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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 | import OAI.Probability.GaussianRegression.Smoothing
namespace OAI
/-!
Exact noiseless projection laws and completed-event bounds.
-/
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open InnerProductSpace
noncomputable section
universe u
namespace NoiselessRegression
section ProjectionCoupling
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
def matrixLabelReference (m : ℕ) : Measure ((Fin m → E) × (Fin m → ℝ)) :=
(Measure.pi (fun _ => stdGaussian E)).prod volume
def projectionLaw (ν : Measure E) (m : ℕ) : Measure ((Fin m → E) × (Fin m → ℝ)) :=
((Measure.pi (fun _ => stdGaussian E)).prod ν).map
(fun z => (z.1, fun j => ⟪z.2, z.1 j⟫))
def gaussianCoupling (ν : Measure E) (m : ℕ) :
Measure ((Fin m → E) × (E × (Fin m → ℝ))) :=
(Measure.pi (fun _ => stdGaussian E)).prod
(ν.prod (Measure.pi (fun _ => gaussianReal 0 1)))
def noisyObservation (m : ℕ) (δ : ℝ)
(z : (Fin m → E) × (E × (Fin m → ℝ))) : (Fin m → E) × (Fin m → ℝ) :=
(z.1, fun j => ⟪z.2.1, z.1 j⟫ + δ * z.2.2 j)
lemma measurable_noisyObservation (m : ℕ) (δ : ℝ) :
Measurable (noisyObservation (E := E) m δ) := by
unfold noisyObservation
fun_prop
lemma gaussianCoupling_map_noisyObservation_zero (ν : Measure E) [SigmaFinite ν] (m : ℕ) :
(gaussianCoupling ν m).map (noisyObservation m 0) = projectionLaw ν m := by
apply Measure.ext_of_lintegral
intro f hf
rw [lintegral_map hf (measurable_noisyObservation m 0),
projectionLaw, lintegral_map hf (by fun_prop)]
dsimp [gaussianCoupling, noisyObservation]
simp only [zero_mul, add_zero]
rw [lintegral_prod _ (by fun_prop), lintegral_prod _ (by fun_prop)]
apply lintegral_congr
intro A
rw [lintegral_prod _ (by fun_prop)]
simp
lemma gaussianCoupling_map_noisyObservation (ν : Measure E) [SigmaFinite ν]
(m : ℕ) (δ : ℝ) (hδ : δ ≠ 0) :
(gaussianCoupling ν m).map (noisyObservation m δ) =
(matrixLabelReference (E := E) m).withDensity
(projectionSmoothing ν (gaussianPDF 0 (NNReal.mk (δ^2) (sq_nonneg δ))) m) := by
apply Measure.ext_of_lintegral
intro f hf
rw [lintegral_map hf (measurable_noisyObservation m δ),
lintegral_withDensity_eq_lintegral_mul _
(measurable_projectionSmoothing _ _ (measurable_gaussianPDF _ _) _) hf]
dsimp [gaussianCoupling, matrixLabelReference, noisyObservation]
rw [lintegral_prod _ (by fun_prop), lintegral_prod _ (by
exact ((measurable_projectionSmoothing _ _ (measurable_gaussianPDF _ _) _).mul hf).aemeasurable)]
apply lintegral_congr
intro A
rw [lintegral_prod _ (by fun_prop)]
have he (p : E) := lintegral_gaussian_shift_scale m δ hδ
(fun j => ⟪p, A j⟫) (fun y => f (A, y)) (hf.comp (measurable_const.prodMk measurable_id))
simp_rw [he]
rw [lintegral_lintegral_swap (by fun_prop)]
apply lintegral_congr
intro y
rw [lintegral_mul_const _ (by fun_prop)]
rfl
lemma matrixLabelReference_outerRegular (m : ℕ) :
(matrixLabelReference (E := E) m).OuterRegular := by
unfold matrixLabelReference
exact (Measure.Regular.of_sigmaCompactSpace_of_isLocallyFiniteMeasure _).toOuterRegular
omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
lemma tendsto_noisyObservation (m : ℕ)
(z : (Fin m → E) × (E × (Fin m → ℝ))) :
Tendsto (fun k : ℕ => noisyObservation m (1 / ((k : ℝ) + 1)) z)
atTop (𝓝 (noisyObservation m 0 z)) := by
apply Filter.Tendsto.prodMk_nhds tendsto_const_nhds
apply tendsto_pi_nhds.mpr
intro j
exact tendsto_const_nhds.add
(tendsto_one_div_add_atTop_nhds_zero_nat.mul_const _)
theorem projectionLaw_set_bound {ν : Measure E} [IsFiniteMeasure ν]
{a : ℝ} (hν : HasLocalMassBound ν a)
(hsupport : ν.support ⊆ Metric.closedBall 0 1)
(m n : ℕ) (hn : 1 ≤ n) (hmargin : (n : ℝ) + m ≤ a)
(S : Set ((Fin m → E) × (Fin m → ℝ))) :
projectionLaw ν m S ≤
(ν Set.univ * (replicaConstant ν a m)^n)^(1 / ((n : ℝ) + 1)) *
(matrixLabelReference (E := E) m S)^(1 - 1 / ((n : ℝ) + 1)) := by
have hp : 1 < (n : ℝ) + 1 := by exact_mod_cast (by omega : 1 < n + 1)
have hα : 0 < 1 - 1 / ((n : ℝ) + 1) := by
exact sub_pos.mpr ((div_lt_one (by positivity)).mpr hp)
have hC : ν Set.univ * (replicaConstant ν a m)^n ≠ ∞ := by
unfold replicaConstant
finiteness
have := matrixLabelReference_outerRegular (E := E) m
apply measure_apply_le_of_isOpen _ _ (ENNReal.rpow_ne_top_of_nonneg (by positivity) hC) hα
intro U hU
rw [← gaussianCoupling_map_noisyObservation_zero ν m]
apply map_apply_open_le_of_tendsto (gaussianCoupling ν m)
(matrixLabelReference (E := E) m) (noisyObservation m 0)
(fun k => noisyObservation m (1 / ((k : ℝ) + 1)))
(measurable_noisyObservation m 0) (fun k => measurable_noisyObservation m _)
(tendsto_noisyObservation m) _ _ (U := U) (hU := hU)
intro k V hV
have hδ : (1 / ((k : ℝ) + 1)) ≠ 0 := by positivity
rw [gaussianCoupling_map_noisyObservation ν m _ hδ]
apply withDensity_apply_le_moment _ _
(measurable_projectionSmoothing _ _ (measurable_gaussianPDF _ _) _) hp _ _ hV.measurableSet
have hv : (NNReal.mk ((1 / ((k : ℝ) + 1)) ^ 2)
(sq_nonneg (1 / ((k : ℝ) + 1)))) ≠ 0 := by
rw [← NNReal.coe_ne_zero]
exact pow_ne_zero 2 hδ
simpa only [← ENNReal.rpow_natCast, Nat.cast_add, Nat.cast_one, matrixLabelReference]
using projectionSmoothing_moment hν hsupport
(gaussianPDF 0 (NNReal.mk ((1 / ((k : ℝ) + 1)) ^ 2)
(sq_nonneg (1 / ((k : ℝ) + 1))))) (measurable_gaussianPDF _ _)
(lintegral_gaussianPDF_eq_one _ hv) m n hmargin
end ProjectionCoupling
end NoiselessRegression
end
end OAI
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