File size: 3,492 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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.JointL1
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
def PointwiseDensity {d : ℕ} (μ : Measure (Space d)) (m : ℕ)
(x : (Fin m → Space d) × (Fin m → ℝ)) : ℝ := by
classical
exact if h : ∃ l : ℝ, Tendsto (fun i : ℕ => GaussianRegularization μ m (1/(i:ℝ)) x)
atTop (𝓝 l) then Classical.choose h else 0
lemma measurable_GaussianRegularization {d : ℕ} (μ : Measure (Space d)) [IsFiniteMeasure μ]
(m : ℕ) (δ : ℝ) : Measurable (GaussianRegularization μ m δ) := by
have he : GaussianRegularization μ m δ = fun x =>
(projectionSmoothing μ (gaussianPDF 0 ⟨δ^2,sq_nonneg δ⟩) m x).toReal :=
funext (GaussianRegularization_eq_toReal μ m δ)
rw [he]
exact (measurable_projectionSmoothing μ _ (measurable_gaussianPDF _ _) m).ennreal_toReal
lemma PointwiseDensity_eq_of_tendsto {d m : ℕ} (μ : Measure (Space d))
(x : (Fin m → Space d) × (Fin m → ℝ)) {l : ℝ}
(h : Tendsto (fun i : ℕ => GaussianRegularization μ m (1/(i:ℝ)) x) atTop (𝓝 l)) :
PointwiseDensity μ m x = l := by
classical
unfold PointwiseDensity
rw [dite_eq_left ⟨l,h⟩]
exact tendsto_nhds_unique (Classical.choose_spec (show ∃ l : ℝ,
Tendsto (fun i : ℕ => GaussianRegularization μ m (1/(i:ℝ)) x) atTop (𝓝 l) from ⟨l,h⟩)) h
lemma measurable_PointwiseDensity {d : ℕ} (μ : Measure (Space d)) [IsFiniteMeasure μ]
(m : ℕ) : Measurable (PointwiseDensity μ m) := by
classical
let S := {x : (Fin m → Space d) × (Fin m → ℝ) | ∃ l : ℝ,
Tendsto (fun i : ℕ => GaussianRegularization μ m (1/(i:ℝ)) x) atTop (𝓝 l)}
have hS : MeasurableSet S :=
measurableSet_exists_tendsto (fun i => measurable_GaussianRegularization μ m _)
apply measurable_of_tendsto_metrizable
(f := fun i => S.indicator (GaussianRegularization μ m (1/(i:ℝ))))
(fun i => (measurable_GaussianRegularization μ m _).indicator hS)
apply tendsto_pi_nhds.mpr
intro x
by_cases hx : x ∈ S
· rcases hx with ⟨l,hl⟩
rw [PointwiseDensity_eq_of_tendsto μ x hl]
simpa only [Set.indicator_of_mem (show x ∈ S from ⟨l,hl⟩)] using hl
· have he : PointwiseDensity μ m x = 0 := by
unfold PointwiseDensity
exact dite_eq_right hx
rw [he]
simpa only [Set.indicator_of_notMem hx] using
(tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0))
lemma GaussianRegularization_nonneg {d : ℕ} (μ : Measure (Space d)) (m : ℕ) (δ : ℝ)
(x : (Fin m → Space d) × (Fin m → ℝ)) : 0 ≤ GaussianRegularization μ m δ x :=
integral_nonneg (fun _ => Finset.prod_nonneg (fun _ _ => gaussianPDFReal_nonneg _ _ _))
lemma PointwiseDensity_nonneg {d : ℕ} (μ : Measure (Space d)) (m : ℕ)
(x : (Fin m → Space d) × (Fin m → ℝ)) : 0 ≤ PointwiseDensity μ m x := by
classical
unfold PointwiseDensity
split_ifs with h
· exact ge_of_tendsto (Classical.choose_spec h)
(Eventually.of_forall (fun i => GaussianRegularization_nonneg μ m _ x))
· exact le_rfl
end MemoryPrecision
end
end OAI
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