# Twincher: Bijective Representation Learning for Robust Inversion of Continuous Systems

arXiv: https://arxiv.org/abs/2605.13470 · HTML: https://mlsanity.com/paper/2605.13470/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 63 of its 114 references that are in the map.
- First posted on arXiv 13 May 2026.

## Builds on (most cited first, 30 of 63)

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- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
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- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [On the Inductive Bias of Neural Tangent Kernels](https://mlsanity.com/paper/1905.12173/) (2019)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Understanding World or Predicting Future? A Comprehensive Survey of World Models](https://mlsanity.com/paper/2411.14499/) (2024)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)

## Abstract (arXiv)

Perception, state estimation, and action planning in physical systems often reduce to inverting a continuous forward map p mapsto y = f(p) (a renderer, a physics simulator, or a learned surrogate) that can be queried but not inverted in closed form. Learned direct inverses are limited by an approximation error that decreases only gradually with the amount of data, whereas iterative solvers that minimize the misfit in observation space can stall at spurious stationary points. We study an intermediate route: learning a representation r(y) in which the composite map p mapsto r(f(p)) is a well-conditioned bijection and which is insensitive to perturbations of y normal to the data manifold. In such coordinates, the inverse problem becomes a square nonlinear system whose least-squares formulation has no spurious interior stationary points, so that Newton-type iterations using the exact forward model can refine the solution to the accuracy permitted by the observations; the learned model must be qualitatively correct rather than accurate. We introduce Twincher, an architecture composed of localized, area-preserving pairwise rotations ("twinches"), and explore training possibilities with Jacobian-level objectives, a domain-growing curriculum, and memory-efficient routines for propagating derivative tensors. On a family of synthetic forward maps with controllable nonconvexity, once the query budget suffices to form a bijective representation, the worst-case residual over 103 test targets reaches machine precision within five refinement steps, whereas a parameter-matched baseline that combines an MLP inverse with Gauss-Newton refinement improves only gradually with the budget. A pose-estimation example from noisy depth maps illustrates inference errors that scale linearly with, and vanish together with, the observation noise. We release an open-source CPU/GPU Python implementation.