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import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.DensityPassage
import OAI.Probability.MemoryPrecision.SharpGaussian
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 MatrixLabelReference (d m : ℕ) : @Measure ((Fin m → Space d) × (Fin m → ℝ))
(@MeasurableSpace.prod (Fin m → Space d) (Fin m → ℝ)
(@MeasurableSpace.pi (Fin m) (fun _ => Space d)
(fun _ => WithLp.measurableSpace
(@OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))) (Fin d → ℝ)))
inferInstance) :=
let exponent : ℝ≥0∞ := @OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))
let euclideanMeasurable := WithLp.measurableSpace exponent (Fin d → ℝ)
let rowMeasurable := @MeasurableSpace.pi (Fin m) (fun _ => Space d)
(fun _ => euclideanMeasurable)
let rowLaw := @Measure.pi (Fin m) (fun _ => Space d) inferInstance
(fun _ => euclideanMeasurable)
(fun _ => @stdGaussian (Space d)
(PiLp.normedAddCommGroup exponent (fun _ : Fin d => ℝ))
(PiLp.innerProductSpace (𝕜 := ℝ) (fun _ : Fin d => ℝ))
inferInstance euclideanMeasurable)
@Measure.prod _ _ rowMeasurable inferInstance rowLaw volume
def ProjectionLaw {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) : @Measure ((Fin m → Space d) × (Fin m → ℝ))
(@MeasurableSpace.prod (Fin m → Space d) (Fin m → ℝ)
(@MeasurableSpace.pi (Fin m) (fun _ => Space d)
(fun _ => WithLp.measurableSpace
(@OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))) (Fin d → ℝ)))
inferInstance) :=
let exponent : ℝ≥0∞ := @OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))
let euclideanMeasurable := WithLp.measurableSpace exponent (Fin d → ℝ)
let rowMeasurable := @MeasurableSpace.pi (Fin m) (fun _ => Space d)
(fun _ => euclideanMeasurable)
let rowLaw := @Measure.pi (Fin m) (fun _ => Space d) inferInstance
(fun _ => euclideanMeasurable)
(fun _ => @stdGaussian (Space d)
(PiLp.normedAddCommGroup exponent (fun _ : Fin d => ℝ))
(PiLp.innerProductSpace (𝕜 := ℝ) (fun _ : Fin d => ℝ))
inferInstance euclideanMeasurable)
@Measure.map ((Fin m → Space d) × Space d) ((Fin m → Space d) × (Fin m → ℝ))
(@MeasurableSpace.prod _ _ rowMeasurable euclideanMeasurable)
(@MeasurableSpace.prod _ _ rowMeasurable inferInstance)
(fun pair => (pair.1, 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)))
pair.2 (pair.1 index)))
(@Measure.prod _ _ rowMeasurable euclideanMeasurable rowLaw μ)
def ProjectedEllipsoid {d m : ℕ} (z : Space d) (R : ℝ) :
Set ((Fin m → Space d) × (Fin m → ℝ)) :=
let exponent : ℝ≥0∞ := @OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))
let euclideanInner := @InnerProductSpace.toInner ℝ (Space d) _
(PiLp.seminormedAddCommGroup exponent (fun _ : Fin d => ℝ))
(PiLp.innerProductSpace (𝕜 := ℝ) (fun _ : Fin d => ℝ))
{pair | ∃ displacement ∈ @Metric.closedBall (Space d)
(PiLp.instPseudoMetricSpace (p := exponent) (β := fun _ : Fin d => ℝ))
(WithLp.toLp exponent (fun _ => (0 : ℝ))) R,
pair.2 = fun index => @inner ℝ (Space d) euclideanInner z (pair.1 index) +
@inner ℝ (Space d) euclideanInner displacement (pair.1 index)}
def ProjectionNormScale (m p : ℕ) (H β R : ℝ) : ℝ :=
let two : ℝ := @OfNat.ofNat ℝ 2 (@instOfNatAtLeastTwo ℝ 2 Real.commRing.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))
(two * Real.pi) ^ (-(m : ℝ) * ((p : ℝ)-1) / (two*(p : ℝ))) * H *
R ^ (β - (m : ℝ)*(1-1/(p : ℝ))) *
(∏ j ∈ Finset.range (p-1), J β m j) ^ (1/(p : ℝ))
lemma isClosed_projectedEllipsoid {d m : ℕ} (z : Space d) (R : ℝ) :
IsClosed (ProjectedEllipsoid (m := m) z R) := by
let K := Metric.closedBall (0 : Space d) R
have hK : IsCompact K := isCompact_closedBall _ _
let : CompactSpace K := isCompact_iff_compactSpace.mp hK
let D : Set (((Fin m → Space d) × (Fin m → ℝ)) × K) :=
{w | w.1.2 = fun j => ⟪z, w.1.1 j⟫ + ⟪(w.2 : Space d), w.1.1 j⟫}
have hD : IsClosed D := isClosed_eq (by fun_prop) (by fun_prop)
have he : ProjectedEllipsoid (m := m) z R = Prod.fst '' D := by
ext Ay
constructor
· rintro ⟨v, hv, he⟩
exact ⟨(Ay, ⟨v,hv⟩), he, rfl⟩
· rintro ⟨⟨Ay,v⟩, hv, rfl⟩
exact ⟨v, v.property, hv⟩
rw [he]
exact isClosedMap_fst_of_compactSpace D hD
lemma projectionLaw_ellipsoid_compl {d m : ℕ} (μ : Measure (Space d)) [IsFiniteMeasure μ]
{z : Space d} {R : ℝ} (hsupport : μ.support ⊆ Metric.closedBall z R) :
projectionLaw μ m (ProjectedEllipsoid z R)ᶜ = 0 := by
rw [projectionLaw, Measure.map_apply (by fun_prop)
(isClosed_projectedEllipsoid z R).measurableSet.compl]
apply measure_mono_null (t := Set.univ ×ˢ μ.supportᶜ)
· rintro ⟨A,s⟩ hs
refine ⟨mem_univ _, ?_⟩
intro hss
apply hs
refine ⟨s-z, ?_, ?_⟩
· change dist (s - z) (0 : Space d) ≤ R
simpa [Metric.mem_closedBall, dist_eq_norm] using hsupport hss
· ext j
simp [inner_sub_left]
· rw [Measure.prod_prod]
simp [μ.measure_compl_support]
lemma projectionNormScale_pos {m p : ℕ} {H β R : ℝ} (hp : 2 ≤ p) (hH : 0 < H)
(hR : 0 < R) (hgap : (m:ℝ)+(p:ℝ)-2 < β) :
0 < ProjectionNormScale m p H β R := by
have hJ : 0 < ∏ j ∈ Finset.range (p-1), J β m j := by
apply Finset.prod_pos
intro j hj
apply J_pos
have hj' : j+2 ≤ p := by have := Finset.mem_range.mp hj; omega
have hj'' : (j:ℝ)+2 ≤ p := by exact_mod_cast hj'
linarith
unfold ProjectionNormScale
positivity
lemma moment_root_eq_scale {m p : ℕ} {H β R : ℝ} (hp : 2 ≤ p) (hH : 0 < H)
(hR : 0 < R) (hgap : (m:ℝ)+(p:ℝ)-2 < β) :
(gaussianPeak ^ ((p-1)*m) * ENNReal.ofReal
(H^p * R^(β*(p:ℝ)-(m:ℝ)*((p-1:ℕ):ℝ)) * ∏ j : Fin (p-1), J β m j)) ^ (1/(p:ℝ)) =
ENNReal.ofReal (ProjectionNormScale m p H β R) := by
have hp0 : (p:ℝ) ≠ 0 := by exact_mod_cast (by omega : p ≠ 0)
have hp1 : 1 ≤ p := by omega
have hprod : 0 < ∏ j : Fin (p-1), J β m j := by
apply Finset.prod_pos
intro j _
apply J_pos
have hj : (j:ℝ)+2 ≤ p := by exact_mod_cast (show (j:ℕ)+2 ≤ p by have := j.isLt; omega)
linarith
have hpi : 0 < 2 * Real.pi := by positivity
rw [ENNReal.mul_rpow_of_nonneg _ _ (by positivity), gaussianPeak,
← ENNReal.ofReal_pow (by positivity), ENNReal.ofReal_rpow_of_pos (by positivity),
ENNReal.ofReal_rpow_of_pos (by positivity), ← ENNReal.ofReal_mul (by positivity)]
congr 1
rw [Real.mul_rpow (by positivity : 0 ≤ H^p * R^(β*p-m*((p-1:ℕ):ℝ))) hprod.le,
Real.mul_rpow (by positivity : 0 ≤ H^p) (by positivity),
← Real.rpow_natCast_mul hH.le, mul_one_div_cancel hp0, Real.rpow_one,
← Real.rpow_mul hR.le,
← Real.rpow_natCast_mul (by positivity : 0 ≤ (Real.sqrt (2*Real.pi))⁻¹),
Real.inv_rpow (by positivity), Real.sqrt_eq_rpow, ← Real.rpow_mul hpi.le,
← Real.rpow_neg hpi.le]
have he1 : -(1/2 * (((p-1:ℕ):ℝ)*(m:ℝ) * (1/(p:ℝ)))) =
-(m:ℝ)*((p:ℝ)-1)/(2*(p:ℝ)) := by rw [Nat.cast_sub hp1]; push_cast; ring
have he2 : (β*(p:ℝ)-(m:ℝ)*((p-1:ℕ):ℝ)) * (1/(p:ℝ)) =
β-(m:ℝ)*(1-1/(p:ℝ)) := by rw [Nat.cast_sub hp1]; push_cast; field_simp
simp only [Nat.cast_mul] at *
rw [he1, he2, Fin.prod_univ_eq_prod_range]
unfold ProjectionNormScale
ring
local instance final_reference_sigmaFinite (d m : ℕ) :
SigmaFinite (matrixLabelReference (E := Space d) m) := by
unfold matrixLabelReference
infer_instance
local instance final_projection_finite {d : ℕ} (μ : Measure (Space d)) [IsFiniteMeasure μ] (m : ℕ) :
IsFiniteMeasure (projectionLaw μ m) := by
unfold projectionLaw
infer_instance
lemma exact_finite_density (d m p : ℕ) (hp : 2 ≤ p) (hm : 1 ≤ m)
(β H R : ℝ) (hgap : (m:ℝ)+(p:ℝ)-2 < β) (hH : 0 < H) (hR : 0 < R)
(z : Space d) (μ : Measure (Space d)) [IsFiniteMeasure μ]
(hsupport : μ.support ⊆ Metric.closedBall z R) (hball : AmbientBallBound μ H β) :
∃ 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 ∧
(∀ᵐ x ∂MatrixLabelReference d m, x ∉ ProjectedEllipsoid z R → F x = 0) ∧
eLpNorm F (p : ℝ≥0∞) (MatrixLabelReference d m) ≤
ENNReal.ofReal (ProjectionNormScale m p H β R) := by
let C := gaussianPeak ^ ((p-1)*m) * ENNReal.ofReal
(H^p * R^(β*(p:ℝ)-(m:ℝ)*((p-1:ℕ):ℝ)) * ∏ j : Fin (p-1), J β m j)
have hC : C ≠ ∞ := by unfold C gaussianPeak; finiteness
have hp1 : p-1+1 = p := by omega
have hmom : ∀ φ : ℝ → ℝ≥0∞, Measurable φ → (∫⁻ x, φ x = 1) →
(∫⁻ Ay, projectionSmoothing μ φ m Ay ^ (p-1+1) ∂matrixLabelReference m) ≤ C := by
intro φ hφ hφ1
have hg : ((p-1:ℕ):ℝ)+(m:ℝ)-1 < β := by
rw [Nat.cast_sub (by omega : 1 ≤ p)]
push_cast
linarith
simpa only [hp1] using sharp_projectionSmoothing_moment hH hR hsupport hball φ hφ hφ1 hm (p-1) hg
have hac := projectionLaw_ac_of_moment μ m (p-1) (by omega) C hC hmom
let G := (projectionLaw μ m).rnDeriv (matrixLabelReference m)
have hG : Measurable G := Measure.measurable_rnDeriv _ _
have hden : (matrixLabelReference m).withDensity G = projectionLaw μ m :=
Measure.withDensity_rnDeriv_eq _ _ hac
have hGtop : ∀ᵐ x ∂matrixLabelReference m, G x ≠ ∞ := Measure.rnDeriv_ne_top _ _
have hGmom : (∫⁻ x, G x^p ∂matrixLabelReference m) ≤ C := by
simpa only [hp1] using projectionLaw_rnDeriv_moment μ m (p-1) (by omega) C hC hmom
let F := fun x => (G x).toReal
have hFG : (fun x => ENNReal.ofReal (F x)) =ᵐ[matrixLabelReference m] G := by
filter_upwards [hGtop] with x hx
exact ENNReal.ofReal_toReal hx
refine ⟨F, hG.ennreal_toReal, fun _ => ENNReal.toReal_nonneg, ?_, ?_, ?_⟩
· change (matrixLabelReference m).withDensity _ = projectionLaw μ m
rw [withDensity_congr_ae hFG, hden]
· have hnull := projectionLaw_ellipsoid_compl μ hsupport (m := m)
rw [← hden, withDensity_apply _ (isClosed_projectedEllipsoid z R).measurableSet.compl] at hnull
have hz := (lintegral_eq_zero_iff' hG.aemeasurable).mp hnull
change ∀ᵐ x ∂(matrixLabelReference m).restrict (ProjectedEllipsoid z R)ᶜ, G x = 0 at hz
rw [ae_restrict_iff' (isClosed_projectedEllipsoid z R).measurableSet.compl] at hz
filter_upwards [hz] with x hx hnot
dsimp [F]
rw [hx hnot]
rfl
· change eLpNorm F (p:ℝ≥0∞) (matrixLabelReference m) ≤ _
rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by exact_mod_cast (by omega : p ≠ 0))
(by finiteness) hG.ennreal_toReal.aestronglyMeasurable]
simp only [ENNReal.toReal_natCast]
have he : (fun x => ‖F x‖ₑ ^ (p:ℝ)) =ᵐ[matrixLabelReference m] (fun x => G x^p) := by
filter_upwards [hFG] with x hx
rw [Real.enorm_eq_ofReal ENNReal.toReal_nonneg, hx, ENNReal.rpow_natCast]
rw [lintegral_congr_ae he]
exact (ENNReal.rpow_le_rpow hGmom (by positivity)).trans_eq
(moment_root_eq_scale hp hH hR hgap)
def FiniteDensityStatement : 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)
1 ≤ d → 1 ≤ m → 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 →
∃ 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 ∧
(∀ᵐ x ∂MatrixLabelReference d m, x ∉ ProjectedEllipsoid z R → F x = 0) ∧
eLpNorm F (p : ℝ≥0∞) (MatrixLabelReference d m) ≤
ENNReal.ofReal (ProjectionNormScale m p H β R)
theorem finite_density : FiniteDensityStatement := by
intro d m p _ hm hp β H R hgap hH hR z μ hμ hs hb
let := hμ
exact exact_finite_density d m p hp hm β H R hgap hH hR z μ hs hb
end MemoryPrecision
end
end OAI
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