recovering group-sparse matrices (two-dimensional---2D---arrays) from compressive measurements. We propose a 2D version of OSCAR (2OSCAR) consisting of the $\ell_1$ norm and the pair-wise $\ell_{\infty}$ norm, which is convex but non-differentiable. We show that the proximity operator of 2OSCAR can be computed based on that of OSCAR. The 2OSCAR problem can thus be efficiently solved by state-of-the-art proximal splitting algorithms. Experiments on group-sparse 2D array recovery show that 2OSCAR regularization solved by the SpaRSA algorithm is the fastest choice, while the PADMM algorithm (with debiasing) yields the most accurate results.

We propose a novel SPARsity and Clustering (SPARC) regularizer, which is a modified version of the previous octagonal shrinkage and clustering algorithm for regression (OSCAR), where, the proposed regularizer consists of a $K$-sparse constraint and a pair-wise $\ell_{\infty}$ norm restricted on the $K$ largest components in magnitude. The proposed regularizer is able to separably enforce $K$-sparsity and encourage the non-zeros to be equal in magnitude. Moreover, it can accurately group the features without shrinking their magnitude. In fact, SPARC is closely related to OSCAR, so that the proximity operator of the former can be efficiently computed based on that of the latter, allowing using proximal splitting algorithms to solve problems with SPARC regularization. Experiments on synthetic data and with benchmark breast cancer data show that SPARC is a competitive group-sparsity inducing regularizer for regression and classification.

Determinantal point processes (DPPs) are well-suited for modeling repulsion and have proven useful in many applications where diversity is desired. While DPPs have many appealing properties, such as efficient sampling, learning the parameters of a DPP is still considered a difficult problem due to the non-convex nature of the likelihood function. In this paper, we propose using Bayesian methods to learn the DPP kernel parameters. These methods are applicable in large-scale and continuous DPP settings even when the exact form of the eigendecomposition is unknown. We demonstrate the utility of our DPP learning methods in studying the progression of diabetic neuropathy based on spatial distribution of nerve fibers, and in studying human perception of diversity in images.

We consider new formulations and methods for sparse quantile regression in the high-dimensional setting. Quantile regression plays an important role in many applications, including outlier-robust exploratory analysis in gene selection. In addition, the sparsity consideration in quantile regression enables the exploration of the entire conditional distribution of the response variable given the predictors and therefore yields a more comprehensive view of the important predictors. We propose a generalized OMP algorithm for variable selection, taking the misfit loss to be either the traditional quantile loss or a smooth version we call quantile Huber, and compare the resulting greedy approaches with convex sparsity-regularized formulations. We apply a recently proposed interior point methodology to efficiently solve all convex formulations as well as convex subproblems in the generalized OMP setting, pro- vide theoretical guarantees of consistent estimation, and demonstrate the performance of our approach using empirical studies of simulated and genomic datasets.

We study the task of retrieving relevant experiments given a query experiment. By experiment, we mean a collection of measurements from a set of `covariates' and the associated `outcomes'. While similar experiments can be retrieved by comparing available `annotations', this approach ignores the valuable information available in the measurements themselves. To incorporate this information in the retrieval task, we suggest employing a retrieval metric that utilizes probabilistic models learned from the measurements. We argue that such a metric is a sensible measure of similarity between two experiments since it permits inclusion of experiment-specific prior knowledge. However, accurate models are often not analytical, and one must resort to storing posterior samples which demands considerable resources. Therefore, we study strategies to select informative posterior samples to reduce the computational load while maintaining the retrieval performance. We demonstrate the efficacy of our approach on simulated data with simple linear regression as the models, and real world datasets.

Statistical and machine-learning algorithms are frequently applied to high-dimensional data. In many of these applications data is scarce, and often much more costly than computation time. We provide the first sample-efficient polynomial-time estimator for high-dimensional spherical Gaussian mixtures. For mixtures of any $k$ $d$-dimensional spherical Gaussians, we derive an intuitive spectral-estimator that uses $\mathcal{O}_k\bigl(\frac{d\log^2d}{\epsilon^4}\bigr)$ samples and runs in time $\mathcal{O}_{k,\epsilon}(d^3\log^5 d)$, both significantly lower than previously known. The constant factor $\mathcal{O}_k$ is polynomial for sample complexity and is exponential for the time complexity, again much smaller than what was previously known. We also show that $\Omega_k\bigl(\frac{d}{\epsilon^2}\bigr)$ samples are needed for any algorithm. Hence the sample complexity is near-optimal in the number of dimensions. We also derive a simple estimator for one-dimensional mixtures that uses $\mathcal{O}\bigl(\frac{k \log \frac{k}{\epsilon} }{\epsilon^2} \bigr)$ samples and runs in time $\widetilde{\mathcal{O}}\left(\bigl(\frac{k}{\epsilon}\bigr)^{3k+1}\right)$. Our other technical contributions include a faster algorithm for choosing a density estimate from a set of distributions, that minimizes the $\ell_1$ distance to an unknown underlying distribution.