Shattering families

This file defines the shattering property and VC-dimension of set families.

Main declarations

• Finset.Shatters: The shattering property.
• Finset.shatterer: The set family of sets shattered by a set family.
• Finset.vcDim: The Vapnik-Chervonenkis dimension.

TODO

• Order-shattering
• Strong shattering

def Finset.Shatters {α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) (s : Finset α) : Prop

A set family 𝒜 shatters a set s if all subsets of s can be obtained as the intersection of s and some element of the set family, and we denote this 𝒜.Shatters s. We also say that s is traced by 𝒜.

Equations

• 𝒜.Shatters s = ∀ ⦃t : Finset α⦄, t ⊆ s → ∃ u ∈ 𝒜, s ∩ u = t

instance Finset.instDecidablePredShatters {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} : DecidablePred 𝒜.Shatters

Equations

• Finset.instDecidablePredShatters _s = Finset.decidableForallOfDecidableSubsets

theorem Finset.Shatters.exists_inter_eq_singleton {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} {a : α} (hs : 𝒜.Shatters s) (ha : a ∈ s) : ∃ t ∈ 𝒜, s ∩ t = {a}

theorem Finset.Shatters.mono_left {α : Type u_1} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} (h : 𝒜 ⊆ ℬ) (h𝒜 : 𝒜.Shatters s) : ℬ.Shatters s

theorem Finset.Shatters.mono_right {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s t : Finset α} (h : t ⊆ s) (hs : 𝒜.Shatters s) : 𝒜.Shatters t

theorem Finset.Shatters.exists_superset {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} (h : 𝒜.Shatters s) : ∃ t ∈ 𝒜, s ⊆ t

theorem Finset.shatters_of_forall_subset {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} (h : ∀ t ⊆ s, t ∈ 𝒜) : 𝒜.Shatters s

theorem Finset.Shatters.nonempty {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} (h : 𝒜.Shatters s) : 𝒜.Nonempty

@[simp]

theorem Finset.shatters_empty {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} : 𝒜.Shatters ∅ ↔ 𝒜.Nonempty

theorem Finset.Shatters.subset_iff {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s t : Finset α} (h : 𝒜.Shatters s) : t ⊆ s ↔ ∃ u ∈ 𝒜, s ∩ u = t

theorem Finset.shatters_iff {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} : 𝒜.Shatters s ↔ image (fun (t : Finset α) => s ∩ t) 𝒜 = s.powerset

theorem Finset.univ_shatters {α : Type u_1} [DecidableEq α] {s : Finset α} [Fintype α] : univ.Shatters s

@[simp]

theorem Finset.shatters_univ {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} [Fintype α] : 𝒜.Shatters univ ↔ 𝒜 = univ

def Finset.shatterer {α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) : Finset (Finset α)

The set family of sets that are shattered by 𝒜.

Equations

• 𝒜.shatterer = Finset.filter (fun (s : Finset α) => 𝒜.Shatters s) (𝒜.biUnion Finset.powerset)

@[simp]

theorem Finset.mem_shatterer {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} : s ∈ 𝒜.shatterer ↔ 𝒜.Shatters s

theorem Finset.shatterer_mono {α : Type u_1} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} (h : 𝒜 ⊆ ℬ) : 𝒜.shatterer ⊆ ℬ.shatterer

theorem Finset.subset_shatterer {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} (h : IsLowerSet ↑𝒜) : 𝒜 ⊆ 𝒜.shatterer

@[simp]

theorem Finset.isLowerSet_shatterer {α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) : IsLowerSet ↑𝒜.shatterer

@[simp]

theorem Finset.shatterer_eq {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} : 𝒜.shatterer = 𝒜 ↔ IsLowerSet ↑𝒜

@[simp]

theorem Finset.shatterer_idem {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} : 𝒜.shatterer.shatterer = 𝒜.shatterer

@[simp]

theorem Finset.shatters_shatterer {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} : 𝒜.shatterer.Shatters s ↔ 𝒜.Shatters s

theorem Finset.Shatters.shatterer {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} : 𝒜.Shatters s → 𝒜.shatterer.Shatters s

Alias of the reverse direction of Finset.shatters_shatterer.

theorem Finset.card_le_card_shatterer {α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) : 𝒜.card ≤ 𝒜.shatterer.card

Pajor's variant of the Sauer-Shelah lemma.

theorem Finset.Shatters.of_compression {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} {a : α} (hs : (Down.compression a 𝒜).Shatters s) : 𝒜.Shatters s

theorem Finset.shatterer_compress_subset_shatterer {α : Type u_1} [DecidableEq α] (a : α) (𝒜 : Finset (Finset α)) : (Down.compression a 𝒜).shatterer ⊆ 𝒜.shatterer

Vapnik-Chervonenkis dimension

def Finset.vcDim {α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) : ℕ

The Vapnik-Chervonenkis dimension of a set family is the maximal size of a set it shatters.

Equations

• 𝒜.vcDim = 𝒜.shatterer.sup Finset.card

theorem Finset.vcDim_mono {α : Type u_1} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} (h𝒜ℬ : 𝒜 ⊆ ℬ) : 𝒜.vcDim ≤ ℬ.vcDim

theorem Finset.Shatters.card_le_vcDim {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} (hs : 𝒜.Shatters s) : s.card ≤ 𝒜.vcDim

theorem Finset.vcDim_compress_le {α : Type u_1} [DecidableEq α] (a : α) (𝒜 : Finset (Finset α)) : (Down.compression a 𝒜).vcDim ≤ 𝒜.vcDim

Down-compressing decreases the VC-dimension.

theorem Finset.card_shatterer_le_sum_vcDim {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} [Fintype α] : 𝒜.shatterer.card ≤ ∑ k ∈ Iic 𝒜.vcDim, (Fintype.card α).choose k

The Sauer-Shelah lemma.