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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 | import Mathlib
import OAI.Probability.SKRatio.Quantization.BinTail
import OAI.Probability.SKRatio.Variational.ScalarCoherent
namespace OAI
noncomputable section
open scoped BigOperators Matrix ENNReal
open MeasureTheory ProbabilityTheory Filter Real
namespace SKRatio.Bins
open Planted Scalar SKRatioClock.Regression MatrixNet Calculus
attribute [local instance] Classical.propDecidable
lemma independent_finite_selection_bound
{Ω A D : Type*} [MeasurableSpace Ω] [MeasurableSpace A]
[Fintype D] [MeasurableSpace D] [MeasurableSingletonClass D]
{μ : Measure Ω} [IsProbabilityMeasure μ] {R : Ω → A} {X : Ω → D}
(hX : Measurable X) (hRX : IndepFun R X μ)
(E : D → Set A) (hE : ∀ d, MeasurableSet (E d))
{B : ℝ} (hB : ∀ d, μ.real (R ⁻¹' E d) ≤ B) :
μ.real {ω | R ω ∈ E (X ω)} ≤ B := by
have hall : (⋃ d : D, X ⁻¹' {d}) = Set.univ := by
ext ω
simp
have hdis : Pairwise (Function.onFun Disjoint (fun d : D => X ⁻¹' {d})) := by
intro d e hde
exact Set.disjoint_left.mpr (fun ω hd he => hde ((Set.mem_singleton_iff.mp hd).symm.trans
(Set.mem_singleton_iff.mp he)))
have hsum : (∑ d, μ.real (X ⁻¹' {d})) = 1 := by
rw [←measureReal_iUnion_fintype hdis (fun d => (MeasurableSet.singleton d).preimage hX),hall]
simp [Measure.real]
have hb (d : D) : μ.real (R ⁻¹' E d ∩ X ⁻¹' {d}) ≤ B*μ.real (X ⁻¹' {d}) := by
have hi := hRX.measure_inter_preimage_eq_mul (E d) {d} (hE d) (MeasurableSet.singleton d)
change (μ _).toReal ≤ _
rw [hi,ENNReal.toReal_mul]
exact mul_le_mul_of_nonneg_right (hB d) ENNReal.toReal_nonneg
calc
_ ≤ μ.real (⋃ d : D, R ⁻¹' E d ∩ X ⁻¹' {d}) := by
apply measureReal_mono _ (measure_ne_top _ _)
intro ω hω
exact Set.mem_iUnion.mpr ⟨X ω,hω,rfl⟩
_ ≤ ∑ d, μ.real (R ⁻¹' E d ∩ X ⁻¹' {d}) := measureReal_iUnion_fintype_le _
_ ≤ ∑ d, B*μ.real (X ⁻¹' {d}) := Finset.sum_le_sum (fun d _ => hb d)
_ = B := by rw [←Finset.mul_sum,hsum,mul_one]
def residualFormBad {α : Type*} {n : ℕ} (β : ℝ) (h : α → ℝ)
(σ : Fin n → α) (K C : ℝ) : Set ((Fin n × Fin n) → ℝ) :=
{M | euclideanOpNorm (fun i j => M (i,j)) ≤ K ∧
∃ p : Fin n → ℝ, (∑ i, p i^2)=1 ∧ C <
p ⬝ᵥ ((fun i j => M (i,j))*ᵥ (fun i => v (h (σ i))*p i))+
correctedForm β (fun i => h (σ i)) p}
lemma residualFormBad_measurable {α : Type*} {n : ℕ}
(β : ℝ) (h : α → ℝ) (σ : Fin n → α) (K C : ℝ) :
MeasurableSet (residualFormBad β h σ K C) := by
have hc : Continuous (fun M : (Fin n × Fin n) → ℝ =>
euclideanOpNorm (fun i j => M (i,j))) := by
have : ContinuousSMul ℝ (EuclideanSpace ℝ (Fin n)) :=
IsBoundedSMul.continuousSMul
have : ContinuousSMul ℝ (EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ (Fin n)) :=
ContinuousLinearMap.continuousSMul
exact continuous_norm.comp
(((Matrix.toEuclideanCLM (𝕜 := ℝ) (n := Fin n)).toAlgEquiv.toLinearEquiv.toLinearMap.continuous_of_finiteDimensional).comp
(by fun_prop))
have ho : IsOpen {M : (Fin n × Fin n) → ℝ |
∃ p : Fin n → ℝ, (∑ i, p i^2)=1 ∧ C <
p ⬝ᵥ ((fun i j => M (i,j))*ᵥ (fun i => v (h (σ i))*p i))+
correctedForm β (fun i => h (σ i)) p} := by
have he : {M : (Fin n × Fin n) → ℝ | ∃ p : Fin n → ℝ, (∑ i, p i^2)=1 ∧ C <
p ⬝ᵥ ((fun i j => M (i,j))*ᵥ (fun i => v (h (σ i))*p i))+
correctedForm β (fun i => h (σ i)) p} =
⋃ p : {p : Fin n → ℝ // (∑ i, p i^2)=1}, {M | C <
p.val ⬝ᵥ ((fun i j => M (i,j))*ᵥ (fun i => v (h (σ i))*p.val i))+
correctedForm β (fun i => h (σ i)) p.val} := by
ext M
simp
rw [he]
apply isOpen_iUnion
intro p
apply isOpen_lt continuous_const
unfold dotProduct Matrix.mulVec
fun_prop
exact (measurableSet_le hc.measurable measurable_const).inter ho.measurableSet
theorem random_binForm_tail {α : Type*} [Fintype α] [DecidableEq α]
[MeasurableSpace α] [MeasurableSingletonClass α]
{β K a C : ℝ} (hβ : 0<β) (hK : 0≤K) (ha : 0<a)
(h : α → ℝ) (q : ℝ → α) (hq : Measurable q) :
∃ N : ℕ, ∀ {n : ℕ} (_hn : 0<n),
(standardArrayLaw (Fin n × Fin n)).real {g |
(∀ d, 2≤count (fun i => q (augmentedField β g i)) d) ∧
(∀ z : Parameters α, finiteV β (binT (fun i => q (augmentedField β g i))) h z≤C) ∧
g ∈ binFormBad β h (fun i => q (augmentedField β g i)) K (C+2*a)} ≤
N*exp (-a^2*(n:ℝ)/(π^2*β^2)) := by
obtain ⟨N,hN⟩ := binForm_tail hβ hK ha h
refine ⟨N,?_⟩
intro n hn
let μ := standardArrayLaw (Fin n × Fin n)
let R := fun g : (Fin n × Fin n) → ℝ => fun p : Fin n × Fin n =>
(fieldProjection n*(β • goe g)*fieldProjection n) p.1 p.2
let X := fun g : (Fin n × Fin n) → ℝ => fun i => q (augmentedField β g i)
let good := fun σ : Fin n → α => (∀ d, 2≤count σ d) ∧
(∀ z : Parameters α, finiteV β (binT σ) h z≤C)
let E := fun σ : Fin n → α => if good σ then residualFormBad β h σ K (C+2*a) else ∅
have hX : Measurable X := by
apply Measurable.of_eval
intro i
apply hq.comp
unfold augmentedField augmented goe
fun_prop
have hi : IndepFun R X μ := by
have ht := (field_residual_independent hn β).comp measurable_id
(show Measurable (fun H : Fin n → ℝ => fun i => q (H i)) from
Measurable.of_eval (fun i => hq.comp (measurable_pi_apply i)))
exact ht
have hE (σ : Fin n → α) : MeasurableSet (E σ) := by
dsimp only [E]
split_ifs
· exact residualFormBad_measurable β h σ K (C+2*a)
· exact MeasurableSet.empty
have hb (σ : Fin n → α) : μ.real (R ⁻¹' E σ) ≤
N*exp (-a^2*(n:ℝ)/(π^2*β^2)) := by
dsimp only [E]
split_ifs with hg
· exact hN hn σ hg.1 C hg.2
· simp only [Set.preimage_empty,measureReal_empty]
positivity
apply (measureReal_mono ?_ (measure_ne_top μ _)).trans
(independent_finite_selection_bound hX hi E hE hb)
intro g hg
change R g ∈ E (X g)
dsimp only [E]
rw [ite_eq_left (show good (X g) from ⟨hg.1,hg.2.1⟩)]
exact hg.2.2
end SKRatio.Bins
end
end OAI
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