| import Mathlib |
| import OAI.Probability.SKRatio.Quantization.BinStability |
| import OAI.Probability.SKRatio.Variational.FieldRadical |
| import OAI.Probability.SKRatio.Variational.ScalarVariance |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped NNReal ENNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real |
| namespace SKRatio.Bins |
| open Planted Scalar |
|
|
| variable {μ : Measure ℝ} [IsProbabilityMeasure μ] |
|
|
| def fieldDiagonal (μ : Measure ℝ) (H : ℝ → ℝ) (x : ℝ) : ℝ := |
| ∫ y, v (H y)^2/(w (H x)+w (H y)) ∂μ |
|
|
| def fieldForm (μ : Measure ℝ) (β : ℝ) (H b r : ℝ → ℝ) : ℝ := |
| (∫ x, (β^2*fieldDiagonal μ H x)*(b x^2+r x^2) ∂μ)+ |
| (∫ x, b x*H x ∂μ)*(∫ x, b x*v (H x) ∂μ)+ |
| (∫ x, b x ∂μ)*(∫ x, b x*(H x*v (H x)) ∂μ)- |
| β^2*(∫ x, b x ∂μ)*(∫ x, b x*v (H x) ∂μ)+ |
| 2*β^2*(∫ x, b x ∂μ)*(∫ x, b x*(m (H x)*v (H x)) ∂μ)+ |
| β^2*(∫ z : ℝ×ℝ, b z.1*kernel (H z.1) (H z.2)*b z.2 ∂μ.prod μ)+ |
| β*(∫ x, r x*sqrt (fieldRadicand μ H b r x) ∂μ) |
|
|
| lemma fieldForm_id (β : ℝ) (b r : ℝ → ℝ) : |
| fieldForm (fieldLaw β) β id b r=variational β b r := rfl |
|
|
| lemma measurable_fieldDiagonal {H : ℝ → ℝ} (hH : Measurable H) : |
| Measurable (fieldDiagonal μ H) := by |
| have hm : StronglyMeasurable (fun z : ℝ×ℝ => v (H z.2)^2/(w (H z.1)+w (H z.2))) := |
| (continuous_diagonal_kernel.measurable.comp |
| ((hH.comp measurable_fst).prodMk (hH.comp measurable_snd))).stronglyMeasurable |
| exact hm.integral_prod_right.measurable |
|
|
| lemma fieldDiagonal_integrable {H : ℝ → ℝ} (hH : Measurable H) (x : ℝ) : |
| Integrable (fun y => v (H y)^2/(w (H x)+w (H y))) μ := by |
| apply (integrable_const (2:ℝ)).mono' |
| (continuous_diagonal_kernel.measurable.comp (measurable_const.prodMk hH)).aestronglyMeasurable |
| exact ae_of_all _ (fun y => by simpa only [norm_eq_abs,Function.comp_apply] using abs_diagonal_kernel_le (H x) (H y)) |
|
|
| lemma abs_fieldDiagonal_le {H : ℝ → ℝ} (hH : Measurable H) (x : ℝ) : |
| |fieldDiagonal μ H x| ≤ 2 := |
| integral_abs_bound (fieldDiagonal_integrable hH x) |
| (fun point => abs_diagonal_kernel_le (H x) (H point)) |
|
|
| lemma fieldDiagonal_error {H H' : ℝ → ℝ} (hH : Measurable H) (hH' : Measurable H') |
| {a : ℝ} (hf : ∀ x y, |v (H y)^2/(w (H x)+w (H y))- |
| v (H' y)^2/(w (H' x)+w (H' y))| ≤ a) (x : ℝ) : |
| |fieldDiagonal μ H x-fieldDiagonal μ H' x| ≤ a := by |
| unfold fieldDiagonal |
| rw [←integral_sub (fieldDiagonal_integrable hH x) (fieldDiagonal_integrable hH' x)] |
| exact integral_abs_bound ((fieldDiagonal_integrable hH x).sub (fieldDiagonal_integrable hH' x)) (hf x) |
|
|
| lemma fieldKernel_integrable {H b : ℝ → ℝ} (hH : Measurable H) (hb : MemLp b 2 μ) : |
| Integrable (fun z : ℝ×ℝ => b z.1*kernel (H z.1) (H z.2)*b z.2) (μ.prod μ) := by |
| have hI := hb.integrable (by norm_num : (1:ENNReal) ≤ 2) |
| apply (hI.norm.mul_prod hI.norm).mono' |
| ((hb.aestronglyMeasurable.comp_fst.mul |
| (continuous_kernel.measurable.comp ((hH.comp measurable_fst).prodMk |
| (hH.comp measurable_snd))).aestronglyMeasurable).mul hb.aestronglyMeasurable.comp_snd) |
| exact ae_of_all _ (fun z => by |
| change |b z.1*kernel (H z.1) (H z.2)*b z.2| ≤ |b z.1| * |b z.2| |
| simp only [abs_mul] |
| have hh := mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_left (abs_kernel_le_one (H z.1) (H z.2)) (abs_nonneg (b z.1))) (abs_nonneg (b z.2)) |
| simpa only [mul_one] using hh) |
|
|
| lemma fieldKernel_error {H H' b : ℝ → ℝ} (hH : Measurable H) (hH' : Measurable H') |
| (hb : MemLp b 2 μ) (hb1 : l2Norm μ b ≤ 1) {a : ℝ} (ha : 0 ≤ a) |
| (hk : ∀ x y, |kernel (H x) (H y)-kernel (H' x) (H' y)| ≤ a) : |
| |(∫ z : ℝ×ℝ, b z.1*kernel (H z.1) (H z.2)*b z.2 ∂μ.prod μ)- |
| (∫ z : ℝ×ℝ, b z.1*kernel (H' z.1) (H' z.2)*b z.2 ∂μ.prod μ)| ≤ a := by |
| have hI := hb.integrable (by norm_num : (1:ENNReal) ≤ 2) |
| rw [←integral_sub (fieldKernel_integrable hH hb) (fieldKernel_integrable hH' hb)] |
| apply abs_integral_le_integral_abs.trans |
| have hi := integral_mono ((fieldKernel_integrable hH hb).sub (fieldKernel_integrable hH' hb)).abs |
| ((hI.abs.mul_prod hI.abs).const_mul a) (fun z : ℝ×ℝ => by |
| change |b z.1*kernel (H z.1) (H z.2)*b z.2-b z.1*kernel (H' z.1) (H' z.2)*b z.2| ≤ a*(|b z.1| * |b z.2|) |
| rw [show b z.1*kernel (H z.1) (H z.2)*b z.2-b z.1*kernel (H' z.1) (H' z.2)*b z.2= |
| (kernel (H z.1) (H z.2)-kernel (H' z.1) (H' z.2))*(b z.1*b z.2) by ring,abs_mul,abs_mul] |
| exact mul_le_mul_of_nonneg_right (hk z.1 z.2) (mul_nonneg (abs_nonneg _) (abs_nonneg _))) |
| rw [integral_const_mul,integral_prod_mul (fun x => |b x|) (fun x => |b x|)] at hi |
| apply hi.trans |
| have hn : 0 ≤ ∫ x, |b x| ∂μ := integral_nonneg (fun x => abs_nonneg (b x)) |
| have hh := (integral_abs_le_l2 hb).trans hb1 |
| nlinarith only [mul_nonneg ha (sub_nonneg.mpr hh), |
| mul_nonneg (mul_nonneg ha hn) (sub_nonneg.mpr hh)] |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma field_pair_difference {b f g : ℝ → ℝ} (hb : MemLp b 2 μ) |
| (hb1 : l2Norm μ b ≤ 1) (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : |
| |(∫ x, b x*f x ∂μ)-(∫ x, b x*g x ∂μ)| ≤ l2Norm μ (fun x => f x-g x) := by |
| have hfi : Integrable (fun x => b x*f x) μ := hb.integrable_mul hf |
| have hgi : Integrable (fun x => b x*g x) μ := hb.integrable_mul hg |
| rw [←integral_sub hfi hgi] |
| simp_rw [←mul_sub] |
| exact field_pair_abs_le hb (hf.sub hg) hb1 |
|
|
| lemma field_pair_v_abs {b H : ℝ → ℝ} (hb : MemLp b 2 μ) (hb1 : l2Norm μ b ≤ 1) |
| (hH : Measurable H) : |∫ x, b x*v (H x) ∂μ| ≤ 1 := |
| (field_pair_abs_le hb (memLp_bounded (continuous_v.measurable.comp hH) |
| (fun x => v_abs_le_one (H x))) hb1).trans |
| (l2Norm_le_bounded (continuous_v.measurable.comp hH) (by norm_num) (fun x => v_abs_le_one (H x))) |
|
|
| lemma mean_abs_le_one {b : ℝ → ℝ} (hb : MemLp b 2 μ) (hb1 : l2Norm μ b ≤ 1) : |
| |∫ x, b x ∂μ| ≤ 1 := (abs_integral_le_integral_abs.trans (integral_abs_le_l2 hb)).trans hb1 |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma memLp_Hv {H : ℝ → ℝ} (hH : Measurable H) (hH2 : MemLp H 2 μ) : |
| MemLp (fun x => H x*v (H x)) 2 μ := |
| memLp_field_mul hH2 (continuous_v.measurable.comp hH) (fun point => v_abs_le_one (H point)) |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma Hv_error {H H' : ℝ → ℝ} (hH : Measurable H) (hH' : Measurable H') |
| (hH2 : MemLp H 2 μ) (hH2' : MemLp H' 2 μ) |
| {a d T : ℝ} (ha : 0 ≤ a) (he : l2Norm μ (fun x => H x-H' x) ≤ d) |
| (hsize : l2Norm μ H' ≤ T) (hv : ∀ x, |v (H x)-v (H' x)| ≤ a) : |
| l2Norm μ (fun x => H x*v (H x)-H' x*v (H' x)) ≤ d+T*a := by |
| have heq : (fun x => H x*v (H x)-H' x*v (H' x))= |
| (fun x => (H x-H' x)*v (H x)+H' x*(v (H x)-v (H' x))) := by funext x; ring |
| rw [heq] |
| have h1 := l2Norm_field_mul (hH2.sub hH2') (continuous_v.measurable.comp hH) |
| (by norm_num : (0:ℝ) ≤ 1) (fun x => v_abs_le_one _) |
| have h2 := l2Norm_field_mul hH2' |
| ((continuous_v.measurable.comp hH).sub (continuous_v.measurable.comp hH')) ha hv |
| have ht := l2Norm_add (memLp_field_mul (hH2.sub hH2') |
| (continuous_v.measurable.comp hH) (fun x => v_abs_le_one _)) |
| (memLp_field_mul hH2' ((continuous_v.measurable.comp hH).sub (continuous_v.measurable.comp hH')) hv) |
| have h3 := mul_le_mul_of_nonneg_left hsize ha |
| simp only [Pi.sub_apply,Pi.sub_def,Function.comp_apply] at h1 h2 ht |
| nlinarith only [h1,h2,ht,h3,he] |
|
|
| lemma field_radical_pair_error {b r H H' : ℝ → ℝ} (hb : MemLp b 2 μ) (hr : MemLp r 2 μ) |
| (hbm : Measurable b) (hrm : Measurable r) (hH : Measurable H) (hH' : Measurable H') |
| (he : (∫ x, b x^2+r x^2 ∂μ)=1) {a : ℝ} (ha : 0 ≤ a) |
| (hv : ∀ x, |v (H x)-v (H' x)| ≤ a) : |
| |(∫ x, r x*sqrt (fieldRadicand μ H b r x) ∂μ)- |
| (∫ x, r x*sqrt (fieldRadicand μ H' b r x) ∂μ)| ≤ 4*sqrt a := by |
| apply bounded_pair_error hr (norm_components hb hr he).2 |
| ((measurable_fieldRadicand hbm hrm hH).sqrt) ((measurable_fieldRadicand hbm hrm hH').sqrt) |
| (C := 2) (D := 2) _ _ (by positivity) (sqrt_fieldRadicand_error hb hr hH hH' he ha hv) |
| · intro x |
| rw [abs_of_nonneg (sqrt_nonneg _)] |
| exact (sqrt_le_iff).mpr ⟨by norm_num,by norm_num only [show (2:ℝ)^2=4 by norm_num]; exact fieldRadicand_le hb hr hH he x⟩ |
| · intro x |
| rw [abs_of_nonneg (sqrt_nonneg _)] |
| exact (sqrt_le_iff).mpr ⟨by norm_num,by norm_num only [show (2:ℝ)^2=4 by norm_num]; exact fieldRadicand_le hb hr hH' he x⟩ |
|
|
| theorem fieldForm_error {b r H H' : ℝ → ℝ} (β : ℝ) (hβ : 0 ≤ β) |
| (hb : MemLp b 2 μ) (hr : MemLp r 2 μ) (hbm : Measurable b) (hrm : Measurable r) |
| (hH : Measurable H) (hH' : Measurable H') (hH2 : MemLp H 2 μ) (hH2' : MemLp H' 2 μ) |
| (he : (∫ x, b x^2+r x^2 ∂μ)=1) {a d T : ℝ} (ha : 0 ≤ a) (hT : 0 ≤ T) |
| (herr : l2Norm μ (fun x => H x-H' x) ≤ d) (hsize : l2Norm μ H' ≤ T) |
| (hv : ∀ x, |v (H x)-v (H' x)| ≤ a) |
| (hmv : ∀ x, |m (H x)*v (H x)-m (H' x)*v (H' x)| ≤ a) |
| (hf : ∀ x y, |v (H y)^2/(w (H x)+w (H y))- |
| v (H' y)^2/(w (H' x)+w (H' y))| ≤ a) |
| (hk : ∀ x y, |kernel (H x) (H y)-kernel (H' x) (H' y)| ≤ a) : |
| |fieldForm μ β H b r-fieldForm μ β H' b r| ≤ |
| 2*d+2*T*a+5*β^2*a+4*β*sqrt a := by |
| have hb1 := (norm_components hb hr he).1 |
| have hV := bounded_pair_error hb hb1 (continuous_v.measurable.comp hH) |
| (continuous_v.measurable.comp hH') (fun x => v_abs_le_one _) (fun x => v_abs_le_one _) ha hv |
| have hmabs (x : ℝ) : |m x*v x| ≤ 1 := by |
| have hm : |m x| ≤ 1 := abs_le.mpr ⟨(m_bounds x).1.le,(m_bounds x).2.le⟩ |
| rw [abs_mul] |
| simpa only [mul_one] using mul_le_mul hm (v_abs_le_one x) (abs_nonneg _) (by norm_num : (0:ℝ) ≤ 1) |
| have hMV := bounded_pair_error hb hb1 |
| ((continuous_m.mul continuous_v).measurable.comp hH) |
| ((continuous_m.mul continuous_v).measurable.comp hH') |
| (fun x => hmabs _) (fun x => hmabs _) ha hmv |
| have hHv := (field_pair_difference hb hb1 (memLp_Hv hH hH2) |
| (memLp_Hv hH' hH2')).trans (Hv_error hH hH' hH2 hH2' ha herr hsize hv) |
| have h₁ := product_error (by norm_num : (0:ℝ) ≤ 1) hT (field_pair_v_abs hb hb1 hH) |
| ((field_pair_abs_le hb hH2' hb1).trans hsize) |
| ((field_pair_difference hb hb1 hH2 hH2').trans herr) hV |
| have h₂ : |(∫ x, b x ∂μ)*(∫ x, b x*(H x*v (H x)) ∂μ)- |
| (∫ x, b x ∂μ)*(∫ x, b x*(H' x*v (H' x)) ∂μ)| ≤ d+T*a := by |
| rw [←mul_sub,abs_mul] |
| exact (mul_le_mul (mean_abs_le_one hb hb1) hHv (abs_nonneg _) (by norm_num : (0:ℝ) ≤ 1)).trans_eq (one_mul _) |
| have h₃ : |β^2*(∫ x, b x ∂μ)*((∫ x, b x*v (H x) ∂μ)- |
| (∫ x, b x*v (H' x) ∂μ))| ≤ β^2*a := by |
| rw [abs_mul,abs_mul,abs_of_nonneg (sq_nonneg β)] |
| exact mul_le_mul ((mul_le_mul_of_nonneg_left (mean_abs_le_one hb hb1) (sq_nonneg β)).trans_eq (mul_one _)) |
| hV (abs_nonneg _) (sq_nonneg β) |
| have h₄ : |2*β^2*(∫ x, b x ∂μ)*((∫ x, b x*(m (H x)*v (H x)) ∂μ)- |
| (∫ x, b x*(m (H' x)*v (H' x)) ∂μ))| ≤ 2*β^2*a := by |
| rw [abs_mul,abs_mul,abs_of_nonneg (by positivity : 0 ≤ 2*β^2)] |
| exact mul_le_mul ((mul_le_mul_of_nonneg_left (mean_abs_le_one hb hb1) (by positivity : 0 ≤ 2*β^2)).trans_eq (mul_one _)) |
| hMV (abs_nonneg _) (by positivity) |
| have hd := energy_error hb hr he (measurable_fieldDiagonal (μ := μ) hH) (measurable_fieldDiagonal (μ := μ) hH') |
| (abs_fieldDiagonal_le hH) (abs_fieldDiagonal_le hH') (fieldDiagonal_error hH hH' hf) |
| have h₅ : |(∫ x, (β^2*fieldDiagonal μ H x)*(b x^2+r x^2) ∂μ)- |
| (∫ x, (β^2*fieldDiagonal μ H' x)*(b x^2+r x^2) ∂μ)| ≤ β^2*a := by |
| simp_rw [mul_assoc,integral_const_mul] |
| rw [←mul_sub,abs_mul,abs_of_nonneg (sq_nonneg β)] |
| exact mul_le_mul_of_nonneg_left hd (sq_nonneg β) |
| have h₆ : |β^2*((∫ z : ℝ×ℝ, b z.1*kernel (H z.1) (H z.2)*b z.2 ∂μ.prod μ)- |
| (∫ z : ℝ×ℝ, b z.1*kernel (H' z.1) (H' z.2)*b z.2 ∂μ.prod μ))| ≤ β^2*a := by |
| rw [abs_mul,abs_of_nonneg (sq_nonneg β)] |
| exact mul_le_mul_of_nonneg_left (fieldKernel_error hH hH' hb hb1 ha hk) (sq_nonneg β) |
| have h₇ : |β*((∫ x, r x*sqrt (fieldRadicand μ H b r x) ∂μ)- |
| (∫ x, r x*sqrt (fieldRadicand μ H' b r x) ∂μ))| ≤ 4*β*sqrt a := by |
| rw [abs_mul,abs_of_nonneg hβ] |
| have hh := mul_le_mul_of_nonneg_left (field_radical_pair_error hb hr hbm hrm hH hH' he ha hv) hβ |
| nlinarith only [hh] |
| unfold fieldForm |
| simp only [Function.comp_apply] at h₁ |
| have bounds₁ := abs_le.mp h₁ |
| have bounds₂ := abs_le.mp h₂ |
| have bounds₃ := abs_le.mp h₃ |
| have bounds₄ := abs_le.mp h₄ |
| have bounds₅ := abs_le.mp h₅ |
| have bounds₆ := abs_le.mp h₆ |
| have bounds₇ := abs_le.mp h₇ |
| exact abs_le.mpr ⟨by nlinarith only [bounds₁.1,bounds₂.1,bounds₃.2,bounds₄.1,bounds₅.1,bounds₆.1,bounds₇.1], |
| by nlinarith only [bounds₁.2,bounds₂.2,bounds₃.1,bounds₄.2,bounds₅.2,bounds₆.2,bounds₇.2]⟩ |
|
|
| end SKRatio.Bins |
|
|
| end |
|
|
| end OAI |
|
|
