Sharp Reconstruction Bounds for Autoencoders Using the Same Forward Map
Theoretical work derives sharp reconstruction-derivative error bounds for autoencoders using the same forward map, validated on a 798,452-point LiDAR forest scan.
For equal odd input and hidden dimensions d≥3, the paper proves the least uniform reconstruction-derivative error among orientation-preserving diffeomorphisms with Jacobian singular values in [m,M] equals max{1−M(M−m)/2,0}, attained by affine maps. A translated radial rotation can nonetheless reconstruct any prescribed ball exactly with singular values near one, motivating additional conditions for finite-data bounds. On a 798,452-point terrestrial LiDAR forest scan, the theoretical bound at input scale 0.05 was 0.155, about 84% of the mean normalized training error of 0.185.
- Sharp uniform bound max{1−M(M−m)/2,0} proven for diffeomorphic autoencoders
- Affine maps attain the bound at every prescribed depth
- Translated radial rotation reconstructs balls exactly, motivating extra finite-data conditions
- Theory matches ~84% of training error on a 798,452-point LiDAR forest scan
Full article138 words · extracted from arxiv.org · click to collapse
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 $6\times10^{-6}$.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.20333