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ARXIV:2603.24493 · THEORETICAL ML · SUBMITTED 02 APR · 02:30 UTC · FRESHNESS STALE
ARXIV:2603.24493THEORETICAL MLSUBMITTED 02 APR · 02:30 UTCFRESHNESS STALERon Holzman · Shay Moran · Alexander Shlimovich · arXiv
This paper theoretically investigates uniform convergence phenomena in product spaces, extending Vapnik--Chervonenkis theory with a focus on linear VC dimension.
Opportunity summary
Pain This paper theoretically investigates uniform convergence phenomena in product spaces, extending Vapnik--Chervonenkis theory with a focus on linear VC dimension.
Evidence 0 refs | 0 sources | 17% coverage
Blocker Evidence unverified
This paper theoretically investigates uniform convergence phenomena in product spaces, extending Vapnik--Chervonenkis theory with a focus on linear VC dimension. In this work, we study uniform convergence phenomena in cartesian product spaces, under assumptions…
Uniform laws of large numbers form a cornerstone of Vapnik--Chervonenkis theory, where they are characterized by the finiteness of the VC dimension. In this work, we study uniform convergence phenomena in cartesian product spaces,…
ScienceToStartup currently rates this 1.0/10 on the public viability pass. We show that, under this assumption, a uniform law of large numbers holds for a family of events if and only if the linear…
Theoretical ML moved forward this cycle; last verified April 2026. Public score 1.0/10.
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This paper theoretically investigates uniform convergence phenomena in product spaces, extending Vapnik--Chervonenkis theory with a focus on linear VC dimension.
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10.48550/arXiv.2603.24493This paper theoretically investigates uniform convergence phenomena in product spaces, extending Vapnik--Chervonenkis theory with a focus on linear VC dimension.
Abstract
Uniform laws of large numbers form a cornerstone of Vapnik--Chervonenkis theory, where they are characterized by the finiteness of the VC dimension. In this work, we study uniform convergence phenomena in cartesian product spaces, under assumptions on the underlying distribution that are compatible with the product structure. Specifically, we assume that the distribution is absolutely continuous with respect to the product of its marginals, a condition that captures many natural settings, including product distributions, sparse mixtures of product distributions, distributions with low mutual information, and more. We show that, under this assumption, a uniform law of large numbers holds for a family of events if and only if the linear VC dimension of the family is finite. The linear VC dimension is defined as the maximum size of a shattered set that lies on an axis-parallel line, namely, a set of vectors that agree on all but at most one coordinate. This dimension is always at most the classical VC dimension, yet it can be arbitrarily smaller. For instance, the family of convex sets in $\mathbb{R}^d$ has linear VC dimension $2$, while its VC dimension is infinite already for $d\ge 2$. Our proofs rely on estimator that departs substantially from the standard empirical mean estimator and exhibits more intricate structure. We show that such deviations from the standard empirical mean estimator are unavoidable in this setting. Throughout the paper, we propose several open questions, with a particular focus on quantitative sample complexity bounds.
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PROBLEM
This paper theoretically investigates uniform convergence phenomena in product spaces, extending Vapnik--Chervonenkis theory with a focus on linear VC dimension. In this work, we study uniform convergence phenomena in cartesian product spaces, under assumptions on the underlying...
METHOD
Uniform laws of large numbers form a cornerstone of Vapnik--Chervonenkis theory, where they are characterized by the finiteness of the VC dimension. In this work, we study uniform convergence phenomena in cartesian product spaces, under assumptions on the underlying distribution...
RESULT
ScienceToStartup currently rates this 1.0/10 on the public viability pass. We show that, under this assumption, a uniform law of large numbers holds for a family of events if and only if the linear VC dimension of the family is finite.
WHY NOW
Theoretical ML moved forward this cycle; last verified April 2026. Public score 1.0/10.
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This paper theoretically investigates uniform convergence phenomena in product spaces, extending Vapnik--Chervonenkis theory with a focus on linear VC dimension. In this work, we study uniform convergence phenomena in cartesian product spaces, under assumptions on the underlying distribution that are compatible with the product structure.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
Uniform laws of large numbers form a cornerstone of Vapnik--Chervonenkis theory, where they are characterized by the finiteness of the VC dimension. In this work, we study uniform convergence phenomena in cartesian product spaces, under assumptions on the underlying distribution that are compatible with the product structure.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
ScienceToStartup currently rates this 1.0/10 on the public viability pass. We show that, under this assumption, a uniform law of large numbers holds for a family of events if and only if the linear VC dimension of the family is finite.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
Theoretical ML moved forward this cycle; last verified April 2026. Public score 1.0/10.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
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This paper theoretically investigates uniform convergence phenomena in product spaces, extending Vapnik--Chervonenkis theory with a focus on linear VC dimension.
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