Efficient density estimation via piecewise polynomial approximation
Efficient density estimation via piecewise polynomial approximation
复制标题
通过分段多项式近似进行有效密度估计
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Xiaorui Sun
中科院分区：
文献类型：
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作者：
Siu On Chan;Ilias Diakonikolas;R. Servedio;Xiaorui Sun
我们给出了一种用于学习单变量概率分布的计算有效的半无形算法，该分布通过分段多项式密度函数近似，让P是间隔I的任意分布，并且假设P是τ-粘液（总变异距离）到由i在t间隔中的未知分区定义的未知概率分布q和在每个间隔上指定Q的未知度d多项式。我们给出了一种从P中绘制É（t（d + 1）/ε2）样品的算法（14τ +ε）在总变化距离中插入P。参数t，d和ε;我们表明，即使对于τ= 0 。对数的混合物的混合物；在大多数情况下，对于所有这些问题，通过单个统一算法在所有参数中，复杂性（达到对数因素）。
We give a computationally efficient semi-agnostic algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let p be an arbitrary distribution over an interval I, and suppose that p is τ-close (in total variation distance) to an unknown probability distribution q that is defined by an unknown partition of I into t intervals and t unknown degree d polynomials specifying q over each of the intervals. We give an algorithm that draws Õ(t(d + 1)/ε2) samples from p, runs in time poly(t, d + 1, 1/ε), and with high probability outputs a piecewise polynomial hypothesis distribution h that is (14τ + ε)-close to p in total variation distance. Our algorithm combines tools from real approximation theory, uniform convergence, linear programming, and dynamic programming. Its sample complexity is simultaneously near optimal in all three parameters t, d and ε; we show that even for τ = 0, any algorithm that learns an unknown t-piecewise degree-d probability distribution over I to accuracy ε must use [EQUATION] samples from the distribution, regardless of its running time. We apply this general algorithm to obtain a wide range of results for many natural density estimation problems over both continuous and discrete domains. These include state-of-the-art results for learning mixtures of log-concave distributions; mixtures of t-modal distributions; mixtures of Monotone Hazard Rate distributions; mixtures of Poisson Binomial Distributions; mixtures of Gaussians; and mixtures of k-monotone densities. Our general technique gives improved results, with provably optimal sample complexities (up to logarithmic factors) in all parameters in most cases, for all these problems via a single unified algorithm.
影响因子：
4.5
作者：
Balabdaoui F;Rufibach K;Wellner JA
通讯作者：
Wellner JA