{"ID":23475265,"CreatedAt":"2026-09-18T01:09:05.407443952Z","UpdatedAt":"2026-09-20T18:11:56.143995915Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20333","arxiv_id":"2609.20333","title":"Sharp Reconstruction Bounds for Autoencoders Using the Same Forward Map","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 $d\\geq 3$, 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 $6\\times10^{-6}$.","short_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 $d\\geq 3$, among orientation-preserving diffeomorphisms whose Jacobian singular values lie in $[m,M]$, we show that the least uniform reconstructio...","url_abs":"https://arxiv.org/abs/2609.20333","url_pdf":"https://arxiv.org/pdf/2609.20333v1","authors":"[\"Patricia Medina\",\"Hy P. G. Lam\"]","published":"2026-09-17T13:04:38Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.DS\"]","methods":"[]","has_code":false}