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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.01510 |
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| _version_ | 1866910120239169536 |
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| author | Frick, Florian Hosseini, Kaave Vasileuski, Aliaksei |
| author_facet | Frick, Florian Hosseini, Kaave Vasileuski, Aliaksei |
| contents | We develop a topological framework for proving lower bounds on sign-rank via $\mathbb{Z}_2$-equivariant topology, and use it to resolve the sign-rank of the Gap Hamming Distance problem up to lower-order terms.
For every (partial) sign matrix $A$, we associate a free $\mathbb{Z}_2$-simplicial complex $S(A)$ and show that sign-rank of $A$ is characterized by the linear analog of $\mathbb{Z}_2$-index of $S(A)$. As a consequence, the classical $\mathbb{Z}_2$-index of $S(A)$ lower bounds the sign-rank of $A$, which reduces sign-rank lower bounds to topological obstructions. This reduction allows us to use various tools from $\mathbb{Z}_2$-equivariant topology, particularly in regimes where classical lower-bound techniques break down.
As the main application, we consider the Gap Hamming Distance function $\mathrm{GHD}_k^n$ (defined for $k < n/2$), which distinguishes pairs of strings in $\{0,1\}^n$ with Hamming distance at most $k$ from pairs with distance at least $n-k$. We prove an essentially tight lower bound and show that for any $k$, \[ \text{sign-rank}(\mathrm{GHD}_k^n) = (1-o_k(1)) 2k. \]
where the $o_k(1)$ term is $O\left(\sqrt{\frac{\log k}{k}}\right)$. This improves on the previous lower bound of Hatami, Hosseini, and Meng (STOC 2023) who proved that sign-rank of $\mathrm{GHD}_k^n$ is at least $Ω(k/\log(n/k))$.
A key technical ingredient is a new analysis of the $\mathbb{Z}_2$-coindex (which lower bounds $\mathbb{Z}_2$-index) of the Vietoris-Rips complex of the hypercube in the sparse regime which yields an essentially tight lower bound. Previously, no results were known in the sparse regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01510 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A $\mathbb{Z}_2$-Topological Framework for Sign-rank Lower Bounds Frick, Florian Hosseini, Kaave Vasileuski, Aliaksei Combinatorics We develop a topological framework for proving lower bounds on sign-rank via $\mathbb{Z}_2$-equivariant topology, and use it to resolve the sign-rank of the Gap Hamming Distance problem up to lower-order terms. For every (partial) sign matrix $A$, we associate a free $\mathbb{Z}_2$-simplicial complex $S(A)$ and show that sign-rank of $A$ is characterized by the linear analog of $\mathbb{Z}_2$-index of $S(A)$. As a consequence, the classical $\mathbb{Z}_2$-index of $S(A)$ lower bounds the sign-rank of $A$, which reduces sign-rank lower bounds to topological obstructions. This reduction allows us to use various tools from $\mathbb{Z}_2$-equivariant topology, particularly in regimes where classical lower-bound techniques break down. As the main application, we consider the Gap Hamming Distance function $\mathrm{GHD}_k^n$ (defined for $k < n/2$), which distinguishes pairs of strings in $\{0,1\}^n$ with Hamming distance at most $k$ from pairs with distance at least $n-k$. We prove an essentially tight lower bound and show that for any $k$, \[ \text{sign-rank}(\mathrm{GHD}_k^n) = (1-o_k(1)) 2k. \] where the $o_k(1)$ term is $O\left(\sqrt{\frac{\log k}{k}}\right)$. This improves on the previous lower bound of Hatami, Hosseini, and Meng (STOC 2023) who proved that sign-rank of $\mathrm{GHD}_k^n$ is at least $Ω(k/\log(n/k))$. A key technical ingredient is a new analysis of the $\mathbb{Z}_2$-coindex (which lower bounds $\mathbb{Z}_2$-index) of the Vietoris-Rips complex of the hypercube in the sparse regime which yields an essentially tight lower bound. Previously, no results were known in the sparse regime. |
| title | A $\mathbb{Z}_2$-Topological Framework for Sign-rank Lower Bounds |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.01510 |