用随机投影逼近混合分布之间的L1-distances
Approximating L1-distances between mixture distributions using random projections
摘要：
本文考虑了一个给定族中每对c概率密度之间的L1-距离的计算问题。我们指出，柯西随机投影技术（Indyk'06）在这种情况下变成随机积分柯西运动。对于分段线性密度，如果可以从函数x->（1，x）的随机积分中采样，则可以从其中采样这些积分。我们给出了一个显式的密度函数的随机积分，并提出了一个有效的采样算法。因此，我们得到了一个有效的算法来近似的L1-距离具有较小的相对误差。对于分段多项式密度，我们展示了如何从随机积分产生的分布中近似采样。这也导致了一个有效的算法来近似L1距离，虽然我们无法获得精确的样本，这取决于参数。
英文摘要：
We consider the problem of computing L1-distances between every pair ofcprobability densities from a given family. We point out that the technique of Cauchy random projections (Indyk'06) 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 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.
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