Sep 17, 2026 · cs.LGJ/K move · Enter open · S save
Patricia Medina, Hy P. G. Lam
New York City College of Technology, CUNY, Brooklyn, NY, USA; The Graduate Center, CUNY, New York, NY, USA. · Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA, USA.
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≥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
6×10−6.