A quadratically convergent proximal algorithm for nonnegative tensor decomposition
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866914755534389248 |
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| author | Vervliet, Nico Themelis, Andreas Patrinos, Panagiotis De Lathauwer, Lieven |
| author_facet | Vervliet, Nico Themelis, Andreas Patrinos, Panagiotis De Lathauwer, Lieven |
| contents | The decomposition of tensors into simple rank-1 terms is key in a variety of applications in signal processing, data analysis and machine learning. While this canonical polyadic decomposition (CPD) is unique under mild conditions, including prior knowledge such as nonnegativity can facilitate interpretation of the components. Inspired by the effectiveness and efficiency of Gauss-Newton (GN) for unconstrained CPD, we derive a proximal, semismooth GN type algorithm for nonnegative tensor factorization. If the algorithm converges to the global optimum, we show that $Q$-quadratic convergence can be obtained in the exact case. Global convergence is achieved via backtracking on the forward-backward envelope function. The $Q$-quadratic convergence is verified experimentally, and we illustrate that using the GN step significantly reduces number of (expensive) gradient computations compared to proximal gradient descent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_03502 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A quadratically convergent proximal algorithm for nonnegative tensor decomposition Vervliet, Nico Themelis, Andreas Patrinos, Panagiotis De Lathauwer, Lieven Optimization and Control 15A69, 49J52, 90C26, 90C53 The decomposition of tensors into simple rank-1 terms is key in a variety of applications in signal processing, data analysis and machine learning. While this canonical polyadic decomposition (CPD) is unique under mild conditions, including prior knowledge such as nonnegativity can facilitate interpretation of the components. Inspired by the effectiveness and efficiency of Gauss-Newton (GN) for unconstrained CPD, we derive a proximal, semismooth GN type algorithm for nonnegative tensor factorization. If the algorithm converges to the global optimum, we show that $Q$-quadratic convergence can be obtained in the exact case. Global convergence is achieved via backtracking on the forward-backward envelope function. The $Q$-quadratic convergence is verified experimentally, and we illustrate that using the GN step significantly reduces number of (expensive) gradient computations compared to proximal gradient descent. |
| title | A quadratically convergent proximal algorithm for nonnegative tensor decomposition |
| topic | Optimization and Control 15A69, 49J52, 90C26, 90C53 |
| url | https://arxiv.org/abs/2003.03502 |