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

2025-09-14

AI Summary

This work addresses the ill-posed nonlinear inverse problem of signal recovery from low-order moments—particularly in super-resolution multi-object detection. We propose a regularized solution framework that jointly integrates fractional diffusion priors with moment estimation. For the first time, score-based diffusion models are introduced as generative priors for solving polynomial systems, where denoising score matching provides stable guidance for ill-conditioned inversion. The method enforces low-order moment constraints—especially third-order moments—and jointly optimizes moment basis estimation and diffusion-based denoising, significantly enhancing reconstruction robustness and accuracy under high noise. Experiments on MNIST demonstrate consistent superiority over classical moment-based methods across all signal-to-noise ratios, substantially broadening the applicability of generative priors to nonlinear inverse problems.

Technology Category

• Search and Optimization: Non-convex Optimization
• Computer Vision: Diffusion Models for Vision
• Machine Learning: Multi-instance/Multi-view Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

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.

Problem

• Solving ill-conditioned polynomial equations from noisy moments
• Stabilizing polynomial recovery using score-based diffusion priors
• Enabling multi-target detection in high-noise regimes

Innovation

• Score-based diffusion priors for polynomial equations
• Stabilizing recovery from noisy statistical features
• Enabling super-resolution multi-target detection feasibility