Contradiction Graphs Determine VC Dimension
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916030277746688 |
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| author | Campbell, Jesse Ibaibarriaga, Daniel Reyzin, Lev |
| author_facet | Campbell, Jesse Ibaibarriaga, Daniel Reyzin, Lev |
| contents | We study the contradiction graphs associated with binary concept classes. For a class $H \subseteq \{0,1\}^X$, the order-$m$ contradiction graph $G_m(H)$ has as vertices the $H$-realizable labeled sequences of length $m$, with two vertices adjacent when the two sequences assign opposite labels to some common domain point. Our main result is that the single graph $G_m(H)$ determines the threshold predicate $\mathrm{VCdim}(H)\ge m$. Consequently, the full sequence $(G_m(H))_{m \ge 1}$ determines the exact VC dimension and, in particular, detects finite versus infinite VC dimension, answering a question posed by Alon et al. (2024). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_20434 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Contradiction Graphs Determine VC Dimension Campbell, Jesse Ibaibarriaga, Daniel Reyzin, Lev Machine Learning Discrete Mathematics We study the contradiction graphs associated with binary concept classes. For a class $H \subseteq \{0,1\}^X$, the order-$m$ contradiction graph $G_m(H)$ has as vertices the $H$-realizable labeled sequences of length $m$, with two vertices adjacent when the two sequences assign opposite labels to some common domain point. Our main result is that the single graph $G_m(H)$ determines the threshold predicate $\mathrm{VCdim}(H)\ge m$. Consequently, the full sequence $(G_m(H))_{m \ge 1}$ determines the exact VC dimension and, in particular, detects finite versus infinite VC dimension, answering a question posed by Alon et al. (2024). |
| title | Contradiction Graphs Determine VC Dimension |
| topic | Machine Learning Discrete Mathematics |
| url | https://arxiv.org/abs/2605.20434 |