paper a day : approximating high-dimensional pdfs by low-dimensional ones

Asymptotically Optimal Approximation of Multidimensional pdf’s by Lower Dimensional pdf’s
Steven Kay
IEEE Transactions on Signal Processing, V. 55 No. 2, Feb. 2007, p. 725–729

The title kind of says it all. The main idea is that if you have a sufficient statistic, then you can create the true probability density function (pdf) of the data from the pdf of the sufficient statistic. However, if there is no sufficient statistic, you’re out of luck, and you’d like to create a low-dimensional pdf that somehow best captures the features you want from the data. This paper proves that a certain pdf created by a projection operation is optimal in that it minimizes the Kullback-Leibler (KL) divergence. Since the KL divergence dictates the error in many hypothesis tests, this projection operation is good in that decisions based on the projected pdf will be close to decisions based on the true pdf.

This is a correspondence item, so it’s short and sweet — equations are given for the projection and it is proved to minimize the KL divergence to the true distribution. Examples are given for cases in which sufficient statistics exist and do not exist, and an application to feature selection for discrimination is given. The benefit is that this theorem provides a way of choosing a “good” feature set based on the KL divergence, even when the true pdf is not known. This is done by estimating an expectation from the observed data (the performance then depends on the convergence speed of the empirical mean to the true mean, which should be exponentially fast in the number of data points).

The formulas are sometimes messy, but it looks like it could be a useful technique. I have this niggling feeling that a “bigger picture” view would be forthcoming from looking at information geometry/differential geometry viewpoint, but my fluency in those techniques is lacking at the moment.

Update: My laziness prevented me from putting up the link. Thanks, Cosma, for keeping me honest!