# Sharp Reconstruction Bounds for Autoencoders Using the Same Forward Map

arXiv: https://arxiv.org/abs/2609.20333 · HTML: https://mlsanity.com/paper/2609.20333/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 13 references that are in the map.
- First posted on arXiv 17 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [Loss Landscapes of Regularized Linear Autoencoders](https://mlsanity.com/paper/1901.08168/) (2019)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [Statistical and structural identifiability in representation learning](https://mlsanity.com/paper/2603.11970/) (2026)
- [Integrating Product Coefficients for Improved 3D LiDAR Data Classification (Part II)](https://mlsanity.com/paper/2510.15219/) (2025)
- [Bi-Lipschitz Autoencoder With Injectivity Guarantee](https://mlsanity.com/paper/2604.06701/) (2026)

## Abstract (arXiv)

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 max1-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.