lean/OAI/Probability/GaussianRegression/Projection.lean

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