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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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.GaussianConvolution
import OAI.Probability.MemoryPrecision.PointwiseDensity
import OAI.Probability.MemoryPrecision.SphericalProjection
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 PiGaussianConvolution {m : ℕ} (ν : Measure (Fin m → ℝ)) (δ : ℝ) (y : Fin m → ℝ) : ℝ :=
∫ x, ∏ j, gaussianPDFReal 0 ⟨δ^2,sq_nonneg δ⟩ (y j - x j) ∂ν
lemma piGaussianConvolution_eq {m : ℕ} (ν : Measure (Fin m → ℝ)) (δ : ℝ) (y : Fin m → ℝ) :
PiGaussianConvolution ν δ y =
GaussianConvolution (ν.map (WithLp.toLp 2)) δ (WithLp.toLp 2 y) := by
unfold GaussianConvolution
rw [integral_map (WithLp.measurable_toLp 2 (Fin m → ℝ)).aemeasurable
((show Continuous (fun x : Space m => GaussianKernel m δ (WithLp.toLp 2 y - x)) from
(gaussianKernel_continuous m δ).comp (continuous_const.sub continuous_id)).aestronglyMeasurable)]
rfl
lemma ae_tendsto_piGaussianConvolution {m : ℕ} (hm : 1 ≤ m) (ν : Measure (Fin m → ℝ))
[IsFiniteMeasure ν] (hν : ν ≪ volume) :
∀ᵐ y ∂(volume : Measure (Fin m → ℝ)), Tendsto (fun δ : ℝ => PiGaussianConvolution ν δ y)
(𝓝[>] (0:ℝ)) (𝓝 ((ν.rnDeriv volume y).toReal)) := by
let e := MeasurableEquiv.toLp 2 (Fin m → ℝ)
have hp : MeasurePreserving e volume volume := PiLp.volume_preserving_toLp (Fin m)
have hac : ν.map e ≪ volume := by
rw [← hp.map_eq]
exact hν.map e.measurable
have ht := ae_tendsto_gaussianConvolution hm (ν.map e) hac
have ht' : ∀ᵐ y ∂(volume : Measure (Fin m → ℝ)),
Tendsto (fun δ : ℝ => GaussianConvolution (ν.map e) δ (e y))
(𝓝[>] (0:ℝ)) (𝓝 (((ν.map e).rnDeriv volume (e y)).toReal)) := by
have ht0 : ∀ᵐ y ∂(volume.map e),
Tendsto (fun δ : ℝ => GaussianConvolution (ν.map e) δ y)
(𝓝[>] (0:ℝ)) (𝓝 (((ν.map e).rnDeriv volume y).toReal)) := by
rwa [hp.map_eq]
exact e.measurableEmbedding.ae_map_iff.mp ht0
have hRN := e.measurableEmbedding.rnDeriv_map ν volume
rw [hp.map_eq] at hRN
filter_upwards [ht', hRN] with y hy hyRN
rw [hyRN] at hy
simpa only [piGaussianConvolution_eq, e, MeasurableEquiv.coe_toLp] using hy
lemma GaussianRegularization_eq_convolution {d m : ℕ} (μ : Measure (Space d))
(A : Fin m → Space d) (δ : ℝ) (y : Fin m → ℝ) :
GaussianRegularization μ m δ (A,y) =
PiGaussianConvolution (μ.map (fun s => fun j => ⟪s,A j⟫)) δ y := by
unfold PiGaussianConvolution
rw [integral_map (by fun_prop)
(((show Continuous (fun x : Fin m → ℝ => ∏ j,
gaussianPDFReal 0 ⟨δ^2,sq_nonneg δ⟩ (y j - x j)) from
by unfold gaussianPDFReal; fun_prop)).aestronglyMeasurable)]
rfl
lemma ae_tendsto_GaussianRegularization {d m : ℕ} (hm : 1 ≤ m) (μ : Measure (Space d))
[IsFiniteMeasure μ] (A : Fin m → Space d)
(hA : μ.map (fun s => fun j => ⟪s,A j⟫) ≪ (volume : Measure (Fin m → ℝ))) :
∀ᵐ y ∂(volume : Measure (Fin m → ℝ)), Tendsto
(fun δ : ℝ => GaussianRegularization μ m δ (A,y)) (𝓝[>] (0:ℝ))
(𝓝 (((μ.map (fun s => fun j => ⟪s,A j⟫)).rnDeriv volume y).toReal)) := by
simpa only [GaussianRegularization_eq_convolution] using
ae_tendsto_piGaussianConvolution hm (μ.map (fun s => fun j => ⟪s,A j⟫)) hA
lemma reciprocal_nat_tendsto :
Tendsto (fun i : ℕ => 1/(i:ℝ)) atTop (𝓝[>] (0:ℝ)) := by
refine tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_atTop_nhds_zero_nat, ?_⟩
filter_upwards [eventually_gt_atTop (0:ℕ)] with i hi
change 0 < 1/(i:ℝ)
positivity
lemma pointwiseDensity_fiber {d m : ℕ} (hm : 1 ≤ m) (μ : Measure (Space d))
[IsFiniteMeasure μ] (A : Fin m → Space d)
(hA : μ.map (fun s => fun j => ⟪s,A j⟫) ≪ (volume : Measure (Fin m → ℝ))) :
(volume : Measure (Fin m → ℝ)).withDensity
(fun y => ENNReal.ofReal (PointwiseDensity μ m (A,y))) =
μ.map (fun s => fun j => ⟪s,A j⟫) := by
let ν := μ.map (fun s => fun j => ⟪s,A j⟫)
have he : (fun y => ENNReal.ofReal (PointwiseDensity μ m (A,y))) =ᵐ[volume]
ν.rnDeriv volume := by
filter_upwards [ae_tendsto_GaussianRegularization hm μ A hA, ν.rnDeriv_lt_top volume]
with y hy hyfin
rw [PointwiseDensity_eq_of_tendsto μ (A,y) (hy.comp reciprocal_nat_tendsto)]
exact ENNReal.ofReal_toReal hyfin.ne
rw [withDensity_congr_ae he]
exact Measure.withDensity_rnDeriv_eq _ _ hA
lemma pointwiseDensity_fullRank {d m : ℕ} (hd : 2 ≤ d) (hm : 1 ≤ m) (hmd : m < d)
(μ : Measure (Space d)) [IsFiniteMeasure μ] (hμ : μ ≪ UniformSphere d)
(A : Fin m → Space d) (hA : LinearIndependent ℝ A) :
(volume : Measure (Fin m → ℝ)).withDensity
(fun y => ENNReal.ofReal (PointwiseDensity μ m (A,y))) =
μ.map (fun s => fun j => ⟪s,A j⟫) :=
pointwiseDensity_fiber hm μ A (spherical_projection_ac hd hmd A hA μ hμ)
end MemoryPrecision
end
end OAI
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