自适应极小极大多元L¹反卷积：噪声滤波下的调和平均速率与1-瓦瑟斯坦等价性
Adaptive minimax multivariate \(L^1\)-deconvolution: harmonic-mean rates under noise filtering, and a \(1\)-Wasserstein equivalence
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中文总结 AI 辅助
该研究针对含各向异性噪声的多元卷积模型，扩展了L¹距离估计器，建立了匹配的极小极大速率，提出了自适应带宽选择方法，还揭示了L¹距离与1-瓦瑟斯坦距离的速率关联。
中文摘要 AI 辅助
我们研究含加性独立噪声的多元卷积模型，涉及多维信号与噪声，旨在从受污染观测中恢复信号的累积分布函数。该噪声为各向异性正则性的普通光滑噪声，信号属于各向异性Nikol'skii密度类。我们将基于积分核密度估计的近似最小L¹距离估计器扩展至多元场景，并建立各向异性Nikol'skii类上L¹风险的匹配上下界。据我们所知，这是首个极小极大速率，而非对现有界的改进。尽管下界方案是经典的，但检验函数的构造是新颖的：它区分无有效分量与至少一个有效分量。该速率由删失机制通过正部决定，该正部是指数调和平均表示中的噪声滤波器。我们进一步提出完全数据驱动、速率自适应的过程，在完整Nikol'skii尺度上选择最优带宽向量。在Nikol'skii乘积密度类上，累积分布函数间的L¹距离与1-瓦瑟斯坦距离共享相同的极小极大速率，尽管通过乘积测度上1-瓦瑟斯坦代价的逐坐标解耦，二者一般不等价。这些发现揭示了两种距离间的联系，其完整理解仍是开放问题。
英文摘要
We study the multivariate convolution model with additive independent noise, multidimensional signal and noise, aiming to recover the signal's cumulative distribution function from contaminated observations. The noise is ordinary smooth with anisotropic regularity, and the signal belongs to an anisotropic Nikol'skii density class. We extend to the multivariate setting an approximate minimum \(L^1\)-distance estimator based on integrated kernel density estimation, and establish matching upper and lower bounds for the \(L^1\)-risk over anisotropic Nikol'skii classes. To our knowledge, this is the first minimax rate, not a sharpening of existing bounds. Although the lower-bound scheme is classical, the test-function construction is novel: it distinguishes no active component from at least one active component. The rate is governed by a censoring mechanism through the positive part, a noise filter in a harmonic-mean representation of the exponent. We further propose a fully data-driven, rate-adaptive procedure selecting an optimal bandwidth vector over the full Nikol'skii scale. On Nikol'skii product density classes, the \(L^1\)-distance between cumulative distribution functions and the \(1\)-Wasserstein distance share the same minimax rate, though not equivalent in general, via coordinate-wise decoupling of the \(1\)-Wasserstein cost on product measures. These findings reveal a link between the two distances, whose full understanding remains an open question.
