lean/OAI/Probability/GaussianRegression/GaussianKernel.lean

import OAI.Probability.GaussianRegression.Gram

namespace OAI

/-!
Gaussian coordinate domination and replica kernel bounds.
-/

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

noncomputable section

universe u

namespace NoiselessRegression

section GaussianDomination
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

lemma orthonormal_gaussian_map {n : ℕ} {u : Fin n → E} (hu : Orthonormal ℝ u) :
    (stdGaussian E).map (fun x => fun i => ⟪u i, x⟫) =
      Measure.pi (fun _ : Fin n => gaussianReal 0 1) := by
  let f : E → EuclideanSpace ℝ (Fin n) := fun x => WithLp.toLp 2 (fun i => ⟪u i, x⟫)
  have hf : Measurable f := by dsimp [f]; fun_prop
  have he : (stdGaussian E).map f = stdGaussian (EuclideanSpace ℝ (Fin n)) := by
    apply Measure.ext_of_charFun
    ext t
    have hi (x : E) : ⟪f x, t⟫ = ⟪x, ∑ i, t i • u i⟫ := by
      simp [f, PiLp.inner_apply, inner_sum, inner_smul_right, real_inner_comm x]
    have hn : ‖∑ i, t i • u i‖ ^ 2 = ‖t‖ ^ 2 := by
      rw [← real_inner_self_eq_norm_sq, hu.inner_sum, EuclideanSpace.real_norm_sq_eq]
      simp [pow_two]
    rw [charFun_apply, integral_map hf.aemeasurable (by fun_prop)]
    simp_rw [hi]
    change charFun (stdGaussian E) (∑ i, t i • u i) = _
    rw [charFun_stdGaussian, charFun_stdGaussian]
    congr 2
    exact_mod_cast congrArg Neg.neg hn
  have h := congrArg (Measure.map (WithLp.ofLp : EuclideanSpace ℝ (Fin n) → (Fin n → ℝ))) he
  rw [← map_pi_eq_stdGaussian] at h
  simpa [Measure.map_map (WithLp.measurable_ofLp _ _) hf,
    Measure.map_map (WithLp.measurable_ofLp _ _) (WithLp.measurable_toLp _ _),
    Function.comp_def, f, WithLp.ofLp_toLp, Measure.map_id] using h

lemma gaussianReal_le_volume : gaussianReal 0 1 ≤ (volume : Measure ℝ) := by
  rw [gaussianReal_of_var_ne_zero _ (by norm_num)]
  conv_rhs => rw [← withDensity_one (μ := (volume : Measure ℝ))]
  apply withDensity_mono
  filter_upwards [] with x
  change ENNReal.ofReal (gaussianPDFReal 0 1 x) ≤ 1
  rw [← ENNReal.ofReal_one]
  apply ENNReal.ofReal_le_ofReal
  unfold gaussianPDFReal
  have hs : 1 ≤ Real.sqrt (2 * Real.pi * (1 : ℝ≥0)) := by
    norm_num only [NNReal.coe_one, mul_one]
    apply (Real.le_sqrt (by norm_num) (by positivity)).mpr
    nlinarith [Real.pi_gt_three]
  have he : Real.exp (-(x - 0)^2 / (2 * (1 : ℝ≥0))) ≤ 1 := by
    apply Real.exp_le_one_iff.mpr
    exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (sq_nonneg _)) (by positivity)
  exact (mul_le_of_le_one_left (by positivity) (inv_le_one_of_one_le₀ hs)).trans he
lemma measure_pi_mono {ι : Type*} [Fintype ι] {X : ι → Type*}
    [∀ i, MeasurableSpace (X i)] {μ ν : ∀ i, Measure (X i)}
    (h : ∀ i, μ i ≤ ν i) : Measure.pi μ ≤ Measure.pi ν := by
  have ho : OuterMeasure.pi (fun i => (μ i).toOuterMeasure) ≤
      OuterMeasure.pi (fun i => (ν i).toOuterMeasure) := by
    apply OuterMeasure.le_pi.mpr
    intro s _
    exact (OuterMeasure.pi_pi_le _ s).trans (Finset.prod_le_prod fun i _ => h i (s i))
  apply Measure.le_iff.mpr
  intro s hs
  rw [Measure.pi_def, Measure.pi_def, toMeasure_apply _ _ hs, toMeasure_apply _ _ hs]
  exact ho s

lemma pi_gaussianReal_le_volume (n : ℕ) :
    Measure.pi (fun _ : Fin n => gaussianReal 0 1) ≤ (volume : Measure (Fin n → ℝ)) :=
  measure_pi_mono (fun _ => gaussianReal_le_volume)
lemma gaussian_pairings_le {n : ℕ} (v : Fin n → E) (hv : LinearIndependent ℝ v) :
    (stdGaussian E).map (fun x => fun i => ⟪v i, x⟫) ≤
      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
    _ ≤ (volume : Measure (Fin n → ℝ)).map (Matrix.toLin' R.transpose) :=
      Measure.map_mono (pi_gaussianReal_le_volume n) hf
    _ = _ := by
      rw [Real.map_matrix_volume_pi_eq_smul_volume_pi hd,
        abs_of_nonneg (inv_nonneg.mpr hpos), hdet]
lemma measure_pi_smul {ι : Type*} [Fintype ι] {X : ι → Type*}
    [∀ i, MeasurableSpace (X i)] (μ : ∀ i, Measure (X i)) [∀ i, SigmaFinite (μ i)]
    (c : ℝ≥0∞) (hc : c ≠ ∞) :
    Measure.pi (fun i => c • μ i) = c ^ Fintype.card ι • Measure.pi μ := by
  lift c to ℝ≥0 using hc
  simp_rw [Measure.coe_nnreal_smul]
  apply Measure.pi_eq
  intro s _
  simp only [Measure.smul_apply, Measure.pi_pi, ENNReal.smul_def, smul_eq_mul, Finset.prod_mul_distrib,
    Finset.prod_const, Finset.card_univ]

lemma 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⟫) ≤
      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 => gaussian_pairings_le v hv)
  rw [measure_pi_smul _ _ ENNReal.ofReal_ne_top] at h
  rw [ENNReal.ofReal_pow (by positivity)]
  simpa only [Fintype.card_fin, volume_pi] using h
end GaussianDomination

section ReplicaKernel
variable {X : Type*} [MeasurableSpace X] (μ : Measure X) [SigmaFinite μ]

lemma lintegral_fin_prod (n : ℕ) (f : Fin n → X → ℝ≥0∞) (hf : ∀ i, Measurable (f i)) :
    ∫⁻ x : Fin n → X, ∏ i, f i (x i) ∂Measure.pi (fun _ => μ) =
      ∏ i, ∫⁻ x, f i x ∂μ := by
  induction n with
  | zero => simp
  | succ n ih =>
    rw [lintegral_pi_snoc μ _ (by fun_prop)]
    simp_rw [Fin.prod_univ_castSucc, Fin.snoc_castSucc, Fin.snoc_last]
    simp_rw [lintegral_const_mul _ (hf (Fin.last n))]
    rw [lintegral_mul_const _ (by fun_prop), ih _ (fun i => hf i.castSucc)]

lemma lintegral_pow_pi (f : X → ℝ≥0∞) (hf : Measurable f) (n : ℕ) :
    (∫⁻ x, f x ∂μ)^n =
      ∫⁻ x : Fin n → X, ∏ i, f (x i) ∂Measure.pi (fun _ => μ) := by
  rw [lintegral_fin_prod μ n _ (fun _ => hf)]
  simp

lemma lintegral_product_differences (φ : ℝ → ℝ≥0∞) (hφ : Measurable φ)
    (hφ1 : ∫⁻ x, φ x = 1) (m n : ℕ) (y : Fin m → ℝ) :
    ∫⁻ z : Fin m → Fin n → ℝ,
      ∏ j, φ (y j) * ∏ i, φ (y j - z j i) = ∏ j, φ (y j) := by
  rw [volume_pi, lintegral_fin_prod (volume : Measure (Fin n → ℝ)) m
    (fun j z => φ (y j) * ∏ i, φ (y j - z i)) (by intro i; fun_prop)]
  apply Finset.prod_congr rfl
  intro j _
  rw [lintegral_const_mul _ (by fun_prop), volume_pi,
    lintegral_fin_prod (volume : Measure ℝ) n
      (fun _ z => φ (y j - z)) (by intro i; fun_prop)]
  simp_rw [lintegral_sub_left_eq_self φ (y j), hφ1]
  simp

end ReplicaKernel

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

lemma 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) ≤
      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 ∂(ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ v i‖)⁻¹ ^ m) •
        (volume : Measure (Fin m → Fin n → ℝ))) :=
      lintegral_mono' (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 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) ≤
      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,
      ENNReal.ofReal ((∏ i, ‖gramSchmidt ℝ (fun i => v i - p) i‖)⁻¹ ^ m) *
        ∏ j, φ (y j) := lintegral_mono fun y =>
          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]

end NoiselessRegression

end

end OAI