# Solving ill-conditioned polynomial equations using score-based priors with application to multi-target detection

arXiv: https://arxiv.org/abs/2509.11397 · HTML: https://mlsanity.com/paper/2509.11397/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 11 of its 25 references that are in the map.
- First posted on arXiv 14 Sep 2025.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 11 of 11)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contraction](https://mlsanity.com/paper/2112.05146/) (2021)
- [Solving Linear Inverse Problems Provably via Posterior Sampling with Latent Diffusion Models](https://mlsanity.com/paper/2307.00619/) (2023)
- [NTIRE 2024 Challenge on Image Super-Resolution (x4): Methods and Results](https://mlsanity.com/paper/2404.09790/) (2024)
- [A complete data processing workflow for CryoET and subtomogram averaging](https://mlsanity.com/paper/1902.03978/) (2019)
- [Two-dimensional multi-target detection: an autocorrelation analysis approach](https://mlsanity.com/paper/2105.06765/) (2021)
- [Expectation-maximization for structure determination directly from cryo-EM micrographs](https://mlsanity.com/paper/2303.02157/) (2023)
- [Score-based diffusion priors for multi-target detection](https://mlsanity.com/paper/2312.08500/) (2023)
- [A note on the sample complexity of multi-target detection](https://mlsanity.com/paper/2501.11980/) (2025)

## Abstract (arXiv)

Recovering signals from low-order moments is a fundamental yet notoriously difficult task in inverse problems. This recovery process often reduces to solving ill-conditioned systems of polynomial equations. In this work, we propose a new framework that integrates score-based diffusion priors with moment-based estimators to regularize and solve these nonlinear inverse problems. This introduces a new role for generative models: stabilizing polynomial recovery from noisy statistical features. As a concrete application, we study the multi-target detection (MTD) model in the high-noise regime. We demonstrate two main results: (i) diffusion priors substantially improve recovery from third-order moments, and (ii) they make the super-resolution MTD problem, otherwise ill-posed, feasible. Numerical experiments on MNIST data confirm consistent gains in reconstruction accuracy across SNR levels. Our results suggest a promising new direction for combining generative priors with nonlinear polynomial inverse problems.