Kaczmarz Kac Walk
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913575599079424 |
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| author | Steinerberger, Stefan |
| author_facet | Steinerberger, Stefan |
| contents | The Kaczmarz method is a way to iteratively solve a linear system of equations $Ax = b$. One interprets the solution $x$ as the point where hyperplanes intersect and then iteratively projects an approximate solution onto these hyperplanes to get better and better approximations. We note a somewhat related idea: one could take two random hyperplanes and project one into the orthogonal complement of the other. This leads to a sequence of linear systems $A^{(k)} x = b^{(k)}$ which is fast to compute, preserves the original solution and whose small singular values grow like $σ_{\ell}(A^{(k)}) \sim \exp(k/n^2) \cdot σ_{\ell}(A)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06614 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kaczmarz Kac Walk Steinerberger, Stefan Numerical Analysis Probability The Kaczmarz method is a way to iteratively solve a linear system of equations $Ax = b$. One interprets the solution $x$ as the point where hyperplanes intersect and then iteratively projects an approximate solution onto these hyperplanes to get better and better approximations. We note a somewhat related idea: one could take two random hyperplanes and project one into the orthogonal complement of the other. This leads to a sequence of linear systems $A^{(k)} x = b^{(k)}$ which is fast to compute, preserves the original solution and whose small singular values grow like $σ_{\ell}(A^{(k)}) \sim \exp(k/n^2) \cdot σ_{\ell}(A)$. |
| title | Kaczmarz Kac Walk |
| topic | Numerical Analysis Probability |
| url | https://arxiv.org/abs/2411.06614 |