File size: 6,384 Bytes
a2ee19a | 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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.AmbientGram
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision
open NoiselessRegression
lemma space_finiteDimensional (d : ℕ) : FiniteDimensional ℝ (Space d) := by
infer_instance
section WeightedImage
variable {X : Type*} [MeasurableSpace X] {d m N : ℕ}
def WeightedBlock (ι : X → Space d) (h : Fin N → X → ℝ≥0∞)
(g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞) (x : X) : ℝ≥0∞ :=
let exponent : ℝ≥0∞ := @OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))
let euclideanMeasurable := @WithLp.measurableSpace exponent (Fin d → ℝ) inferInstance
let rowMeasurable := @MeasurableSpace.pi (Fin m) (fun _ => Space d)
(fun _ => euclideanMeasurable)
@MeasureTheory.lintegral (Fin m → Space d) rowMeasurable
(@Measure.pi (Fin m) (fun _ => Space d) inferInstance (fun _ => euclideanMeasurable)
(fun _ => @stdGaussian (Space d)
(PiLp.normedAddCommGroup exponent (fun _ : Fin d => ℝ))
(PiLp.innerProductSpace (𝕜 := ℝ) (fun _ : Fin d => ℝ))
(space_finiteDimensional d) euclideanMeasurable))
(fun rows => ∑ state, g state (rows, fun index => @inner ℝ (Space d)
(@InnerProductSpace.toInner ℝ (Space d) _
(PiLp.seminormedAddCommGroup exponent (fun _ : Fin d => ℝ))
(PiLp.innerProductSpace (𝕜 := ℝ) (fun _ : Fin d => ℝ)))
(ι x) (rows index)) * h state x)
lemma measurable_weightedBlock {ι : X → Space d} (hι : Measurable ι)
{h : Fin N → X → ℝ≥0∞} (hh : ∀ w, Measurable (h w))
{g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞}
(hg : ∀ w, Measurable (g w)) : Measurable (WeightedBlock ι h g) := by
apply Measurable.lintegral_prod_left
exact Finset.measurable_sum _ (fun w _ => ((hg w).comp (by fun_prop)).mul ((hh w).comp measurable_snd))
omit [MeasurableSpace X] in
lemma weightedBlock_le_one (ι : X → Space d) {h : Fin N → X → ℝ≥0∞}
(hh1 : ∀ w x, h w x ≤ 1)
{g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞}
(hg1 : ∀ x, ∑ w, g w x ≤ 1) (x : X) : WeightedBlock ι h g x ≤ 1 := by
unfold WeightedBlock
calc
_ ≤ ∫⁻ _A : Fin m → Space d, (1:ℝ≥0∞) ∂Measure.pi (fun _ => stdGaussian (Space d)) := by
apply lintegral_mono
intro A
exact (Finset.sum_le_sum (fun w _ => mul_le_of_le_one_right' (hh1 w x))).trans (hg1 _)
_ = 1 := by simp
lemma weighted_image_finite (μ : Measure X) [IsFiniteMeasure μ] (ι : X → Space d)
(h : X → ℝ≥0∞) (hh1 : ∀ x, h x ≤ 1) :
IsFiniteMeasure ((μ.withDensity h).map ι) := by
let := isFiniteMeasure_withDensity (μ := μ) (ne_of_lt ((lintegral_mono hh1).trans_lt (by simp)))
infer_instance
lemma weighted_image_ball (μ : Measure X) {ι : X → Space d} (hι : Measurable ι)
(h : X → ℝ≥0∞) (z : Space d) (r : ℝ) :
(μ.withDensity h).map ι (Metric.closedBall z r) =
∫⁻ x in {x | dist (ι x) z ≤ r}, h x ∂μ := by
rw [Measure.map_apply hι Metric.isClosed_closedBall.measurableSet,
withDensity_apply _ (hι Metric.isClosed_closedBall.measurableSet)]
rfl
lemma projectionLaw_lintegral (ν : Measure (Space d)) [SFinite ν]
(g : ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞) (hg : Measurable g) :
(∫⁻ x, g x ∂projectionLaw ν m) =
∫⁻ s, ∫⁻ A, g (A,fun j => ⟪s,A j⟫)
∂Measure.pi (fun _ : Fin m => stdGaussian (Space d)) ∂ν := by
have hobs : Measurable (fun x : (Fin m → Space d) × Space d => g (x.1,fun j => ⟪x.2,x.1 j⟫)) :=
hg.comp (by fun_prop)
rw [projectionLaw, lintegral_map hg (by fun_prop)]
exact lintegral_prod_symm _ hobs.aemeasurable
lemma weightedBlock_mass_identity (μ : Measure X) [IsFiniteMeasure μ]
{ι : X → Space d} (hι : Measurable ι)
{h : Fin N → X → ℝ≥0∞} (hh : ∀ w, Measurable (h w)) (hh1 : ∀ w x, h w x ≤ 1)
{g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞}
(hg : ∀ w, Measurable (g w)) {B : Set (Space d)} (hB : MeasurableSet B) :
(∫⁻ x in ι ⁻¹' B, WeightedBlock ι h g x ∂μ) =
∑ w, ∫⁻ x, g w x ∂projectionLaw (((μ.withDensity (h w)).map ι).restrict B) m := by
classical
let (w : Fin N) : IsFiniteMeasure (μ.withDensity (h w)) :=
isFiniteMeasure_withDensity (ne_of_lt ((lintegral_mono (hh1 w)).trans_lt (by simp)))
let G (w : Fin N) (x : X) (A : Fin m → Space d) := g w (A, fun j => ⟪ι x,A j⟫) * h w x
have hG (w) : Measurable (Function.uncurry (G w)) :=
((hg w).comp (by fun_prop)).mul ((hh w).comp measurable_fst)
have he (w) : (∫⁻ u, g w u ∂projectionLaw (((μ.withDensity (h w)).map ι).restrict B) m) =
∫⁻ x in ι ⁻¹' B, ∫⁻ A, G w x A
∂Measure.pi (fun _ : Fin m => stdGaussian (Space d)) ∂μ := by
have hobs : Measurable (fun x : (Fin m → Space d) × Space d =>
g w (x.1,fun j => ⟪x.2,x.1 j⟫)) := (hg w).comp (by fun_prop)
have htech : Measurable (fun x : (Fin m → Space d) × X =>
g w (x.1,fun j => ⟪ι x.2,x.1 j⟫)) := (hg w).comp (by fun_prop)
rw [projectionLaw_lintegral _ _ (hg w), Measure.restrict_map hι hB,
lintegral_map hobs.lintegral_prod_left hι,
restrict_withDensity (hι hB), lintegral_withDensity_eq_lintegral_mul _ (hh w)
htech.lintegral_prod_left]
apply lintegral_congr
intro x
simp only [Pi.mul_apply]
rw [← lintegral_const_mul' _ _ (ne_of_lt ((hh1 w x).trans_lt (by simp)))]
apply lintegral_congr
intro A
exact mul_comm _ _
simp_rw [he]
rw [← lintegral_finsetSum _ (fun w _ => (hG w).lintegral_prod_right)]
apply lintegral_congr
intro x
unfold WeightedBlock
exact lintegral_finsetSum _ (fun w _ => (hG w).comp (measurable_const.prodMk measurable_id))
end WeightedImage
end MemoryPrecision
end
end OAI
|