# Positive Competitive Networks for Sparse Reconstruction

arXiv: https://arxiv.org/abs/2311.03821 · HTML: https://mlsanity.com/paper/2311.03821/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Nov 2023.

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

- [An Energy-Based Mechanism for Compositional Behavior](https://mlsanity.com/paper/2512.04745/) (2025)
- [Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization](https://mlsanity.com/paper/2604.05042/) (2026)
- [Similarity Matching Networks: Hebbian Learning and Convergence Over Multiple Time Scales](https://mlsanity.com/paper/2506.06134/) (2025)
- [Predictive Coding with Bayesian Priors via Proximal Gradients](https://mlsanity.com/paper/2606.08374/) (2026)

## Abstract (arXiv)

We propose and analyze a continuous-time firing-rate neural network, the positive firing-rate competitive network (pfcn), to tackle sparse reconstruction problems with non-negativity constraints. These problems, which involve approximating a given input stimulus from a dictionary using a set of sparse (active) neurons, play a key role in a wide range of domains, including for example neuroscience, signal processing, and machine learning. First, by leveraging the theory of proximal operators, we relate the equilibria of a family of continuous-time firing-rate neural networks to the optimal solutions of sparse reconstruction problems. Then, we prove that the pfcn is a positive system and give rigorous conditions for the convergence to the equilibrium. Specifically, we show that the convergence: (i) only depends on a property of the dictionary; (ii) is linear-exponential, in the sense that initially the convergence rate is at worst linear and then, after a transient, it becomes exponential. We also prove a number of technical results to assess the contractivity properties of the neural dynamics of interest. Our analysis leverages contraction theory to characterize the behavior of a family of firing-rate competitive networks for sparse reconstruction with and without non-negativity constraints. Finally, we validate the effectiveness of our approach via a numerical example.