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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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import Mathlib
import OAI.Probability.LogConcave.Analysis.PrimitivePotentialModeBounds
import OAI.Probability.LogConcave.Sampling.UnitUniform
namespace OAI
section
section
noncomputable section
open MeasureTheory Filter
open scoped ENNReal NNReal Topology
section UpperProof
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open Function MeasureTheory Set Filter
open scoped Topology NNReal
namespace LogConcaveSampling
open scoped RealInnerProductSpace
def gaussianProposalPotential (d : ℕ) (x : Point d) : ℝ := inner ℝ x x / 2
lemma gaussianProposalPotential_eq (d : ℕ) (x : Point d) :
gaussianProposalPotential d x = ‖x‖^2/2 := by
simp [gaussianProposalPotential]
lemma gaussianProposalPotential_hasGradientAt (d : ℕ) (x : Point d) :
HasGradientAt (gaussianProposalPotential d) x x := by
apply hasGradientAt_iff_hasFDerivAt.mpr
have hd := ((hasFDerivAt_id x).inner ℝ (hasFDerivAt_id x)).const_smul (1/2 : ℝ)
convert! hd using 1
· ext y
simp only [gaussianProposalPotential,Pi.smul_apply,smul_eq_mul,id_eq]
ring
· ext y
change inner ℝ x y = (1/2 : ℝ)*(inner ℝ x y+inner ℝ y x)
rw [real_inner_comm y x]
ring
@[simp] lemma gaussianProposalPotential_gradient (d : ℕ) :
gradient (gaussianProposalPotential d) = id :=
gradient_eq (gaussianProposalPotential_hasGradientAt d)
lemma gaussianProposalPotential_admissible (d : ℕ) :
Admissible (gaussianProposalPotential d) := by
refine ⟨(contDiff_id.inner ℝ contDiff_id).div_const 2, ?_, ?_, ?_⟩
· simp [gaussianProposalPotential]
· simp
· intro x v
simp only [gaussianProposalPotential_gradient,fderiv_id,ContinuousLinearMap.id_apply,
real_inner_self_eq_norm_sq]
constructor
· rfl
· nlinarith [sq_nonneg ‖v‖]
lemma partition_comparison {d : ℕ} {V : Point d → ℝ} (hV : Admissible V) :
(2:ℝ)^(- (d:ℤ)) * (partition (gaussianProposalPotential d)).toReal ≤
(partition V).toReal ∧
(partition V).toReal ≤ (partition (gaussianProposalPotential d)).toReal := by
have hB := gaussianProposalPotential_admissible d
rw [partition_toReal hV.integrable_exp_neg,partition_toReal hB.integrable_exp_neg]
constructor
· have hs := Measure.integral_comp_smul_of_nonneg (volume : Measure (Point d))
(fun x => Real.exp (-gaussianProposalPotential d x)) (2:ℝ) (hR := by norm_num)
simp only [finrank_euclideanSpace_fin,smul_eq_mul] at hs
have hh : Integrable (fun x : Point d => Real.exp (-gaussianProposalPotential d ((2:ℝ) • x))) := by
convert integrable_gaussian_envelope d (b := 2) (by norm_num) using 1
ext x
rw [gaussianProposalPotential_eq,norm_smul,Real.norm_of_nonneg (by norm_num)]
congr 1
ring
rw [zpow_neg,zpow_natCast,← hs]
apply integral_mono hh hV.integrable_exp_neg
intro x
apply Real.exp_le_exp.mpr
rw [gaussianProposalPotential_eq,norm_smul,Real.norm_of_nonneg (by norm_num)]
have ht := (hV.quadratic_bounds x).2
nlinarith [sq_nonneg ‖x‖]
· apply integral_mono hV.integrable_exp_neg hB.integrable_exp_neg
intro x
apply Real.exp_le_exp.mpr
rw [gaussianProposalPotential_eq]
linarith [(hV.quadratic_bounds x).1]
def gaussianAcceptance {d : ℕ} (V : Point d → ℝ) (x : Point d) : ℝ :=
Real.exp (-V x+gaussianProposalPotential d x)
lemma gaussianAcceptance_bounds {d : ℕ} {V : Point d → ℝ} (hV : Admissible V) (x : Point d) :
0 ≤ gaussianAcceptance V x ∧ gaussianAcceptance V x ≤ 1 := by
refine ⟨(Real.exp_pos _).le, ?_⟩
apply Real.exp_le_one_iff.mpr
rw [gaussianProposalPotential_eq]
linarith [(hV.quadratic_bounds x).1]
@[fun_prop] lemma measurable_gaussianAcceptance {d : ℕ} {V : Point d → ℝ} (hV : Admissible V) :
Measurable (gaussianAcceptance V) :=
(Real.continuous_exp.comp (hV.smooth.continuous.neg.add
(gaussianProposalPotential_admissible d).smooth.continuous)).measurable
lemma gaussian_accepted_measure {d : ℕ} {V : Point d → ℝ} (hV : Admissible V) :
(((gibbs (gaussianProposalPotential d)).prod unitUniform).restrict
(acceptanceEvent (gaussianAcceptance V))).map Prod.fst =
ENNReal.ofReal ((partition V).toReal/(partition (gaussianProposalPotential d)).toReal) • gibbs V := by
have hB := gaussianProposalPotential_admissible d
have := hB.isProbabilityMeasure_gibbs
rw [accepted_measure _ (measurable_gaussianAcceptance hV)
(fun x => (gaussianAcceptance_bounds hV x).1) (fun x => (gaussianAcceptance_bounds hV x).2)]
rw [gibbs,withDensity_smul_measure,← withDensity_mul]
· have he : gibbsDensity (gaussianProposalPotential d) *
(fun x => ENNReal.ofReal (gaussianAcceptance V x)) = gibbsDensity V := by
funext x
simp only [Pi.mul_apply,gibbsDensity,gaussianAcceptance,← ENNReal.ofReal_mul (Real.exp_pos _).le,
← Real.exp_add]
congr 2
ring
rw [he,gibbs,smul_smul]
congr 1
rw [ENNReal.ofReal_div_of_pos (ENNReal.toReal_pos hB.partition_pos.ne' hB.partition_ne_top),
ENNReal.ofReal_toReal hV.partition_ne_top,ENNReal.ofReal_toReal hB.partition_ne_top,div_eq_mul_inv]
rw [mul_right_comm,ENNReal.mul_inv_cancel hV.partition_pos.ne' hV.partition_ne_top,one_mul]
· exact (Real.continuous_exp.comp hB.smooth.continuous.neg).measurable.ennreal_ofReal
· exact (measurable_gaussianAcceptance hV).ennreal_ofReal
end LogConcaveSampling
end UpperProof
end
end
end
end OAI
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