Inverse Density as an Inverse Problem: The Fredholm Equation Approach

In this paper 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. It may also be useful for separating the underlying geometry of a space, say a manifold, from the density function defined on it. Our approach is based on reformulating the problem of estimating $\frac{q}{p}$ 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 kernel-based 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 in the case of densities defined on $\R^d$, compact domains in $\R^d$ and smooth $d$-dimensional sub-manifolds of the Euclidean space. We also show experimental results including applications to classification and semi-supervised learning within the covariate shift framework and demonstrate some encouraging experimental comparisons. We also show how the parameters of our algorithms can be chosen in a completely unsupervised manner.Comment: Fixing a few typos in last versio

Beyond Hartigan Consistency: Merge Distortion Metric for Hierarchical Clustering

Hierarchical clustering is a popular method for analyzing data which associates a tree to a dataset. Hartigan consistency has been used extensively as a framework to analyze such clustering algorithms from a statistical point of view. Still, as we show in the paper, a tree which is Hartigan consistent with a given density can look very different than the correct limit tree. Specifically, Hartigan consistency permits two types of undesirable configurations which we term over-segmentation and improper nesting. Moreover, Hartigan consistency is a limit property and does not directly quantify difference between trees. In this paper we identify two limit properties, separation and minimality, which address both over-segmentation and improper nesting and together imply (but are not implied by) Hartigan consistency. We proceed to introduce a merge distortion metric between hierarchical clusterings and show that convergence in our distance implies both separation and minimality. We also prove that uniform separation and minimality imply convergence in the merge distortion metric. Furthermore, we show that our merge distortion metric is stable under perturbations of the density. Finally, we demonstrate applicability of these concepts by proving convergence results for two clustering algorithms. First, we show convergence (and hence separation and minimality) of the recent robust single linkage algorithm of Chaudhuri and Dasgupta (2010). Second, we provide convergence results on manifolds for topological split tree clustering