File size: 7,269 Bytes
2f887db | 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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 | import Mathlib
import OAI.Computability.MinUncut.Model
namespace OAI
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
open scoped Topology NNReal
open scoped BigOperators
open MeasureTheory ProbabilityTheory Polynomial Filter
open scoped BigOperators Topology
open MeasureTheory ProbabilityTheory WithLp
open scoped BigOperators RealInnerProductSpace
open scoped BigOperators
namespace MinUncut.Box
variable {A : Type*} [Fintype A] [Nonempty A]
def vertex {m : ℕ} (b : Fin m → Bool) (x y : Fin m → A) : Fin m → A :=
fun i => if b i then x i else y i
def moment {m : ℕ} (f : (Fin m → A) → ℝ) : ℝ :=
𝔼 x, 𝔼 y, ∏ b : Fin m → Bool, f (vertex b x y)
def correlation {m : ℕ} (f : (Fin m → A) → ℝ)
(a : Fin m → (Fin m → A) → ℝ) : ℝ :=
𝔼 x, f x * ∏ i, a i x
def Local {m : ℕ} (a : Fin m → (Fin m → A) → ℝ) : Prop :=
∀ i x z, a i (Function.update x i z) = a i x
omit [Nonempty A] in
lemma expect_cons {m : ℕ} (f : (Fin (m+1) → A) → ℝ) :
(𝔼 x, f x) = 𝔼 z, 𝔼 x, f (Fin.cons z x) := by
rw [← Finset.expect_product', Finset.univ_product_univ]
exact Fintype.expect_equiv (Fin.consEquiv (fun _ : Fin (m+1) => A)).symm _ _ (fun x => congrArg f (Fin.cons_self_tail x).symm)
lemma prod_cons {m : ℕ} (f : (Fin (m+1) → Bool) → ℝ) :
(∏ b, f b) = ∏ z : Bool, ∏ b, f (Fin.cons z b) := by
rw [← Finset.prod_product' ]
rw [Finset.univ_product_univ]
exact Fintype.prod_equiv (Fin.consEquiv (fun _ : Fin (m+1) => Bool)).symm _ _ (fun x => congrArg f (Fin.cons_self_tail x).symm)
omit [Fintype A] [Nonempty A] in
@[simp] lemma vertex_cons {m : ℕ} (b : Bool) (bs : Fin m → Bool)
(x y : A) (xs ys : Fin m → A) :
vertex (Fin.cons b bs) (Fin.cons x xs) (Fin.cons y ys) =
Fin.cons (if b then x else y) (vertex bs xs ys) := by
ext i
refine Fin.cases ?_ (fun j => ?_) i <;> simp [vertex]
omit [Nonempty A] in
lemma moment_succ {m : ℕ} (f : (Fin (m+1) → A) → ℝ) :
moment f = 𝔼 z, 𝔼 w, moment (fun x => f (Fin.cons z x) * f (Fin.cons w x)) := by
unfold moment
rw [expect_cons]
conv_lhs => arg 2; ext z; arg 2; ext x; rw [expect_cons]
simp_rw [prod_cons, Fintype.prod_bool, vertex_cons]
simp only [Bool.false_eq_true, ↓reduceIte, Finset.prod_mul_distrib]
apply Finset.expect_congr rfl
intro z _
rw [Finset.expect_comm]
lemma expect_pow_two_pow {T : Type*} [Fintype T] [Nonempty T] (f : T → ℝ) (m : ℕ) :
(𝔼 t, f t)^(2^m) ≤ 𝔼 t, (f t)^(2^m) := by
cases m with
| zero => simp
| succ m =>
have he : Even (2^(m+1)) := ⟨2^m, by rw [pow_succ]; omega⟩
have h := Real.pow_arith_mean_le_arith_mean_pow_of_even Finset.univ
(fun _ : T => (Fintype.card T : ℝ)⁻¹) f (fun _ _ => by positivity)
(by simp [Fintype.card_ne_zero]) he
simpa only [Finset.expect_eq_sum_div_card, Finset.card_univ, ← Finset.mul_sum,
← Finset.sum_mul, div_eq_mul_inv, mul_comm] using h
lemma expect_sq_le {T : Type*} [Fintype T] [Nonempty T] (f : T → ℝ) :
(𝔼 t, f t)^2 ≤ 𝔼 t, (f t)^2 := by
simpa using expect_pow_two_pow f 1
omit [Fintype A] [Nonempty A] in
lemma face_zero {m : ℕ} (a : Fin (m+1) → (Fin (m+1) → A) → ℝ)
(ha : Local a) (z w : A) (x : Fin m → A) :
a 0 (Fin.cons z x) = a 0 (Fin.cons w x) := by
simpa only [Fin.update_cons_zero] using ha 0 (Fin.cons w x) z
lemma first_step {m : ℕ} (f : (Fin (m+1) → A) → ℝ)
(a : Fin (m+1) → (Fin (m+1) → A) → ℝ)
(ha : Local a) (hb : ∀ i x, |a i x| ≤ 1) :
(correlation f a)^2 ≤ 𝔼 z, 𝔼 w,
correlation (fun x => f (Fin.cons z x)*f (Fin.cons w x))
(fun i x => a i.succ (Fin.cons z x)*a i.succ (Fin.cons w x)) := by
let z₀ := Classical.choice (inferInstance : Nonempty A)
let g (x : Fin m → A) : ℝ := 𝔼 z, f (Fin.cons z x) * ∏ i : Fin m, a i.succ (Fin.cons z x)
have hc : correlation f a = 𝔼 x, a 0 (Fin.cons z₀ x) * g x := by
unfold correlation
rw [expect_cons, Finset.expect_comm]
apply Finset.expect_congr rfl
intro x _
simp only [Fin.prod_univ_succ]
simp_rw [face_zero a ha _ z₀ x]
dsimp [g]
rw [Finset.mul_expect]
apply Finset.expect_congr rfl
intro z _
ring
rw [hc]
calc
_ ≤ 𝔼 x, (a 0 (Fin.cons z₀ x)*g x)^2 := expect_sq_le _
_ ≤ 𝔼 x, (g x)^2 := by
apply Finset.expect_le_expect
intro x _
rw [mul_pow]
have hs : (a 0 (Fin.cons z₀ x))^2 ≤ 1 := by
nlinarith [sq_abs (a 0 (Fin.cons z₀ x)), hb 0 (Fin.cons z₀ x), abs_nonneg (a 0 (Fin.cons z₀ x))]
simpa only [one_mul] using mul_le_mul_of_nonneg_right hs (sq_nonneg (g x))
_ = _ := by
dsimp [g, correlation]
simp only [sq, Finset.expect_mul_expect]
rw [Finset.expect_comm]
apply Finset.expect_congr rfl
intro z _
rw [Finset.expect_comm]
apply Finset.expect_congr rfl
intro w _
apply Finset.expect_congr rfl
intro x _
rw [Finset.prod_mul_distrib]
ring
theorem box_cs (m : ℕ) (f : (Fin m → A) → ℝ)
(a : Fin m → (Fin m → A) → ℝ) (ha : Local a) (hb : ∀ i x, |a i x|≤1) :
(correlation f a)^(2^m) ≤ moment f := by
induction m with
| zero =>
have hf : ∀ x : Fin 0 → A, f x = f (fun i => Fin.elim0 i) :=
fun x => congrArg f (Subsingleton.elim _ _)
simp [correlation, moment, hf]
| succ m ih =>
let g := fun z w (x : Fin m → A) => f (Fin.cons z x)*f (Fin.cons w x)
let b := fun z w (i : Fin m) (x : Fin m → A) =>
a i.succ (Fin.cons z x)*a i.succ (Fin.cons w x)
have h₁ := first_step f a ha hb
have hlocal (z w : A) : Local (b z w) := by
intro i x t
dsimp [b]
simp only [Fin.cons_update, ha i.succ]
have hbound (z w : A) (i : Fin m) (x : Fin m → A) : |b z w i x|≤1 := by
dsimp [b]
rw [abs_mul]
simpa using mul_le_mul (hb i.succ (Fin.cons z x)) (hb i.succ (Fin.cons w x))
(abs_nonneg _) (by norm_num : (0:ℝ)≤1)
calc
_ = ((correlation f a)^2)^(2^m) := by rw [← pow_mul, pow_succ]; congr 1; omega
_ ≤ (𝔼 z, 𝔼 w, correlation (g z w) (b z w))^(2^m) :=
pow_le_pow_left₀ (sq_nonneg _) h₁ _
_ ≤ 𝔼 z, (𝔼 w, correlation (g z w) (b z w))^(2^m) := expect_pow_two_pow _ m
_ ≤ 𝔼 z, 𝔼 w, (correlation (g z w) (b z w))^(2^m) :=
Finset.expect_le_expect (fun z _ => expect_pow_two_pow _ m)
_ ≤ 𝔼 z, 𝔼 w, moment (g z w) := by
apply Finset.expect_le_expect
intro z _
apply Finset.expect_le_expect
intro w _
exact ih _ _ (hlocal z w) (hbound z w)
_ = _ := (moment_succ f).symm
lemma abs_correlation_cs {m : ℕ} (hm : 0 < m) (f : (Fin m → A) → ℝ)
(a : Fin m → (Fin m → A) → ℝ) (ha : Local a) (hb : ∀ i x, |a i x|≤1) :
|correlation f a|^(2^m) ≤ moment f := by
have he : Even (2^m) := by
obtain ⟨k,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hm)
exact ⟨2^k, by rw [pow_succ]; omega⟩
simpa only [he.pow_abs] using box_cs m f a ha hb
end MinUncut.Box
end
end OAI
|