Approximating L1-distances Between Mixture Distributions Using Random Projections

We consider the problem of computing L1-distances between every pair of probability densities from a given family, a problem motivated by density estimation [15]. We point out that the technique of Cauchy random projections [10] in this context turns into stochastic integrals with respect to Cauchy motion. For piecewise-linear densities these integrals can be sampled from if one can sample from the stochastic integral of the function x → (1, x). We give an explicit density function for this stochastic integral and present an efficient (exact) sampling algorithm. As a consequence we obtain an efficient algorithm to approximate the L1-distances with a small relative error. For piecewise-polynomial densities we show how to approximately sample from the distributions resulting from the stochastic integrals. This also results in an efficient algorithm to approximate the L1-distances, although our inability to get exact samples worsens the dependence on the parameters.