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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 | import Mathlib
import OAI.Probability.GaussianInformation.Kernel.ProjectionMoments
namespace OAI
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
namespace KernelPotential
open Filter
open scoped Topology NNReal
namespace Projection
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal Topology
open InnerProductSpace Matrix MeasureTheory Set
open scoped RealInnerProductSpace ENNReal
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal
open Support _root_.OAI.KernelPotential.Geometry
lemma fixed_center_replicas {k q d : ℕ} (hk : 0 < k) (η : ℝ) (hη : 0 < η)
(v : Fin q → EuclideanSpace ℝ (Fin d)) :
(∫⁻ X, ∏ i, boxKernel η (projectVector X (v i)) 0 ∂gaussianRows k d) ≤
gramInverseProduct k v := by
classical
let D := {X : Fin k → Fin d → ℝ | ∀ r i, |(rowMatrix v *ᵥ X r) i| ≤ η}
have hD : MeasurableSet D := by
dsimp only [D]
simp only [Set.ofPred_forall]
apply MeasurableSet.iInter
intro r
apply MeasurableSet.iInter
intro i
exact measurableSet_le (by fun_prop) measurable_const
have hpoint : ∀ X, (∏ i, boxKernel η (projectVector X (v i)) 0) ≤
D.indicator (fun _ => boxScale k η ^ q) X := by
intro X
by_cases hX : X ∈ D
· rw [indicator_of_mem hX]
calc
_ ≤ ∏ _i : Fin q, boxScale k η := Finset.prod_le_prod
(fun _ _ => boxKernel_le η _ _)
_ = _ := by simp
· rw [indicator_of_notMem hX]
change ¬ ∀ r i, |(rowMatrix v *ᵥ X r) i| ≤ η at hX
push Not at hX
obtain ⟨r,i,hi⟩ := hX
apply le_of_eq
apply Finset.prod_eq_zero (Finset.mem_univ i)
by_contra hh
have hm := boxKernel_ne_zero_mem hh
have hx := hm r
simp only [Pi.zero_apply, zero_sub, abs_neg] at hx
rw [projectVector_rowMatrix] at hx
exact (not_le.mpr hi) hx
calc
_ ≤ ∫⁻ X, D.indicator (fun _ => boxScale k η ^ q) X ∂gaussianRows k d := lintegral_mono hpoint
_ = boxScale k η ^ q * gaussianRows k d D := by rw [lintegral_indicator hD,lintegral_const,Measure.restrict_apply_univ]
_ ≤ boxScale k η ^ q * ((ENNReal.ofReal (Real.sqrt (rowMatrix v * (rowMatrix v)ᵀ).det))⁻¹^k *
ENNReal.ofReal ((2*η)^(q*k))) := by
apply mul_le_mul_right
simpa only [gaussianRows,Fintype.card_fin] using
gaussian_multirow_box_le_extended (ρ := Fin k) (by simpa using hk) (rowMatrix v) η hη
_ = gramInverseProduct k v := by
rw [rowMatrix_gram, ← gramInverseProduct_eq_inverseVolume]
have hc : boxScale k η ^ q * ENNReal.ofReal ((2*η)^(q*k)) = 1 := by
rw [Nat.mul_comm q k,pow_mul,ENNReal.ofReal_pow (by positivity),boxScale,
← mul_pow, ENNReal.inv_mul_cancel (by positivity) ENNReal.ofReal_ne_top,one_pow]
calc
_ = gramInverseProduct k v * (boxScale k η ^ q * ENNReal.ofReal ((2*η)^(q*k))) := by ac_rfl
_ = _ := by rw [hc,mul_one]
lemma fixed_center_moment {α : Type*} [MeasurableSpace α] {k d : ℕ}
(μ : Measure α) [SigmaFinite μ] (a : α → EuclideanSpace ℝ (Fin d))
(ha : Measurable a) (hk : 0 < k) (p : ℕ) (M : ℝ≥0∞)
(hcond : ∀ V : AffineSubspace ℝ (EuclideanSpace ℝ (Fin d)),
(V : Set (EuclideanSpace ℝ (Fin d))).Nonempty →
Module.finrank ℝ V.direction ≤ p →
(∫⁻ z,(ENNReal.ofReal (infDist (a z) (V : Set (EuclideanSpace ℝ (Fin d)))))⁻¹^k ∂μ) ≤ M)
(q : ℕ) (hq : q ≤ p+1) (η : ℝ) (hη : 0 < η) :
(∫⁻ X, (∫⁻ z,boxKernel η (projectVector X (a z)) 0 ∂μ)^q ∂gaussianRows k d) ≤ M^q := by
have : IsProbabilityMeasure (gaussianRows k d) := by unfold gaussianRows; infer_instance
have hm : Measurable (fun x : (Fin k → Fin d → ℝ) × α =>
boxKernel η (projectVector x.1 (a x.2)) 0) :=
(measurable_zero_kernel (k := k) (d := d) η).comp
(f := fun x : (Fin k → Fin d → ℝ) × α => (x.1,a x.2))
(measurable_fst.prodMk (ha.comp measurable_snd))
have hrep := measurable_replicated (m := q) _ hm
calc
_ = ∫⁻ X, ∫⁻ z : Fin q → α, ∏ i,boxKernel η (projectVector X (a (z i))) 0
∂Measure.pi (fun _ => μ) ∂gaussianRows k d := by
apply lintegral_congr
intro X
exact (lintegral_pi_product μ _ (hm.comp (measurable_const.prodMk measurable_id)) q).symm
_ = ∫⁻ z : Fin q → α, ∫⁻ X, ∏ i,boxKernel η (projectVector X (a (z i))) 0
∂gaussianRows k d ∂Measure.pi (fun _ => μ) := lintegral_lintegral_swap hrep.aemeasurable
_ ≤ ∫⁻ z : Fin q → α, gramInverseProduct k (fun i => a (z i)) ∂Measure.pi (fun _ => μ) :=
lintegral_mono (fun z => fixed_center_replicas hk η hη _)
_ ≤ M^q := by
simpa only [sub_zero] using lintegral_gramInverseProduct_le μ a ha k p M hcond q hq 0
lemma fresh_rows_kernel_le {k d : ℕ} (hk : 0 < k) (η : ℝ) (hη : 0 < η)
(x : EuclideanSpace ℝ (Fin d)) :
(∫⁻ Z, boxKernel η (projectVector Z x) 0 ∂gaussianRows k d) ≤
(ENNReal.ofReal ‖x‖)⁻¹^k := by
have h := fixed_center_replicas hk η hη (fun _ : Fin 1 => x)
have he : gramInverseProduct k (fun _ : Fin 1 => x) = (ENNReal.ofReal ‖x‖)⁻¹^k := by
simp only [gramInverseProduct, Fin.prod_univ_one]
rw [show (0 : Fin 1) = ⊥ from rfl, gramSchmidt_bot]
simpa only [Fin.prod_univ_succ, Fin.prod_univ_zero, mul_one, he] using h
noncomputable def mixedCubeAverage {k d : ℕ} (μ : Measure (EuclideanSpace ℝ (Fin d)))
(η : ℝ) (X Z : Fin k → Fin d → ℝ) : ℝ≥0∞ :=
∫⁻ x, boxKernel η (projectVector X x) 0 * boxKernel η (projectVector Z x) 0 ∂μ
lemma auxiliary_average_le {k d : ℕ} (μ : Measure (EuclideanSpace ℝ (Fin d)))
[SigmaFinite μ] (hk : 0 < k) (η : ℝ) (hη : 0 < η) (X : Fin k → Fin d → ℝ) :
(∫⁻ Z,mixedCubeAverage μ η X Z ∂gaussianRows k d) ≤
∫⁻ x,boxKernel η (projectVector X x) 0 * (ENNReal.ofReal ‖x‖)⁻¹^k ∂μ := by
have : IsProbabilityMeasure (gaussianRows k d) := by unfold gaussianRows; infer_instance
have hm : Measurable (fun x : (Fin k → Fin d → ℝ) × EuclideanSpace ℝ (Fin d) =>
boxKernel η (projectVector X x.2) 0 * boxKernel η (projectVector x.1 x.2) 0) :=
((measurable_zero_kernel (k := k) (d := d) η).comp
(f := fun x : (Fin k → Fin d → ℝ) × EuclideanSpace ℝ (Fin d) => (X,x.2))
(measurable_const.prodMk measurable_snd)).mul
(measurable_zero_kernel (k := k) (d := d) η)
unfold mixedCubeAverage
rw [lintegral_lintegral_swap hm.aemeasurable]
apply lintegral_mono
intro x
dsimp only
have hmX : Measurable (fun Z : Fin k → Fin d → ℝ => boxKernel η (projectVector Z x) 0) :=
(measurable_zero_kernel (k := k) (d := d) η).comp
(f := fun Z : Fin k → Fin d → ℝ => (Z,x)) (measurable_id.prodMk measurable_const)
rw [lintegral_const_mul _ hmX]
exact mul_le_mul_right (fresh_rows_kernel_le hk η hη x) _
theorem mixed_fixed_origin_moment {k d n : ℕ}
(μ : Measure (EuclideanSpace ℝ (Fin d))) [SigmaFinite μ]
{B D : ℝ} (hB : 0 ≤ B) (hD : 0 ≤ D) (hk : 0 < k) (hn : 2*k < n)
(hsupport : ∀ᵐ x ∂μ, ‖x‖ ≤ 2)
(hballB : ∀ z r, 0 < r → μ (ball z r) ≤ ENNReal.ofReal (B*r^(2*k)))
(hballD : ∀ z r, 0 < r → μ (ball z r) ≤ ENNReal.ofReal (D*r^n))
(L : ℕ) (hDB : D ≤ B*2^L) (η : ℝ) (hη : 0 < η) :
(∫⁻ X, (∫⁻ Z, mixedCubeAverage μ η X Z ∂gaussianRows k d)^k ∂gaussianRows k d) ≤
ENNReal.ofReal ((9^(k-1)*3^n*2^(n+2*k+2))*B*((L:ℝ)+2))^k := by
let w : EuclideanSpace ℝ (Fin d) → ℝ≥0∞ := fun x => (ENNReal.ofReal ‖x‖)⁻¹^k
have hw : Measurable w := measurable_norm.ennreal_ofReal.inv.pow_const _
have hz : ∀ᵐ x ∂μ, ‖x‖ ≠ 0 := by
apply ae_iff.mpr
simpa only [not_not] using ball_bound_zero_null μ (by omega : 0 < 2*k) hB hballB
have : SigmaFinite (μ.withDensity w) := SigmaFinite.withDensity_of_ne_top (by
filter_upwards [hz] with x hx
exact ENNReal.pow_ne_top (ENNReal.inv_ne_top.mpr (by
exact (ENNReal.ofReal_pos.mpr (lt_of_le_of_ne (norm_nonneg x) hx.symm)).ne')))
let M : ℝ≥0∞ := ENNReal.ofReal ((9^(k-1)*3^n*2^(n+2*k+2))*B*((L:ℝ)+2))
have hcond : ∀ V : AffineSubspace ℝ (EuclideanSpace ℝ (Fin d)),
(V : Set (EuclideanSpace ℝ (Fin d))).Nonempty → Module.finrank ℝ V.direction ≤ k-1 →
(∫⁻ x,(ENNReal.ofReal (infDist x (V : Set (EuclideanSpace ℝ (Fin d)))))⁻¹^k
∂μ.withDensity w) ≤ M := by
intro V hV hdim
have hg : Measurable (fun x : EuclideanSpace ℝ (Fin d) =>
(ENNReal.ofReal (infDist x (V : Set (EuclideanSpace ℝ (Fin d)))))⁻¹^k) :=
(continuous_infDist_pt _).measurable.ennreal_ofReal.inv.pow_const _
rw [lintegral_withDensity_eq_lintegral_mul μ hw hg]
change (∫⁻ x, (ENNReal.ofReal ‖x‖)⁻¹^k *
(ENNReal.ofReal (infDist x (V : Set (EuclideanSpace ℝ (Fin d)))))⁻¹^k ∂μ) ≤ M
apply (mixed_integral_logarithmic μ hB hD hk hn hsupport hballB hballD V hV hdim L hDB).trans
apply ENNReal.ofReal_le_ofReal
have hp : (9:ℝ)^(Module.finrank ℝ V.direction) ≤ 9^(k-1) :=
pow_le_pow_right₀ (by norm_num) hdim
exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hp (by positivity))
(by positivity)) hB) (by positivity)
calc
_ ≤ ∫⁻ X, (∫⁻ x,boxKernel η (projectVector X x) 0 * w x ∂μ)^k ∂gaussianRows k d := by
apply lintegral_mono
intro X
exact pow_le_pow_left' (auxiliary_average_le μ hk η hη X) k
_ = ∫⁻ X,(∫⁻ x,boxKernel η (projectVector X x) 0 ∂μ.withDensity w)^k ∂gaussianRows k d := by
apply lintegral_congr
intro X
congr 1
have hg : Measurable (fun x : EuclideanSpace ℝ (Fin d) => boxKernel η (projectVector X x) 0) :=
(measurable_zero_kernel (k := k) (d := d) η).comp
(f := fun x : EuclideanSpace ℝ (Fin d) => (X,x)) (measurable_const.prodMk measurable_id)
rw [lintegral_withDensity_eq_lintegral_mul μ hw hg]
apply lintegral_congr
intro x
exact mul_comm _ _
_ ≤ M^k := fixed_center_moment (μ.withDensity w) id measurable_id hk (k-1) M hcond k (by omega) η hη
end Projection
end KernelPotential
end OAI
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