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Poster
Inverse Density as an Inverse Problem: the Fredholm Equation Approach
Qichao Que · Mikhail Belkin
Sat Dec 07 07:00 PM  11:59 PM (PST) @ Harrah's Special Events Center, 2nd Floor
We address the problem of estimating the ratio $\frac{q}{p}$ where $p$ is a density function and $q$ is another density, or, more generally an arbitrary function. Knowing or approximating this ratio is needed in various problems of inference and integration, in particular, when one needs to average a function with respect to one probability distribution, given a sample from another. It is often referred as {\it importance sampling} in statistical inference and is also closely related to the problem of {\it covariate shift} in transfer learning as well as to various MCMC methods. Our approach is based on reformulating the problem of estimating the ratio as an inverse problem in terms of an integral operator corresponding to a kernel, and thus reducing it to an integral equation, known as the Fredholm problem of the first kind. This formulation, combined with the techniques of regularization and kernel methods, leads to a principled kernelbased framework for constructing algorithms and for analyzing them theoretically. The resulting family of algorithms (FIRE, for Fredholm Inverse Regularized Estimator) is flexible, simple and easy to implement. We provide detailed theoretical analysis including concentration bounds and convergence rates for the Gaussian kernel for densities defined on $\R^d$ and smooth $d$dimensional submanifolds of the Euclidean space. Model selection for unsupervised or semisupervised inference is generally a difficult problem. Interestingly, it turns out that in the density ratio estimation setting, when samples from both distributions are available, there are simple completely unsupervised methods for choosing parameters. We call this model selection mechanism CDCV for CrossDensity CrossValidation. Finally, we show encouraging experimental results including applications to classification within the covariate shift framework.
Author Information
Qichao Que (The Ohio State University)
Mikhail Belkin (Ohio State University)
Related Events (a corresponding poster, oral, or spotlight)

2013 Spotlight: Inverse Density as an Inverse Problem: the Fredholm Equation Approach »
Sat. Dec 7th 07:48  07:52 PM Room Harvey's Convention Center Floor, CC
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