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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2506.06338 |
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| _version_ | 1866912435281068032 |
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| author | Fang, Max Chicky |
| author_facet | Fang, Max Chicky |
| contents | The Kruithof iterative scaling process, which adjusts matrices to meet target row and column sums, is a longstanding problem that lacks a general closed form for its limit. While Nathanson derived the closed form for the Sinkhorn limit of $2\times 2$ matrices when target row and column sums are 1, and recent work by Rowland and Wu has advanced understanding of Sinkhorn limits for $3\times 3$, and general $n\times m$ matrices through polynomials, a "generalized Sinkhorn limit" (i.e. the original "Kruithof limit", with arbitrary target sums) remains elusive. Here, we derive the closed form for the generalized Sinkhorn limit of $2\times 2$ matrices, and discuss how this approach can be extended to larger matrices. More significantly, we prove that for any positive $n \times m$ matrix and positive target row and column sums, each entry in the generalized Sinkhorn limit is algebraic over the input data with degree at most $\binom{n+m-2}{n-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06338 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Closed Form of a Generalized Sinkhorn Limit Fang, Max Chicky General Mathematics The Kruithof iterative scaling process, which adjusts matrices to meet target row and column sums, is a longstanding problem that lacks a general closed form for its limit. While Nathanson derived the closed form for the Sinkhorn limit of $2\times 2$ matrices when target row and column sums are 1, and recent work by Rowland and Wu has advanced understanding of Sinkhorn limits for $3\times 3$, and general $n\times m$ matrices through polynomials, a "generalized Sinkhorn limit" (i.e. the original "Kruithof limit", with arbitrary target sums) remains elusive. Here, we derive the closed form for the generalized Sinkhorn limit of $2\times 2$ matrices, and discuss how this approach can be extended to larger matrices. More significantly, we prove that for any positive $n \times m$ matrix and positive target row and column sums, each entry in the generalized Sinkhorn limit is algebraic over the input data with degree at most $\binom{n+m-2}{n-1}$. |
| title | Closed Form of a Generalized Sinkhorn Limit |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2506.06338 |