lean/OAI/Probability/GaussianRegression/Caps.lean

import OAI.Probability.GaussianRegression.Stream
import OAI.Probability.GaussianRegression.Sphere

namespace OAI

/-!
Spherical cap mass bounds along sequential routes.
-/

open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open InnerProductSpace
open scoped Pointwise

noncomputable section

universe u

namespace NoiselessRegression

section StreamCaps
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E]
variable {J : Type*} [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J] [Nonempty J]

def capWeight (y : E) (ε : ℝ) : E → ℝ≥0∞ :=
  (Metric.closedBall y ε).indicator (fun _ => 1)

omit [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [Nontrivial E] in
lemma measurable_capWeight (y : E) (ε : ℝ) : Measurable (capWeight y ε) :=
  measurable_const.indicator Metric.isClosed_closedBall.measurableSet

omit [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] in
lemma capWeight_le_one (y s : E) (ε : ℝ) : capWeight y ε s ≤ 1 := by
  classical
  simp only [capWeight, Set.indicator_apply]
  split <;> simp

lemma capWeight_mass (y : E) {ε : ℝ} (hε : 0 < ε) (a : ℕ)
    (ha : 2*a ≤ Module.finrank ℝ E - 1) :
    PolynomialMass ((ambientSphereLaw (E := E)).withDensity (capWeight y ε)) a
      (ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E) * ENNReal.ofReal (ε^a)) := by
  change PolynomialMass ((ambientSphereLaw (E := E)).withDensity ((Metric.closedBall y ε).indicator 1)) a _
  rw [withDensity_indicator_one Metric.isClosed_closedBall.measurableSet]
  exact fun z r hr => terminal_cap_mass y z hε hr a ha

lemma ambient_mass_bound {f : E → ℝ≥0∞} {a : ℕ} {C : ℝ≥0∞}
    (h : PolynomialMass ((ambientSphereLaw (E := E)).withDensity f) a C) :
    (∫⁻ s, f s ∂ambientSphereLaw) ≤ C := by
  rw [← setLIntegral_univ, ← withDensity_apply _ MeasurableSet.univ]
  apply (measure_mono_ae ?_).trans (by simpa using h 0 1 zero_lt_one)
  filter_upwards [(withDensity_absolutelyContinuous (ambientSphereLaw (E := E)) f).ae_le
    (ambientSphereLaw_ae_norm (E := E))] with s hs
  intro _
  change dist s 0 ≤ 1
  simp [dist_zero_right, hs]

lemma continuation_cap_integral_bound {δ : ℕ → J → (E × ℝ) → J}
    (hδ : ∀ n j, Measurable (δ n j)) (y : ℕ → J → E)
    {ε : ℝ} (hε : 0 < ε) {a q : ℕ} (ha : 2*a ≤ Module.finrank ℝ E - 1)
    (hq : 2 ≤ q) (hqa : 2*q ≤ a) (n k : ℕ) (j : J) :
    (∫⁻ s, continuation δ (fun n j => capWeight (y n j) ε) n k j s ∂ambientSphereLaw) ≤
      (ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E) * ENNReal.ofReal (ε^a)) *
        streamFactor (E := E) (J := J) q ^ (k/q+1) := by
  apply ambient_mass_bound
  apply continuation_mass_bound _ hδ (fun n j => measurable_capWeight _ _)
    (fun n j s => capWeight_le_one _ _ _)
    (mul_ne_zero (by positivity) (by positivity)) (ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top)
    (fun n j => capWeight_mass (y n j) hε a ha)
    (Module.finrank_pos (R := ℝ) (M := E)) (by omega) hq hqa

lemma route_cap_bound {δ : ℕ → J → (E × ℝ) → J}
    (hδ : ∀ n j, Measurable (δ n j)) (y : J → E)
    {ε : ℝ} (hε : 0 < ε) {a q : ℕ} (ha : 2*a ≤ Module.finrank ℝ E - 1)
    (hq : 2 ≤ q) (hqa : 2*q ≤ a) (k : ℕ) (j : J) :
    ((sphereLaw (E := E)).prod (Measure.pi (fun _ : Fin k => stdGaussian E)))
      {z | dist z.1.val (y (streamRoute δ 0 k j (z.2,fun i => ⟪z.1.val,z.2 i⟫))) ≤ ε} ≤
      (ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E) * ENNReal.ofReal (ε^a)) *
        streamFactor (E := E) (J := J) q ^ (k/q+1) := by
  let w : ℕ → J → E → ℝ≥0∞ := fun _ j => capWeight (y j) ε
  have hV := measurable_continuation (w := w) hδ (fun _ j => measurable_capWeight (y j) ε) 0 k j
  have he (s : E) : continuation δ w 0 k j s =
      ∫⁻ A : Fin k → E, capWeight (y (streamRoute δ 0 k j (A,fun i => ⟪s,A i⟫))) ε s
        ∂Measure.pi (fun _ => stdGaussian E) := by
    simpa [w, blockValue, continuation] using continuation_unroll (w := w) hδ 0 k 0 j s
  let S : Set ((Metric.sphere (0 : E) 1) × (Fin k → E)) :=
    {z | dist z.1.val (y (streamRoute δ 0 k j (z.2,fun i => ⟪z.1.val,z.2 i⟫))) ≤ ε}
  have hS : MeasurableSet S := by
    apply measurableSet_le _ measurable_const
    exact measurable_subtype_coe.comp measurable_fst |>.dist
      ((measurable_of_finite y).comp ((measurable_streamRoute hδ 0 k j).comp (by fun_prop)))
  change ((sphereLaw (E := E)).prod _) S ≤ _
  rw [← one_mul (((sphereLaw (E := E)).prod _) S), ← lintegral_indicator_const hS 1,
    lintegral_prod _ (measurable_const.indicator hS).aemeasurable]
  have hf : ∀ s : Metric.sphere (0 : E) 1,
      (∫⁻ A, S.indicator (fun _ => (1 : ℝ≥0∞)) (s,A) ∂Measure.pi (fun _ => stdGaussian E)) =
        continuation δ w 0 k j s.val := by
    intro s
    rw [he]
    apply lintegral_congr
    intro A
    classical
    change (if dist s.val (y (streamRoute δ 0 k j (A,fun i => ⟪s.val,A i⟫))) ≤ ε then 1 else 0) = _
    unfold capWeight
    rw [Set.indicator_apply]
    rfl
  simp_rw [hf]
  rw [← lintegral_map hV measurable_subtype_coe]
  exact continuation_cap_integral_bound hδ (fun _ => y) hε ha hq hqa 0 k j

end StreamCaps

end NoiselessRegression

end

end OAI