We propose a novel application of the Simultaneous Orthogonal Matching Pursuit (S-OMP) procedure for sparsistant variable selection in ultra-high dimensional multi-task regression problems. Screening of variables, as introduced in \cite{fan08sis}, is an efficient and highly scalable way to remove many irrelevant variables from the set of all variables, while retaining all the relevant variables. S-OMP can be applied to problems with hundreds of thousands of variables and once the number of variables is reduced to a manageable size, a more computationally demanding procedure can be used to identify the relevant variables for each of the regression outputs. To our knowledge, this is the first attempt to utilize relatedness of multiple outputs to perform fast screening of relevant variables. As our main theoretical contribution, we prove that, asymptotically, S-OMP is guaranteed to reduce an ultra-high number of variables to below the sample size without losing true relevant variables. We also provide formal evidence that a modified Bayesian information criterion (BIC) can be used to efficiently determine the number of iterations in S-OMP. We further provide empirical evidence on the benefit of variable selection using multiple regression outputs jointly, as opposed to performing variable selection for each output separately. The finite sample performance of S-OMP is demonstrated on extensive simulation studies, and on a genetic association mapping problem. $Keywords$ Adaptive Lasso; Greedy forward regression; Orthogonal matching pursuit; Multi-output regression; Multi-task learning; Simultaneous orthogonal matching pursuit; Sure screening; Variable selection

A new distance function dist(A,B) for fuzzy sets A and B is introduced. It is based on the descriptive complexity, i.e., the number of bits (on average) that are needed to describe an element in the symmetric difference of the two sets. The distance gives the amount of additional information needed to describe any one of the two sets given the other. We prove its mathematical properties and perform pattern clustering on data based on this distance.

A learning algorithm based on primary school teaching and learning is presented. The methodology is to continuously evaluate a student and to give them training on the examples for which they repeatedly fail, until, they can correctly answer all types of questions. This incremental learning procedure produces better learning curves by demanding the student to optimally dedicate their learning time on the failed examples. When used in machine learning, the algorithm is found to train a machine on a data with maximum variance in the feature space so that the generalization ability of the network improves. The algorithm has interesting applications in data mining, model evaluations and rare objects discovery.

The goal of cross-domain object matching (CDOM) is to find correspondence between two sets of objects in different domains in an unsupervised way. Photo album summarization is a typical application of CDOM, where photos are automatically aligned into a designed frame expressed in the Cartesian coordinate system. CDOM is usually formulated as finding a mapping from objects in one domain (photos) to objects in the other domain (frame) so that the pairwise dependency is maximized. A state-of-the-art CDOM method employs a kernel-based dependency measure, but it has a drawback that the kernel parameter needs to be determined manually. In this paper, we propose alternative CDOM methods that can naturally address the model selection problem. Through experiments on image matching, unpaired voice conversion, and photo album summarization tasks, the effectiveness of the proposed methods is demonstrated.

Suppose a given observation matrix can be decomposed as the sum of a low-rank matrix and a sparse matrix (outliers), and the goal is to recover these individual components from the observed sum. Such additive decompositions have applications in a variety of numerical problems including system identification, latent variable graphical modeling, and principal components analysis. We study conditions under which recovering such a decomposition is possible via a combination of $\ell_1$ norm and trace norm minimization. We are specifically interested in the question of how many outliers are allowed so that convex programming can still achieve accurate recovery, and we obtain stronger recovery guarantees than previous studies. Moreover, we do not assume that the spatial pattern of outliers is random, which stands in contrast to related analyses under such assumptions via matrix completion.

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg, in which for example one obtains an existence and uniqueness theorem instead of a non-existence result. We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.

We consider rules for discarding predictors in lasso regression and related problems, for computational efficiency. El Ghaoui et al (2010) propose "SAFE" rules that guarantee that a coefficient will be zero in the solution, based on the inner products of each predictor with the outcome. In this paper we propose strong rules that are not foolproof but rarely fail in practice. These can be complemented with simple checks of the Karush- Kuhn-Tucker (KKT) conditions to provide safe rules that offer substantial speed and space savings in a variety of statistical convex optimization problems.

A large set of signals can sometimes be described sparsely using a dictionary, that is, every element can be represented as a linear combination of few elements from the dictionary. Algorithms for various signal processing applications, including classification, denoising and signal separation, learn a dictionary from a set of signals to be represented. Can we expect that the representation found by such a dictionary for a previously unseen example from the same source will have L_2 error of the same magnitude as those for the given examples? We assume signals are generated from a fixed distribution, and study this questions from a statistical learning theory perspective. We develop generalization bounds on the quality of the learned dictionary for two types of constraints on the coefficient selection, as measured by the expected L_2 error in representation when the dictionary is used. For the case of l_1 regularized coefficient selection we provide a generalization bound of the order of O(sqrt(np log(m lambda)/m)), where n is the dimension, p is the number of elements in the dictionary, lambda is a bound on the l_1 norm of the coefficient vector and m is the number of samples, which complements existing results. For the case of representing a new signal as a combination of at most k dictionary elements, we provide a bound of the order O(sqrt(np log(m k)/m)) under an assumption on the level of orthogonality of the dictionary (low Babel function). We further show that this assumption holds for most dictionaries in high dimensions in a strong probabilistic sense. Our results further yield fast rates of order 1/m as opposed to 1/sqrt(m) using localized Rademacher complexity. We provide similar results in a general setting using kernels with weak smoothness requirements.

This paper discusses the topic of dimensionality reduction for $k$-means clustering. We prove that any set of $n$ points in $d$ dimensions (rows in a matrix $A \in \RR^{n \times d}$) can be projected into $t = \Omega(k / \eps^2)$ dimensions, for any $\eps \in (0,1/3)$, in $O(n d \lceil \eps^{-2} k/ \log(d) \rceil )$ time, such that with constant probability the optimal $k$-partition of the point set is preserved within a factor of $2+\eps$. The projection is done by post-multiplying $A$ with a $d \times t$ random matrix $R$ having entries $+1/\sqrt{t}$ or $-1/\sqrt{t}$ with equal probability. A numerical implementation of our technique and experiments on a large face images dataset verify the speed and the accuracy of our theoretical results.

While traditional research on text clustering has largely focused on grouping documents by topic, it is conceivable that a user may want to cluster documents along other dimensions, such as the author's mood, gender, age, or sentiment. Without knowing the user's intention, a clustering algorithm will only group documents along the most prominent dimension, which may not be the one the user desires. To address the problem of clustering documents along the user-desired dimension, previous work has focused on learning a similarity metric from data manually annotated with the user's intention or having a human construct a feature space in an interactive manner during the clustering process. With the goal of reducing reliance on human knowledge for fine-tuning the similarity function or selecting the relevant features required by these approaches, we propose a novel active clustering algorithm, which allows a user to easily select the dimension along which she wants to cluster the documents by inspecting only a small number of words. We demonstrate the viability of our algorithm on a variety of commonly-used sentiment datasets.