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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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.BallAverages
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_translated_ball_mixture (m : ℕ) (hm : 1 ≤ m) {δ : ℝ} (hδ : 0 < δ)
(y x : Space m) :
ENNReal.ofReal (GaussianKernel m δ (y-x)) =
∫⁻ r in Ioi 0, (Metric.ball y r).indicator
(fun _ => ENNReal.ofReal ((r/δ^2)*GaussianKernel m δ (r • CoordinateUnit m))) x := by
rw [gaussianKernel_ball_mixture m hm hδ]
apply lintegral_congr
intro r
have he : y-x ∈ Metric.ball (0 : Space m) r ↔ x ∈ Metric.ball y r := by
simp only [Metric.mem_ball, dist_eq_norm, sub_zero, norm_sub_rev]
by_cases hx : x ∈ Metric.ball y r
· rw [Set.indicator_of_mem (he.mpr hx), Set.indicator_of_mem hx]
· rw [Set.indicator_of_notMem (mt he.mp hx), Set.indicator_of_notMem hx]
def GaussianConvolution {m : ℕ} (ν : Measure (Space m)) (δ : ℝ) (y : Space m) : ℝ :=
∫ x, GaussianKernel m δ (y-x) ∂ν
lemma gaussianConvolution_shell {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m))
[IsFiniteMeasure ν] {δ : ℝ} (hδ : 0 < δ) (y : Space m) :
GaussianConvolution ν δ y = ∫ r in Ioi 0,
((r/δ^2)*GaussianKernel m δ (r • CoordinateUnit m)) * (ν (Metric.ball y r)).toReal := by
have hk : Measurable (GaussianKernel m δ) := (gaussianKernel_continuous m δ).measurable
have hmeas : Measurable (fun z : Space m × ℝ =>
(Metric.ball y z.2).indicator
(fun _ => ENNReal.ofReal ((z.2/δ^2)*GaussianKernel m δ (z.2 • CoordinateUnit m))) z.1) := by
have he : (fun z : Space m × ℝ => (Metric.ball y z.2).indicator
(fun _ => ENNReal.ofReal ((z.2/δ^2)*GaussianKernel m δ (z.2 • CoordinateUnit m))) z.1) =
{z : Space m × ℝ | dist z.1 y < z.2}.indicator
(fun z => ENNReal.ofReal ((z.2/δ^2)*GaussianKernel m δ (z.2 • CoordinateUnit m))) := by
ext z
by_cases hz : z.1 ∈ Metric.ball y z.2
· rw [Set.indicator_of_mem hz, Set.indicator_of_mem
(show z ∈ {z : Space m × ℝ | dist z.1 y < z.2} from hz)]
· rw [Set.indicator_of_notMem hz, Set.indicator_of_notMem
(show z ∉ {z : Space m × ℝ | dist z.1 y < z.2} from hz)]
rw [he]
exact (by fun_prop : Measurable (fun z : Space m × ℝ =>
ENNReal.ofReal ((z.2/δ^2)*GaussianKernel m δ (z.2 • CoordinateUnit m)))).indicator
(measurableSet_lt (by fun_prop) (by fun_prop))
unfold GaussianConvolution
rw [integral_eq_lintegral_of_nonneg_ae
(Eventually.of_forall (fun _ => gaussianKernel_nonneg m δ _))
((hk.comp (by fun_prop)).aestronglyMeasurable)]
simp_rw [gaussianKernel_translated_ball_mixture m hm hδ y]
rw [lintegral_lintegral_swap hmeas.aemeasurable]
simp_rw [lintegral_indicator Metric.isOpen_ball.measurableSet, setLIntegral_const]
have hmass : Measurable (fun r : ℝ => (ν (Metric.ball y r)).toReal) :=
(show Monotone (fun r : ℝ => ν (Metric.ball y r)) from
fun a b h => measure_mono (Metric.ball_subset_ball h)).measurable.ennreal_toReal
rw [integral_eq_lintegral_of_nonneg_ae
((ae_restrict_mem measurableSet_Ioi).mono (fun r hr =>
mul_nonneg (mul_nonneg (div_nonneg hr.le (sq_nonneg δ)) (gaussianKernel_nonneg m δ _))
ENNReal.toReal_nonneg)) (by fun_prop)]
congr 1
apply lintegral_congr_ae
filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr
rw [ENNReal.ofReal_mul (mul_nonneg (div_nonneg hr.le (sq_nonneg δ))
(gaussianKernel_nonneg m δ _)), ENNReal.ofReal_toReal (measure_ne_top _ _)]
lemma gaussianConvolution_mixture {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m))
[IsFiniteMeasure ν] {δ : ℝ} (hδ : 0 < δ) (y : Space m) :
GaussianConvolution ν δ y = ∫ r in Ioi 0, RadialMixture m δ r * BallAverage ν y r := by
rw [gaussianConvolution_shell hm ν hδ y]
apply integral_congr_ae
filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr
have hr0 : 0 < r := hr
unfold RadialMixture BallAverage
field_simp [(unitBallVolume_pos m).ne', hr0.ne']
lemma gaussianConvolution_scaled_mixture {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m))
[IsFiniteMeasure ν] {δ : ℝ} (hδ : 0 < δ) (y : Space m) :
GaussianConvolution ν δ y = ∫ r in Ioi 0, RadialMixture m 1 r * BallAverage ν y (δ*r) := by
rw [gaussianConvolution_mixture hm ν hδ y]
have he := integral_comp_mul_left_Ioi'
(fun r => RadialMixture m δ r * BallAverage ν y r) 0 hδ
simp only [mul_zero, smul_eq_mul] at he
rw [← he, ← integral_const_mul]
apply integral_congr_ae
exact Eventually.of_forall (fun radius => by
dsimp only
rw [radialMixture_scale m hδ]
field_simp [hδ.ne'])
lemma ae_tendsto_gaussianConvolution {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Space m))
[IsFiniteMeasure ν] (hν : ν ≪ volume) :
∀ᵐ y ∂(volume : Measure (Space m)), Tendsto (fun δ : ℝ => GaussianConvolution ν δ y)
(𝓝[>] (0:ℝ)) (𝓝 ((ν.rnDeriv volume y).toReal)) := by
filter_upwards [ae_tendsto_ballAverage hm ν hν] with y hy
apply (tendsto_radial_average hm ν y ENNReal.toReal_nonneg hy).congr'
filter_upwards [self_mem_nhdsWithin] with δ hδ
exact (gaussianConvolution_scaled_mixture hm ν hδ y).symm
end MemoryPrecision
end
end OAI
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