| import OAI.Combinatorics.ExpectationThreshold.SelectorTruncationCompact |
| import OAI.Combinatorics.ExpectationThreshold.SelectorTruncationSlope |
|
|
| namespace OAI |
|
|
| /-! |
| # Maximal weighted truncation |
|
|
| This is the maximal truncation lemma used by the multiscale selector. The |
| cutoff is obtained from the actual compact admissible set. Its heavy-set |
| bound follows from the exact affine increase on the next interval between |
| weights, rather than an assumed slope or regularity certificate. |
| -/ |
|
|
| namespace TalagrandThreshold.Selector |
|
|
| open scoped BigOperators |
|
|
| variable {α : Type*} [Fintype α] [DecidableEq α] |
|
|
| / |
| the multiplied cardinality bound. -/ |
| theorem exists_maximal_truncation (w : α → ℝ) (A : Finset α) (m d : ℝ) |
| (hw : ∀ x, 0 ≤ w x) (hsum : ∑ x, w x = 1) |
| (hm : 0 ≤ m) (hd0 : 0 < d) (hd1 : d < 1) |
| (hne : ∃ u, Admissible w A m d u) : |
| ∃ ε, Admissible w A m d ε ∧ |
| (∀ u, Admissible w A m d u → u ≤ ε) ∧ |
| d * ((heavySet w ε).card : ℝ) ≤ m := by |
| obtain ⟨ε, hε, hmax⟩ := exists_greatest_admissible w A m d hne |
| exact ⟨ε, hε, hmax, |
| maximal_admissible_card_bound w A m d ε hw hsum hm hd0 hd1 hε hmax⟩ |
|
|
| / |
| theorem exists_maximal_truncation_div (w : α → ℝ) (A : Finset α) (m d : ℝ) |
| (hw : ∀ x, 0 ≤ w x) (hsum : ∑ x, w x = 1) |
| (hm : 0 ≤ m) (hd0 : 0 < d) (hd1 : d < 1) |
| (hne : ∃ u, Admissible w A m d u) : |
| ∃ ε, Admissible w A m d ε ∧ |
| (∀ u, Admissible w A m d u → u ≤ ε) ∧ |
| ((heavySet w ε).card : ℝ) ≤ m / d := by |
| obtain ⟨ε, hε, hmax, hcard⟩ := |
| exists_maximal_truncation w A m d hw hsum hm hd0 hd1 hne |
| refine ⟨ε, hε, hmax, (le_div_iff₀ hd0).2 ?_⟩ |
| simpa only [mul_comm] using hcard |
|
|
| end TalagrandThreshold.Selector |
|
|
| end OAI |
|
|
