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ea76aac | 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 | import Mathlib
import OAI.Computability.MinUncut.Estimates.ExpectResampleRow
namespace OAI
section
noncomputable section
open scoped BigOperators
namespace MinUncut.BoxGaussian
open MeasureTheory ProbabilityTheory GaussianHermite
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
def singletonEmbed (x : ι) : Unit → ι := fun _ => x
omit [Fintype ι] [DecidableEq ι] in
lemma singletonEmbed_injective (x : ι) : Function.Injective (singletonEmbed x) := by
intro a b _
exact Subsingleton.elim a b
omit [Fintype ι] in
lemma update_join_singleton (x : ι) (C : Unit → ℝ) (D : Outside (singletonEmbed x) → ℝ) (t : ℝ) :
Function.update (join (singletonEmbed x) (singletonEmbed_injective x) C D) x t =
join (singletonEmbed x) (singletonEmbed_injective x) (fun _ => t) D := by
funext y
by_cases hy : y=x
· subst y
simpa only [Function.update_self, singletonEmbed] using
(join_at (singletonEmbed x) (singletonEmbed_injective x) (fun _ => t) D Unit.unit).symm
· rw [Function.update_of_ne hy]
have hout : y∉Set.range (singletonEmbed x) := by
rintro ⟨a,ha⟩
exact hy ha.symm
rw [join_other _ _ _ _ ⟨y,hout⟩,join_other _ _ _ _ ⟨y,hout⟩]
lemma integral_singleton (v : ℝ → ℝ) :
(∫ C : Unit → ℝ, v (C Unit.unit) ∂γpi Unit) = ∫ t, v t ∂γ := by
exact (measurePreserving_funUnique γ Unit).integral_comp
(MeasurableEquiv.funUnique Unit ℝ).measurableEmbedding v
lemma centered_resample_memLp (x : ι) (u : (ι → ℝ) → ℝ) (hu : Measurable u)
{A : ℝ} (hA : 0≤A) :
MemLp (fun C => Slice.centeredClip γ A (fun t => u (Function.update C x t)) (C x)) 2 (γpi ι) := by
have hupdate : Measurable (fun p : (ι → ℝ) × ℝ => Function.update p.1 x p.2) := by
apply Measurable.of_eval
intro y
by_cases hy : y=x
· subst y
simpa only [Function.update_self, singletonEmbed] using measurable_snd
· simp only [Function.update_of_ne hy]
exact (measurable_pi_apply y).comp measurable_fst
have hm : Measurable (fun p : (ι → ℝ) × ℝ => Slice.clip A (u (Function.update p.1 x p.2))) :=
(measurable_const.max ((hu.comp hupdate).min measurable_const))
have hmean : Measurable (fun C => ∫ t, Slice.clip A (u (Function.update C x t)) ∂γ) :=
hm.stronglyMeasurable.integral_prod_right.measurable
have hd : Measurable (fun C => Slice.centeredClip γ A (fun t => u (Function.update C x t)) (C x)) := by
unfold Slice.centeredClip
simp only [Function.update_eq_self]
exact (measurable_const.max (hu.min measurable_const)).sub hmean
apply MemLp.of_bound hd.aestronglyMeasurable (2*A)
filter_upwards [] with C
rw [Real.norm_eq_abs]
unfold Slice.centeredClip
exact (abs_sub _ _).trans (by
have hc := Slice.clip_abs_le hA (u (Function.update C x (C x)))
have hm : |∫ t, Slice.clip A (u (Function.update C x t)) ∂γ|≤A := by
refine (abs_integral_le_integral_abs).trans ?_
exact (integral_mono_of_nonneg (ae_of_all _ (fun _ => abs_nonneg _))
(integrable_const A) (ae_of_all _ (fun t => Slice.clip_abs_le hA _))).trans_eq
(by simp)
linarith)
omit [DecidableEq ι] in
lemma integral_split_le {κ : Type*} [Fintype κ] (μ : Measure ℝ) [SigmaFinite μ]
(e : κ → ι) (he : Function.Injective e) (u v : (ι → ℝ) → ℝ)
(hu : Integrable u (Measure.pi (fun _ : ι => μ)))
(hv : Integrable v (Measure.pi (fun _ : ι => μ))) {a : ℝ}
(h : ∀ D : Outside e → ℝ,
(∫ C : κ → ℝ, u (join e he C D) ∂Measure.pi (fun _ : κ => μ)) ≤
a*(∫ C : κ → ℝ, v (join e he C D) ∂Measure.pi (fun _ : κ => μ))) :
(∫ C, u C ∂Measure.pi (fun _ : ι => μ)) ≤ a*(∫ C, v C ∂Measure.pi (fun _ : ι => μ)) := by
have hup := ((join_preserves μ e he).integrable_comp_emb (joinEquiv e he).measurableEmbedding).mpr hu
have hvp := ((join_preserves μ e he).integrable_comp_emb (joinEquiv e he).measurableEmbedding).mpr hv
rw [integral_split μ e he u hu,integral_split μ e he v hv, ← integral_const_mul]
exact integral_mono hup.integral_prod_right (hvp.integral_prod_right.const_mul a) h
attribute [local irreducible] Finset.univ Fintype.piFinset
lemma centered_resample_loss (x : ι) (u : (ι → ℝ) → ℝ) (hum : Measurable u)
(hu : MemLp u 2 (γpi ι))
(hv : ∀ C, MemLp (fun t => u (Function.update C x t)) 2 γ)
(hz : ∀ C, (∫ t, u (Function.update C x t) ∂γ)=0)
{A : ℝ} (hA : 0<A) :
(∫ C, |u C-Slice.centeredClip γ A (fun t => u (Function.update C x t)) (C x)| ∂γpi ι) ≤
2*(∫ C, (u C)^2 ∂γpi ι)/A := by
let e := singletonEmbed x
let he := singletonEmbed_injective x
let w := fun C => Slice.centeredClip γ A (fun t => u (Function.update C x t)) (C x)
have hw : MemLp w 2 (γpi ι) := centered_resample_memLp x u hum hA.le
have hl : Integrable (fun C => |u C-w C|) (γpi ι) := (hu.sub hw).integrable (by norm_num) |>.abs
have hq : Integrable (fun C => (u C)^2) (γpi ι) := hu.integrable_sq
have hs D : (∫ C : Unit → ℝ, |u (join e he C D)-w (join e he C D)| ∂γpi Unit) ≤
2*(∫ C : Unit → ℝ, (u (join e he C D))^2 ∂γpi Unit)/A := by
let C₀ := join e he (fun _ => 0) D
let v := fun t => u (Function.update C₀ x t)
have huC (C : Unit → ℝ) : u (join e he C D)=v (C Unit.unit) := by
dsimp only [v,C₀,e,he]
rw [update_join_singleton]
have hwC (C : Unit → ℝ) : w (join e he C D)=Slice.centeredClip γ A v (C Unit.unit) := by
dsimp only [w]
have hh : (fun t => u (Function.update (join e he C D) x t))=v := by
funext t
dsimp only [v,C₀,e,he]
rw [update_join_singleton,update_join_singleton]
rw [hh]
congr 1
exact join_at e he C D Unit.unit
simp_rw [huC,hwC]
rw [integral_singleton (fun t => |v t-Slice.centeredClip γ A v t|),
integral_singleton (fun t => (v t)^2)]
exact Slice.centeredClip_loss hA (hv C₀) (hz C₀)
change (∫ C, |u C-w C| ∂γpi ι) ≤ _
rw [← div_mul_eq_mul_div]
apply integral_split_le γ e he _ _ hl hq
intro D
simpa only [div_mul_eq_mul_div] using hs D
end MinUncut.BoxGaussian
end
end
end OAI
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