Variable (feature, gene, model, which we use interchangeably) selections for regression with high-dimensional BIGDATA have found many applications in bioinformatics, computational biology, image processing, and engineering. One appealing approach is the L0 regularized regression which penalizes the number of nonzero features in the model directly. L0 is known as the most essential sparsity measure and has nice theoretical properties, while the popular L1 regularization is only a best convex relaxation of L0. Therefore, it is natural to expect that L0 regularized regression performs better than LASSO. However, it is well-known that L0 optimization is NP-hard and computationally challenging. Instead of solving the L0 problems directly, most publications so far have tried to solve an approximation problem that closely resembles L0 regularization. In this paper, we propose an efficient EM algorithm (L0EM) that directly solves the L0 optimization problem. $L_0$EM is efficient with high dimensional data. It also provides a natural solution to all Lp p in [0,2] problems. The regularized parameter can be either determined through cross-validation or AIC and BIC. Theoretical properties of the L0-regularized estimator are given under mild conditions that permit the number of variables to be much larger than the sample size. We demonstrate our methods through simulation and high-dimensional genomic data. The results indicate that L0 has better performance than LASSO and L0 with AIC or BIC has similar performance as computationally intensive cross-validation. The proposed algorithms are efficient in identifying the non-zero variables with less-bias and selecting biologically important genes and pathways with high dimensional BIGDATA.

Motivation: Algorithms that discover variables which are causally related to a target may inform the design of experiments. With observational gene expression data, many methods discover causal variables by measuring each variable's degree of statistical dependence with the target using dependence measures (DMs). However, other methods measure each variable's ability to explain the statistical dependence between the target and the remaining variables in the data using conditional dependence measures (CDMs), since this strategy is guaranteed to find the target's direct causes, direct effects, and direct causes of the direct effects in the infinite sample limit. In this paper, we design a new algorithm in order to systematically compare the relative abilities of DMs and CDMs in discovering causal variables from gene expression data. Results: The proposed algorithm using a CDM is sample efficient, since it consistently outperforms other state-of-the-art local causal discovery algorithms when samples sizes are small. However, the proposed algorithm using a CDM outperforms the proposed algorithm using a DM only when sample sizes are above several hundred. These results suggest that accurate causal discovery from gene expression data using current CDM-based algorithms requires datasets with at least several hundred samples. Availability: The proposed algorithm is freely available at https://github.com/ericstrobl/DvCD.

Scaling feature values is an important step in numerous machine learning tasks. Different features can have different value ranges and some form of a feature scaling is often required in order to learn an accurate classifier. However, feature scaling is conducted as a preprocessing task prior to learning. This is problematic in an online setting because of two reasons. First, it might not be possible to accurately determine the value range of a feature at the initial stages of learning when we have observed only a few number of training instances. Second, the distribution of data can change over the time, which render obsolete any feature scaling that we perform in a pre-processing step. We propose a simple but an effective method to dynamically scale features at train time, thereby quickly adapting to any changes in the data stream. We compare the proposed dynamic feature scaling method against more complex methods for estimating scaling parameters using several benchmark datasets for binary classification. Our proposed feature scaling method consistently outperforms more complex methods on all of the benchmark datasets and improves classification accuracy of a state-of-the-art online binary classifier algorithm.

We consider the problem of sparse coding, where each sample consists of a sparse linear combination of a set of dictionary atoms, and the task is to learn both the dictionary elements and the mixing coefficients. Alternating minimization is a popular heuristic for sparse coding, where the dictionary and the coefficients are estimated in alternate steps, keeping the other fixed. Typically, the coefficients are estimated via $\ell_1$ minimization, keeping the dictionary fixed, and the dictionary is estimated through least squares, keeping the coefficients fixed. In this paper, we establish local linear convergence for this variant of alternating minimization and establish that the basin of attraction for the global optimum (corresponding to the true dictionary and the coefficients) is $\order{1/s^2}$, where $s$ is the sparsity level in each sample and the dictionary satisfies RIP. Combined with the recent results of approximate dictionary estimation, this yields provable guarantees for exact recovery of both the dictionary elements and the coefficients, when the dictionary elements are incoherent.

We propose a general algorithm for approximating nonstandard Bayesian posterior distributions. The algorithm minimizes the Kullback-Leibler divergence of an approximating distribution to the intractable posterior distribution. Our method can be used to approximate any posterior distribution, provided that it is given in closed form up to the proportionality constant. The approximation can be any distribution in the exponential family or any mixture of such distributions, which means that it can be made arbitrarily precise. Several examples illustrate the speed and accuracy of our approximation method in practice.

We introduce kLog, a novel approach to statistical relational learning. Unlike standard approaches, kLog does not represent a probability distribution directly. It is rather a language to perform kernel-based learning on expressive logical and relational representations. kLog allows users to specify learning problems declaratively. It builds on simple but powerful concepts: learning from interpretations, entity/relationship data modeling, logic programming, and deductive databases. Access by the kernel to the rich representation is mediated by a technique we call graphicalization: the relational representation is first transformed into a graph --- in particular, a grounded entity/relationship diagram. Subsequently, a choice of graph kernel defines the feature space. kLog supports mixed numerical and symbolic data, as well as background knowledge in the form of Prolog or Datalog programs as in inductive logic programming systems. The kLog framework can be applied to tackle the same range of tasks that has made statistical relational learning so popular, including classification, regression, multitask learning, and collective classification. We also report about empirical comparisons, showing that kLog can be either more accurate, or much faster at the same level of accuracy, than Tilde and Alchemy. kLog is GPLv3 licensed and is available at http://klog.dinfo.unifi.it along with tutorials.

In mixture model-based clustering applications, it is common to fit several models from a family and report clustering results from only the `best' one. In such circumstances, selection of this best model is achieved using a model selection criterion, most often the Bayesian information criterion. Rather than throw away all but the best model, we average multiple models that are in some sense close to the best one, thereby producing a weighted average of clustering results. Two (weighted) averaging approaches are considered: averaging the component membership probabilities and averaging models. In both cases, Occam's window is used to determine closeness to the best model and weights are computed within a Bayesian model averaging paradigm. In some cases, we need to merge components before averaging; we introduce a method for merging mixture components based on the adjusted Rand index. The effectiveness of our model-based clustering averaging approaches is illustrated using a family of Gaussian mixture models on real and simulated data.

Learning structured outputs with general structures is computationally challenging, except for tree-structured models. Thus we propose an efficient boosting-based algorithm AdaBoost.MRF for this task. The idea is based on the realization that a graph is a superimposition of trees. Different from most existing work, our algorithm can handle partial labelling, and thus is particularly attractive in practice where reliable labels are often sparsely observed. In addition, our method works exclusively on trees and thus is guaranteed to converge. We apply the AdaBoost.MRF algorithm to an indoor video surveillance scenario, where activities are modelled at multiple levels.

We study the problem of collaborative filtering where ranking information is available. Focusing on the core of the collaborative ranking process, the user and their community, we propose new models for representation of the underlying permutations and prediction of ranks. The first approach is based on the assumption that the user makes successive choice of items in a stage-wise manner. In particular, we extend the Plackett-Luce model in two ways - introducing parameter factoring to account for user-specific contribution, and modelling the latent community in a generative setting. The second approach relies on log-linear parameterisation, which relaxes the discrete-choice assumption, but makes learning and inference much more involved. We propose MCMC-based learning and inference methods and derive linear-time prediction algorithms.

This paper considers the problem of clustering a partially observed unweighted graph---i.e., one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organize the nodes into disjoint clusters so that there is relatively dense (observed) connectivity within clusters, and sparse across clusters. We take a novel yet natural approach to this problem, by focusing on finding the clustering that minimizes the number of "disagreements"---i.e., the sum of the number of (observed) missing edges within clusters, and (observed) present edges across clusters. Our algorithm uses convex optimization; its basis is a reduction of disagreement minimization to the problem of recovering an (unknown) low-rank matrix and an (unknown) sparse matrix from their partially observed sum. We evaluate the performance of our algorithm on the classical Planted Partition/Stochastic Block Model. Our main theorem provides sufficient conditions for the success of our algorithm as a function of the minimum cluster size, edge density and observation probability; in particular, the results characterize the tradeoff between the observation probability and the edge density gap. When there are a constant number of clusters of equal size, our results are optimal up to logarithmic factors.