# Gradient Descent Provably Solves Nonlinear Tomographic Reconstruction

arXiv: https://arxiv.org/abs/2310.03956 · HTML: https://mlsanity.com/paper/2310.03956/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 31 references that are in the map.
- First posted on arXiv 6 Oct 2023.

## Builds on (most cited first, 8 of 8)

- [Plenoxels: Radiance Fields without Neural Networks](https://mlsanity.com/paper/2112.05131/) (2021)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [The Landscape of Empirical Risk for Non-convex Losses](https://mlsanity.com/paper/1607.06534/) (2016)
- [Learning ReLUs via Gradient Descent](https://mlsanity.com/paper/1705.04591/) (2017)
- [Structured signal recovery from quadratic measurements: Breaking sample complexity barriers via nonconvex optimization](https://mlsanity.com/paper/1702.06175/) (2017)
- [Sharp Time--Data Tradeoffs for Linear Inverse Problems](https://mlsanity.com/paper/1507.04793/) (2015)
- [Isometric sketching of any set via the Restricted Isometry Property](https://mlsanity.com/paper/1506.03521/) (2015)
- [Fast and Reliable Parameter Estimation from Nonlinear Observations](https://mlsanity.com/paper/1610.07108/) (2016)

## Built on by (most cited first, 3 of 3)

- [A Local Polyak-Lojasiewicz and Descent Lemma of Gradient Descent For Overparametrized Linear Models](https://mlsanity.com/paper/2505.11664/) (2025)
- [Accurate, provable and fast polychromatic tomographic reconstruction: A variational inequality approach](https://mlsanity.com/paper/2503.19925/) (2025)
- [Perfusion Imaging and Single Material Reconstruction in Polychromatic Photon Counting CT](https://mlsanity.com/paper/2602.02713/) (2026)

## Abstract (arXiv)

In computed tomography (CT), the forward model consists of a linear Radon transform followed by an exponential nonlinearity based on the attenuation of light according to the Beer-Lambert Law. Conventional reconstruction often involves inverting this nonlinearity and then solving a linear inverse problem. However, this nonlinear measurement preprocessing is poorly conditioned in the vicinity of high-density materials, such as metal. This preprocessing makes CT reconstruction methods numerically sensitive and susceptible to artifacts near high-density regions. In this paper, we study a technique where the signal is directly reconstructed from raw measurements through the nonlinear forward model. Though this optimization is nonconvex, we show that gradient descent provably converges to the global optimum at a geometric rate, perfectly reconstructing the underlying signal with a near minimal number of random measurements. We also prove similar results in the under-determined setting where the number of measurements is significantly smaller than the dimension of the signal. This is achieved by enforcing prior structural information about the signal through constraints on the optimization variables. We illustrate the benefits of direct nonlinear CT reconstruction with cone-beam CT experiments on synthetic and real 3D volumes, in which metal artifacts are reduced compared to standard linear reconstruction methods. Our experiments also demonstrate that logarithmic preprocessing alone is sufficient to produce metal artifacts, even in the absence of other causes such as beam hardening.