Computing Bouligand stationary points efficiently in low-rank optimization
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arXiv
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| Format: | Preprint |
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2024
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| author | Olikier, Guillaume Absil, P. -A. |
| author_facet | Olikier, Guillaume Absil, P. -A. |
| contents | This paper considers the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on the algebraic variety of all $m$-by-$n$ real matrices of rank at most $r$. Several definitions of stationarity exist for this nonconvex problem. Among them, Bouligand stationarity is the strongest necessary condition for local optimality. Only a handful of algorithms generate a sequence in the variety whose accumulation points are provably Bouligand stationary. Among them, the most parsimonious with (truncated) singular value decompositions (SVDs) or eigenvalue decompositions can still require a truncated SVD of a matrix whose rank can be as large as $\min\{m, n\}-r+1$ if the gradient does not have low rank, which is computationally prohibitive in the typical case where $r \ll \min\{m, n\}$. This paper proposes a first-order algorithm that generates a sequence in the variety whose accumulation points are Bouligand stationary while requiring SVDs of matrices whose smaller dimension is always at most $r$. A standard measure of Bouligand stationarity converges to zero along the bounded subsequences at a rate at least $O(1/\sqrt{i+1})$, where $i$ is the iteration counter. Furthermore, a rank-increasing scheme based on the proposed algorithm is presented, which can be of interest if the parameter $r$ is potentially overestimated. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_12298 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing Bouligand stationary points efficiently in low-rank optimization Olikier, Guillaume Absil, P. -A. Optimization and Control Numerical Analysis 14M12, 65K10, 90C26, 90C30, 40A05 This paper considers the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on the algebraic variety of all $m$-by-$n$ real matrices of rank at most $r$. Several definitions of stationarity exist for this nonconvex problem. Among them, Bouligand stationarity is the strongest necessary condition for local optimality. Only a handful of algorithms generate a sequence in the variety whose accumulation points are provably Bouligand stationary. Among them, the most parsimonious with (truncated) singular value decompositions (SVDs) or eigenvalue decompositions can still require a truncated SVD of a matrix whose rank can be as large as $\min\{m, n\}-r+1$ if the gradient does not have low rank, which is computationally prohibitive in the typical case where $r \ll \min\{m, n\}$. This paper proposes a first-order algorithm that generates a sequence in the variety whose accumulation points are Bouligand stationary while requiring SVDs of matrices whose smaller dimension is always at most $r$. A standard measure of Bouligand stationarity converges to zero along the bounded subsequences at a rate at least $O(1/\sqrt{i+1})$, where $i$ is the iteration counter. Furthermore, a rank-increasing scheme based on the proposed algorithm is presented, which can be of interest if the parameter $r$ is potentially overestimated. |
| title | Computing Bouligand stationary points efficiently in low-rank optimization |
| topic | Optimization and Control Numerical Analysis 14M12, 65K10, 90C26, 90C30, 40A05 |
| url | https://arxiv.org/abs/2409.12298 |