In many real world applications we do not have access to fully-labeled training data, but only to a list of possible labels. This is the case, e.g., when learning visual classifiers from images downloaded from the web, using just their text captions or tags as learning oracles. In general, these problems can be very difficult. However most of the time there exist different implicit sources of information, coming from the relations between instances and labels, which are usually dismissed. In this paper, we propose a semi-supervised framework to model this kind of problems. Each training sample is a bag containing multi-instances, associated with a set of candidate labeling vectors. Each labeling vector encodes the possible labels for the instances in the bag, with only one being fully correct. The use of the labeling vectors provides a principled way not to exclude any information. We propose a large margin discriminative formulation, and an efficient algorithm to solve it. Experiments conducted on artificial datasets and a real-world images and captions dataset show that our approach achieves performance comparable to SVM trained with the ground-truth labels, and outperforms other baselines.

We consider multivariate regression problems involving high-dimensional predictor and response spaces. To efficiently address such problems, we propose a variable selection method, Multivariate Group Orthogonal Matching Pursuit, which extends the standard Orthogonal Matching Pursuit technique to account for arbitrary sparsity patterns induced by domain-specific groupings over both input and output variables, while also taking advantage of the correlation that may exist between the multiple outputs. We illustrate the utility of this framework for inferring causal relationships over a collection of high-dimensional time series variables. When applied to time-evolving social media content, our models yield a new family of causality-based influence measures that may be seen as an alternative to PageRank. Theoretical guarantees, extensive simulations and empirical studies confirm the generality and value of our framework.

Arithmetic circuits (ACs) exploit context-specific independence and determinism to allow exact inference even in networks with high treewidth. In this paper, we introduce the first ever approximate inference methods using ACs, for domains where exact inference remains intractable. We propose and evaluate a variety of techniques based on exact compilation, forward sampling, AC structure learning, Markov network parameter learning, variational inference, and Gibbs sampling. In experiments on eight challenging real-world domains, we find that the methods based on sampling and learning work best: one such method (AC2-F) is faster and usually more accurate than loopy belief propagation, mean field, and Gibbs sampling; another (AC2-G) has a running time similar to Gibbs sampling but is consistently more accurate than all baselines.

Several motor related Brain Computer Interfaces (BCIs) have been developed over the years that use activity decoded from the contralateral hemisphere to operate devices. Contralateralprimary motor cortex is also the region most severely affected by hemispheric stroke. Recent studies have identified ipsilateral cortical activity in planning of motor movements and its potential implications for a stroke relevant BCI.The most fundamental functional loss after a hemispheric stroke is the loss of fine motor control of the hand. Thus, whether ipsilateral cortex encodes finger movements is critical to the potential feasibility of BCI approaches in the future. This study uses ipsilateral cortical signals from humans (using ECoG) to decode finger movements. We demonstrate, for the first time, successful finger movement detection using machine learning algorithms. Our results show high decoding accuracies in all cases which are always above chance. We also show that significant accuracies can be achieved with the use of only a fraction of all the features recorded and that these core features are consistent with previous physiological findings.The results of this study have substantial implications for advancing neuroprosthetic approaches to stroke populations not currently amenable to existing BCI techniques.

We consider the tree structured group Lasso where the structure over the features can be represented as a tree with leaf nodes as features and internal nodes as clusters of the features. The structured regularization with a pre-defined tree structure is based on a group-Lasso penalty, where one group is defined for each node in the tree. Such a regularization can help uncover the structured sparsity, which is desirable for applications with some meaningful tree structures on the features. However, the tree structured group Lasso is challenging to solve due to the complex regularization. In this paper, we develop an efficient algorithm for the tree structured group Lasso. One of the key steps in the proposed algorithm is to solve the Moreau-Yosida regularization associated with the grouped tree structure. The main technical contributions of this paper include (1) we show that the associated Moreau-Yosida regularization admits an analytical solution, and (2) we develop an efficient algorithm for determining the effective interval for the regularization parameter. Our experimental results on the AR and JAFFE face data sets demonstrate the efficiency and effectiveness of the proposed algorithm.

We consider the following sparse signal recovery (or feature selection) problem: given a design matrix $X\in \mathbb{R}^{n\times m}$ $(m\gg n)$ and a noisy observation vector $y\in \mathbb{R}^{n}$ satisfying $y=X\beta^*+\epsilon$ where $\epsilon$ is the noise vector following a Gaussian distribution $N(0,\sigma^2I)$, how to recover the signal (or parameter vector) $\beta^*$ when the signal is sparse? The Dantzig selector has been proposed for sparse signal recovery with strong theoretical guarantees. In this paper, we propose a multi-stage Dantzig selector method, which iteratively refines the target signal $\beta^*$. We show that if $X$ obeys a certain condition, then with a large probability the difference between the solution $\hat\beta$ estimated by the proposed method and the true solution $\beta^*$ measured in terms of the $l_p$ norm ($p\geq 1$) is bounded as \begin{equation*} \|\hat\beta-\beta^*\|_p\leq \left(C(s-N)^{1/p}\sqrt{\log m}+\Delta\right)\sigma, \end{equation*} $C$ is a constant, $s$ is the number of nonzero entries in $\beta^*$, $\Delta$ is independent of $m$ and is much smaller than the first term, and $N$ is the number of entries of $\beta^*$ larger than a certain value in the order of $\mathcal{O}(\sigma\sqrt{\log m})$. The proposed method improves the estimation bound of the standard Dantzig selector approximately from $Cs^{1/p}\sqrt{\log m}\sigma$ to $C(s-N)^{1/p}\sqrt{\log m}\sigma$ where the value $N$ depends on the number of large entries in $\beta^*$. When $N=s$, the proposed algorithm achieves the oracle solution with a high probability. In addition, with a large probability, the proposed method can select the same number of correct features under a milder condition than the Dantzig selector.

We propose a new nonparametric learning method based on multivariate dyadic regression trees (MDRTs). Unlike traditional dyadic decision trees (DDTs) or classification and regression trees (CARTs), MDRTs are constructed using penalized empirical risk minimization with a novel sparsity-inducing penalty. Theoretically, we show that MDRTs can simultaneously adapt to the unknown sparsity and smoothness of the true regression functions, and achieve the nearly optimal rates of convergence (in a minimax sense) for the class of $(\alpha, C)$-smooth functions. Empirically, MDRTs can simultaneously conduct function estimation and variable selection in high dimensions. To make MDRTs applicable for large-scale learning problems, we propose a greedy heuristics. The superior performance of MDRTs are demonstrated on both synthetic and real datasets.

A challenging problem in estimating high-dimensional graphical models is to choose the regularization parameter in a data-dependent way. The standard techniques include $K$-fold cross-validation ($K$-CV), Akaike information criterion (AIC), and Bayesian information criterion (BIC). Though these methods work well for low-dimensional problems, they are not suitable in high dimensional settings. In this paper, we present StARS: a new stability-based method for choosing the regularization parameter in high dimensional inference for undirected graphs. The method has a clear interpretation: we use the least amount of regularization that simultaneously makes a graph sparse and replicable under random sampling. This interpretation requires essentially no conditions. Under mild conditions, we show that StARS is partially sparsistent in terms of graph estimation: i.e. with high probability, all the true edges will be included in the selected model even when the graph size asymptotically increases with the sample size. Empirically, the performance of StARS is compared with the state-of-the-art model selection procedures, including $K$-CV, AIC, and BIC, on both synthetic data and a real microarray dataset. StARS outperforms all competing procedures.

Undirected graphical models encode in a graph $G$ the dependency structure of a random vector $Y$. In many applications, it is of interest to model $Y$ given another random vector $X$ as input. We refer to the problem of estimating the graph $G(x)$ of $Y$ conditioned on $X=x$ as ``graph-valued regression''. In this paper, we propose a semiparametric method for estimating $G(x)$ that builds a tree on the $X$ space just as in CART (classification and regression trees), but at each leaf of the tree estimates a graph. We call the method ``Graph-optimized CART'', or Go-CART. We study the theoretical properties of Go-CART using dyadic partitioning trees, establishing oracle inequalities on risk minimization and tree partition consistency. We also demonstrate the application of Go-CART to a meteorological dataset, showing how graph-valued regression can provide a useful tool for analyzing complex data.

This paper proposes a principled extension of the traditional single-layer flat sparse coding scheme, where a two-layer coding scheme is derived based on theoretical analysis of nonlinear functional approximation that extends recent results for local coordinate coding. The two-layer approach can be easily generalized to deeper structures in a hierarchical multiple-layer manner. Empirically, it is shown that the deep coding approach yields improved performance in benchmark datasets.