We study the excess minimum risk in statistical inference, defined as the difference between the minimum expected loss in estimating a random variable from an observed feature vector and the minimum expected loss in estimating the same random variable from a transformation (statistic) of the feature vector. After characterizing lossless transformations, i.e., transformations for which the excess risk is zero for all loss functions, we construct a partitioning test statistic for the hypothesis that a given transformation is lossless and show that for i.i.d. data the test is strongly consistent. More generally, we develop information-theoretic upper bounds on the excess risk that uniformly hold over fairly general classes of loss functions. Based on these bounds, we introduce the notion of a delta-lossless transformation and give sufficient conditions for a given transformation to be universally delta-lossless. Applications to classification, nonparametric regression, portfolio strategies, information bottleneck, and deep learning, are also surveyed.

A privacy-constrained information extraction problem is considered where for a pair of correlated discrete random variables $(X,Y)$ governed by a given joint distribution, an agent observes $Y$ and wants to convey to a potentially public user as much information about $Y$ as possible without compromising the amount of information revealed about $X$. To this end, the so-called {\em rate-privacy function} is introduced to quantify the maximal amount of information (measured in terms of mutual information) that can be extracted from $Y$ under a privacy constraint between $X$ and the extracted information, where privacy is measured using either mutual information or maximal correlation. Properties of the rate-privacy function are analyzed and information-theoretic and estimation-theoretic interpretations of it are presented for both the mutual information and maximal correlation privacy measures. It is also shown that the rate-privacy function admits a closed-form expression for a large family of joint distributions of $(X,Y)$. Finally, the rate-privacy function under the mutual information privacy measure is considered for the case where $(X,Y)$ has a joint probability density function by studying the problem where the extracted information is a uniform quantization of $Y$ corrupted by additive Gaussian noise. The asymptotic behavior of the rate-privacy function is studied as the quantization resolution grows without bound and it is observed that not all of the properties of the rate-privacy function carry over from the discrete to the continuous case.

Principal curves have been defined as "self consistent" smooth curves which pass through the "middle" of a d-dimensional probability distribution ordata cloud. Recently, we [1] have offered a new approach by defining principal curves as continuous curves of a given length which minimize the expected squared distance between the curve and points of the space randomly chosen according to a given distribution. The new definition made it possible to carry out a theoretical analysis of learning principal curves from training data. In this paper we propose a practical construction based on the new definition. Simulation results demonstrate that the new algorithm compares favorably with previous methods both in terms of performance and computational complexity.

In this paper we apply the method of complexity regularization to derive estimationbounds for nonlinear function estimation using a single hidden layer radial basis function network.

In this paper we apply the method of complexity regularization to derive estimation bounds for nonlinear function estimation using a single hidden layer radial basis function network.