# An efficient Quasi-Newton method for nonlinear inverse problems via learned singular values

arXiv: https://arxiv.org/abs/2012.07676 · HTML: https://mlsanity.com/paper/2012.07676/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2020.

## Built on by (most cited first, 4 of 4)

- [Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems](https://mlsanity.com/paper/2103.15138/) (2021)
- [Electrical Impedance Tomography: A Fair Comparative Study on Deep Learning and Analytic-based Approaches](https://mlsanity.com/paper/2310.18636/) (2023)
- [Deep Inertia Lp Half-Quadratic Splitting Unrolling Network for Sparse View CT Reconstruction](https://mlsanity.com/paper/2408.06600/) (2024)
- [Graph convolutional networks enable fast hemorrhagic stroke monitoring with electrical impedance tomography](https://mlsanity.com/paper/2412.07888/) (2024)

## Abstract (arXiv)

Solving complex optimization problems in engineering and the physical sciences requires repetitive computation of multi-dimensional function derivatives. Commonly, this requires computationally-demanding numerical differentiation such as perturbation techniques, which ultimately limits the use for time-sensitive applications. In particular, in nonlinear inverse problems Gauss-Newton methods are used that require iterative updates to be computed from the Jacobian. Computationally more efficient alternatives are Quasi-Newton methods, where the repeated computation of the Jacobian is replaced by an approximate update. Here we present a highly efficient data-driven Quasi-Newton method applicable to nonlinear inverse problems. We achieve this, by using the singular value decomposition and learning a mapping from model outputs to the singular values to compute the updated Jacobian. This enables a speed-up expected of Quasi-Newton methods without accumulating roundoff errors, enabling time-critical applications and allowing for flexible incorporation of prior knowledge necessary to solve ill-posed problems. We present results for the highly non-linear inverse problem of electrical impedance tomography with experimental data.