| import Mathlib |
| import OAI.Probability.SKRatio.Variational.FieldL2 |
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| namespace OAI |
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| noncomputable section |
| open scoped NNReal ENNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real |
| namespace SKRatio.Bins |
| open Planted Scalar |
|
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| variable {μ : Measure ℝ} [IsProbabilityMeasure μ] |
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| def fieldRadicand (μ : Measure ℝ) (H b r : ℝ → ℝ) (x : ℝ) : ℝ := |
| (∫ y, ((v (H x)+v (H y))*r y)^2 ∂μ)+ |
| (∫ y, ((v (H x)+v (H y))*b y-(∫ t, (v (H x)+v (H t))*b t ∂μ))^2 ∂μ) |
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| omit [IsProbabilityMeasure μ] in |
| lemma fieldRadicand_nonneg (H b r : ℝ → ℝ) (x : ℝ) : |
| 0 ≤ fieldRadicand μ H b r x := |
| add_nonneg (integral_nonneg (fun _ => sq_nonneg _)) |
| (integral_nonneg (fun _ => sq_nonneg _)) |
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| omit [IsProbabilityMeasure μ] in |
| lemma memLp_sum_v_field {b H : ℝ → ℝ} (hb : MemLp b 2 μ) (hH : Measurable H) |
| (x : ℝ) : MemLp (fun y => (v (H x)+v (H y))*b y) 2 μ := by |
| have hh := memLp_field_mul hb (u := fun y => v (H x)+v (H y)) |
| (measurable_const.add (continuous_v.measurable.comp hH)) (fun y => abs_v_sum_le (H x) (H y)) |
| simpa only [mul_comm] using hh |
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| omit [IsProbabilityMeasure μ] in |
| lemma field_energy_integrable {b r H : ℝ → ℝ} (hb : MemLp b 2 μ) (hr : MemLp r 2 μ) |
| (hH : Measurable H) (x : ℝ) : |
| Integrable (fun y => (v (H x)+v (H y))^2*(b y^2+r y^2)) μ := by |
| have he (y : ℝ) : (v (H x)+v (H y))^2*(b y^2+r y^2)= |
| ((v (H x)+v (H y))*b y)^2+((v (H x)+v (H y))*r y)^2 := by ring |
| simp_rw [he] |
| exact (memLp_sum_v_field hb hH x).integrable_sq.add (memLp_sum_v_field hr hH x).integrable_sq |
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|
| lemma fieldRadicand_expansion {b r H : ℝ → ℝ} (hb : MemLp b 2 μ) (hr : MemLp r 2 μ) |
| (hH : Measurable H) (x : ℝ) : fieldRadicand μ H b r x = |
| (∫ y, (v (H x)+v (H y))^2*(b y^2+r y^2) ∂μ)- |
| (∫ y, (v (H x)+v (H y))*b y ∂μ)^2 := by |
| rw [fieldRadicand,integral_centered_sq (memLp_sum_v_field hb hH x)] |
| have he (y : ℝ) : (v (H x)+v (H y))^2*(b y^2+r y^2)= |
| ((v (H x)+v (H y))*b y)^2+((v (H x)+v (H y))*r y)^2 := by ring |
| simp_rw [he] |
| rw [integral_add (memLp_sum_v_field hb hH x).integrable_sq |
| (memLp_sum_v_field hr hH x).integrable_sq] |
| ring |
|
|
| lemma fieldRadicand_le {b r H : ℝ → ℝ} (hb : MemLp b 2 μ) (hr : MemLp r 2 μ) |
| (hH : Measurable H) (he : (∫ x, b x^2+r x^2 ∂μ)=1) (x : ℝ) : |
| fieldRadicand μ H b r x ≤ 4 := by |
| rw [fieldRadicand_expansion hb hr hH] |
| have ih : Integrable (fun y => b y^2+r y^2) μ := hb.integrable_sq.add hr.integrable_sq |
| have hi := integral_mono (field_energy_integrable hb hr hH x) (ih.const_mul (4:ℝ)) |
| (fun y => mul_le_mul_of_nonneg_right (v_sum_sq_le (H x) (H y)) |
| (add_nonneg (sq_nonneg _) (sq_nonneg _))) |
| rw [integral_const_mul,he,mul_one] at hi |
| nlinarith only [hi,sq_nonneg (∫ y, (v (H x)+v (H y))*b y ∂μ)] |
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| lemma measurable_fieldRadicand {b r H : ℝ → ℝ} (hb : Measurable b) (hr : Measurable r) |
| (hH : Measurable H) : Measurable (fieldRadicand μ H b r) := by |
| have hv := continuous_v.measurable.comp hH |
| have hm : Measurable (fun x => ∫ t, (v (H x)+v (H t))*b t ∂μ) := |
| (((hv.comp measurable_fst).add (hv.comp measurable_snd)).mul |
| (hb.comp measurable_snd)).stronglyMeasurable.integral_prod_right.measurable |
| have h1 : Measurable (fun x => ∫ y, ((v (H x)+v (H y))*r y)^2 ∂μ) := |
| ((((hv.comp measurable_fst).add (hv.comp measurable_snd)).mul |
| (hr.comp measurable_snd)).pow_const 2).stronglyMeasurable.integral_prod_right.measurable |
| have h2 : Measurable (fun x => ∫ y, |
| ((v (H x)+v (H y))*b y-(∫ t, (v (H x)+v (H t))*b t ∂μ))^2 ∂μ) := |
| (((((hv.comp measurable_fst).add (hv.comp measurable_snd)).mul |
| (hb.comp measurable_snd)).sub (hm.comp measurable_fst)).pow_const 2).stronglyMeasurable.integral_prod_right.measurable |
| exact h1.add h2 |
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|
| lemma abs_square_sub_le {x y C e : ℝ} (hC : 0 ≤ C) (hx : |x| ≤ C) (hy : |y| ≤ C) |
| (he : |x-y| ≤ e) : |x^2-y^2| ≤ 2*C*e := by |
| rw [show x^2-y^2=(x-y)*(x+y) by ring,abs_mul] |
| have hsum : |x+y| ≤ 2*C := (abs_add_le _ _).trans (by linarith) |
| nlinarith only [mul_le_mul hsum he (abs_nonneg _) (mul_nonneg (by norm_num) hC)] |
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| lemma fieldRadicand_error {b r H H' : ℝ → ℝ} (hb : MemLp b 2 μ) (hr : MemLp r 2 μ) |
| (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) : |
| |fieldRadicand μ H b r x-fieldRadicand μ H' b r x| ≤ 16*a := by |
| let f := fun y => v (H x)+v (H y) |
| let g := fun y => v (H' x)+v (H' y) |
| have hf : Measurable f := measurable_const.add (continuous_v.measurable.comp hH) |
| have hg : Measurable g := measurable_const.add (continuous_v.measurable.comp hH') |
| have hf2 (y : ℝ) : |f y| ≤ 2 := abs_v_sum_le _ _ |
| have hg2 (y : ℝ) : |g y| ≤ 2 := abs_v_sum_le _ _ |
| have hfg (y : ℝ) : |f y-g y| ≤ 2*a := by |
| have heq : f y-g y=(v (H x)-v (H' x))+(v (H y)-v (H' y)) := by dsimp [f,g]; ring |
| rw [heq] |
| exact (abs_add_le _ _).trans (by linarith [hv x,hv y]) |
| have hf_sq (y : ℝ) : |f y^2| ≤ 4 := by rw [abs_of_nonneg (sq_nonneg _)]; exact v_sum_sq_le _ _ |
| have hg_sq (y : ℝ) : |g y^2| ≤ 4 := by rw [abs_of_nonneg (sq_nonneg _)]; exact v_sum_sq_le _ _ |
| have he_sq (y : ℝ) : |f y^2-g y^2| ≤ 8*a := by |
| have hh := abs_square_sub_le (by norm_num : (0:ℝ) ≤ 2) (hf2 y) (hg2 y) (hfg y) |
| nlinarith only [hh] |
| have he' := energy_error hb hr he (hf.pow_const 2) (hg.pow_const 2) hf_sq hg_sq he_sq |
| have hb1 := (norm_components hb hr he).1 |
| have hp := bounded_pair_error hb hb1 hf hg hf2 hg2 (show 0 ≤ 2*a by positivity) hfg |
| have hpf : |∫ y, b y*f y ∂μ| ≤ 2 := (field_pair_abs_le hb (memLp_bounded hf hf2) hb1).trans |
| (l2Norm_le_bounded hf (by norm_num) hf2) |
| have hpg : |∫ y, b y*g y ∂μ| ≤ 2 := (field_pair_abs_le hb (memLp_bounded hg hg2) hb1).trans |
| (l2Norm_le_bounded hg (by norm_num) hg2) |
| have hps := abs_square_sub_le (by norm_num : (0:ℝ) ≤ 2) hpf hpg hp |
| rw [fieldRadicand_expansion hb hr hH,fieldRadicand_expansion hb hr hH'] |
| change |(∫ y, f y^2*(b y^2+r y^2) ∂μ)-(∫ y, f y*b y ∂μ)^2- |
| ((∫ y, g y^2*(b y^2+r y^2) ∂μ)-(∫ y, g y*b y ∂μ)^2)| ≤ _ |
| simp_rw [mul_comm (f _) (b _),mul_comm (g _) (b _)] |
| have heq : (∫ y, f y^2*(b y^2+r y^2) ∂μ)-(∫ y, b y*f y ∂μ)^2- |
| ((∫ y, g y^2*(b y^2+r y^2) ∂μ)-(∫ y, b y*g y ∂μ)^2)= |
| ((∫ y, f y^2*(b y^2+r y^2) ∂μ)-(∫ y, g y^2*(b y^2+r y^2) ∂μ))- |
| ((∫ y, b y*f y ∂μ)^2-(∫ y, b y*g y ∂μ)^2) := by ring |
| rw [heq] |
| exact (abs_sub _ _).trans (by linarith only [he',hps]) |
| |
| lemma sqrt_fieldRadicand_error {b r H H' : ℝ → ℝ} (hb : MemLp b 2 μ) (hr : MemLp r 2 μ) |
| (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) : |
| |sqrt (fieldRadicand μ H b r x)-sqrt (fieldRadicand μ H' b r x)| ≤ 4*sqrt a := by |
| have hh := sqrt_difference (fieldRadicand_nonneg H b r x) (fieldRadicand_nonneg H' b r x) |
| (fieldRadicand_error hb hr hH hH' he ha hv x) |
| simpa only [sqrt_mul (by norm_num : (0:ℝ) ≤ 16),show sqrt (16:ℝ)=4 by norm_num] using hh |
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| end SKRatio.Bins |
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| end |
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| end OAI |
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