File size: 8,585 Bytes
7d9f027 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 | import Mathlib
import OAI.Probability.SKRatio.Matrices.RandomMatrixNet
namespace OAI
noncomputable section
open scoped BigOperators Topology
open Set Metric
namespace SKRatio.MatrixNet
attribute [local instance] Classical.propDecidable
lemma masked_matrix_norm_le {n : ℕ} (A : Matrix (Fin n) (Fin n) ℝ)
(s t : Fin n → Prop) :
‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ)
(fun i j => if s i ∧ t j then A i j else 0)‖ ≤
‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A‖ := by
let D (s : Fin n → Prop) : Matrix (Fin n) (Fin n) ℝ :=
Matrix.diagonal (fun i => if s i then 1 else 0)
have hde (s : Fin n → Prop) :
‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (D s)‖ ≤ 1 := by
apply matrix_diagonal_opNorm_le _ 1 zero_le_one
intro i
split_ifs <;> norm_num
have he : (fun i j => if s i ∧ t j then A i j else 0) = D s*A*D t := by
ext i j
simp only [D,Matrix.diagonal_mul,Matrix.mul_diagonal]
split_ifs <;> simp_all
rw [he,map_mul,map_mul]
calc
_ ≤ (‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (D s)‖*
‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A‖)*
‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (D t)‖ :=
(norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
_ ≤ (1*‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A‖)*1 :=
mul_le_mul (mul_le_mul_of_nonneg_right (hde s) (norm_nonneg _)) (hde t)
(norm_nonneg _) (by positivity)
_ = _ := by ring
lemma bin_kernel_decomposition {n : ℕ} {κ : Type*} [Fintype κ] [DecidableEq κ]
(A : Matrix (Fin n) (Fin n) ℝ) (b : Fin n → κ) (G : κ → κ → ℝ) :
Matrix.of (fun i j => A i j*G (b i) (b j)) =
∑ a : κ, ∑ c : κ, G a c • Matrix.of (fun i j => if b i=a ∧ b j=c then A i j else 0) := by
apply Matrix.ext
intro i j
simp only [Matrix.sum_apply,Matrix.of_apply,Matrix.smul_apply,smul_eq_mul]
simp only [mul_ite,mul_zero, ite_and,
Finset.sum_ite_irrel,Finset.sum_const_zero,Finset.sum_ite_eq,
Finset.mem_univ,ite_true]
ring
lemma euclideanCLM_smul {n : ℕ} (a : ℝ) (A : Matrix (Fin n) (Fin n) ℝ) :
Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (a • A) =
a • Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A := by
exact map_smul (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ)) a A
lemma masked_smul_norm_le {n : ℕ} (a : ℝ) (A : Matrix (Fin n) (Fin n) ℝ)
(s t : Fin n → Prop) :
‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ)
(a • Matrix.of (fun i j => if s i ∧ t j then A i j else 0))‖ ≤
|a| * ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A‖ := by
rw [euclideanCLM_smul,norm_smul,Real.norm_eq_abs]
exact mul_le_mul_of_nonneg_left (masked_matrix_norm_le A s t) (abs_nonneg a)
lemma bin_kernel_norm_le {n : ℕ} {κ : Type*} [Fintype κ] [DecidableEq κ]
(A : Matrix (Fin n) (Fin n) ℝ) (b : Fin n → κ) (G : κ → κ → ℝ) :
‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.of (fun i j => A i j*G (b i) (b j)))‖ ≤
(∑ a : κ, ∑ c : κ, |G a c|) * ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A‖ := by
let T := Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ)
let P (a c : κ) : Matrix (Fin n) (Fin n) ℝ :=
G a c • Matrix.of (fun i j => if b i=a ∧ b j=c then A i j else 0)
have hmap : T (∑ a, ∑ c, P a c) = ∑ a, ∑ c, T (P a c) := by
rw [map_sum]
exact Finset.sum_congr rfl (fun a _ => map_sum T _ _)
have hnormsum : ‖∑ a, ∑ c, T (P a c)‖ ≤ ∑ a, ∑ c, ‖T (P a c)‖ :=
(norm_sum_le _ _).trans (Finset.sum_le_sum (fun a _ => norm_sum_le _ _))
have hbnd : ∑ a, ∑ c, ‖T (P a c)‖ ≤ ∑ a, ∑ c, |G a c| *‖T A‖ := by
dsimp only [T,P]
apply Finset.sum_le_sum
intro a _
apply Finset.sum_le_sum
intro c _
convert masked_smul_norm_le (G a c) A (fun i => b i=a) (fun j => b j=c) using 1
try rfl
apply congrArg (fun M : Matrix (Fin n) (Fin n) ℝ =>
‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) M‖)
apply Matrix.ext
intro i j
simp only [Matrix.smul_apply,Matrix.of_apply,smul_eq_mul]
split_ifs <;> rfl
rw [bin_kernel_decomposition A b G]
change ‖T (∑ a, ∑ c, P a c)‖ ≤ _
rw [hmap]
apply (hnormsum.trans hbnd).trans_eq
simp only [T, Finset.sum_mul]
lemma kernel_error_norm_le {n : ℕ} (A : Matrix (Fin n) (Fin n) ℝ)
(G G' : Fin n → Fin n → ℝ) {a C : ℝ} (ha : 0≤a) (hC : 0≤C)
(hr : ∀ i, ∑ j, |A i j| ≤ C) (hc : ∀ j, ∑ i, |A i j| ≤ C)
(he : ∀ i j, |G i j-G' i j| ≤ a) :
‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (fun i j => A i j*(G i j-G' i j))‖ ≤ a*C := by
apply matrix_opNorm_le_schur _ (a*C) (mul_nonneg ha hC)
· intro i
calc
_ ≤ ∑ j, a*|A i j| := Finset.sum_le_sum (fun j _ => by
rw [abs_mul,mul_comm]
exact mul_le_mul_of_nonneg_right (he i j) (abs_nonneg _))
_ = a*(∑ j, |A i j|) := (Finset.mul_sum ..).symm
_ ≤ _ := mul_le_mul_of_nonneg_left (hr i) ha
· intro j
calc
_ ≤ ∑ i, a*|A i j| := Finset.sum_le_sum (fun i _ => by
rw [abs_mul,mul_comm]
exact mul_le_mul_of_nonneg_right (he i j) (abs_nonneg _))
_ = a*(∑ i, |A i j|) := (Finset.mul_sum ..).symm
_ ≤ _ := mul_le_mul_of_nonneg_left (hc j) ha
lemma compact_kernel_approximation {X : Type*} [PseudoMetricSpace X] [CompactSpace X]
(G : X × X → ℝ) (hG : Continuous G) {a : ℝ} (ha : 0<a) :
∃ (s : Finset X) (b : X → s), ∀ x y, |G (x,y)-G (b x,b y)| ≤ a := by
classical
obtain ⟨δ,hδ,hcont⟩ := Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous hG) a ha
obtain ⟨s,_,hs,hcov⟩ := finite_cover_balls_of_compact (X := X) isCompact_univ hδ
have hnear (x : X) : ∃ y : {z // z ∈ hs.toFinset}, dist x y < δ := by
obtain ⟨y,hy,hxy⟩ := Set.mem_iUnion₂.mp (hcov (Set.mem_univ x))
exact ⟨⟨y,hs.mem_toFinset.mpr hy⟩,hxy⟩
choose b hb using hnear
refine ⟨hs.toFinset,b,?_⟩
intro x y
have hh := hcont (show dist (x,y) ((b x: X),(b y: X)) < δ by
simpa only [Prod.dist_eq,max_lt_iff] using And.intro (hb x) (hb y))
exact (Real.dist_eq _ _ ▸ hh).le
theorem continuous_kernel_insertion {X : Type*} [PseudoMetricSpace X] [CompactSpace X]
(G : X × X → ℝ) (hG : Continuous G) {C u : ℝ} (hC : 0≤C) (hu : 0<u) :
∃ δ : ℝ, 0<δ ∧ ∀ {n : ℕ} (A : Matrix (Fin n) (Fin n) ℝ),
(∀ i, ∑ j, |A i j| ≤ C) → (∀ j, ∑ i, |A i j| ≤ C) →
‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A‖ ≤ δ →
∀ x : Fin n → X,
‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (fun i j => A i j*G (x i,x j))‖ ≤ u := by
classical
let a := u/(2*(C+1))
have ha : 0<a := by dsimp [a]; positivity
obtain ⟨s,b,hb⟩ := compact_kernel_approximation G hG ha
let B : ℝ := ∑ k : s, ∑ l : s, |G (k,l)|
have hB : 0≤B := Finset.sum_nonneg (fun _ _ => Finset.sum_nonneg (fun _ _ => abs_nonneg _))
let δ := u/(2*(B+1))
have hd : 0<δ := by dsimp [δ]; positivity
refine ⟨δ,hd,?_⟩
intro n A hr hc hsmall x
have hbin := bin_kernel_norm_le A (fun i => b (x i)) (fun k l => G (k,l))
have herr := kernel_error_norm_le A (fun i j => G (x i,x j))
(fun i j => G (b (x i),b (x j))) ha.le hC hr hc (fun i j => hb (x i) (x j))
have hdecomp : Matrix.of (fun i j => A i j*G (x i,x j)) =
Matrix.of (fun i j => A i j*G (b (x i),b (x j))) +
Matrix.of (fun i j => A i j*(G (x i,x j)-G (b (x i),b (x j)))) := by
apply Matrix.ext
intro i j
change A i j*G (x i,x j) = A i j*G (b (x i),b (x j)) + A i j*(G (x i,x j)-G (b (x i),b (x j)))
ring
have hBδ : B*δ ≤ u/2 := by
dsimp [δ]
rw [←mul_div_assoc]
apply (div_le_iff₀ (by positivity : (0:ℝ)<2*(B+1))).mpr
nlinarith
have haC : a*C ≤ u/2 := by
dsimp [a]
rw [div_mul_eq_mul_div]
apply (div_le_iff₀ (by positivity : (0:ℝ)<2*(C+1))).mpr
nlinarith
calc
_ ≤ B*‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A‖+a*C := by
change ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.of (fun i j => A i j*G (x i,x j)))‖ ≤ _
rw [hdecomp,map_add]
exact (norm_add_le _ _).trans (add_le_add hbin herr)
_ ≤ B*δ+a*C := add_le_add_left (mul_le_mul_of_nonneg_left hsmall hB) _
_ ≤ u := by linarith only [hBδ,haC]
end SKRatio.MatrixNet
end
end OAI
|