研究自动编码器重构界限的新理论
Sharp Reconstruction Bounds for Autoencoders Using the Same Forward Map
这是关于自动编码器重构理论的新研究，给出了更精确的误差上界，对理解这类模型有理论价值。
研究在输入坐标置零后应用相同前向映射的自动编码器，证明了当输入和隐藏维度为奇数且大于等于3时，其最小均匀重构导数误差的上界为 max{1-M(M-m)/2,0}，其中M和m是雅可比矩阵奇异值范围的上限和下限。在输入尺度为0.05时，对798,452个点的陆地激光雷达森林扫描测试，理论平均上界为0.155，约84%于平均归一化训练误差0.185，添加一个隐藏坐标可将平均重构误差降低至6×10⁻⁶以下。
Sharp Reconstruction Bounds for Autoencoders Using the Same Forward Map
We study reconstruction in autoencoders that apply the same forward map before and after setting the observed coordinates to zero. For equal odd input and hidden dimensions $d\geq 3$, among orientation-preserving diffeomorphisms whose Jacobian singular values lie in $[m,M]$, we show that the least uniform reconstruction-derivative error is $\max\{1-M(M-m)/2,0\}$, with affine maps attaining this sharp bound at every prescribed depth. A translated radial rotation can nevertheless reconstruct any prescribed ball exactly with singular values arbitrarily close to one, motivating additional conditions for a finite-data bound. We test this prediction on a 798,452-point terrestrial LiDAR forest scan. At input scale $0.05$, the mean theoretical bound is $0.155$, about $84\%$ of the mean normalized training error $0.185$ across four spatial regions, two depths, and three seeds. At this scale, adding one hidden coordinate reduces the mean reconstruction error below $6\times10^{-6}$.
