内在谱块变化下紧致流形上的密度估计
Density Estimation on Compact Manifolds under Intrinsic Spectral Block Variation
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中文总结 AI 辅助
该研究针对紧致连通黎曼流形上的非参数密度估计，提出内在谱块变分模型，构造坐标无关的块收缩估计器与块惩罚指数谱筛，建立相关风险界，得到尊重几何且与特征基无关的稀疏密度估计理论。
中文摘要 AI 辅助
我们针对紧致连通黎曼流形上的非参数密度估计，引入了一种内在谱稀疏模型。我们没有对任意选取的拉普拉斯-贝尔特拉米（Laplace–Beltrami）特征基中的系数进行惩罚，而是将每个完整特征空间分组，并测量其谱分量的希尔伯特范数。由此得到的块变分空间与基无关，且具有等距不变性。我们建立了该空间的结构性质、原子性质和非线性逼近性质，并阐明了它与索伯列夫（Sobolev）空间、贝索夫（Besov）空间及逐系数谱ℓ¹类的关系。随后，我们构造了一种与坐标无关的块收缩估计器，并证明了一个非渐近的、依赖信号的L²- oracle不等式，该不等式可适配未知的可检测特征空间集合。在多项式谱增长条件下，风险理论将谱块的数量与其重数分离开来，并呈现两种状态：一种由单个高维特征空间驱动，另一种由累积谱复杂度驱动。在匹配的谱增长和非退化假设下，对应的极小极大下界表明，这种重数依赖性是内在的，对球面和旋转群SO(3)有明确的影响。最后，我们针对对数密度开发了一种正的、归一化的块惩罚指数谱筛，并推导了似然oracle不等式以及期望的库尔贝克-莱布勒（Kullback–Leibler）散度、赫林格（Hellinger）距离和L²风险界。该框架提供了一种尊重几何的稀疏密度估计理论，且在特征基变化下保持不变。
英文摘要
We introduce an intrinsic spectral sparsity model for nonparametric density estimation on compact connected Riemannian manifolds. Instead of penalizing coefficients in an arbitrarily chosen Laplace--Beltrami eigenbasis, we group each complete eigenspace and measure the Hilbert norm of its spectral component. The resulting block-variation space is basis independent and isometry invariant. We establish its structural, atomic, and nonlinear approximation properties and clarify its relation to Sobolev, Besov, and coefficientwise spectral $\ell^1$ classes. We then construct a coordinate-free block-shrinkage estimator and prove a nonasymptotic signal-dependent $L^2$-oracle inequality that adapts to the unknown set of detectable eigenspaces. Under polynomial spectral growth, the risk theory separates the number of spectral blocks from their multiplicities and exhibits two regimes: one driven by a single high-dimensional eigenspace and the other by cumulative spectral complexity. Under matching spectral-growth and nondegeneracy assumptions, corresponding minimax lower bounds show that this multiplicity dependence is intrinsic, with sharp consequences for spheres and the rotation group $SO(3)$. Finally, we develop a positive, normalized, block-penalized exponential spectral sieve for log-densities and derive likelihood oracle inequalities together with expected Kullback--Leibler, Hellinger, and $L^2$ risk bounds. The resulting framework provides a geometry-respecting theory of sparse density estimation that remains invariant under changes of eigenbasis.
