Geometry of fibers of the multiplication map of deep linear neural networks
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913607228325888 |
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| author | Lehalleur, Simon Pepin Rimányi, Richárd |
| author_facet | Lehalleur, Simon Pepin Rimányi, Richárd |
| contents | We study the geometry of the algebraic set of tuples of composable matrices which multiply to a fixed matrix, using tools from the theory of quiver representations. In particular, we determine its codimension $C$ and the number $θ$ of its top-dimensional irreducible components. Our solution is presented in three forms: a Poincaré series in equivariant cohomology, a quadratic integer program, and an explicit formula. In the course of the proof, we establish a surprising property: $C$ and $θ$ are invariant under arbitrary permutations of the dimension vector. We also show that the real log-canonical threshold of the function taking a tuple to the square Frobenius norm of its product is $C/2$. These results are motivated by the study of deep linear neural networks in machine learning and Bayesian statistics (singular learning theory) and show that deep linear networks are in a certain sense ``mildly singular". |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19920 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometry of fibers of the multiplication map of deep linear neural networks Lehalleur, Simon Pepin Rimányi, Richárd Algebraic Geometry Representation Theory Machine Learning 16G20 05E14 62F15 62R01 We study the geometry of the algebraic set of tuples of composable matrices which multiply to a fixed matrix, using tools from the theory of quiver representations. In particular, we determine its codimension $C$ and the number $θ$ of its top-dimensional irreducible components. Our solution is presented in three forms: a Poincaré series in equivariant cohomology, a quadratic integer program, and an explicit formula. In the course of the proof, we establish a surprising property: $C$ and $θ$ are invariant under arbitrary permutations of the dimension vector. We also show that the real log-canonical threshold of the function taking a tuple to the square Frobenius norm of its product is $C/2$. These results are motivated by the study of deep linear neural networks in machine learning and Bayesian statistics (singular learning theory) and show that deep linear networks are in a certain sense ``mildly singular". |
| title | Geometry of fibers of the multiplication map of deep linear neural networks |
| topic | Algebraic Geometry Representation Theory Machine Learning 16G20 05E14 62F15 62R01 |
| url | https://arxiv.org/abs/2411.19920 |