| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.BoxKernels |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| namespace KernelPotential |
| open Filter |
| open scoped Topology NNReal |
| namespace Projection |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Topology |
| open InnerProductSpace Matrix MeasureTheory Set |
| open scoped RealInnerProductSpace ENNReal |
| namespace Support |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
|
|
| lemma gram_linearCombination {ι : Type*} [Fintype ι] (g : ι → E) |
| (T : Matrix ι ι ℝ) : |
| Matrix.gram ℝ (fun j => ∑ i, T i j • g i) = Tᵀ * Matrix.gram ℝ g * T := by |
| ext i j |
| simp only [Matrix.gram_apply, sum_inner, inner_sum, real_inner_smul_left, |
| real_inner_smul_right, Matrix.mul_apply, Matrix.transpose_apply, Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro k _ |
| apply Finset.sum_congr rfl |
| intro l _ |
| ring |
|
|
| noncomputable def gramTriangular {q : ℕ} (f : Fin q → E) : Matrix (Fin q) (Fin q) ℝ := |
| fun i j => (if i = j then 1 else 0) + |
| (if i < j then ⟪gramSchmidt ℝ f i, f j⟫ / ‖gramSchmidt ℝ f i‖ ^ 2 else 0) |
|
|
| lemma gramTriangular_expansion {q : ℕ} (f : Fin q → E) (j : Fin q) : |
| f j = ∑ i, gramTriangular f i j • gramSchmidt ℝ f i := by |
| rw [gramSchmidt_def'' ℝ f j] |
| have hfin : (Finset.univ.filter fun i : Fin q => i < j) = Finset.Iio j := by |
| ext i |
| simp |
| simp only [gramTriangular, add_smul, Finset.sum_add_distrib, ite_smul, one_smul, |
| zero_smul, Finset.sum_ite_eq', Finset.mem_univ, ite_true] |
| rw [← Finset.sum_filter, hfin] |
| simp only [RCLike.ofReal_real_eq_id, id_eq] |
| |
| lemma gramTriangular_det {q : ℕ} (f : Fin q → E) : (gramTriangular f).det = 1 := by |
| have htri : (gramTriangular f).IsUpperTriangular := by |
| intro i j hij |
| change j < i at hij |
| simp only [gramTriangular, ite_eq_right (ne_of_gt hij), ite_eq_right (not_lt.mpr (le_of_lt hij)), |
| zero_add] |
| rw [Matrix.det_of_isUpperTriangular htri] |
| simp [gramTriangular] |
| |
| lemma gram_gramSchmidt {q : ℕ} (f : Fin q → E) : |
| Matrix.gram ℝ (gramSchmidt ℝ f) = |
| Matrix.diagonal (fun i => ‖gramSchmidt ℝ f i‖ ^ 2) := by |
| ext i j |
| by_cases h : i = j |
| · subst j |
| simp |
| · simp [h, gramSchmidt_orthogonal ℝ f h] |
|
|
| lemma gram_det_eq_prod_gramSchmidt {q : ℕ} (f : Fin q → E) : |
| (Matrix.gram ℝ f).det = (∏ i, ‖gramSchmidt ℝ f i‖)^2 := by |
| have he : f = (fun j => ∑ i, gramTriangular f i j • gramSchmidt ℝ f i) := |
| funext (gramTriangular_expansion f) |
| calc |
| (Matrix.gram ℝ f).det = |
| ((gramTriangular f)ᵀ * Matrix.gram ℝ (gramSchmidt ℝ f) * gramTriangular f).det := by |
| conv_lhs => rw [he] |
| rw [gram_linearCombination] |
| _ = (∏ i, ‖gramSchmidt ℝ f i‖)^2 := by |
| rw [Matrix.det_mul, Matrix.det_mul, Matrix.det_transpose, |
| gramTriangular_det, one_mul, mul_one, gram_gramSchmidt, Matrix.det_diagonal, |
| Finset.prod_pow] |
|
|
| lemma sqrt_gram_det_eq_prod_gramSchmidt {q : ℕ} (f : Fin q → E) : |
| Real.sqrt (Matrix.gram ℝ f).det = ∏ i, ‖gramSchmidt ℝ f i‖ := by |
| rw [gram_det_eq_prod_gramSchmidt, Real.sqrt_sq (Finset.prod_nonneg fun _ _ => norm_nonneg _)] |
|
|
| lemma gramSchmidt_real_def {q : ℕ} (f : Fin q → E) (j : Fin q) : |
| gramSchmidt ℝ f j = f j - ∑ i ∈ Finset.Iio j, |
| (⟪gramSchmidt ℝ f i, f j⟫ / ‖gramSchmidt ℝ f i‖ ^ 2) • gramSchmidt ℝ f i := by |
| apply eq_sub_iff_add_eq.mpr |
| simpa only [RCLike.ofReal_real_eq_id, id_eq] using (gramSchmidt_def'' ℝ f j).symm |
|
|
| lemma measurable_gramSchmidt {q : ℕ} [MeasurableSpace E] [BorelSpace E] |
| [SecondCountableTopology E] (j : Fin q) : |
| Measurable (fun f : Fin q → E => gramSchmidt ℝ f j) := by |
| induction j using (wellFounded_lt (α := Fin q)).induction with |
| | h j ih => |
| simp_rw [gramSchmidt_real_def (j := j)] |
| refine (measurable_pi_apply j).sub (Finset.measurable_fun_sum _ fun i hi => ?_) |
| have hmi := ih i (Finset.mem_Iio.mp hi) |
| exact ((hmi.inner (measurable_pi_apply j)).div (hmi.norm.pow_const 2)).smul hmi |
|
|
| lemma gramSchmidt_prefix_congr {q : ℕ} {f g : Fin q → E} (j : Fin q) |
| (h : ∀ i ≤ j, f i = g i) : gramSchmidt ℝ f j = gramSchmidt ℝ g j := by |
| induction j using (wellFounded_lt (α := Fin q)).induction with |
| | h j ih => |
| rw [gramSchmidt_real_def, gramSchmidt_real_def, h j le_rfl] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i hi |
| have hgi := ih i (Finset.mem_Iio.mp hi) (fun a ha => h a (ha.trans (Finset.mem_Iio.mp hi).le)) |
| rw [hgi] |
|
|
| lemma gramSchmidt_residual_mem_span {q : ℕ} (f : Fin q → E) (j : Fin q) : |
| f j - gramSchmidt ℝ f j ∈ Submodule.span ℝ (f '' Set.Iio j) := by |
| rw [gramSchmidt_real_def, sub_sub_cancel] |
| apply Submodule.sum_mem |
| intro i hi |
| apply Submodule.smul_mem |
| rw [← span_gramSchmidt_Iio ℝ f j] |
| exact Submodule.subset_span (mem_image_of_mem _ (Finset.mem_Iio.mp hi)) |
|
|
| lemma infDist_span_le_gramSchmidt {q : ℕ} (f : Fin q → E) (j : Fin q) : |
| Metric.infDist (f j) (Submodule.span ℝ (f '' Set.Iio j) : Set E) ≤ |
| ‖gramSchmidt ℝ f j‖ := by |
| simpa [dist_eq_norm] using |
| (Metric.infDist_le_dist_of_mem (x := f j) (gramSchmidt_residual_mem_span f j)) |
|
|
| end Support |
|
|
| end Projection |
| end KernelPotential |
|
|
| end OAI |
|
|
