| import Mathlib |
| import OAI.Probability.LogConcave.JetEstimates.PolyC1 |
| import OAI.Probability.LogConcave.Analysis.GaussianProposalPotential |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open MeasureTheory Function |
| open scoped RealInnerProductSpace |
|
|
| lemma compact_gibbs_ibp {d : ℕ} {H g : Point d → ℝ} |
| (hH : ContDiff ℝ 1 H) (hi : Integrable (fun z => Real.exp (-H z))) |
| (hg : ContDiff ℝ 1 g) (hgc : HasCompactSupport g) (v : Point d) : |
| (∫z,directional v g z ∂gibbs H)= |
| ∫z,(inner ℝ (gradient H z) v)*g z ∂gibbs H := by |
| have he : ContDiff ℝ 1 (fun z => Real.exp (-H z)) := hH.neg.exp |
| have hdg : Continuous (directional v g) := (hg.continuous_fderiv (by norm_num)).clm_apply continuous_const |
| have hdgc : HasCompactSupport (directional v g) := hgc.fderiv_apply ℝ v |
| have hD (z : Point d) : fderiv ℝ (fun z => Real.exp (-H z)) z v= |
| -(Real.exp (-H z)*inner ℝ (gradient H z) v) := by |
| change (fderiv ℝ (fun z => Real.exp ((-H) z)) z) v=_ |
| rw [((hH.differentiable (by norm_num) z).hasFDerivAt.neg.exp).fderiv] |
| simp only [smul_apply,neg_apply,smul_eq_mul,←toDual_gradient, |
| InnerProductSpace.toDual_apply_apply,Pi.neg_apply] |
| ring |
| have hh := integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable (μ:=volume) |
| (f:=fun z => Real.exp (-H z)) (g:=g) (v:=v) |
| (((he.continuous_fderiv (by norm_num)).clm_apply continuous_const).mul hg.continuous |
| |>.integrable_of_hasCompactSupport hgc.mul_left) |
| ((he.continuous.mul hdg).integrable_of_hasCompactSupport hdgc.mul_left) |
| ((he.continuous.mul hg.continuous).integrable_of_hasCompactSupport hgc.mul_left) |
| (fun z _ => he.differentiable (by norm_num) z) |
| (fun z _ => hg.differentiable (by norm_num) z) |
| simp only [hD,neg_mul,mul_assoc,integral_neg,neg_neg] at hh |
| rw [integral_gibbs_vector hH.continuous hi,integral_gibbs_vector hH.continuous hi] |
| simpa only [smul_eq_mul,directional] using congrArg (fun a => (∫z,Real.exp (-H z))⁻¹*a) hh |
|
|
| lemma adjointCoordinate_compact_pairing {d : ℕ} {H f g : Point d → ℝ} |
| (hH : ContDiff ℝ 1 H) (hi : Integrable (fun z => Real.exp (-H z))) |
| (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g) (hgc : HasCompactSupport g) (v : Point d) : |
| (∫z,adjointCoordinate H v f z*g z ∂gibbs H)= |
| ∫z,f z*directional v g z ∂gibbs H := by |
| have hdf : Continuous (directional v f) := (hf.continuous_fderiv (by norm_num)).clm_apply continuous_const |
| have hdg : Continuous (directional v g) := (hg.continuous_fderiv (by norm_num)).clm_apply continuous_const |
| have hdH : Continuous (directional v H) := (hH.continuous_fderiv (by norm_num)).clm_apply continuous_const |
| let := probability_gibbs_of_partition (partition_pos_of_continuous hH.continuous).ne' |
| (partition_ne_top_of_integrable hi) |
| have h1 : Integrable (fun z => directional v f z*g z) (gibbs H) := |
| (hdf.mul hg.continuous).integrable_of_hasCompactSupport hgc.mul_left |
| have h2 : Integrable (fun z => f z*directional v g z) (gibbs H) := |
| (hf.continuous.mul hdg).integrable_of_hasCompactSupport (hgc.fderiv_apply ℝ v).mul_left |
| have h3 : Integrable (fun z => (directional v H z*f z)*g z) (gibbs H) := |
| ((hdH.mul hf.continuous).mul hg.continuous).integrable_of_hasCompactSupport hgc.mul_left |
| have hh := compact_gibbs_ibp hH hi (hf.mul hg) hgc.mul_left v |
| rw [directional_mul (hf.differentiable (by norm_num)) (hg.differentiable (by norm_num)),integral_add h1 h2] at hh |
| simp only [←directional_gradient,mul_assoc] at h3 ⊢ hh |
| change (∫z,(-directional v f z+directional v H z*f z)*g z ∂gibbs H)=_ |
| simp_rw [add_mul,neg_mul,mul_assoc] |
| rw [integral_add (f:=fun z => -(directional v f z*g z)) h1.neg h3,integral_neg] |
| linarith |
| lemma tensorAdjoint_compact_pairing {d : ℕ} {ι : Type*} [Fintype ι] |
| {H g : Point d → ℝ} {V : ι → Point d → ℝ} |
| (hH : ContDiff ℝ 1 H) (hi : Integrable (fun z => Real.exp (-H z))) |
| (hV : ∀i,ContDiff ℝ 1 (V i)) (hg : ContDiff ℝ 1 g) (hgc : HasCompactSupport g) |
| (b : ι → Point d) : |
| (∫z,tensorAdjoint H b V z*g z ∂gibbs H)= |
| ∫z,∑i,V i z*directional (b i) g z ∂gibbs H := by |
| let := probability_gibbs_of_partition (partition_pos_of_continuous hH.continuous).ne' |
| (partition_ne_top_of_integrable hi) |
| have h1 (i : ι) : Integrable (fun z => adjointCoordinate H (b i) (V i) z*g z) (gibbs H) := |
| ((((hV i).continuous_fderiv (by norm_num)).clm_apply continuous_const).neg.add |
| (((hH.continuous_fderiv (by norm_num)).clm_apply continuous_const).mul (hV i).continuous) |
| |>.mul hg.continuous).integrable_of_hasCompactSupport hgc.mul_left |
| have h2 (i : ι) : Integrable (fun z => V i z*directional (b i) g z) (gibbs H) := |
| ((hV i).continuous.mul ((hg.continuous_fderiv (by norm_num)).clm_apply continuous_const)).integrable_of_hasCompactSupport |
| (hgc.fderiv_apply ℝ (b i)).mul_left |
| simp only [tensorAdjoint,Finset.sum_mul] |
| rw [integral_finsetSum Finset.univ (fun i _ => h1 i),integral_finsetSum Finset.univ (fun i _ => h2 i)] |
| exact Finset.sum_congr rfl (fun i _ => adjointCoordinate_compact_pairing hH hi (hV i) hg hgc (b i)) |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling |
| open MeasureTheory ProbabilityTheory EulerDensity |
| open scoped RealInnerProductSpace ENNReal |
|
|
| lemma standardGaussian_eq_gibbs (d : ℕ) : |
| stdGaussian (Point d)=gibbs (gaussianProposalPotential d) := by |
| have he : (fun z : Point d => ENNReal.ofReal (Real.exp (-‖z‖^2/2)))= |
| gibbsDensity (gaussianProposalPotential d) := by |
| funext z |
| simp only [gibbsDensity,gaussianProposalPotential_eq,neg_div] |
| have hd := standard_density d |
| rw [he] at hd |
| have hp := congrArg (fun μ : Measure (Point d) => μ Set.univ) hd |
| rw [measure_univ,Measure.smul_apply,withDensity_apply _ MeasurableSet.univ, |
| Measure.restrict_univ,smul_eq_mul] at hp |
| change (1:ℝ≥0∞)=gaussianCoefficient d*partition (gaussianProposalPotential d) at hp |
| rw [hd,ENNReal.eq_inv_of_mul_eq_one_left hp.symm] |
| rfl |
|
|
| lemma compact_gaussian_ibp {d : ℕ} {g : Point d → ℝ} |
| (hg : ContDiff ℝ 1 g) (hgc : HasCompactSupport g) (v : Point d) : |
| (∫z,fderiv ℝ g z v ∂stdGaussian (Point d))= |
| ∫z,inner ℝ z v*g z ∂stdGaussian (Point d) := by |
| rw [standardGaussian_eq_gibbs] |
| have hH := (gaussianProposalPotential_admissible d).smooth.of_le (by norm_num : (1:WithTop ℕ∞)≤2) |
| simpa only [directional,gaussianProposalPotential_gradient,id_eq] using |
| compact_gibbs_ibp hH (gaussianProposalPotential_admissible d).integrable_exp_neg hg hgc v |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
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