Sharp Reconstruction Bounds for Autoencoders Using the Same Forward Map

arXiv:2609.20333v1 Announce Type: new Abstract: 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 dgeq 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 6imes10^{-6}.

Source: arXiv cs.LG | 2026-09-18