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7d9f027 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | import Mathlib
import OAI.Probability.SKRatio.Quantization.BinFunctional
namespace OAI
noncomputable section
open scoped BigOperators Matrix
open MeasureTheory ProbabilityTheory Filter Real
namespace SKRatio.Bins
open Planted Scalar SKRatioClock.Regression MatrixNet Calculus
variable {α : Type*} [Fintype α] [DecidableEq α] {n : ℕ}
attribute [local instance] Classical.propDecidable
lemma norm_of_sum_sq_one {p : Fin n → ℝ} (hp : ∑ i, p i^2=1) :
‖WithLp.toLp 2 p‖=1 := by
have he := EuclideanSpace.real_norm_sq_eq (WithLp.toLp 2 p)
simp only [hp] at he
nlinarith only [he,norm_nonneg (WithLp.toLp 2 p)]
lemma quadratic_vector_difference (M : Matrix (Fin n) (Fin n) ℝ)
{p q : Fin n → ℝ} (hp : ∑ i, p i^2=1) (hq : ∑ i, q i^2=1) :
|quadratic M p-quadratic M q| ≤
2*euclideanOpNorm M*‖WithLp.toLp 2 (fun i => p i-q i)‖ := by
let A := Matrix.toEuclideanCLM (𝕜 := ℝ) M
let P : EuclideanSpace ℝ (Fin n) := WithLp.toLp 2 p
let Q : EuclideanSpace ℝ (Fin n) := WithLp.toLp 2 q
have hP : ‖P‖=1 := norm_of_sum_sq_one hp
have hQ : ‖Q‖=1 := norm_of_sum_sq_one hq
have he : inner ℝ P (A P)-inner ℝ Q (A Q) =
inner ℝ (P-Q) (A P)+inner ℝ Q (A (P-Q)) := by
simp only [map_sub,inner_sub_left,inner_sub_right]
ring
rw [quadratic_inner,quadratic_inner]
change |inner ℝ P (A P)-inner ℝ Q (A Q)|≤_
rw [he]
have h1 := (abs_real_inner_le_norm (P-Q) (A P)).trans
(mul_le_mul_of_nonneg_left (A.le_opNorm P) (norm_nonneg _))
have h2 := (abs_real_inner_le_norm Q (A (P-Q))).trans
(mul_le_mul_of_nonneg_left (A.le_opNorm (P-Q)) (norm_nonneg _))
rw [hP] at h1
rw [hQ] at h2
have ht := abs_add_le (inner ℝ (P-Q) (A P)) (inner ℝ Q (A (P-Q)))
change |inner ℝ (P-Q) (A P)+inner ℝ Q (A (P-Q))|≤2*‖A‖*‖P-Q‖
nlinarith only [ht,h1,h2]
lemma variance_bilinear_difference (M : Matrix (Fin n) (Fin n) ℝ)
(v : Fin n → ℝ) (hv : ∀ i, |v i|≤1)
{p q : Fin n → ℝ} (hp : ∑ i, p i^2=1) (hq : ∑ i, q i^2=1) :
|p ⬝ᵥ (M*ᵥ(fun i => v i*p i))-q ⬝ᵥ (M*ᵥ(fun i => v i*q i))|≤
2*euclideanOpNorm M*‖WithLp.toLp 2 (fun i => p i-q i)‖ := by
have he (p : Fin n → ℝ) : p ⬝ᵥ (M*ᵥ(fun i => v i*p i))=
quadratic (M*Matrix.diagonal v) p := by
simp only [quadratic,Matrix.mul_diagonal,dotProduct,Matrix.mulVec,Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
ring
rw [he,he]
apply (quadratic_vector_difference _ hp hq).trans
have hd := matrix_diagonal_opNorm_le v 1 (by norm_num) hv
have hn : euclideanOpNorm (M*Matrix.diagonal v)≤euclideanOpNorm M := by
unfold euclideanOpNorm
rw [map_mul]
exact (norm_mul_le _ _).trans ((mul_le_mul_of_nonneg_left hd (norm_nonneg _)).trans_eq (mul_one _))
gcongr
abbrev WeightBox (α : Type*) := α → Set.Icc (0:ℝ) 1
lemma binT_le_one [Nonempty (Fin n)] (σ : Fin n → α) (a : α) : binT σ a≤1 := by
have hs : binT σ a^2≤1 := by
rw [←binT_unit σ]
exact Finset.single_le_sum (fun i _ => sq_nonneg (binT σ i)) (Finset.mem_univ a)
nlinarith only [hs]
def binBox [Nonempty (Fin n)] (σ : Fin n → α) : WeightBox α :=
fun a => ⟨binT σ a,binT_nonneg σ a,binT_le_one σ a⟩
omit [DecidableEq α] in
lemma finiteA_parameter_tolerance (β : ℝ) (h : α → ℝ) {a : ℝ} (ha : 0<a) :
∃ δ : ℝ, 0<δ ∧ ∀ (t : WeightBox α) (z z' : Parameters α),
‖z.val-z'.val‖<δ →
|finiteA β (fun d => t d) h z-finiteA β (fun d => t d) h z'|<a := by
have hc : Continuous (fun q : WeightBox α × Parameters α =>
finiteA β (fun d => q.1 d) h q.2) := by
unfold finiteA moment
have hb (d : α) := (continuous_paramB d).comp (continuous_snd (X := WeightBox α))
have hr (d : α) := (continuous_paramR d).comp (continuous_snd (X := WeightBox α))
have ht (d : α) : Continuous (fun q : WeightBox α × Parameters α => (q.1 d:ℝ)) :=
continuous_subtype_val.comp ((continuous_apply d).comp continuous_fst)
fun_prop
obtain ⟨δ,hδ,hcont⟩ := Metric.uniformContinuous_iff.mp
(CompactSpace.uniformContinuous_of_continuous hc) a ha
refine ⟨δ,hδ,fun t z z' hh => ?_⟩
apply hcont (a := (t,z)) (b := (t,z'))
simp only [Prod.dist_eq,dist_self,Subtype.dist_eq,dist_eq_norm,max_lt_iff]
exact ⟨hδ,hh⟩
theorem binForm_finite_net (β : ℝ) (h : α → ℝ) {K a : ℝ} (hK : 0≤K) (ha : 0<a) :
∃ F : Finset (Parameters α), ∀ {n : ℕ} (_ : 0<n) (σ : Fin n → α),
(∀ d, 2≤count σ d) → ∀ M : Matrix (Fin n) (Fin n) ℝ,
euclideanOpNorm M≤K → ∀ (z : Parameters α) (u : Directions σ),
∃ z' ∈ F,
(vectorOf z u) ⬝ᵥ (M*ᵥ(fun i => v (h (σ i))*vectorOf z u i))+
finiteA β (binT σ) h z ≤
(vectorOf z' u) ⬝ᵥ (M*ᵥ(fun i => v (h (σ i))*vectorOf z' u i))+
finiteA β (binT σ) h z'+a := by
obtain ⟨δ,hδ,hA⟩ := finiteA_parameter_tolerance β h (show 0<a/2 by positivity)
let η := min δ (a/(4*K+1))
have hη : 0<η := lt_min hδ (by positivity)
obtain ⟨F,hF⟩ := parameters_finite_net (α := α) hη
refine ⟨F,?_⟩
intro n hn σ hc M hM z u
have : Nonempty (Fin n) := Fin.pos_iff_nonempty.mp hn
have hc' : ∀ d, 0<count σ d := fun d => lt_of_lt_of_le (by norm_num) (hc d)
obtain ⟨z',hz',hnear⟩ := hF z
refine ⟨z',hz',?_⟩
have hA' := (le_abs_self _).trans (hA (binBox σ) z z' (hnear.trans_le (min_le_left _ _))).le
have hdiff := variance_bilinear_difference M (fun i => v (h (σ i)))
(fun i => by rw [abs_of_nonneg (v_pos _).le]; exact v_le_one _)
(vectorOf_unit hc' z u) (vectorOf_unit hc' z' u)
rw [vectorOf_difference hc'] at hdiff
have hkd := mul_le_mul_of_nonneg_right hM (norm_nonneg (z.val-z'.val))
have hηd := (lt_div_iff₀ (by positivity : 0<4*K+1)).mp (hnear.trans_le (min_le_right _ _))
have habs := le_abs_self ((vectorOf z u) ⬝ᵥ (M*ᵥ (fun i => v (h (σ i))*vectorOf z u i))-
(vectorOf z' u) ⬝ᵥ (M*ᵥ (fun i => v (h (σ i))*vectorOf z' u i)))
change finiteA β (binT σ) h z-finiteA β (binT σ) h z'≤a/2 at hA'
nlinarith only [habs,hdiff,hkd,hηd,hA',norm_nonneg (z.val-z'.val)]
end SKRatio.Bins
end
end OAI
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