Alphabeta MathHow statement and proof provenance work
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Shattering and the Vapnik–Chervonenkis dimension of a set family
Definition
Let and let .
The trace of on is
The family shatters when
The VC dimension of is the greatest integer such that some subset with is shattered. For the empty family we set by convention.
This maximum is well defined: every shattered set is a subset of the finite set , so its size lies in , and for any nonempty family the empty set is shattered because every trace on is .
Remarks
- A one-element family has VC dimension : it shatters and no singleton.
Depends on
- The cardinality $\lvert A\rvert$ of a finite set
- A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$, and equality holds if and only if $B = A$
- $\lvert\mathcal{P}(A)\rvert = 2^{\lvert A\rvert}$ for finite $A$
- The natural numbers $\mathbb{N}$ (von Neumann)
- Order on the natural numbers
Used by
- All subsets of [4] of size at most 2: VC dimension 2 and exactly ∑_i≤2C(4, i)=11 members Example
- FALSE: a family on [n] of VC dimension at most d has at most nᵈ members False statement
- Every set shattered by Sⱼ(F) is shattered by F Lemma
- If F does not shatter T then x_T agrees on {v_F:F inF} with a combination of the x_S for S⊊ T Lemma
- If F is closed under taking subsets then F shatters every F inF Lemma
- The shifting proof of Sauer–Shelah uses no field and no vector space Remark
- A second proof of Sauer–Shelah, from the multilinear polynomial space Theorem
- Sauer–Shelah: a family on [n] of VC dimension at most d has at most ∑ᵢ₌₀ᵈC(n, i) members Theorem
Dependency tree · two levels
25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §7.4 (standard reference, not scraped)