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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.AmbientGram
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 EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision
open NoiselessRegression InnerProductSpace
def gaussianPeak : ℝ≥0∞ := ENNReal.ofReal ((Real.sqrt (2 * Real.pi))⁻¹)
lemma sharp_gaussianReal_le_volume :
gaussianReal 0 1 ≤ gaussianPeak • (volume : Measure ℝ) := by
rw [gaussianReal_of_var_ne_zero _ (by norm_num)]
calc
_ ≤ (volume : Measure ℝ).withDensity (fun _ => gaussianPeak) := by
apply withDensity_mono
filter_upwards [] with x
apply ENNReal.ofReal_le_ofReal
unfold gaussianPDFReal
simp only [NNReal.coe_one, mul_one, sub_zero]
exact mul_le_of_le_one_right (by positivity)
(Real.exp_le_one_iff.mpr (div_nonpos_of_nonpos_of_nonneg
(neg_nonpos.mpr (sq_nonneg x)) (by positivity)))
_ = _ := by rw [withDensity_const]
lemma sharp_pi_gaussianReal_le_volume (n : ℕ) :
Measure.pi (fun _ : Fin n => gaussianReal 0 1) ≤
gaussianPeak ^ n • (volume : Measure (Fin n → ℝ)) := by
have h := measure_pi_mono (fun _ : Fin n => sharp_gaussianReal_le_volume)
rw [measure_pi_smul _ _ (show gaussianPeak ≠ ∞ from ENNReal.ofReal_ne_top)] at h
simpa only [Fintype.card_fin, volume_pi] using h
section Gaussian
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
local instance sharp_reference_sigmaFinite (m : ℕ) :
SigmaFinite (matrixLabelReference (E := E) m) := by
unfold matrixLabelReference
infer_instance
lemma sharp_gaussian_pairings_le {n : ℕ} (v : Fin n → E) (hv : LinearIndependent ℝ v) :
(stdGaussian E).map (fun x => fun i => ⟪v i, x⟫) ≤
(gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹)) •
(volume : Measure (Fin n → ℝ)) := by
let u := gramSchmidtNormed ℝ v
let R : Matrix (Fin n) (Fin n) ℝ :=
Matrix.diagonal (fun i => ‖gramSchmidt ℝ v i‖) * gsUpper n v
have hu : Orthonormal ℝ u := gramSchmidtNormed_orthonormal hv
have hn (i : Fin n) : ‖gramSchmidt ℝ v i‖ ≠ 0 :=
norm_ne_zero_iff.mpr (gramSchmidt_ne_zero i hv)
have hexp (j : Fin n) : ∑ i, R i j • u i = v j := by
rw [← gsUpper_expansion n v j]
apply Finset.sum_congr rfl
intro i _
simp only [R, u, Matrix.diagonal_mul, gramSchmidtNormed, smul_smul]
congr 1
simp [hn i, mul_comm]
have hdet : R.transpose.det = ∏ i, ‖gramSchmidt ℝ v i‖ := by
dsimp only [R]
rw [Matrix.det_transpose, Matrix.det_mul, Matrix.det_diagonal, gsUpper_det, mul_one]
have hd : R.transpose.det ≠ 0 := by
rw [hdet]
exact Finset.prod_ne_zero_iff.mpr (fun i _ => hn i)
have hpos : 0 ≤ R.transpose.det := by rw [hdet]; positivity
have hf : Measurable (Matrix.toLin' R.transpose) :=
(Matrix.toLin' R.transpose).continuous_of_finiteDimensional.measurable
have heq : (fun x : E => fun j => ⟪v j, x⟫) =
(Matrix.toLin' R.transpose) ∘ (fun x => fun i => ⟪u i, x⟫) := by
funext x j
rw [← hexp j]
simp [Matrix.toLin'_apply, Matrix.mulVec, dotProduct, sum_inner,
inner_smul_left]
rw [heq, ← Measure.map_map hf (by fun_prop), orthonormal_gaussian_map hu]
calc
_ ≤ (gaussianPeak ^ n • (volume : Measure (Fin n → ℝ))).map
(Matrix.toLin' R.transpose) :=
Measure.map_mono (sharp_pi_gaussianReal_le_volume n) hf
_ = _ := by
rw [Measure.map_smul _ hf.aemeasurable, Real.map_matrix_volume_pi_eq_smul_volume_pi hd,
abs_of_nonneg (inv_nonneg.mpr hpos), hdet, smul_smul]
lemma sharp_gaussian_rows_pairings_le {n : ℕ} (v : Fin n → E)
(hv : LinearIndependent ℝ v) (m : ℕ) :
(Measure.pi (fun _ : Fin m => stdGaussian E)).map
(fun A => fun j i => ⟪v i, A j⟫) ≤
((gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹)) ^ m) •
(volume : Measure (Fin m → Fin n → ℝ)) := by
rw [Measure.pi_map_pi (fun _ => (by fun_prop : Measurable
(fun x : E => fun i => ⟪v i, x⟫)).aemeasurable)]
have h := measure_pi_mono (fun _ : Fin m => sharp_gaussian_pairings_le v hv)
rw [measure_pi_smul _ _ (ENNReal.mul_ne_top (ENNReal.pow_ne_top
(show gaussianPeak ≠ ∞ from ENNReal.ofReal_ne_top)) ENNReal.ofReal_ne_top)] at h
simpa only [Fintype.card_fin, volume_pi] using h
lemma sharp_gaussian_rows_kernel_bound {n : ℕ} (v : Fin n → E) (hv : LinearIndependent ℝ v)
(φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) (hφ1 : ∫⁻ z, φ z = 1)
(m : ℕ) (y : Fin m → ℝ) :
∫⁻ A : Fin m → E, (∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i, A j⟫)
∂Measure.pi (fun _ => stdGaussian E) ≤
((gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹)) ^ m) * ∏ j, φ (y j) := by
let H : (Fin m → Fin n → ℝ) → ℝ≥0∞ :=
fun z => ∏ j, φ (y j) * ∏ i, φ (y j - z j i)
have hH : Measurable H := by dsimp [H]; fun_prop
have heq (A : Fin m → E) :
(∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i, A j⟫) =
H (fun j i => ⟪v i, A j⟫) := by
dsimp [H]
rw [Finset.prod_mul_distrib, Finset.prod_comm]
simp_rw [heq]
rw [← lintegral_map hH (by fun_prop)]
calc
_ ≤ ∫⁻ z, H z ∂(((gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹)) ^ m) •
(volume : Measure (Fin m → Fin n → ℝ))) :=
lintegral_mono' (sharp_gaussian_rows_pairings_le v hv m) (fun _ => le_rfl)
_ = _ := by
rw [lintegral_smul_measure]
change _ * (∫⁻ z : Fin m → Fin n → ℝ, ∏ j, φ (y j) * ∏ i, φ (y j - z j i)) = _
rw [lintegral_product_differences φ hφ hφ1]
lemma sharp_gaussian_replica_kernel_bound {n : ℕ} (p : E) (v : Fin n → E)
(hv : LinearIndependent ℝ (fun i => v i - p))
(φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) (hφ1 : ∫⁻ z, φ z = 1) (m : ℕ) :
∫⁻ A : Fin m → E, ∫⁻ y : Fin m → ℝ,
(∏ j, φ (y j - ⟪p, A j⟫)) * ∏ i, ∏ j, φ (y j - ⟪v i, A j⟫)
∂volume ∂Measure.pi (fun _ => stdGaussian E) ≤
((gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ (fun i => v i - p) i‖)⁻¹)) ^ m) := by
have ht (A : Fin m → E) :
(∫⁻ y : Fin m → ℝ, (∏ j, φ (y j - ⟪p, A j⟫)) *
∏ i, ∏ j, φ (y j - ⟪v i, A j⟫)) =
∫⁻ y : Fin m → ℝ, (∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i - p, A j⟫) := by
have he (y : Fin m → ℝ) (i : Fin n) (j : Fin m) :
y j - ⟪p, A j⟫ - ⟪v i - p, A j⟫ = y j - ⟪v i, A j⟫ := by
rw [inner_sub_left]
ring
simpa only [Pi.sub_apply, he] using
lintegral_sub_right_eq_self
(fun y : Fin m → ℝ => (∏ j, φ (y j)) * ∏ i, ∏ j, φ (y j - ⟪v i - p, A j⟫))
(fun j => ⟪p, A j⟫)
simp_rw [ht]
rw [lintegral_lintegral_swap (by fun_prop)]
calc
_ ≤ ∫⁻ y : Fin m → ℝ,
((gaussianPeak ^ n * ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ (fun i => v i - p) i‖)⁻¹)) ^ m) *
∏ j, φ (y j) := lintegral_mono fun y =>
sharp_gaussian_rows_kernel_bound (fun i => v i - p) hv φ hφ hφ1 m y
_ = _ := by
rw [lintegral_const_mul _ (by fun_prop), volume_pi,
lintegral_fin_prod volume m (fun _ => φ) (fun _ => hφ)]
simp [hφ1]
lemma sharp_projectionSmoothing_moment {ν : Measure E} [IsFiniteMeasure ν]
{H β R : ℝ} {z : E} (hH : 0 < H) (hR : 0 < R)
(hsupport : ν.support ⊆ Metric.closedBall z R)
(hball : ∀ x : E, ∀ r : ℝ, 0 < r →
ν (Metric.closedBall x r) ≤ ENNReal.ofReal (H*r^β))
(φ : ℝ → ℝ≥0∞) (hφ : Measurable φ) (hφ1 : ∫⁻ x, φ x = 1)
{m : ℕ} (hm : 0 < m) (n : ℕ) (hgap : (n:ℝ)+(m:ℝ)-1 < β) :
(∫⁻ Ay : (Fin m → E) × (Fin m → ℝ),
(projectionSmoothing ν φ m Ay)^(n+1) ∂matrixLabelReference m) ≤
gaussianPeak ^ (n*m) * ENNReal.ofReal
(H^(n+1) * R^(β*(n+1:ℕ)-(m:ℝ)*(n:ℝ)) * ∏ j : Fin n, J β m j) := by
simp_rw [projectionSmoothing_pow ν φ hφ m n]
rw [lintegral_lintegral_swap (by fun_prop)]
calc
_ ≤ ∫⁻ w : E × (Fin n → E), gaussianPeak ^ (n*m) * replicaWeight m w.1 w.2
∂ν.prod (Measure.pi (fun _ => ν)) := by
apply lintegral_mono_ae
filter_upwards [ae_joint_replica_independent hH hR hsupport hball hm n hgap] with w hw
rw [matrixLabelReference, lintegral_prod _ (by fun_prop)]
convert sharp_gaussian_replica_kernel_bound w.1 w.2 hw φ hφ hφ1 m using 1
rw [mul_pow, ← pow_mul, replicaWeight_eq_det, sqrt_det_gram,
ENNReal.ofReal_pow (by positivity)]
_ = gaussianPeak ^ (n*m) *
(∫⁻ w : E × (Fin n → E), replicaWeight m w.1 w.2
∂ν.prod (Measure.pi (fun _ => ν))) :=
lintegral_const_mul _ (measurable_joint_replicaWeight m n)
_ ≤ _ := mul_le_mul' le_rfl
(joint_replica_integral_exact hH hR hsupport hball hm n hgap)
end Gaussian
end MemoryPrecision
end
end OAI
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