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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 | import Mathlib
import OAI.Probability.SKRatio.Matrices.GaussianCoordinateMemLp
import OAI.Probability.SKRatio.Matrices.IntegralAbsSet
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory Real Set
open scoped NNReal ENNReal
open Filter
open scoped Topology
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius NNReal
noncomputable section
open MeasureTheory ProbabilityTheory Real Set Filter
open MeasureTheory.Measure
open scoped ENNReal NNReal MeasureTheory Topology
open MeasureTheory
noncomputable section
noncomputable section
open MeasureTheory Set NormedSpace
open scoped Topology
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius
noncomputable section
open Set Real
open scoped Topology
noncomputable section
open Matrix Set Filter
open scoped Topology Matrix.Norms.Frobenius
noncomputable section
open Matrix NormedSpace ContinuousLinearMap
open scoped Matrix.Norms.Frobenius
noncomputable section
open Matrix
noncomputable section
open MeasureTheory ProbabilityTheory Real Set
open scoped ENNReal NNReal
namespace SKRatioGaussian
open Matrix MeasureTheory ProbabilityTheory Real
open scoped BigOperators Matrix.Norms.Frobenius NNReal ENNReal
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
omit [DecidableEq ι] in
lemma abs_mul_entry_bound (M K : Matrix ι ι ℝ) (C : ℝ) (hK : ∀ a b, |K a b| ≤ C) (i : ι) :
|(M*K) i i| ≤ C*∑ b, |M i b| := by
rw [Matrix.mul_apply]
apply (Finset.abs_sum_le_sum_abs _ _).trans
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro b _
rw [abs_mul,mul_comm C]
exact mul_le_mul_of_nonneg_left (hK b i) (abs_nonneg _)
omit [DecidableEq ι] in
theorem goe_diagonal_inverse_loop {r : ℝ} (hr : 0 ≤ r) (a c : ι → ℝ) (i : ι)
{K : (MatrixCoordinates ι → ℝ) → Matrix ι ι ℝ} {C : ℝ}
{s : Set (MatrixCoordinates ι → ℝ)} (hK : Continuous K)
(hC : ∀ g a b, |K g a b| ≤ C) (hs : MeasurableSet s)
(he : ∀ g ∈ s, K g i i = 1+a i*((goeMatrix r g*K g) i i-c i*K g i i)) :
let μ := Measure.pi (fun _ : MatrixCoordinates ι => gaussianReal 0 1)
|(∫ g, K g i i ∂μ)-
(1+a i*((∫ g, (goeMatrix r g*K g) i i ∂μ)-c i*(∫ g, K g i i ∂μ)))| ≤
(C+1+|a i*c i| *C)*μ.real sᶜ+
(|a i| *C)*(Fintype.card ι:ℝ)*sqrt (2*r)*sqrt (μ.real sᶜ) := by
intro μ
have hC0 : 0 ≤ C := (abs_nonneg _).trans (hC 0 i i)
have hKi (a b : ι) : Integrable (fun g => K g a b) μ :=
Integrable.mono' (integrable_const C) (hK.matrix_elem a b).aestronglyMeasurable
(ae_of_all _ (fun g => by simpa only [Real.norm_eq_abs] using hC g a b))
have hW (b : ι) : Integrable (fun g => goeMatrix r g i b) μ :=
(goeEntry_memLp r i b).integrable (by norm_num)
have hWK (b : ι) : Integrable (fun g => goeMatrix r g i b*K g b i) μ := by
simpa only [mul_comm] using (hW b).bdd_mul (hK.matrix_elem b i).aestronglyMeasurable
(ae_of_all _ (fun g => by simpa only [Real.norm_eq_abs] using hC g b i))
have hWI : Integrable (fun g => (goeMatrix r g*K g) i i) μ :=
integrable_finsetSum _ (fun b _ => hWK b)
let A := C+1+|a i*c i| *C
let B := |a i| *C
let H := fun g => A+B*∑ b, |goeMatrix r g i b|
have hHA : Integrable (fun g => ∑ b, |goeMatrix r g i b|) μ :=
integrable_finsetSum _ (fun b _ => (hW b).norm)
have hHI : Integrable H μ := (integrable_const A).add (hHA.const_mul B)
have hGI : Integrable (fun g => 1+a i*((goeMatrix r g*K g) i i-c i*K g i i)) μ :=
(integrable_const 1).add ((hWI.sub ((hKi i i).const_mul _)).const_mul _)
have hb (g : MatrixCoordinates ι → ℝ) :
‖K g i i-(1+a i*((goeMatrix r g*K g) i i-c i*K g i i))‖ ≤ H g := by
rw [Real.norm_eq_abs]
have hid : K g i i-(1+a i*((goeMatrix r g*K g) i i-c i*K g i i)) =
(K g i i-1)+(a i*c i)*K g i i+(-a i)*(goeMatrix r g*K g) i i := by ring
rw [hid]
apply (abs_add_le _ _).trans
apply (add_le_add (abs_add_le _ _) le_rfl).trans
have hk1 : |K g i i-1| ≤ C+1 := (abs_sub _ _).trans (by simpa using add_le_add_right (hC g i i) 1)
have hk2 : |(a i*c i)*K g i i| ≤ |a i*c i| *C := by
rw [abs_mul]; exact mul_le_mul_of_nonneg_left (hC g i i) (abs_nonneg _)
have hk3 : |(-a i)*(goeMatrix r g*K g) i i| ≤ |a i| *(C*∑ b, |goeMatrix r g i b|) := by
rw [abs_mul,abs_neg]
exact mul_le_mul_of_nonneg_left (abs_mul_entry_bound _ _ C (hC g) i) (abs_nonneg _)
exact (add_le_add (add_le_add hk1 hk2) hk3).trans_eq (by dsimp [H,A,B]; ring)
have hraw := integral_exceptional_equal_bound hs.compl (hKi i i) hGI hHI
(fun g hg => he g (by simpa using hg)) (fun g _ => hb g)
have hbad (b : ι) : (∫ g in sᶜ, |goeMatrix r g i b| ∂μ) ≤
sqrt (2*r)*sqrt (μ.real sᶜ) := by
apply (integral_abs_on_set_le (goeEntry_memLp r i b) hs.compl).trans
exact mul_le_mul_of_nonneg_right (sqrt_le_sqrt (goeEntry_integral_sq_le hr i b)) (sqrt_nonneg _)
have hHr : (∫ g in sᶜ, H g ∂μ) ≤ A*μ.real sᶜ+B*(Fintype.card ι:ℝ)*sqrt (2*r)*sqrt (μ.real sᶜ) := by
rw [show H = fun g => A+B*∑ b, |goeMatrix r g i b| from rfl,
integral_add (integrable_const A) ((hHA.const_mul B).mono_measure Measure.restrict_le_self),
integral_const,integral_const_mul]
have hWa (b : ι) : Integrable (fun g => |goeMatrix r g i b|) (μ.restrict sᶜ) := by
simpa only [Real.norm_eq_abs] using (hW b).norm.mono_measure Measure.restrict_le_self
rw [integral_finsetSum _ (fun b _ => hWa b)]
simp only [measureReal_restrict_apply_univ,smul_eq_mul]
calc
_ ≤ μ.real sᶜ*A+B*(∑ _b : ι, sqrt (2*r)*sqrt (μ.real sᶜ)) :=
add_le_add le_rfl (mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun b _ => hbad b))
(mul_nonneg (abs_nonneg _) hC0))
_ = _ := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; ring
have heI : (∫ g, 1+a i*((goeMatrix r g*K g) i i-c i*K g i i) ∂μ) =
1+a i*((∫ g, (goeMatrix r g*K g) i i ∂μ)-c i*(∫ g, K g i i ∂μ)) := by
have ht := integral_add (integrable_const (1:ℝ))
((hWI.sub ((hKi i i).const_mul (c i))).const_mul (a i))
change (∫ g, 1+a i*((goeMatrix r g*K g) i i-c i*K g i i) ∂μ) = _ at ht
rw [ht,integral_const_mul]
simp only [Pi.sub_apply]
rw [integral_sub hWI ((hKi i i).const_mul (c i)),integral_const_mul]
simp
rw [heI] at hraw
exact hraw.trans hHr
end SKRatioGaussian
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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