用随机投影+稀疏化实现最优密度与模式估计
Optimal rates for density and mode estimation with expand-and-sparsify representations
- 通过随机线性映射再保留最大值实现稀疏表示
- 密度估计达到极小极大最优的∞范数收敛率
- 模式估计在温和条件下达最优率，适合生物感知建模
expand-and-sparsify 表示是一类理论模型，模拟动物感官系统中的稀疏表示现象。其将输入 $x \in \mathbb{R}^d$ 通过随机线性投影映射到更高维 $m \gg d$，再保留其中 $k \ll m$ 个最大值，得到 $\{0,1\}^m$ 中的 $k$-稀疏向量。本文研究该表示在两个基础统计问题上的适用性：密度估计与模式估计。对于密度估计，我们证明对 expand-and-sparsify 表示进行简单线性变换即可获得 $\ell_{\infty}$ 范数下极小极大最优的收敛率。对于模式估计，在密度估计基础上设计的简单算法可在温和条件下以最优率（含对数因子）恢复单峰或多峰。
原文摘要 · Abstract (English)
Expand-and-sparsify representations are a class of theoretical models that capture sparse representation phenomena observed in the sensory systems of many animals. At a high level, these representations map an input $x \in \mathbb{R}^d$ to a much higher dimension $m \gg d$ via random linear projections before zeroing out all but the $k \ll m$ largest entries. The result is a $k$-sparse vector in $\{0,1\}^m$. We study the suitability of this representation for two fundamental statistical problems: density estimation and mode estimation. For density estimation, we show that a simple linear function of the expand-and-sparsify representation produces an estimator with minimax-optimal $\ell_{\infty}$ convergence rates. In mode estimation, we provide simple algorithms on top of our density estimator that recover single or multiple modes at optimal rates up to logarithmic factors under mild conditions.
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