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
From the original article
1 min read at arxiv.orgWe study reconstruction in autoencoders that apply the same forward map before and after setting the observed coordinates to zero.
By Patricia Medina · arxiv.org
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