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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.RadialMixture
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 gaussianKernel_scale (m : ℕ) {δ : ℝ} (hδ : 0 < δ) (y : Space m) :
GaussianKernel m δ (δ • y) = δ⁻¹ ^ m * GaussianKernel m 1 y := by
have he (a : ℝ) : gaussianPDFReal 0 ⟨δ^2,sq_nonneg δ⟩ (δ*a) =
δ⁻¹ * gaussianPDFReal 0 1 a := by
have h := gaussianPDFReal_inv_mul (μ := 0) (v := 1) hδ.ne' (δ*a)
simp only [inv_mul_cancel_left₀ hδ.ne', mul_zero, mul_one, abs_of_pos hδ] at h
rw [h]
field_simp
rfl
unfold GaussianKernel
simp only [PiLp.smul_apply, smul_eq_mul, one_pow]
simp_rw [he]
simp only [Finset.prod_mul_distrib, Finset.prod_const, Finset.card_univ, Fintype.card_fin]
rfl
lemma radialMixture_scale (m : ℕ) {δ : ℝ} (hδ : 0 < δ) (r : ℝ) :
RadialMixture m δ (δ*r) = δ⁻¹ * RadialMixture m 1 r := by
unfold RadialMixture
rw [mul_smul, gaussianKernel_scale m hδ]
simp only [one_pow, div_one, mul_pow, inv_pow]
field_simp
def BallAverage {m : ℕ} (ν : Measure (Space m)) (y : Space m) (r : ℝ) : ℝ :=
(ν (Metric.ball y r)).toReal / (UnitBallVolume m * r^m)
lemma ballAverage_measurable {m : ℕ} (ν : Measure (Space m)) (y : Space m) :
Measurable (BallAverage ν y) := by
have h : Monotone (fun r : ℝ => ν (Metric.ball y r)) := fun r t h =>
measure_mono (Metric.ball_subset_ball h)
exact h.measurable.ennreal_toReal.div (by fun_prop)
lemma ballAverage_nonneg {m : ℕ} (ν : Measure (Space m)) (y : Space m) {r : ℝ}
(hr : 0 < r) : 0 ≤ BallAverage ν y r := by
exact div_nonneg ENNReal.toReal_nonneg (mul_nonneg (unitBallVolume_pos m).le
(pow_nonneg hr.le _))
lemma ac_ball_eq_closedBall {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m))
(hν : ν ≪ volume) (y : Space m) (r : ℝ) :
ν (Metric.ball y r) = ν (Metric.closedBall y r) := by
have : NeZero m := ⟨by omega⟩
apply measure_eq_measure_of_null_sdiff Metric.ball_subset_closedBall
rw [Metric.closedBall_sdiff_ball]
exact hν (Measure.addHaar_sphere volume y r)
lemma ae_tendsto_ballAverage {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m))
[IsFiniteMeasure ν] (hν : ν ≪ volume) :
∀ᵐ y ∂(volume : Measure (Space m)), Tendsto (BallAverage ν y)
(𝓝[>] (0:ℝ)) (𝓝 ((ν.rnDeriv volume y).toReal)) := by
have : NeZero m := ⟨by omega⟩
filter_upwards [Besicovitch.ae_tendsto_rnDeriv ν volume, ν.rnDeriv_lt_top volume] with y hy hyfin
have ht := (ENNReal.continuousAt_toReal hyfin.ne).tendsto.comp hy
apply ht.congr'
filter_upwards [self_mem_nhdsWithin] with r hr
unfold BallAverage
simp only [Function.comp_apply]
rw [ENNReal.toReal_div, Measure.addHaar_closedBall_eq_addHaar_ball volume y r,
← ac_ball_eq_closedBall hm ν hν y r, volume_ball_eq m hm y hr,
ENNReal.toReal_ofReal (mul_nonneg (unitBallVolume_pos m).le (pow_nonneg hr.le _))]
lemma ballAverage_bounded {m : ℕ} (ν : Measure (Space m)) [IsFiniteMeasure ν]
(y : Space m) {f : ℝ} (hf : 0 ≤ f)
(ht : Tendsto (BallAverage ν y) (𝓝[>] (0:ℝ)) (𝓝 f)) :
∃ B : ℝ, 0 ≤ B ∧ ∀ r : ℝ, 0 < r → ‖BallAverage ν y r‖ ≤ B := by
rcases Metric.tendsto_nhdsWithin_nhds.mp ht 1 zero_lt_one with ⟨b,hb,hnear⟩
let C : ℝ := max (f+1) ((ν Set.univ).toReal / (UnitBallVolume m * b^m))
refine ⟨C, le_trans (by linarith : 0 ≤ f+1) (le_max_left _ _), ?_⟩
intro r hr
rw [Real.norm_eq_abs, abs_of_nonneg (ballAverage_nonneg ν y hr)]
by_cases h : r < b
· have he := hnear hr (by simpa [Real.dist_eq, abs_of_pos hr] using h)
have he' : BallAverage ν y r < f+1 := by
rw [Real.dist_eq] at he
have := (abs_lt.mp he).2
linarith
exact he'.le.trans (le_max_left _ _)
· apply le_trans ?_ (le_max_right _ _)
unfold BallAverage
have hvol : 0 < UnitBallVolume m := unitBallVolume_pos m
apply div_le_div₀ (by positivity) ?_ (by positivity) ?_
· exact (ENNReal.toReal_le_toReal (measure_ne_top _ _) (measure_ne_top _ _)).mpr
(measure_mono (Set.subset_univ _))
· exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hb.le (le_of_not_gt h) _)
(unitBallVolume_pos m).le
lemma tendsto_radial_average {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m)) [IsFiniteMeasure ν]
(y : Space m) {f : ℝ} (hf : 0 ≤ f)
(ht : Tendsto (BallAverage ν y) (𝓝[>] (0:ℝ)) (𝓝 f)) :
Tendsto (fun δ : ℝ => ∫ r in Ioi 0, RadialMixture m 1 r * BallAverage ν y (δ*r))
(𝓝[>] (0:ℝ)) (𝓝 f) := by
obtain ⟨B,hB,hbound⟩ := ballAverage_bounded ν y hf ht
have hw : IntegrableOn (RadialMixture m 1) (Ioi 0) :=
integrable_of_integral_eq_one (radialMixture_integral m hm zero_lt_one)
have hl := tendsto_integral_filter_of_dominated_convergence
(μ := volume.restrict (Ioi 0)) (l := 𝓝[>] (0:ℝ))
(F := fun δ r => RadialMixture m 1 r * BallAverage ν y (δ*r))
(f := fun r => RadialMixture m 1 r * f)
(fun r => RadialMixture m 1 r * B)
(Eventually.of_forall (fun δ =>
((radialMixture_measurable m 1).mul ((ballAverage_measurable ν y).comp
(by fun_prop))).aestronglyMeasurable))
(show ∀ᶠ δ in 𝓝[>] (0:ℝ), ∀ᵐ r ∂volume.restrict (Ioi 0),
‖RadialMixture m 1 r * BallAverage ν y (δ*r)‖ ≤ RadialMixture m 1 r * B from by
filter_upwards [self_mem_nhdsWithin] with δ hδ
filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr
rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (radialMixture_nonneg m 1 hr)]
exact mul_le_mul_of_nonneg_left (hbound _ (mul_pos hδ hr)) (radialMixture_nonneg m 1 hr))
(hw.mul_const B) (show ∀ᵐ r ∂volume.restrict (Ioi 0),
Tendsto (fun δ => RadialMixture m 1 r * BallAverage ν y (δ*r))
(𝓝[>] (0:ℝ)) (𝓝 (RadialMixture m 1 r * f)) from by
filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr
apply Tendsto.const_mul
apply ht.comp
apply tendsto_nhdsWithin_iff.mpr
refine ⟨?_, ?_⟩
· simpa using ((continuous_id.mul_const r).tendsto (0:ℝ)).mono_left nhdsWithin_le_nhds
· filter_upwards [self_mem_nhdsWithin] with δ hδ
change 0 < δ*r
exact mul_pos hδ hr)
simpa only [integral_mul_const, radialMixture_integral m hm zero_lt_one, one_mul] using hl
end MemoryPrecision
end
end OAI
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