File size: 3,925 Bytes
dd14c26 | 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 | import Mathlib
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling.GradientIntegral
open MeasureTheory Set Function
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
[MeasurableSpace E] [BorelSpace E] {μ : Measure E} [IsFiniteMeasure μ]
omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
lemma gradient_bound {φ : E → ℝ} (hφ : ContDiff ℝ 1 φ) (hφc : HasCompactSupport φ) :
∃C : ℝ,0≤C ∧ ∀y,‖fderiv ℝ φ y‖≤C := by
obtain ⟨C,hC⟩ := (hφc.fderiv ℝ).exists_bound_of_continuousOn
(hφ.continuous_fderiv (by norm_num)).continuousOn
refine ⟨max C 0,le_max_right _ _,fun y => ?_⟩
by_cases hy : y∈tsupport (fderiv ℝ φ)
· exact (hC y hy).trans (le_max_left _ _)
· rw [image_eq_zero_of_notMem_tsupport hy,norm_zero]
exact le_max_right _ _
lemma continuous_gradient_expectation {φ : E → ℝ} (hφ : ContDiff ℝ 1 φ)
(hφc : HasCompactSupport φ) {X : ℝ × E → E →L[ℝ] E} (hX : Continuous X) (u : E) :
Continuous (fun t => ∫y,fderiv ℝ φ y (X (t,y) u) ∂μ) := by
have hc : Continuous (fun p : ℝ × E => fderiv ℝ φ p.2 (X p u)) :=
((hφ.continuous_fderiv (by norm_num)).comp continuous_snd).clm_apply (hX.clm_apply continuous_const)
have he (t : ℝ) : (∫y in tsupport φ,fderiv ℝ φ y (X (t,y) u) ∂μ)=
∫y,fderiv ℝ φ y (X (t,y) u) ∂μ := by
apply setIntegral_eq_integral_of_forall_compl_eq_zero
intro y hy
rw [(HasFDerivAt.of_notMem_tsupport ℝ hy).fderiv]
rfl
simpa only [he] using
(continuous_parametric_integral_of_continuous (μ:=μ) (f:=fun t y => fderiv ℝ φ y (X (t,y) u)) hc hφc)
lemma gradient_time_space_integrable {φ : E → ℝ} (hφ : ContDiff ℝ 1 φ)
(hφc : HasCompactSupport φ) {X : ℝ × E → E →L[ℝ] E} (hX : Continuous X)
{T C : ℝ} (hC : 0≤C) (hbound : ∀t∈Icc (0:ℝ) T,∀y,‖X (t,y)‖≤C) (u : E) :
Integrable (fun p : ℝ × E => fderiv ℝ φ p.2 (X p u)) ((volume.restrict (Icc 0 T)).prod μ) := by
obtain ⟨D,_,hDb⟩ := gradient_bound hφ hφc
have hc : Continuous (fun p : ℝ × E => fderiv ℝ φ p.2 (X p u)) :=
((hφ.continuous_fderiv (by norm_num)).comp continuous_snd).clm_apply (hX.clm_apply continuous_const)
apply (integrable_const (D*C*‖u‖)).mono' hc.aestronglyMeasurable
have ha : ∀ᵐ p ∂((volume.restrict (Icc (0:ℝ) T)).prod μ),p.1∈Icc (0:ℝ) T :=
Measure.quasiMeasurePreserving_fst.ae (ae_restrict_mem measurableSet_Icc)
filter_upwards [ha] with p hp
calc
‖fderiv ℝ φ p.2 (X p u)‖≤‖fderiv ℝ φ p.2‖*‖X p u‖ := ContinuousLinearMap.le_opNorm _ _
_≤‖fderiv ℝ φ p.2‖*(C*‖u‖) := mul_le_mul_of_nonneg_left
((ContinuousLinearMap.le_opNorm _ _).trans (mul_le_mul_of_nonneg_right (hbound p.1 hp p.2) (norm_nonneg _)))
(norm_nonneg _)
_≤D*(C*‖u‖) := mul_le_mul_of_nonneg_right (hDb p.2) (mul_nonneg hC (norm_nonneg _))
_=D*C*‖u‖ := by ring
theorem gradient_exchange {φ : E → ℝ} (hφ : ContDiff ℝ 1 φ) (hφc : HasCompactSupport φ)
{X : ℝ × E → E →L[ℝ] E} (hX : Continuous X) {T C : ℝ}
(hC : 0≤C) (hbound : ∀t∈Icc (0:ℝ) T,∀y,‖X (t,y)‖≤C) (u : E) :
(∫t in Icc (0:ℝ) T,∫y,fderiv ℝ φ y (X (t,y) u) ∂μ)=
∫y,fderiv ℝ φ y ((∫t in Icc (0:ℝ) T,X (t,y)) u) ∂μ := by
rw [integral_integral_swap (gradient_time_space_integrable (μ:=μ) hφ hφc hX hC hbound u)]
apply integral_congr_ae
filter_upwards [] with y
have hi : IntegrableOn (fun t => X (t,y)) (Icc (0:ℝ) T) :=
(hX.comp (continuous_id.prodMk continuous_const)).continuousOn.integrableOn_compact isCompact_Icc
have hh := ((fderiv ℝ φ y).comp (ContinuousLinearMap.apply ℝ E u)).integral_comp_comm hi
exact hh
end LogConcaveSampling.GradientIntegral
end
end
end
end OAI
|