File size: 1,910 Bytes
e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 | 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]
/-- A nonempty admissible set has a greatest cutoff, whose heavy set satisfies
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⟩
/-- The division form of the maximal truncation lemma. -/
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
|