| 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 |
|
|
