| 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 | |