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import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.DensityUniqueness
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 RadialMixtureRepresentation (m : ℕ) : Prop :=
∀ δ : ℝ, 0 < δ →
(∀ r : ℝ, 0 < r → 0 ≤ RadialMixture m δ r) ∧
Measurable (RadialMixture m δ) ∧
(∫ r in Set.Ioi 0, RadialMixture m δ r) = 1 ∧
(∀ y : Space m, GaussianKernel m δ y =
∫ r in Set.Ioi ‖y‖, RadialMixture m δ r / (UnitBallVolume m * r^m))
def FullRankDensityFormula {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) : Prop :=
let exponent : ℝ≥0∞ := @OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))
∀ A : Fin m → Space d,
@LinearIndependent (Fin m) ℝ (Space d) A Real.commRing.toSemiring
(WithLp.instAddCommGroup exponent (Fin d → ℝ)).toAddCommMonoid
(WithLp.instModule (p := exponent) (K := ℝ) (V := Fin d → ℝ)) →
(volume : Measure (Fin m → ℝ)).withDensity
(fun y => ENNReal.ofReal (PointwiseDensity μ m (A,y))) =
@Measure.map (Space d) (Fin m → ℝ)
(WithLp.measurableSpace exponent (Fin d → ℝ)) inferInstance
(fun signal => fun 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 (A index)) μ
def PointwiseDensityProperties {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 p : ℕ) (H β R : ℝ) : Prop :=
letI : MeasurableSpace (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 reference : Measure ((Fin m → Space d) × (Fin m → ℝ)) := MatrixLabelReference d m
RadialMixtureRepresentation m ∧
(∀ x, 0 ≤ PointwiseDensity μ m x) ∧
Measurable (PointwiseDensity μ m) ∧
FullRankDensityFormula μ m ∧
reference.withDensity (fun x => ENNReal.ofReal (PointwiseDensity μ m x)) =
ProjectionLaw μ m ∧
(∀ F : ((Fin m → Space d) × (Fin m → ℝ)) → ℝ,
Measurable F → (∀ x, 0 ≤ F x) →
reference.withDensity (fun x => ENNReal.ofReal (F x)) =
ProjectionLaw μ m → PointwiseDensity μ m =ᵐ[reference] F) ∧
(∫⁻ x, ENNReal.ofReal (PointwiseDensity μ m x)^p ∂reference) ≤
ENNReal.ofReal (ProjectionNormScale m p H β R)^p
def PointwiseDensityStatement : 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, 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) (WithLp.measurableSpace exponent (Fin d → ℝ)),
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 → PointwiseDensityProperties μ m p H β R
theorem pointwise_density : PointwiseDensityStatement := by
intro d m p hd hm hmd hp β H R hgap hH hR z μ hμ hs hb hac
let : IsFiniteMeasure μ := hμ
obtain ⟨F,hFm,hF0,hden,hFs,hFnorm⟩ :=
exact_finite_density d m p hp hm β H R hgap hH hR z μ hs hb
have he := pointwiseDensity_ae_eq hm μ F hFm hF0 hden
refine ⟨?_, PointwiseDensity_nonneg μ m, measurable_PointwiseDensity μ m,
?_, ?_, pointwiseDensity_ae_eq hm μ, ?_⟩
· intro δ hδ
exact ⟨fun r hr => radialMixture_nonneg m δ hr, radialMixture_measurable m δ,
radialMixture_integral m hm hδ, radialMixture_tail m hm hδ⟩
· intro A hA
exact pointwiseDensity_fullRank hd hm hmd μ hac A hA
· exact (withDensity_congr_ae (he.fun_comp ENNReal.ofReal)).trans hden
· have hp0 : (p:ℝ) ≠ 0 := by exact_mod_cast (by omega : p ≠ 0)
have hpE : (p:ℝ≥0∞) ≠ 0 := by exact_mod_cast (by omega : p ≠ 0)
have hpow := ENNReal.rpow_le_rpow hFnorm (show 0 ≤ (p:ℝ) by positivity)
rw [eLpNorm_eq_lintegral_rpow_enorm_toReal hpE (by finiteness)
hFm.aestronglyMeasurable] at hpow
simp only [ENNReal.toReal_natCast] at hpow
rw [← ENNReal.rpow_mul, one_div_mul_cancel hp0, ENNReal.rpow_one] at hpow
simp only [ENNReal.rpow_natCast] at hpow
have hi : (fun x => ENNReal.ofReal (PointwiseDensity μ m x)^p) =ᵐ[MatrixLabelReference d m]
(fun x => ‖F x‖ₑ^p) := by
filter_upwards [he] with x hx
rw [hx, Real.enorm_eq_ofReal (hF0 x)]
rw [lintegral_congr_ae hi]
exact hpow
end MemoryPrecision
end
end OAI
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