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import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.FiniteDensity
import OAI.Probability.MemoryPrecision.KernelModel
import OAI.Probability.MemoryPrecision.LabelTranslations
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 UniformSphere (d : ℕ) : @Measure (Space d)
(WithLp.measurableSpace
(@OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))) (Fin d → ℝ)) :=
let euclideanMeasurable := WithLp.measurableSpace
(@OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))) (Fin d → ℝ)
@Measure.map (Kernel.Sphere d) (Space d)
(MeasurableSpace.comap Subtype.val euclideanMeasurable) euclideanMeasurable
Subtype.val (Kernel.UniformSphere d)
def GaussianRegularization {d : ℕ}
(μ : @Measure (Space d)
(WithLp.measurableSpace
(@OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))) (Fin d → ℝ)))
(m : ℕ) (δ : ℝ)
(Ay : (Fin m → Space d) × (Fin m → ℝ)) : ℝ :=
let exponent : ℝ≥0∞ := @OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))
letI : MeasurableSpace (Space d) := WithLp.measurableSpace exponent (Fin d → ℝ)
∫ signal, ∏ index, gaussianPDFReal 0 ⟨δ^2, sq_nonneg δ⟩
(Ay.2 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)))
signal (Ay.1 index)) ∂μ
lemma gaussianPDF_zero_sub_comm (v : ℝ≥0) (a b : ℝ) :
gaussianPDF 0 v (a-b) = gaussianPDF 0 v (b-a) := by
simp only [gaussianPDF, gaussianPDFReal, sub_zero]
congr 4
ring
lemma GaussianRegularization_eq_toReal {d : ℕ} (μ : Measure (Space d))
[IsFiniteMeasure μ] (m : ℕ) (δ : ℝ)
(x : (Fin m → Space d) × (Fin m → ℝ)) :
GaussianRegularization μ m δ x =
(projectionSmoothing μ (gaussianPDF 0 ⟨δ^2,sq_nonneg δ⟩) m x).toReal := by
let v : ℝ≥0 := ⟨δ^2,sq_nonneg δ⟩
change (∫ s, ∏ j, gaussianPDFReal 0 v (x.2 j - ⟪s,x.1 j⟫) ∂μ) = _
have hn : ∀ s : Space d, 0 ≤ ∏ j, gaussianPDFReal 0 v (x.2 j - ⟪s,x.1 j⟫) :=
fun s => Finset.prod_nonneg (fun _ _ => gaussianPDFReal_nonneg _ _ _)
have hm : Measurable (fun s : Space d => ∏ j, gaussianPDFReal 0 v (x.2 j - ⟪s,x.1 j⟫)) := by
fun_prop
rw [integral_eq_lintegral_of_nonneg_ae (Eventually.of_forall hn) hm.aestronglyMeasurable]
unfold projectionSmoothing
congr 1
apply lintegral_congr
intro s
rw [ENNReal.ofReal_prod_of_nonneg (fun _ _ => gaussianPDFReal_nonneg _ _ _)]
rfl
lemma GaussianRegularization_ae_eq_average {d : ℕ} (μ : Measure (Space d))
[IsFiniteMeasure μ] (m : ℕ)
(F : ((Fin m → Space d) × (Fin m → ℝ)) → ℝ)
(hF : Measurable F) (hF0 : ∀ x, 0 ≤ F x)
(hden : (MatrixLabelReference d m).withDensity (fun x => ENNReal.ofReal (F x)) =
ProjectionLaw μ m) (δ : ℝ) (hδ : δ ≠ 0) :
GaussianRegularization μ m δ =ᵐ[MatrixLabelReference d m]
averageTranslate (Measure.pi (fun _ : Fin m => gaussianReal 0 1)) δ F := by
have he := projectionSmoothing_ae_eq_fiberSmoothing μ m
(fun x => ENNReal.ofReal (F x)) hF.ennreal_ofReal hden
(gaussianPDF 0 ⟨δ^2,sq_nonneg δ⟩) (measurable_gaussianPDF _ _)
filter_upwards [he] with x hx
rw [GaussianRegularization_eq_toReal, hx]
unfold averageTranslate
rw [integral_eq_lintegral_of_nonneg_ae (Eventually.of_forall (fun u => hF0 _))
((hF.comp (by fun_prop)).aestronglyMeasurable)]
congr 1
change fiberSmoothing volume (fun x => ENNReal.ofReal (F x))
(fun y : Fin m → ℝ => ∏ j, gaussianPDF 0 _ (y j)) x =
∫⁻ u : Fin m → ℝ, ENNReal.ofReal (F (x.1,fun j => x.2 j + δ * u j))
∂Measure.pi (fun _ => gaussianReal 0 1)
rw [lintegral_gaussian_shift_scale m δ hδ x.2 (fun y => ENNReal.ofReal (F (x.1,y)))
(by fun_prop)]
apply lintegral_congr
intro y
congr 1
apply Finset.prod_congr rfl
intro j _
exact gaussianPDF_zero_sub_comm _ _ _
lemma memLp_density_of_finite {X : Type*} [MeasurableSpace X]
(ν₀ η : Measure X) [IsFiniteMeasure η] {F : X → ℝ}
(hFm : Measurable F) (hF0 : ∀ x, 0 ≤ F x)
(hden : ν₀.withDensity (fun x => ENNReal.ofReal (F x)) = η) :
MemLp F 1 ν₀ := by
rw [memLp_iff, eLpNorm_one_eq_lintegral_enorm hFm.aestronglyMeasurable]
have he : (fun x => ‖F x‖ₑ) = (fun x => ENNReal.ofReal (F x)) := by
funext x
exact Real.enorm_eq_ofReal (hF0 x)
rw [he]
have he2 := congrArg (fun ν : Measure X => ν Set.univ) hden
rw [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at he2
rw [he2]
exact measure_lt_top _ _
lemma density_ae_unique_real {X : Type*} [MeasurableSpace X]
(ν₀ : Measure X) [SigmaFinite ν₀] {F G : X → ℝ}
(hFm : Measurable F) (hF0 : ∀ x, 0 ≤ F x)
(hGm : Measurable G) (hG0 : ∀ x, 0 ≤ G x)
(hden : ν₀.withDensity (fun x => ENNReal.ofReal (F x)) =
ν₀.withDensity (fun x => ENNReal.ofReal (G x))) : F =ᵐ[ν₀] G := by
have he := (withDensity_eq_iff_of_sigmaFinite hFm.ennreal_ofReal.aemeasurable
hGm.ennreal_ofReal.aemeasurable).mp hden
filter_upwards [he] with x hx
exact (ENNReal.ofReal_eq_ofReal_iff (hF0 x) (hG0 x)).mp hx
def JointL1Statement : Prop :=
let two : ℝ := @OfNat.ofNat ℝ 2 (@instOfNatAtLeastTwo ℝ 2 Real.commRing.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))
let exponent : ℝ≥0∞ := @OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))
∀ d m p : ℕ,
letI : MeasurableSpace (Space d) := WithLp.measurableSpace exponent (Fin d → ℝ)
2 ≤ d → 1 ≤ m → m < d → 2 ≤ p →
∀ β H R : ℝ, (m : ℝ)+(p : ℝ)-two < β → 0 < H → 0 < R →
∀ z : Space d, ∀ μ : Measure (Space d), IsFiniteMeasure μ →
@Measure.support (Space d)
(PiLp.topologicalSpace (p := exponent) (β := fun _ : Fin d => ℝ))
(WithLp.measurableSpace exponent (Fin d → ℝ)) μ ⊆
@Metric.closedBall (Space d)
(PiLp.instPseudoMetricSpace (p := exponent) (β := fun _ : Fin d => ℝ)) z R →
AmbientBallBound μ H β →
μ ≪ UniformSphere d →
∀ F : ((Fin m → Space d) × (Fin m → ℝ)) → ℝ,
Measurable F → (∀ x, 0 ≤ F x) →
(MatrixLabelReference d m).withDensity (fun x => ENNReal.ofReal (F x)) =
ProjectionLaw μ m →
Tendsto (fun δ => eLpNorm (fun Ay => GaussianRegularization μ m δ Ay - F Ay)
1 (MatrixLabelReference d m)) (𝓝[Set.Ioi 0] 0) (𝓝 0) ∧
eLpNorm F (p : ℝ≥0∞) (MatrixLabelReference d m) ≤
ENNReal.ofReal (ProjectionNormScale m p H β R)
theorem joint_l1 : JointL1Statement := by
intro d m p hd hm hmd hp β H R hgap hH hR z μ hμ hs hb hac F hFm hF0 hden
have : IsFiniteMeasure μ := hμ
have : IsFiniteMeasure (ProjectionLaw μ m) := by unfold ProjectionLaw; infer_instance
have : SigmaFinite (MatrixLabelReference d m) := by unfold MatrixLabelReference; infer_instance
have hF := memLp_density_of_finite (MatrixLabelReference d m) (ProjectionLaw μ m) hFm hF0 hden
constructor
· have ht := tendsto_eLpNorm_averageTranslate_sub
(Measure.pi (fun _ : Fin m => stdGaussian (Space d)))
(volume : Measure (Fin m → ℝ))
(Measure.pi (fun _ : Fin m => gaussianReal 0 1)) hFm hF
apply (ht.mono_left nhdsWithin_le_nhds).congr'
filter_upwards [self_mem_nhdsWithin] with δ hδ
apply eLpNorm_congr_ae
filter_upwards [GaussianRegularization_ae_eq_average μ m F hFm hF0 hden δ
(ne_of_gt hδ)] with x hx
exact congrArg (fun t => t - F x) hx.symm
· obtain ⟨G,hGm,hG0,hGden,hGs,hGnorm⟩ :=
exact_finite_density d m p hp hm β H R hgap hH hR z μ hs hb
have he := density_ae_unique_real (MatrixLabelReference d m) hFm hF0 hGm hG0
(hden.trans hGden.symm)
exact (eLpNorm_congr_ae he).symm ▸ hGnorm
end MemoryPrecision
end
end OAI
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