We introduce a novel framework for estimating vector fields using sparse basis field expansions (S-FLEX). The notion of basis fields, which are an extension of scalar basis functions, arises naturally in our framework from a rotational invariance requirement. We consider a regression setting as well as inverse problems. All variants discussed lead to second-order cone programming formulations. While our framework is generally applicable to any type of vector field, we focus in this paper on applying it to solving the EEG/MEG inverse problem. It is shown that significantly more precise and neurophysiologically more plausible location and shape estimates of cerebral current sources from EEG/MEG measurements become possible with our method when comparing to the state-of-the-art.

This paper investigates a new machine learning strategy called translated learning. Unlikemany previous learning tasks, we focus on how to use labeled data from one feature space to enhance the classification of other entirely different learning spaces. For example, we might wish to use labeled text data to help learn a model for classifying image data, when the labeled images are difficult to obtain. Animportant aspect of translated learning is to build a "bridge" to link one feature space (known as the "source space") to another space (known as the "target space")through a translator in order to migrate the knowledge from source to target. The translated learning solution uses a language model to link the class labels to the features in the source spaces, which in turn is translated to the features inthe target spaces. Finally, this chain of linkages is completed by tracing back to the instances in the target spaces. We show that this path of linkage can be modeled using a Markov chain and risk minimization. Through experiments on the text-aided image classification and cross-language classification tasks, we demonstrate that our translated learning framework can greatly outperform many state-of-the-art baseline methods.

Confidence-weighted (CW) learning [6], an online learning method for linear classifiers, maintains a Gaussian distributions over weight vectors, with a covariance matrix that represents uncertainty about weights and correlations. Confidence constraints ensure that a weight vector drawn from the hypothesis distribution correctly classifies examples with a specified probability. Within this framework, we derive a new convex form of the constraint and analyze it in the mistake bound model. Empirical evaluation with both synthetic and text data shows our version of CW learning achieves lower cumulative and out-of-sample errors than commonly used first-order and second-order online methods.

The coding of information by neural populations depends critically on the statistical dependencies between neuronal responses. However, there is no simple model that combines the observations that (1) marginal distributions over single-neuron spike counts are often approximately Poisson; and (2) joint distributions over the responses of multiple neurons are often strongly dependent. Here, we show that both marginal and joint properties of neural responses can be captured using Poisson copula models. Copulas are joint distributions that allow random variables with arbitrary marginals to be combined while incorporating arbitrary dependencies between them. Different copulas capture different kinds of dependencies, allowing for a richer and more detailed description of dependencies than traditional summary statistics, such as correlation coefficients. We explore a variety of Poisson copula models for joint neural response distributions, and derive an efficient maximum likelihood procedure for estimating them. We apply these models to neuronal data collected in and macaque motor cortex, and quantify the improvement in coding accuracy afforded by incorporating the dependency structure between pairs of neurons.

Aiming towards the development of a general clustering theory, we discuss abstract axiomatization for clustering. In this respect, we follow up on the work of Kelinberg, (Kleinberg) that showed an impossibility result for such axiomatization. We argue that an impossibility result is not an inherent feature of clustering, but rather, to a large extent, it is an artifact of the specific formalism used in Kleinberg. As opposed to previous work focusing on clustering functions, we propose to address clustering quality measures as the primitive object to be axiomatized. We show that principles like those formulated in Kleinberg's axioms can be readily expressed in the latter framework without leading to inconsistency. A clustering-quality measure is a function that, given a data set and its partition into clusters, returns a non-negative real number representing how `strong' or `conclusive' the clustering is. We analyze what clustering-quality measures should look like and introduce a set of requirements (`axioms') that express these requirement and extend the translation of Kleinberg's axioms to our framework. We propose several natural clustering quality measures, all satisfying the proposed axioms. In addition, we show that the proposed clustering quality can be computed in polynomial time.

In this paper we present two transductive bounds on the risk of the majority vote estimated over partially labeled training sets. Our first bound is tight when the additional unlabeled training data are used in the cases where the voted classifier makes its errors on low margin observations and where the errors of the associated Gibbs classifier can accurately be estimated. In semi-supervised learning, considering the margin as an indicator of confidence constitutes the working hypothesis of algorithms which search the decision boundary on low density regions. In this case, we propose a second bound on the joint probability that the voted classifier makes an error over an example having its margin over a fixed threshold. As an application we are interested on self-learning algorithms which assign iteratively pseudo-labels to unlabeled training examples having margin above a threshold obtained from this bound. Empirical results on different datasets show the effectiveness of our approach compared to the same algorithm and the TSVM in which the threshold is fixed manually.

Given $n$ noisy samples with $p$ dimensions, where $n \ll p$, we show that the multi-stage thresholding procedures can accurately estimate a sparse vector $\beta \in \R^p$ in a linear model, under the restricted eigenvalue conditions (Bickel-Ritov-Tsybakov 09). Thus our conditions for model selection consistency are considerably weaker than what has been achieved in previous works. More importantly, this method allows very significant values of $s$, which is the number of non-zero elements in the true parameter $\beta$. For example, it works for cases where the ordinary Lasso would have failed. Finally, we show that if $X$ obeys a uniform uncertainty principle and if the true parameter is sufficiently sparse, the Gauss-Dantzig selector (Cand\{e}s-Tao 07) achieves the $\ell_2$ loss within a logarithmic factor of the ideal mean square error one would achieve with an oracle which would supply perfect information about which coordinates are non-zero and which are above the noise level, while selecting a sufficiently sparse model.

We propose Dirichlet-Bernoulli Alignment (DBA), a generative model for corpora in which each pattern (e.g., a document) contains a set of instances (e.g., paragraphs in the document) and belongs to multiple classes. By casting predefined classes as latent Dirichlet variables (i.e., instance level labels), and modeling the multi-label of each pattern as Bernoulli variables conditioned on the weighted empirical average of topic assignments, DBA automatically aligns the latent topics discovered from data to human-defined classes. DBA is useful for both pattern classification and instance disambiguation, which are tested on text classification and named entity disambiguation for web search queries respectively.

We discuss the framework of Transductive Support Vector Machine (TSVM) from the perspective of the regularization strength induced by the unlabeled data. In this framework, SVM and TSVM can be regarded as a learning machine without regularization and one with full regularization from the unlabeled data, respectively. Therefore, to supplement this framework of the regularization strength, it is necessary to introduce data-dependant partial regularization. To this end, we reformulate TSVM into a form with controllable regularization strength, which includes SVM and TSVM as special cases. Furthermore, we introduce a method of adaptive regularization that is data dependant and is based on the smoothness assumption. Experiments on a set of benchmark data sets indicate the promising results of the proposed work compared with state-of-the-art TSVM algorithms.

We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as L1-norm for sparsity. We develop a new online algorithm, the regularized dual averaging method, that can explicitly exploit the regularization structure in an online setting. In particular, at each iteration, the learning variables are adjusted by solving a simple optimization problem that involves the running average of all past subgradients of the loss functions and the whole regularization term, not just its subgradient. This method achieves the optimal convergence rate and often enjoys a low complexity per iteration similar as the standard stochastic gradient method. Computational experiments are presented for the special case of sparse online learning using L1-regularization.