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9b6e014 | 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 | import Mathlib
import OAI.Probability.BinarySweep.MatrixBounds.SchattenTriangle
namespace OAI
noncomputable section
section
open scoped BigOperators Classical Matrix.Norms.L2Operator
namespace BinaryCoordinateSweeps.TraceHolder
variable {ι α β : Type*} [Fintype ι] [DecidableEq ι] [Fintype α] [Fintype β]
lemma projection_sandwich_split (X Y P : M ι) (R : α → M ι) (C : β → M ι)
(hR : ∀a, R a*R a=R a) (hC : ∀b, C b*C b=C b)
(hsR : ∑a, R a=1) (hsC : ∑b, C b=1) :
Y*P*X = ∑b, ∑a, (Y*C b)*(C b*P*R a)*(R a*X) := by
have he (a : α) (b : β) : (Y*C b)*(C b*P*R a)*(R a*X)=Y*C b*P*R a*X := by
simp only [mul_assoc,← mul_assoc (C b) (C b),hC b,
← mul_assoc (R a) (R a),hR a]
simp_rw [he]
simp only [← Finset.sum_mul,← Finset.mul_sum,hsR,hsC,mul_one]
theorem schatten_projection_split {q : ℕ} (hq : 0<q)
(X Y P : M ι) (R : α → M ι) (C : β → M ι)
(hR : ∀a, R a*R a=R a) (hC : ∀b, C b*C b=C b)
(hsR : ∑a, R a=1) (hsC : ∑b, C b=1) :
schatten q (Y*P*X) ≤ ∑b, ∑a,
schatten q (Y*C b)*‖C b*P*R a‖*schatten q (R a*X) := by
rw [projection_sandwich_split X Y P R C hR hC hsR hsC]
apply (schatten_sum_le hq _).trans
apply Finset.sum_le_sum
intro b _
apply (schatten_sum_le hq _).trans
exact Finset.sum_le_sum fun a _ => schatten_sandwich hq _ _ _
theorem matrixMoment_projection_split_bound {q : ℕ} (hq : 0<q)
(X Y P : M ι) (R : α → M ι) (C : β → M ι)
(hR : ∀a, R a*R a=R a) (hC : ∀b, C b*C b=C b)
(hsR : ∑a, R a=1) (hsC : ∑b, C b=1)
(B : ℝ) (hB : 0≤B)
(hterm : ∀a b, matrixMoment q (Y*C b)*‖C b*P*R a‖^(2*q)*
matrixMoment q (R a*X) ≤ B^(2*q)) :
matrixMoment q (Y*P*X) ≤ ((Fintype.card α*Fintype.card β:ℕ):ℝ)^(2*q)*B^(2*q) := by
have ht (a : α) (b : β) :
schatten q (Y*C b)*‖C b*P*R a‖*schatten q (R a*X) ≤ B := by
apply (pow_le_pow_iff_left₀
(mul_nonneg (mul_nonneg (schatten_nonneg _ _) (norm_nonneg _)) (schatten_nonneg _ _))
hB (by omega : 2*q≠0)).mp
simpa only [mul_pow,schatten_pow hq] using hterm a b
have hs : schatten q (Y*P*X) ≤ ((Fintype.card α*Fintype.card β:ℕ):ℝ)*B := by
apply (schatten_projection_split hq X Y P R C hR hC hsR hsC).trans
calc
_ ≤ ∑b : β, ∑a : α, B := Finset.sum_le_sum fun b _ => Finset.sum_le_sum fun a _ => ht a b
_ = _ := by simp [mul_comm,mul_assoc]
have hh := pow_le_pow_left₀ (schatten_nonneg _ _) hs (2*q)
simpa only [schatten_pow hq,mul_pow] using hh
end BinaryCoordinateSweeps.TraceHolder
end
open scoped BigOperators Classical Matrix.Norms.L2Operator
namespace BinaryCoordinateSweeps.TraceHolder
lemma overlap_power_le {x c L : ℝ} {q : ℕ} (hx : 0≤x) (hx1 : x≤1)
(hc : 0≤c) (hcq : c≤q) (hq : 0<q) (hL : x^2≤Real.exp L) :
x^(2*q)≤Real.exp (c*L) := by
by_cases hz : x=0
· rw [hz,zero_pow (by omega : 2*q≠0)]
exact (Real.exp_pos _).le
have hp : 0<x := lt_of_le_of_ne hx (Ne.symm hz)
have hl : Real.log x≤0 := Real.log_nonpos hp.le hx1
have he : 2*Real.log x≤L := by
have hh := Real.log_le_log (sq_pos_of_pos hp) hL
simpa only [Real.log_pow,Real.log_exp,Nat.cast_ofNat] using hh
calc
x^(2*q) = Real.exp (((2*q:ℕ):ℝ)*Real.log x) := by
rw [Real.exp_nat_mul,Real.exp_log hp]
_ ≤ Real.exp (c*L) := by
apply Real.exp_le_exp.mpr
calc
((2*q:ℕ):ℝ)*Real.log x = (q:ℝ)*(2*Real.log x) := by push_cast; ring
_ ≤ c*(2*Real.log x) := mul_le_mul_of_nonpos_right hcq (by linarith)
_ ≤ c*L := mul_le_mul_of_nonneg_left he hc
variable {ι α β : Type*} [Fintype ι] [DecidableEq ι] [Fintype α] [Fintype β]
theorem matrixMoment_dense_exponential {q : ℕ} (hq : 0<q)
(X Y P : M ι) (R : α → M ι) (C : β → M ι)
(hR : ∀a, R a*R a=R a) (hC : ∀b, C b*C b=C b)
(hsR : ∑a, R a=1) (hsC : ∑b, C b=1)
(FR : α → ℝ) (FC : β → ℝ) (c F H Echild Eoverlap : ℝ)
(hc : 0≤c) (hcq : c≤q)
(hchild : ∀a b, matrixMoment q (Y*C b)*matrixMoment q (R a*X) ≤
Real.exp (-c*(FR a+FC b)+Echild))
(hoverlap : ∀a b, ‖C b*P*R a‖^2≤Real.exp (FR a+FC b-F+H+Eoverlap))
(hcontract : ∀a b, ‖C b*P*R a‖≤1) :
matrixMoment q (Y*P*X) ≤ ((Fintype.card α*Fintype.card β:ℕ):ℝ)^(2*q)*
Real.exp (-c*F+Echild+c*H+c*Eoverlap) := by
let B := -c*F+Echild+c*H+c*Eoverlap
have hq0 : ((2*q:ℕ):ℝ)≠0 := by exact_mod_cast (by omega : 2*q≠0)
have he : (Real.exp (B/((2*q:ℕ):ℝ)))^(2*q)=Real.exp B := by
rw [←Real.exp_nat_mul]
congr 1
field_simp
have ht (a : α) (b : β) :
matrixMoment q (Y*C b)*‖C b*P*R a‖^(2*q)*matrixMoment q (R a*X) ≤
(Real.exp (B/((2*q:ℕ):ℝ)))^(2*q) := by
rw [he]
calc
_ = (matrixMoment q (Y*C b)*matrixMoment q (R a*X))*‖C b*P*R a‖^(2*q) := by ring
_ ≤ Real.exp (-c*(FR a+FC b)+Echild)*
Real.exp (c*(FR a+FC b-F+H+Eoverlap)) :=
mul_le_mul (hchild a b)
(overlap_power_le (norm_nonneg _) (hcontract a b) hc hcq hq (hoverlap a b))
(pow_nonneg (norm_nonneg _) _) (Real.exp_pos _).le
_ = Real.exp B := by rw [←Real.exp_add]; congr 1; dsimp [B]; ring
have hh := matrixMoment_projection_split_bound hq X Y P R C hR hC hsR hsC
(Real.exp (B/((2*q:ℕ):ℝ))) (Real.exp_pos _).le ht
rw [he] at hh
exact hh
end BinaryCoordinateSweeps.TraceHolder
end
end OAI
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