Dimensionality reduction is a topic of recent interest. In this paper, we present the classification constrained dimensionality reduction (CCDR) algorithm to account for label information. The algorithm can account for multiple classes as well as the semi-supervised setting. We present an out-of-sample expressions for both labeled and unlabeled data. For unlabeled data, we introduce a method of embedding a new point as preprocessing to a classifier. For labeled data, we introduce a method that improves the embedding during the training phase using the out-of-sample extension. We investigate classification performance using the CCDR algorithm on hyper-spectral satellite imagery data. We demonstrate the performance gain for both local and global classifiers and demonstrate a 10% improvement of the $k$-nearest neighbors algorithm performance. We present a connection between intrinsic dimension estimation and the optimal embedding dimension obtained using the CCDR algorithm.

This work concerns estimation of multidimensional nonlinear regression models using multilayer perceptron (MLP). The main problem with such model is that we have to know the covariance matrix of the noise to get optimal estimator. however we show that, if we choose as cost function the logarithm of the determinant of the empirical error covariance matrix, we get an asymptotically optimal estimator.

This work concerns testing the number of parameters in one hidden layer multilayer perceptron (MLP). For this purpose we assume that we have identifiable models, up to a finite group of transformations on the weights, this is for example the case when the number of hidden units is know. In this framework, we show that we get a simple asymptotic distribution, if we use the logarithm of the determinant of the empirical error covariance matrix as cost function.

We consider the problems of clustering, classification, and visualization of high-dimensional data when no straightforward Euclidean representation exists. Typically, these tasks are performed by first reducing the high-dimensional data to some lower dimensional Euclidean space, as many manifold learning methods have been developed for this task. In many practical problems however, the assumption of a Euclidean manifold cannot be justified. In these cases, a more appropriate assumption would be that the data lies on a statistical manifold, or a manifold of probability density functions (PDFs). In this paper we propose using the properties of information geometry in order to define similarities between data sets using the Fisher information metric. We will show this metric can be approximated using entirely non-parametric methods, as the parameterization of the manifold is generally unknown. Furthermore, by using multi-dimensional scaling methods, we are able to embed the corresponding PDFs into a low-dimensional Euclidean space. This not only allows for classification of the data, but also visualization of the manifold. As a whole, we refer to our framework as Fisher Information Non-parametric Embedding (FINE), and illustrate its uses on a variety of practical problems, including bio-medical applications and document classification.

In this paper we consider the problem of grouped variable selection in high-dimensional regression using $\ell_1-\ell_q$ regularization ($1\leq q \leq \infty$), which can be viewed as a natural generalization of the $\ell_1-\ell_2$ regularization (the group Lasso). The key condition is that the dimensionality $p_n$ can increase much faster than the sample size $n$, i.e. $p_n \gg n$ (in our case $p_n$ is the number of groups), but the number of relevant groups is small. The main conclusion is that many good properties from $\ell_1-$regularization (Lasso) naturally carry on to the $\ell_1-\ell_q$ cases ($1 \leq q \leq \infty$), even if the number of variables within each group also increases with the sample size. With fixed design, we show that the whole family of estimators are both estimation consistent and variable selection consistent under different conditions. We also show the persistency result with random design under a much weaker condition. These results provide a unified treatment for the whole family of estimators ranging from $q=1$ (Lasso) to $q=\infty$ (iCAP), with $q=2$ (group Lasso)as a special case. When there is no group structure available, all the analysis reduces to the current results of the Lasso estimator ($q=1$).

We study the problem of partitioning a small sample of $n$ individuals from a mixture of $k$ product distributions over a Boolean cube $\{0, 1\}^K$ according to their distributions. Each distribution is described by a vector of allele frequencies in $\R^K$. Given two distributions, we use $\gamma$ to denote the average $\ell_2^2$ distance in frequencies across $K$ dimensions, which measures the statistical divergence between them. We study the case assuming that bits are independently distributed across $K$ dimensions. This work demonstrates that, for a balanced input instance for $k = 2$, a certain graph-based optimization function returns the correct partition with high probability, where a weighted graph $G$ is formed over $n$ individuals, whose pairwise hamming distances between their corresponding bit vectors define the edge weights, so long as $K = \Omega(\ln n/\gamma)$ and $Kn = \tilde\Omega(\ln n/\gamma^2)$. The function computes a maximum-weight balanced cut of $G$, where the weight of a cut is the sum of the weights across all edges in the cut. This result demonstrates a nice property in the high-dimensional feature space: one can trade off the number of features that are required with the size of the sample to accomplish certain tasks like clustering.

We study the efficiency of V-fold cross-validation (VFCV) for model selection from the non-asymptotic viewpoint, and suggest an improvement on it, which we call ``V-fold penalization''. Considering a particular (though simple) regression problem, we prove that VFCV with a bounded V is suboptimal for model selection, because it ``overpenalizes'' all the more that V is large. Hence, asymptotic optimality requires V to go to infinity. However, when the signal-to-noise ratio is low, it appears that overpenalizing is necessary, so that the optimal V is not always the larger one, despite of the variability issue. This is confirmed by some simulated data. In order to improve on the prediction performance of VFCV, we define a new model selection procedure, called ``V-fold penalization'' (penVF). It is a V-fold subsampling version of Efron's bootstrap penalties, so that it has the same computational cost as VFCV, while being more flexible. In a heteroscedastic regression framework, assuming the models to have a particular structure, we prove that penVF satisfies a non-asymptotic oracle inequality with a leading constant that tends to 1 when the sample size goes to infinity. In particular, this implies adaptivity to the smoothness of the regression function, even with a highly heteroscedastic noise. Moreover, it is easy to overpenalize with penVF, independently from the V parameter. A simulation study shows that this results in a significant improvement on VFCV in non-asymptotic situations.

Recent research has studied the role of sparsity in high dimensional regression and signal reconstruction, establishing theoretical limits for recovering sparse models from sparse data. This line of work shows that $\ell_1$-regularized least squares regression can accurately estimate a sparse linear model from $n$ noisy examples in $p$ dimensions, even if $p$ is much larger than $n$. In this paper we study a variant of this problem where the original $n$ input variables are compressed by a random linear transformation to $m \ll n$ examples in $p$ dimensions, and establish conditions under which a sparse linear model can be successfully recovered from the compressed data. A primary motivation for this compression procedure is to anonymize the data and preserve privacy by revealing little information about the original data. We characterize the number of random projections that are required for $\ell_1$-regularized compressed regression to identify the nonzero coefficients in the true model with probability approaching one, a property called ``sparsistence.'' In addition, we show that $\ell_1$-regularized compressed regression asymptotically predicts as well as an oracle linear model, a property called ``persistence.'' Finally, we characterize the privacy properties of the compression procedure in information-theoretic terms, establishing upper bounds on the mutual information between the compressed and uncompressed data that decay to zero.

We discuss two parameterizations of models for marginal independencies for discrete distributions which are representable by bi-directed graph models, under the global Markov property. Such models are useful data analytic tools especially if used in combination with other graphical models. The first parameterization, in the saturated case, is also known as the multivariate logistic transformation, the second is a variant that allows, in some (but not all) cases, variation independent parameters. An algorithm for maximum likelihood fitting is proposed, based on an extension of the Aitchison and Silvey method.

We give an overview of two approaches to probability theory where lower and upper probabilities, rather than probabilities, are used: Walley's behavioural theory of imprecise probabilities, and Shafer and Vovk's game-theoretic account of probability. We show that the two theories are more closely related than would be suspected at first sight, and we establish a correspondence between them that (i) has an interesting interpretation, and (ii) allows us to freely import results from one theory into the other. Our approach leads to an account of probability trees and random processes in the framework of Walley's theory. We indicate how our results can be used to reduce the computational complexity of dealing with imprecision in probability trees, and we prove an interesting and quite general version of the weak law of large numbers.