Learning Exponential Families in High-Dimensions: Strong Convexity and Sparsity

The versatility of exponential families, along with their attendant convexity properties, make them a popular and effective statistical model. A central issue is learning these models in high-dimensions when the optimal parameter vector is sparse. This work characterizes a certain strong convexity property of general exponential families, which allows their generalization ability to be quantified. In particular, we show how this property can be used to analyze generic exponential families under L1 regularization

Entropic Projections and Dominating Points

Generalized entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers. They appear in many areas of applied mathematics such as statistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugate duality and functional analysis, criteria are derived for their existence. Representations of the generalized entropic projections are obtained: they are the ``measure component" of some extended entropy minimization problem.Comment: ESAIM P&S (2011) to appea