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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | 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
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