lean/OAI/Probability/GaussianRegression/Gram.lean

import OAI.Probability.GaussianRegression.Model

namespace OAI

/-!
Gram determinants, orthogonal distances, and product integration.
-/

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

noncomputable section

universe u

namespace NoiselessRegression

section GramFactorization
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  {ι : Type*} [Fintype ι]
lemma gram_linear_combination (w : ι → E) (T : Matrix ι ι ℝ) :
    Matrix.gram ℝ (fun j => ∑ i, (T i j) • w i) =
      T.transpose * Matrix.gram ℝ w * T := by
  ext a b
  simp only [Matrix.gram_apply, sum_inner, inner_sum, real_inner_smul_left,
    real_inner_smul_right, Matrix.mul_apply, Matrix.transpose_apply]
  simp only [Finset.sum_mul]
  congr 1
  ext j
  congr 1
  ext i
  ring

variable (n : ℕ)
def gsUpper (v : Fin n → E) : Matrix (Fin n) (Fin n) ℝ := fun i j =>
  (if i = j then 1 else 0) +
    if i < j then ⟪gramSchmidt ℝ v i, v j⟫ / ‖gramSchmidt ℝ v i‖ ^ 2 else 0

lemma gsUpper_expansion (v : Fin n → E) (j : Fin n) :
    ∑ i, (gsUpper n v i j) • gramSchmidt ℝ v i = v j := by
  simp only [gsUpper, add_smul, Finset.sum_add_distrib]
  simp only [ite_smul, one_smul, zero_smul, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
  conv_rhs => rw [gramSchmidt_def'' ℝ v j]
  congr 1
  rw [← Finset.sum_filter]
  congr 1
  ext i
  simp

lemma gsUpper_det (v : Fin n → E) : (gsUpper n v).det = 1 := by
  have htri : (gsUpper n v).IsUpperTriangular := by
    intro i j hij
    change j < i at hij
    simp [gsUpper, (ne_of_gt hij), (not_lt_of_ge hij.le)]
  rw [Matrix.det_of_isUpperTriangular htri]
  simp [gsUpper]

lemma gram_gramSchmidt_diagonal (v : Fin n → E) :
    Matrix.gram ℝ (gramSchmidt ℝ v) = Matrix.diagonal (fun i => ‖gramSchmidt ℝ v i‖ ^ 2) := by
  ext i j
  by_cases hij : i = j
  · subst j
    simp [Matrix.gram_apply]
  · simp [Matrix.gram_apply, hij,
      gramSchmidt_orthogonal ℝ v hij]

theorem det_gram_eq_prod_gramSchmidt (v : Fin n → E) :
    (Matrix.gram ℝ v).det = ∏ i, ‖gramSchmidt ℝ v i‖ ^ 2 := by
  have hv : (fun j => ∑ i, (gsUpper n v i j) • gramSchmidt ℝ v i) = v :=
    funext (gsUpper_expansion n v)
  have h := gram_linear_combination (gramSchmidt ℝ v) (gsUpper n v)
  rw [hv] at h
  rw [h, Matrix.det_mul, Matrix.det_mul, Matrix.det_transpose, gsUpper_det,
    one_mul, mul_one, gram_gramSchmidt_diagonal, Matrix.det_diagonal]
end GramFactorization

section GramDistances
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {n : ℕ}
omit [FiniteDimensional ℝ E] in
lemma sub_gramSchmidt_mem_span (v : Fin n → E) (j : Fin n) :
    v j - gramSchmidt ℝ v j ∈ Submodule.span ℝ (v '' Set.Iio j) := by
  rw [← span_gramSchmidt_Iio ℝ v j]
  conv => arg 2; lhs; rw [gramSchmidt_def'' ℝ v j]
  rw [add_sub_cancel_left]
  exact Submodule.sum_mem _ fun i hi => Submodule.smul_mem _ _
    (Submodule.subset_span ⟨i, Finset.mem_Iio.mp hi, rfl⟩)

omit [FiniteDimensional ℝ E] in
lemma gramSchmidt_mem_span_orthogonal (v : Fin n → E) (j : Fin n) :
    gramSchmidt ℝ v j ∈ (Submodule.span ℝ (v '' Set.Iio j))ᗮ := by
  rw [Submodule.mem_orthogonal']
  intro w hw
  induction hw using Submodule.span_induction with
  | mem x hx =>
    rcases hx with ⟨i, hi, rfl⟩
    exact gramSchmidt_inv_triangular ℝ v hi
  | zero => simp
  | add x y _ _ hx hy => simp [inner_add_right, hx, hy]
  | smul c x _ hx => simp [inner_smul_right, hx]

lemma gramSchmidt_eq_sub_starProjection (v : Fin n → E) (j : Fin n) :
    gramSchmidt ℝ v j = v j -
      (Submodule.span ℝ (v '' Set.Iio j)).starProjection (v j) := by
  have h := (Submodule.span ℝ (v '' Set.Iio j)).eq_starProjection_of_mem_orthogonal
    (u := v j) (sub_gramSchmidt_mem_span v j)
    (by simpa using gramSchmidt_mem_span_orthogonal v j)
  rw [h, sub_sub_cancel]

lemma norm_gramSchmidt_eq_infDist (v : Fin n → E) (j : Fin n) :
    ‖gramSchmidt ℝ v j‖ = Metric.infDist (v j)
      (Submodule.span ℝ (v '' Set.Iio j) : Set E) := by
  rw [gramSchmidt_eq_sub_starProjection, Submodule.starProjection_minimal,
    Metric.infDist_eq_iInf]
  simp only [dist_eq_norm]
  rfl
lemma infDist_affine_mk' (p x : E) (K : Submodule ℝ E) :
    Metric.infDist x (AffineSubspace.mk' p K : Set E) =
      Metric.infDist (x - p) (K : Set E) := by
  rw [← EuclideanGeometry.dist_orthogonalProjection_eq_infDist,
    EuclideanGeometry.orthogonalProjection_apply_mem _ (AffineSubspace.self_mem_mk' p K),
    dist_eq_norm, Metric.infDist_eq_iInf]
  simp only [dist_eq_norm]
  change ‖x - _‖ = ⨅ y : K, ‖(x - p) - y‖
  rw [← Submodule.starProjection_minimal]
  simp only [AffineSubspace.direction_mk', vsub_eq_sub, vadd_eq_add,
    Submodule.coe_orthogonalProjectionOnto_apply]
  congr 1
  abel

lemma norm_gramSchmidt_sub_eq_affine_infDist (p : E) (t : Fin n → E) (j : Fin n) :
    ‖gramSchmidt ℝ (fun i => t i - p) j‖ =
      Metric.infDist (t j) (AffineSubspace.mk' p
        (Submodule.span ℝ ((fun i => t i - p) '' Set.Iio j)) : Set E) := by
  rw [infDist_affine_mk', norm_gramSchmidt_eq_infDist]

omit [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] in
lemma image_Iio_castSucc (v : Fin (n+1) → E) (j : Fin n) :
    v '' Set.Iio j.castSucc = (fun i => v i.castSucc) '' Set.Iio j := by
  ext x
  constructor
  · rintro ⟨i, hi, rfl⟩
    have hin : i.val < n := lt_trans hi j.isLt
    exact ⟨⟨i.val, hin⟩, hi, rfl⟩
  · rintro ⟨i, hi, rfl⟩
    exact ⟨i.castSucc, hi, rfl⟩

omit [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] in
lemma image_Iio_last (v : Fin (n+1) → E) :
    v '' Set.Iio (Fin.last n) = Set.range (fun i : Fin n => v i.castSucc) := by
  ext x
  constructor
  · rintro ⟨i, hi, rfl⟩
    exact ⟨⟨i.val, hi⟩, rfl⟩
  · rintro ⟨i, rfl⟩
    exact ⟨i.castSucc, i.isLt, rfl⟩

lemma gramSchmidt_castSucc (v : Fin (n+1) → E) (j : Fin n) :
    gramSchmidt ℝ v j.castSucc = gramSchmidt ℝ (fun i => v i.castSucc) j := by
  rw [gramSchmidt_eq_sub_starProjection, gramSchmidt_eq_sub_starProjection,
    image_Iio_castSucc]

variable [MeasurableSpace E] [BorelSpace E]
lemma measurable_gramSchmidt (j : Fin n) :
    Measurable (fun v : Fin n → E => gramSchmidt ℝ v j) := by
  refine (wellFounded_lt : WellFounded ((· < ·) : Fin n → Fin n → Prop)).induction
    (C := fun j : Fin n => Measurable (fun v : Fin n → E => gramSchmidt ℝ v j)) j ?_
  intro j ih
  have heq : (fun v : Fin n → E => gramSchmidt ℝ v j) =
      fun v => v j - ∑ i ∈ Finset.Iio j,
        (⟪gramSchmidt ℝ v i, v j⟫ / ‖gramSchmidt ℝ v i‖ ^ 2) • gramSchmidt ℝ v i := by
    ext v
    rw [gramSchmidt_def]
    simp only [Submodule.starProjection_singleton]
    rfl
  rw [heq]
  apply (measurable_pi_apply j).sub
  apply Finset.measurable_sum
  intro i hi
  have hi' := ih i (Finset.mem_Iio.mp hi)
  exact ((hi'.inner (measurable_pi_apply j)).div (hi'.norm.pow_const 2)).smul hi'
end GramDistances

variable {E : Type*} [MeasurableSpace E] {n : ℕ} (ν : Measure E) [SigmaFinite ν]
lemma lintegral_pi_snoc (f : (Fin (n+1) → E) → ℝ≥0∞) (hf : Measurable f) :
    ∫⁻ v, f v ∂Measure.pi (fun _ => ν) =
      ∫⁻ v : Fin n → E, ∫⁻ x, f (Fin.snoc v x) ∂ν ∂Measure.pi (fun _ => ν) := by
  have hp := (measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) => ν) (Fin.last n)).symm
  rw [← hp.lintegral_comp_emb (MeasurableEquiv.measurableEmbedding _) f,
    lintegral_prod_symm]
  · congr 1
    ext v
    congr 1
    ext x
    congr 1
    simp only [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv]
    exact Fin.insertNth_last' x v
  · exact (hf.comp (MeasurableEquiv.measurable _)).aemeasurable

end NoiselessRegression

end

end OAI