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f18e7de | 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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMasks
import Mathlib.Algebra.BigOperators.Field
namespace OAI
/-! Finite orbit symmetries, masks and exact recovery operations. -/
namespace MatrixMultiplication.InheritedMasks
open MatrixMultiplication.Foundation
open scoped BigOperators
section Pushforward
variable {A B : Type*} [Fintype A] [DecidableEq B]
def pushforwardLaw (statistic : A → B) (ν : A → ℝ) (b : B) : ℝ :=
∑ a with statistic a = b, ν a
theorem pushforwardLaw_nonneg (statistic : A → B) (ν : A → ℝ)
(hν : ∀ a, 0 ≤ ν a) (b : B) : 0 ≤ pushforwardLaw statistic ν b := by
exact Finset.sum_nonneg fun a _ => hν a
theorem pushforwardLaw_le_one (statistic : A → B) (ν : A → ℝ)
(hν : ∀ a, 0 ≤ ν a) (hsum : ∑ a, ν a = 1) (b : B) :
pushforwardLaw statistic ν b ≤ 1 := by
calc
pushforwardLaw statistic ν b ≤ ∑ a, ν a :=
Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
(fun a _ _ => hν a)
_ = 1 := hsum
end Pushforward
section Windows
variable {P A B : Type*} [Fintype P]
[Fintype A] [DecidableEq A] [DecidableEq B]
theorem wordPopulation_statistic (w : P → A) (statistic : A → B) (b : B) :
wordPopulation (statistic ∘ w) b =
∑ a with statistic a = b, wordPopulation w a := by
simpa only [wordPopulation, Fintype.card_subtype, Finset.mem_filter,
Finset.mem_univ, true_and, Function.comp_apply] using
(Finset.sum_card_fiberwise_eq_card_filter (Finset.univ : Finset P)
(Finset.univ.filter fun a => statistic a = b) w).symm
theorem empiricalLaw_statistic (w : P → A) (statistic : A → B) (b : B) :
empiricalLaw (statistic ∘ w) b =
pushforwardLaw statistic (empiricalLaw w) b := by
simp only [empiricalLaw, wordPopulation_statistic, pushforwardLaw,
Nat.cast_sum, Finset.sum_div]
omit [Fintype A] in
theorem typeWindow_mono (ν : A → ℝ) (w : P → A) {τ η : ℝ}
(hτη : τ ≤ η) (hw : typeWindow ν τ w) : typeWindow ν η w := by
intro a
exact (hw a).trans hτη
theorem typeWindow_statistic_coordinate (ν : A → ℝ) (τ : ℝ) (w : P → A)
(statistic : A → B) (hw : typeWindow ν τ w) (b : B) :
|empiricalLaw (statistic ∘ w) b - pushforwardLaw statistic ν b| ≤
((Finset.univ.filter fun a => statistic a = b).card : ℝ) * τ := by
rw [empiricalLaw_statistic]
unfold pushforwardLaw
rw [← Finset.sum_sub_distrib]
calc
_ ≤ ∑ a with statistic a = b, |empiricalLaw w a - ν a| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ _a ∈ Finset.univ.filter (fun a => statistic a = b), τ :=
Finset.sum_le_sum fun a _ => hw a
_ = _ := by simp only [Finset.sum_const, nsmul_eq_mul]
theorem typeWindow_statistic (ν : A → ℝ) (τ : ℝ) (w : P → A)
(statistic : A → B) (hτ : 0 ≤ τ) (hw : typeWindow ν τ w) :
typeWindow (pushforwardLaw statistic ν) (Fintype.card A * τ)
(statistic ∘ w) := by
intro b
calc
_ ≤ ((Finset.univ.filter fun a => statistic a = b).card : ℝ) * τ :=
typeWindow_statistic_coordinate ν τ w statistic hw b
_ ≤ Fintype.card A * τ :=
mul_le_mul_of_nonneg_right (Nat.cast_le.mpr (Finset.card_le_univ _)) hτ
end Windows
end MatrixMultiplication.InheritedMasks
end OAI
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