费希尔宽度：局部学习几何与各向异性恢复
Fisher Widths: Local Learning Geometry and Anisotropic Recovery
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中文总结 AI 辅助
研究统计流形上由费希尔度量和逆费希尔度量诱导的高斯宽度复杂性，通过费希尔宽度衡量局部参数波动，逆费希尔宽度捕捉各向异性高斯测量影响，得到相关估计及两者精确关系，揭示费希尔各向异性对复杂性的作用。
中文摘要 AI 辅助
我们通过一对泛函研究统计流形上的高斯宽度复杂性：由费希尔度量诱导的原始费希尔宽度\(w_G(T)=w(G^{1/2}T)\)和由逆费希尔度量诱导的逆费希尔宽度\(w_{G^{-1}}(T)=w(G^{-1/2}T)\)。这两个宽度发挥互补的统计作用。在学习方面，费希尔宽度衡量费希尔信息诱导几何中局部参数波动的大小。对于费希尔正则损失，我们证明在足够小的费希尔球上达到尺度\(w_G(H_r)/\sqrt{n}\)。在恢复方面，逆费希尔宽度捕捉协方差由逆费希尔信息确定的各向异性高斯测量的影响。对于稀疏恢复，所得几何不仅取决于稀疏性，还取决于费希尔谱中活动坐标的位置。我们得到了相应统计维度的双边估计，以及支持敏感恢复估计和具有不同曲率轮廓的支持的自然排序。最后，我们建立了原始费希尔宽度和逆费希尔宽度之间的精确关系。在任何共同的紧致坐标集\(T\)上，它们满足\(w_G(T)w_{G^{-1}}(T)\geq w(T)^2\)。因此，费希尔各向异性可能将复杂性从一种几何转移到另一种几何，但相对于欧几里得尺度不能同时减小两个宽度。
英文摘要
We study Gaussian-width complexity on statistical manifolds through a pair of functionals: the primal Fisher width $w_G(T) = w(G^{1/2}T)$, induced by the Fisher metric, and the inverse-Fisher width $w_{G^{-1}}(T) = w(G^{-1/2}T)$, induced by the inverse Fisher metric. The two widths play complementary statistical roles. On the learning side, the Fisher width measures the size of local parameter fluctuations in the geometry induced by the Fisher information. For Fisher-regular losses, we prove that the scale \(w_G(H_r)/\sqrt n\) is attained on sufficiently small Fisher balls. On the recovery side, the inverse-Fisher width captures the effect of anisotropic Gaussian measurements whose covariance is determined by the inverse Fisher information. For sparse recovery, the resulting geometry depends not only on sparsity but also on the position of the active coordinates in the Fisher spectrum. We obtain a two-sided estimate for the corresponding statistical dimension, together with support-sensitive recovery estimates and a natural ordering of supports with different curvature profiles. Finally, we establish a sharp relation between the primal and inverse-Fisher widths. On any common compact coordinate set $T$, they satisfy \[ w_G(T)w_{G^{-1}}(T)\geq w(T)^2. \] Thus, Fisher anisotropy may transfer complexity from one geometry to the other, but cannot reduce both widths relative to the Euclidean scale.
发表机构
- FPT University（FPT大学）
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