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Sharp Reconstruction Bounds for Autoencoders Using the Same Forward Map

  • Patricia Medina
  • Hy P. G. Lam
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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 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.

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Cite

BibTeXRIS
BibTeX
@misc{medina2026sharp,
  title = {{Sharp Reconstruction Bounds for Autoencoders Using the Same Forward Map}},
  author = {Patricia Medina and Hy P. G. Lam},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20333},
  eprint = {2609.20333},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20333},
}
RIS
TY  - UNPB
TI  - Sharp Reconstruction Bounds for Autoencoders Using the Same Forward Map
AU  - Patricia Medina
AU  - Hy P. G. Lam
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20333
UR  - https://arxiv.org/abs/2609.20333
AB  - 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.
ER  -