We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low--rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is thus implicitly suited for thin--tailed continuous data. Arguably, common applications of matrix completion require estimators for (a) heterogeneous data--types, such as skewed--continuous, count, binary, etc., (b) for heterogeneous noise models (beyond Gaussian), which capture varied uncertainty in the measurements, and (c) heterogeneous structural constraints beyond low--rank, such as block--sparsity, or a superposition structure of low--rank plus elementwise sparseness, among others. In this paper, we provide a vastly unified framework for generalized matrix completion by considering a matrix completion setting wherein the matrix entries are sampled from any member of the rich family of exponential family distributions; and impose general structural constraints on the underlying matrix, as captured by a general regularizer $\mathcal{R}(.)$. We propose a simple convex regularized $M$--estimator for the generalized framework, and provide a unified and novel statistical analysis for this general class of estimators. We finally corroborate our theoretical results on simulated datasets.

High-throughput mRNA sequencing (RNA-Seq) is widely used for transcript quantification of gene isoforms. Since RNA-Seq data alone is often not sufficient to accurately identify the read origins from the isoforms for quantification, we propose to explore protein domain-domain interactions as prior knowledge for integrative analysis with RNA-seq data. We introduce a Network-based method for RNA-Seq-based Transcript Quantification (Net-RSTQ) to integrate protein domain-domain interaction network with short read alignments for transcript abundance estimation. Based on our observation that the abundances of the neighboring isoforms by domain-domain interactions in the network are positively correlated, Net-RSTQ models the expression of the neighboring transcripts as Dirichlet priors on the likelihood of the observed read alignments against the transcripts in one gene. The transcript abundances of all the genes are then jointly estimated with alternating optimization of multiple EM problems. In simulation Net-RSTQ effectively improved isoform transcript quantifications when isoform co-expressions correlate with their interactions. qRT-PCR results on 25 multi-isoform genes in a stem cell line, an ovarian cancer cell line, and a breast cancer cell line also showed that Net-RSTQ estimated more consistent isoform proportions with RNA-Seq data. In the experiments on the RNA-Seq data in The Cancer Genome Atlas (TCGA), the transcript abundances estimated by Net-RSTQ are more informative for patient sample classification of ovarian cancer, breast cancer and lung cancer. All experimental results collectively support that Net-RSTQ is a promising approach for isoform quantification.

Ridge regularized linear models (RRLMs), such as ridge regression and the SVM, are a popular group of methods that are used in conjunction with coefficient hypothesis testing to discover explanatory variables with a significant multivariate association to a response. However, many investigators are reluctant to draw causal interpretations of the selected variables due to the incomplete knowledge of the capabilities of RRLMs in causal inference. Under reasonable assumptions, we show that a modified form of RRLMs can get very close to identifying a subset of the Markov boundary by providing a worst-case bound on the space of possible solutions. The results hold for any convex loss, even when the underlying functional relationship is nonlinear, and the solution is not unique. Our approach combines ideas in Markov boundary and sufficient dimension reduction theory. Experimental results show that the modified RRLMs are competitive against state-of-the-art algorithms in discovering part of the Markov boundary from gene expression data.

In this paper we propose a strategy for semi-supervised image classification that leverages unsupervised representation learning and co-training. The strategy, that is called CURL from Co-trained Unsupervised Representation Learning, iteratively builds two classifiers on two different views of the data. The two views correspond to different representations learned from both labeled and unlabeled data and differ in the fusion scheme used to combine the image features. To assess the performance of our proposal, we conducted several experiments on widely used data sets for scene and object recognition. We considered three scenarios (inductive, transductive and self-taught learning) that differ in the strategy followed to exploit the unlabeled data. As image features we considered a combination of GIST, PHOG, and LBP as well as features extracted from a Convolutional Neural Network. Moreover, two embodiments of CURL are investigated: one using Ensemble Projection as unsupervised representation learning coupled with Logistic Regression, and one based on LapSVM. The results show that CURL clearly outperforms other supervised and semi-supervised learning methods in the state of the art.

In the co-sparse analysis model a set of filters is applied to a signal out of the signal class of interest yielding sparse filter responses. As such, it may serve as a prior in inverse problems, or for structural analysis of signals that are known to belong to the signal class. The more the model is adapted to the class, the more reliable it is for these purposes. The task of learning such operators for a given class is therefore a crucial problem. In many applications, it is also required that the filter responses are obtained in a timely manner, which can be achieved by filters with a separable structure. Not only can operators of this sort be efficiently used for computing the filter responses, but they also have the advantage that less training samples are required to obtain a reliable estimate of the operator. The first contribution of this work is to give theoretical evidence for this claim by providing an upper bound for the sample complexity of the learning process. The second is a stochastic gradient descent (SGD) method designed to learn an analysis operator with separable structures, which includes a novel and efficient step size selection rule. Numerical experiments are provided that link the sample complexity to the convergence speed of the SGD algorithm.

We develop a sequential low-complexity inference procedure for Dirichlet process mixtures of Gaussians for online clustering and parameter estimation when the number of clusters are unknown a-priori. We present an easily computable, closed form parametric expression for the conditional likelihood, in which hyperparameters are recursively updated as a function of the streaming data assuming conjugate priors. Motivated by large-sample asymptotics, we propose a novel adaptive low-complexity design for the Dirichlet process concentration parameter and show that the number of classes grow at most at a logarithmic rate. We further prove that in the large-sample limit, the conditional likelihood and data predictive distribution become asymptotically Gaussian. We demonstrate through experiments on synthetic and real data sets that our approach is superior to other online state-of-the-art methods.

Dynamic ensemble selection (DES) techniques work by estimating the level of competence of each classifier from a pool of classifiers. Only the most competent ones are selected to classify a given test sample. Hence, the key issue in DES is the criterion used to estimate the level of competence of the classifiers in predicting the label of a given test sample. In order to perform a more robust ensemble selection, we proposed the META-DES framework using meta-learning, where multiple criteria are encoded as meta-features and are passed down to a meta-classifier that is trained to estimate the competence level of a given classifier. In this technical report, we present a step-by-step analysis of each phase of the framework during training and test. We show how each set of meta-features is extracted as well as their impact on the estimation of the competence level of the base classifier. Moreover, an analysis of the impact of several factors in the system performance, such as the number of classifiers in the pool, the use of different linear base classifiers, as well as the size of the validation data. We show that using the dynamic selection of linear classifiers through the META-DES framework, we can solve complex non-linear classification problems where other combination techniques such as AdaBoost cannot.

One significant challenge to scaling entity resolution algorithms to massive datasets is understanding how performance changes after moving beyond the realm of small, manually labeled reference datasets. Unlike traditional machine learning tasks, when an entity resolution algorithm performs well on small hold-out datasets, there is no guarantee this performance holds on larger hold-out datasets. We prove simple bounding properties between the performance of a match function on a small validation set and the performance of a pairwise entity resolution algorithm on arbitrarily sized datasets. Thus, our approach enables optimization of pairwise entity resolution algorithms for large datasets, using a small set of labeled data.

The general perception is that kernel methods are not scalable, and neural nets are the methods of choice for nonlinear learning problems. Or have we simply not tried hard enough for kernel methods? Here we propose an approach that scales up kernel methods using a novel concept called "doubly stochastic functional gradients". Our approach relies on the fact that many kernel methods can be expressed as convex optimization problems, and we solve the problems by making two unbiased stochastic approximations to the functional gradient, one using random training points and another using random functions associated with the kernel, and then descending using this noisy functional gradient. We show that a function produced by this procedure after $t$ iterations converges to the optimal function in the reproducing kernel Hilbert space in rate $O(1/t)$, and achieves a generalization performance of $O(1/\sqrt{t})$. This doubly stochasticity also allows us to avoid keeping the support vectors and to implement the algorithm in a small memory footprint, which is linear in number of iterations and independent of data dimension. Our approach can readily scale kernel methods up to the regimes which are dominated by neural nets. We show that our method can achieve competitive performance to neural nets in datasets such as 8 million handwritten digits from MNIST, 2.3 million energy materials from MolecularSpace, and 1 million photos from ImageNet.

Semi-Non-negative Matrix Factorization is a technique that learns a low-dimensional representation of a dataset that lends itself to a clustering interpretation. It is possible that the mapping between this new representation and our original data matrix contains rather complex hierarchical information with implicit lower-level hidden attributes, that classical one level clustering methodologies can not interpret. In this work we propose a novel model, Deep Semi-NMF, that is able to learn such hidden representations that allow themselves to an interpretation of clustering according to different, unknown attributes of a given dataset. We also present a semi-supervised version of the algorithm, named Deep WSF, that allows the use of (partial) prior information for each of the known attributes of a dataset, that allows the model to be used on datasets with mixed attribute knowledge. Finally, we show that our models are able to learn low-dimensional representations that are better suited for clustering, but also classification, outperforming Semi-Non-negative Matrix Factorization, but also other state-of-the-art methodologies variants.