| import Mathlib |
| import OAI.Probability.SKRatio.Variational.ScalarHilbert |
|
|
| namespace OAI |
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|
| noncomputable section |
| open scoped NNReal ENNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real |
| namespace SKRatio.Bins |
| open Planted Scalar |
|
|
| variable {μ : Measure ℝ} [IsProbabilityMeasure μ] |
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| lemma memLp_bounded {f : ℝ → ℝ} (hf : Measurable f) {C : ℝ} |
| (hC : ∀ x, |f x| ≤ C) : MemLp f 2 μ := |
| MemLp.of_bound hf.aestronglyMeasurable C (ae_of_all _ (fun x => by |
| simpa only [norm_eq_abs] using hC x)) |
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| lemma l2Norm_le_bounded {f : ℝ → ℝ} (hf : Measurable f) {C : ℝ} |
| (hC : 0 ≤ C) (hfC : ∀ x, |f x| ≤ C) : l2Norm μ f ≤ C := by |
| have hi := integral_mono (memLp_bounded hf hfC (μ := μ)).integrable_sq |
| (integrable_const (C^2)) (fun x => show f x^2 ≤ C^2 by |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hC).mpr (hfC x)) |
| simp only [integral_const,probReal_univ,smul_eq_mul,one_mul] at hi |
| have hs := l2Norm_sq (μ := μ) f |
| nlinarith only [hi,hs,l2Norm_nonneg (μ := μ) f,hC] |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma memLp_field_mul {f u : ℝ → ℝ} (hf : MemLp f 2 μ) |
| (hu : Measurable u) {C : ℝ} (hC : ∀ x, |u x| ≤ C) : |
| MemLp (fun x => f x*u x) 2 μ := by |
| apply hf.of_le_mul (hf.aestronglyMeasurable.mul hu.aestronglyMeasurable) (c := C) |
| exact ae_of_all _ (fun x => by |
| change |f x*u x| ≤ C*|f x| |
| rw [abs_mul] |
| nlinarith only [mul_le_mul_of_nonneg_left (hC x) (abs_nonneg (f x))]) |
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|
| omit [IsProbabilityMeasure μ] in |
| lemma l2Norm_field_mul {f u : ℝ → ℝ} (hf : MemLp f 2 μ) |
| (hu : Measurable u) {C : ℝ} (hC : 0 ≤ C) (huC : ∀ x, |u x| ≤ C) : |
| l2Norm μ (fun x => f x*u x) ≤ C*l2Norm μ f := by |
| have hi := integral_mono (memLp_field_mul hf hu huC).integrable_sq |
| (hf.integrable_sq.const_mul (C^2)) (fun x => show (f x*u x)^2 ≤ C^2*f x^2 by |
| have hh : u x^2 ≤ C^2 := by |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hC).mpr (huC x) |
| simpa only [mul_pow,mul_comm] using mul_le_mul_of_nonneg_left hh (sq_nonneg (f x))) |
| rw [integral_const_mul,←l2Norm_sq,←l2Norm_sq] at hi |
| have hn := l2Norm_nonneg (μ := μ) (fun x => f x*u x) |
| have hn' : 0 ≤ C*l2Norm μ f := mul_nonneg hC (l2Norm_nonneg f) |
| nlinarith only [hi,hn,hn'] |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma norm_components {b r : ℝ → ℝ} (hb : MemLp b 2 μ) (hr : MemLp r 2 μ) |
| (he : (∫ x, b x^2+r x^2 ∂μ)=1) : l2Norm μ b ≤ 1 ∧ l2Norm μ r ≤ 1 := by |
| rw [integral_add hb.integrable_sq hr.integrable_sq,←l2Norm_sq,←l2Norm_sq] at he |
| constructor <;> nlinarith only [he,sq_nonneg (l2Norm μ b),sq_nonneg (l2Norm μ r)] |
|
|
| lemma integral_abs_le_l2 {f : ℝ → ℝ} (hf : MemLp f 2 μ) : |
| (∫ x, |f x| ∂μ) ≤ l2Norm μ f := by |
| have hh : |∫ x, |f x| * (1:ℝ) ∂μ| ≤ l2Norm μ (fun x => |f x|)*l2Norm μ (fun _ => (1:ℝ)) := |
| abs_integral_mul_le hf.abs (memLp_const (1:ℝ) (μ := μ) (p := 2)) |
| have he : l2Norm μ (fun x => |f x|)=l2Norm μ f := by simp only [l2Norm,sq_abs] |
| have ho : l2Norm μ (fun _ : ℝ => (1:ℝ))=1 := by |
| simp only [l2Norm,one_pow,integral_const,probReal_univ,smul_eq_mul,one_mul,sqrt_one] |
| simp only [mul_one,he,ho] at hh |
| rw [abs_of_nonneg (integral_nonneg (fun x : ℝ => abs_nonneg (f x)))] at hh |
| exact hh |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma field_pair_abs_le {b f : ℝ → ℝ} (hb : MemLp b 2 μ) (hf : MemLp f 2 μ) |
| (hb1 : l2Norm μ b ≤ 1) : |∫ x, b x*f x ∂μ| ≤ l2Norm μ f := |
| (abs_integral_mul_le hb hf).trans (by |
| simpa only [one_mul] using mul_le_mul_of_nonneg_right hb1 (l2Norm_nonneg f)) |
|
|
| lemma integral_abs_bound {f : ℝ → ℝ} (hf : Integrable f μ) {C : ℝ} |
| (hC : ∀ x, |f x| ≤ C) : |∫ x, f x ∂μ| ≤ C := by |
| apply abs_integral_le_integral_abs.trans |
| simpa only [integral_const,probReal_univ,smul_eq_mul,one_mul] using |
| integral_mono hf.abs (integrable_const C) hC |
|
|
| lemma bounded_pair_error {b f g : ℝ → ℝ} (hb : MemLp b 2 μ) |
| (hb1 : l2Norm μ b ≤ 1) (hf : Measurable f) (hg : Measurable g) |
| {C D a : ℝ} (hfC : ∀ x, |f x| ≤ C) (hgD : ∀ x, |g x| ≤ D) |
| (ha : 0 ≤ a) (he : ∀ x, |f x-g x| ≤ a) : |
| |(∫ x, b x*f x ∂μ)-(∫ x, b x*g x ∂μ)| ≤ a := by |
| have hfi : Integrable (fun x => b x*f x) μ := hb.integrable_mul (memLp_bounded hf hfC) |
| have hgi : Integrable (fun x => b x*g x) μ := hb.integrable_mul (memLp_bounded hg hgD) |
| rw [←integral_sub hfi hgi] |
| simp_rw [←mul_sub] |
| exact (field_pair_abs_le hb ((memLp_bounded hf hfC).sub (memLp_bounded hg hgD)) hb1).trans |
| (l2Norm_le_bounded (hf.sub hg) ha he) |
|
|
| omit [IsProbabilityMeasure μ] in |
| lemma energy_error {b r f g : ℝ → ℝ} (hb : MemLp b 2 μ) (hr : MemLp r 2 μ) |
| (he : (∫ x, b x^2+r x^2 ∂μ)=1) |
| (hf : Measurable f) (hg : Measurable g) {C D a : ℝ} |
| (hfC : ∀ x, |f x| ≤ C) (hgD : ∀ x, |g x| ≤ D) (hfg : ∀ x, |f x-g x| ≤ a) : |
| |(∫ x, f x*(b x^2+r x^2) ∂μ)-(∫ x, g x*(b x^2+r x^2) ∂μ)| ≤ a := by |
| have ih : Integrable (fun x => b x^2+r x^2) μ := hb.integrable_sq.add hr.integrable_sq |
| have ihf := ih.bdd_mul hf.aestronglyMeasurable (ae_of_all _ (fun x => by simpa only [norm_eq_abs] using hfC x)) |
| have ihg := ih.bdd_mul hg.aestronglyMeasurable (ae_of_all _ (fun x => by simpa only [norm_eq_abs] using hgD x)) |
| rw [←integral_sub ihf ihg] |
| apply abs_integral_le_integral_abs.trans |
| have hi := integral_mono (ihf.sub ihg).abs (ih.const_mul a) (fun x => by |
| simp only [Pi.sub_apply,←sub_mul,abs_mul,abs_of_nonneg (add_nonneg (sq_nonneg (b x)) (sq_nonneg (r x)))] |
| exact mul_le_mul_of_nonneg_right (hfg x) (add_nonneg (sq_nonneg _) (sq_nonneg _))) |
| simpa only [integral_const_mul,Pi.add_apply,Pi.sub_apply,he,mul_one] using hi |
|
|
| lemma sqrt_difference {s t e : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) |
| (he : |s-t| ≤ e) : |sqrt s-sqrt t| ≤ sqrt e := by |
| have he0 := (abs_nonneg (s-t)).trans he |
| have hs2 := sq_sqrt hs |
| have ht2 := sq_sqrt ht |
| have he2 := sq_sqrt he0 |
| have habs : (sqrt s-sqrt t)^2 ≤ e := by |
| rcases le_total s t with hst|hts |
| · rw [abs_of_nonpos (sub_nonpos.mpr hst)] at he |
| have hh := sqrt_le_sqrt hst |
| nlinarith only [he,hh,hs2,ht2,mul_nonneg (sqrt_nonneg s) (sub_nonneg.mpr hh)] |
| · rw [abs_of_nonneg (sub_nonneg.mpr hts)] at he |
| have hh := sqrt_le_sqrt hts |
| nlinarith only [he,hh,hs2,ht2,mul_nonneg (sqrt_nonneg t) (sub_nonneg.mpr hh)] |
| exact (sq_le_sq₀ (abs_nonneg _) (sqrt_nonneg e)).mp (by simpa only [sq_abs,he2] using habs) |
|
|
| end SKRatio.Bins |
|
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| end |
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| end OAI |
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