| 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 |
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| noncomputable section |
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|
| universe u |
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|
| 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 |
|
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| end |
|
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| end OAI |
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