Quantized minimax estimation over Sobolev ellipsoids
Quantized minimax estimation over Sobolev ellipsoids
复制标题
索博列夫椭球上的量化极小极大估计
DOI：
10.1093/imaiai/iax007
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
J. Lafferty
中科院分区：
文献类型：
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作者：
Yuancheng Zhu;J. Lafferty
我们制定的存储或通信约束下的极大极小估计的概念，并证明了一个扩展的非参数估计Sobolev椭球的平斯克定理。放置用于编码任何估计的比特数的限制，我们给出了严格的下限和上限的超额风险由于量化的比特数，信号大小和噪声水平。这建立了Sobolev空间的量化约束下的存储和风险之间的帕累托最优权衡。我们的结果和证明技术结合联合收割机元素的率失真理论和极大极小分析。建议的量化估计方案，这表明可扩展性的下限，是自适应的，在通常的统计意义上，实现最佳的量化极大极小速率的Sobolev空间的平滑参数的知识。它在计算意义上也是自适应的，因为它仅在观察数据之后构造代码，以动态地将更多码字分配给估计信号大小较大的块。包括模拟，说明量化的统计风险的影响。非参数估计，极小极大界，率失真理论，约束估计，Sobolev椭球
We formulate the notion of minimax estimation under storage or communication constraints, and prove an extension to Pinsker’s theorem for nonparametric estimation over Sobolev ellipsoids. Placing limits on the number of bits used to encode any estimator, we give tight lower and upper bounds on the excess risk due to quantization in terms of the number of bits, the signal size, and the noise level. This establishes the Pareto optimal tradeoff between storage and risk under quantization constraints for Sobolev spaces. Our results and proof techniques combine elements of rate distortion theory and minimax analysis. The proposed quantized estimation scheme, which shows achievability of the lower bounds, is adaptive in the usual statistical sense, achieving the optimal quantized minimax rate without knowledge of the smoothness parameter of the Sobolev space. It is also adaptive in a computational sense, as it constructs the code only after observing the data, to dynamically allocate more codewords to blocks where the estimated signal size is large. Simulations are included that illustrate the effect of quantization on statistical risk. nonparametric estimation, minimax bounds, rate distortion theory, constrained estimation, Sobolev ellipsoid