Positive Competitive Networks for Sparse Reconstruction
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910378322034688 |
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| author | Centorrino, Veronica Gokhale, Anand Davydov, Alexander Russo, Giovanni Bullo, Francesco |
| author_facet | Centorrino, Veronica Gokhale, Anand Davydov, Alexander Russo, Giovanni Bullo, Francesco |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_03821 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Positive Competitive Networks for Sparse Reconstruction Centorrino, Veronica Gokhale, Anand Davydov, Alexander Russo, Giovanni Bullo, Francesco Neurons and Cognition Systems and Control Optimization and Control 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. |
| title | Positive Competitive Networks for Sparse Reconstruction |
| topic | Neurons and Cognition Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2311.03821 |