揭示费舍尔度量下学习与恢复的几何特性，统一分析复杂度边界。
Fisher Widths: Local Learning Geometry and Anisotropic Recovery
- 引入费舍尔宽度与逆费舍尔宽度，刻画参数局部波动与测量方向的非各向同性。
- 证明在小费舍尔球内可达到尺度 $w_G(H_r)/\sqrt{n}$，并给出稀疏恢复的支撑敏感估计。
- 建立两者乘积不低于欧氏复杂度平方，揭示几何复杂度的传递机制。
我们通过一对泛函研究统计流形上的高斯宽度复杂度：由费舍尔度量诱导的原始费舍尔宽度 $w_G(T) = w(G^{1/2}T)$，以及由逆费舍尔度量诱导的逆费舍尔宽度 $w_{G^{-1}}(T) = w(G^{-1/2}T)$。前者衡量费舍尔信息几何下局部参数波动规模，对费舍尔正则损失，证明在足够小的费舍尔球上可达尺度 $w_G(H_r)/\sqrt{n}$；后者捕获由逆费舍尔信息决定协方差的各向异性高斯测量效应。在稀疏恢复中，几何不仅依赖稀疏性，还受活跃坐标在费舍尔谱中的位置影响。我们得到统计维度的双侧估计、支撑敏感的恢复精度，并确立支持按曲率分布的自然排序。最终，我们在任意共域紧致坐标集 $T$ 上建立尖锐关系：$w_G(T)w_{G^{-1}}(T) \geq w(T)^2$，表明费舍尔各向异性可在几何间转移复杂度，但无法同时降低两者的复杂度相对于欧氏尺度。
原文摘要 · Abstract (English)
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.
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