4 research outputs found
Inverse Density as an Inverse Problem: The Fredholm Equation Approach
In this paper we address the problem of estimating the ratio
where is a density function and 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
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
, compact domains in and smooth -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
Revisiting Kernelized Locality-Sensitive Hashing for Improved Large-Scale Image Retrieval
We present a simple but powerful reinterpretation of kernelized
locality-sensitive hashing (KLSH), a general and popular method developed in
the vision community for performing approximate nearest-neighbor searches in an
arbitrary reproducing kernel Hilbert space (RKHS). Our new perspective is based
on viewing the steps of the KLSH algorithm in an appropriately projected space,
and has several key theoretical and practical benefits. First, it eliminates
the problematic conceptual difficulties that are present in the existing
motivation of KLSH. Second, it yields the first formal retrieval performance
bounds for KLSH. Third, our analysis reveals two techniques for boosting the
empirical performance of KLSH. We evaluate these extensions on several
large-scale benchmark image retrieval data sets, and show that our analysis
leads to improved recall performance of at least 12%, and sometimes much
higher, over the standard KLSH method.Comment: 15 page