| import OAI.Probability.PosteriorReplicas.Model |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace |
| noncomputable section |
| universe u |
| namespace PosteriorRegression |
| open scoped Topology |
| open InnerProductSpace |
| 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 |
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| end PosteriorRegression |
| end |
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| end OAI |
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