Importance Sampling in High Dimensions via Hashing

Moses Charikar
Stanford University

Given a set of points in d dimensions, imagine putting a Gaussian distribution around each of them. How quickly can we evaluate the sum of these Gaussian densities at a new point? This computational problem (and its generalization for densities other than the Gaussian) is called kernel density estimation. This problem arises as a basic primitive in statistics (non-parametric density estimation) and machine learning (kernel methods).

The batch version of this question (compute the sum of n kernels at m given query points) is addressed by the celebrated fast multiple method from the late 80s which has linear or near-linear complexity for points in 2 and 3 dimensions. The high dimensional case has been challenging because at a high level, typical space partitioning approaches have an exponential dependence on the dimension.

In this talk, I will show that locality sensitive hashing (introduced in the late 90s for the approximate nearest neighbor problem) can be adapted to perform importance sampling, yielding unbiased estimators for kernel density in high dimensions. I will discuss extensions of this framework where the Gaussian density is replaced by a general function of inner products.

Joint work with Paris Syminelakis.

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