| import Mathlib |
| import OAI.Probability.LogConcave.Model |
| import OAI.Probability.LogConcave.Sampling.LintegralFinNatProd |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section UpperProof |
| namespace LogConcaveSampling |
|
|
| theorem map_withDensity_of_measurePreserving {E F : Type*} |
| [MeasurableSpace E] [MeasurableSpace F] {μ : Measure E} {ν : Measure F} |
| (e : E ≃ᵐ F) (he : MeasurePreserving e μ ν) {f : E → ℝ≥0∞} (hf : Measurable f) : |
| (μ.withDensity f).map e = ν.withDensity (fun y => f (e.symm y)) := by |
| ext s hs |
| rw [Measure.map_apply e.measurable hs, withDensity_apply _ (hs.preimage e.measurable), |
| withDensity_apply _ hs] |
| have h := he.setLIntegral_comp_preimage hs (hf.comp e.symm.measurable) |
| simpa only [Function.comp_apply, MeasurableEquiv.symm_apply_apply] using h |
|
|
| end LogConcaveSampling |
|
|
| namespace LogConcaveSampling |
| namespace EulerDensity |
| open ProbabilityTheory MeasureTheory WithLp |
| open scoped ENNReal NNReal RealInnerProductSpace |
|
|
| def gaussianCoefficient (d : ℕ) : ℝ≥0∞ := |
| ∏ _ : Fin d, ENNReal.ofReal ((Real.sqrt (2*Real.pi))⁻¹) |
|
|
| lemma standard_pi_density (d : ℕ) : |
| Measure.pi (fun _ : Fin d => gaussianReal 0 1) = |
| (volume : Measure (Fin d → ℝ)).withDensity |
| (fun x => gaussianCoefficient d*ENNReal.ofReal (Real.exp (-(∑ k, (x k)^2)/2))) := by |
| have hg : gaussianReal 0 1 = volume.withDensity (gaussianPDF 0 1) := |
| gaussianReal_of_var_ne_zero _ (by norm_num) |
| have : IsProbabilityMeasure (volume.withDensity (gaussianPDF 0 1)) := by rw [← hg]; infer_instance |
| simp_rw [hg] |
| rw [pi_withDensity_fin _ _ (fun _ => measurable_gaussianPDF _ _)] |
| congr 1 |
| ext x |
| have he (z : ℝ) : gaussianPDF 0 1 z = |
| ENNReal.ofReal ((Real.sqrt (2*Real.pi))⁻¹)*ENNReal.ofReal (Real.exp (-z^2/2)) := by |
| simp [gaussianPDF,gaussianPDFReal] |
| simp_rw [he] |
| rw [Finset.prod_mul_distrib] |
| congr 1 |
| rw [← ENNReal.ofReal_prod_of_nonneg (fun _ _ => (Real.exp_pos _).le),← Real.exp_sum] |
| congr 2 |
| rw [← Finset.sum_div,Finset.sum_neg_distrib] |
|
|
| theorem standard_density (d : ℕ) : |
| stdGaussian (Point d) = gaussianCoefficient d • |
| (volume : Measure (Point d)).withDensity |
| (fun z => ENNReal.ofReal (Real.exp (-‖z‖^2/2))) := by |
| rw [← map_pi_eq_stdGaussian,standard_pi_density] |
| let e : (Fin d → ℝ) ≃ᵐ Point d := MeasurableEquiv.toLp 2 (Fin d → ℝ) |
| change Measure.map e _ = _ |
| rw [map_withDensity_of_measurePreserving e (PiLp.volume_preserving_toLp (Fin d)) (by fun_prop)] |
| have he : (fun y : Point d => gaussianCoefficient d* |
| ENNReal.ofReal (Real.exp (-(∑ k, (e.symm y k)^2)/2))) = |
| gaussianCoefficient d • (fun z : Point d => ENNReal.ofReal (Real.exp (-‖z‖^2/2))) := by |
| funext y |
| simp only [Pi.smul_apply,smul_eq_mul] |
| congr 3 |
| change -(∑ k, (y k)^2)/2 = -‖y‖^2/2 |
| rw [EuclideanSpace.norm_sq_eq] |
| simp [Real.norm_eq_abs,sq_abs] |
| rw [he,withDensity_smul] |
| fun_prop |
|
|
| lemma gaussianCoefficient_positive (d : ℕ) : 0 < gaussianCoefficient d := by |
| apply pos_iff_ne_zero.mpr |
| apply Finset.prod_ne_zero_iff.mpr |
| intro i _ |
| exact ENNReal.ofReal_ne_zero_iff.mpr (by positivity) |
|
|
| lemma gaussianCoefficient_finite (d : ℕ) : gaussianCoefficient d ≠ ⊤ := by |
| exact ENNReal.prod_ne_top (fun _ _ => ENNReal.ofReal_ne_top) |
|
|
| lemma standard_partition (d : ℕ) : |
| (gaussianCoefficient d).toReal*(∫ z : Point d, Real.exp (-‖z‖^2/2)) = 1 := by |
| have he := congrArg (fun μ : Measure (Point d) => μ Set.univ) (standard_density d) |
| rw [measure_univ,Measure.smul_apply,withDensity_apply _ MeasurableSet.univ, |
| Measure.restrict_univ,smul_eq_mul] at he |
| have hh := congrArg ENNReal.toReal he |
| rw [ENNReal.toReal_one,ENNReal.toReal_mul] at hh |
| rw [integral_eq_lintegral_of_nonneg_ae (Filter.Eventually.of_forall (fun _ => (Real.exp_pos _).le)) |
| (by fun_prop)] |
| exact hh.symm |
|
|
| def shear {d : ℕ} (T : Point d → Point d) (hT : Measurable T) : |
| Point d × Point d ≃ᵐ Point d × Point d where |
| toFun p := (p.1,T p.1+p.2) |
| invFun p := (p.1,p.2-T p.1) |
| left_inv p := by ext <;> simp |
| right_inv p := by ext <;> simp |
| measurable_toFun := measurable_fst.prodMk ((hT.comp measurable_fst).add measurable_snd) |
| measurable_invFun := measurable_fst.prodMk (measurable_snd.sub (hT.comp measurable_fst)) |
|
|
| lemma shear_measurePreserving {d : ℕ} {T : Point d → Point d} (hT : Measurable T) : |
| MeasurePreserving (shear T hT) ((volume : Measure (Point d)).prod volume) |
| ((volume : Measure (Point d)).prod volume) := by |
| exact (MeasurePreserving.id volume).skew_product ((hT.comp measurable_fst).add measurable_snd) |
| (Filter.Eventually.of_forall (fun x => map_add_left_eq_self volume (T x))) |
|
|
| end EulerDensity |
| end LogConcaveSampling |
|
|
| namespace LogConcaveSampling |
| namespace EulerDensity |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal NNReal RealInnerProductSpace |
|
|
| lemma map_withDensity_equiv {E F : Type*} [MeasurableSpace E] [MeasurableSpace F] |
| (μ : Measure E) (e : E ≃ᵐ F) {f : E → ℝ≥0∞} (hf : Measurable f) : |
| (μ.withDensity f).map e = (μ.map e).withDensity (fun y => f (e.symm y)) := |
| map_withDensity_of_measurePreserving e ⟨e.measurable,rfl⟩ hf |
|
|
| def noiseCoefficient (d : ℕ) (σ : ℝ) : ℝ≥0∞ := |
| gaussianCoefficient d*ENNReal.ofReal |(σ^d)⁻¹| |
|
|
| lemma scaled_standard_density (d : ℕ) {σ : ℝ} (hσ : σ ≠ 0) : |
| (stdGaussian (Point d)).map (fun z => σ • z) = |
| noiseCoefficient d σ • (volume : Measure (Point d)).withDensity |
| (fun z => ENNReal.ofReal (Real.exp (-‖z‖^2/(2*σ^2)))) := by |
| let e := (Homeomorph.smulOfNeZero (α := Point d) σ hσ).toMeasurableEquiv |
| rw [standard_density,Measure.map_smul _ (by fun_prop)] |
| change gaussianCoefficient d • (Measure.map e _) = _ |
| rw [map_withDensity_equiv volume e (by fun_prop)] |
| have he : (volume : Measure (Point d)).map e = |
| ENNReal.ofReal |(σ^d)⁻¹| • volume := by |
| convert! Measure.map_addHaar_smul (volume : Measure (Point d)) hσ using 1 |
| simp [Point] |
| rw [he,withDensity_smul_measure,smul_smul] |
| congr 2 |
| funext z |
| have hz : e.symm z = σ⁻¹ • z := by |
| exact congrFun (Homeomorph.smulOfNeZero_symm_apply hσ) z |
| rw [hz,norm_smul,Real.norm_eq_abs,mul_pow,sq_abs] |
| congr 2 |
| field_simp |
|
|
| lemma noiseCoefficient_positive (d : ℕ) {σ : ℝ} (hσ : σ ≠ 0) : |
| 0 < noiseCoefficient d σ := by |
| apply ENNReal.mul_pos_iff.mpr |
| exact ⟨gaussianCoefficient_positive d,ENNReal.ofReal_pos.mpr (abs_pos.mpr (inv_ne_zero (pow_ne_zero _ hσ)))⟩ |
|
|
| lemma noiseCoefficient_finite (d : ℕ) (σ : ℝ) : noiseCoefficient d σ ≠ ⊤ := |
| ENNReal.mul_ne_top (gaussianCoefficient_finite d) ENNReal.ofReal_ne_top |
|
|
| theorem joint_density {d : ℕ} {H : Point d → ℝ} (hH : Measurable H) |
| (T : Point d → Point d) (hT : Measurable T) {σ : ℝ} (hσ : σ ≠ 0) : |
| ((gibbs H).prod (stdGaussian (Point d))).map (fun p => (p.1,T p.1+σ • p.2)) = |
| ((partition H)⁻¹*noiseCoefficient d σ) • |
| ((volume : Measure (Point d)).prod volume).withDensity |
| (fun p => ENNReal.ofReal (Real.exp (-H p.1-‖p.2-T p.1‖^2/(2*σ^2)))) := by |
| have : SFinite (gibbs H) := by unfold gibbs; infer_instance |
| have hmap : ((gibbs H).prod (stdGaussian (Point d))).map |
| (fun p => (p.1,T p.1+σ • p.2)) = |
| (((gibbs H).prod ((stdGaussian (Point d)).map (fun z => σ • z))).map (shear T hT)) := by |
| have hj := Measure.map_prod_map (gibbs H) (stdGaussian (Point d)) measurable_id |
| (show Measurable (fun z : Point d => σ • z) by fun_prop) |
| simp only [Measure.map_id] at hj |
| rw [hj,Measure.map_map (shear T hT).measurable (by fun_prop)] |
| rfl |
| rw [hmap,scaled_standard_density d hσ] |
| unfold gibbs |
| rw [Measure.prod_smul_left,Measure.prod_smul_right,smul_smul, |
| prod_withDensity (by unfold gibbsDensity; fun_prop) (by fun_prop), |
| Measure.map_smul _ (shear T hT).measurable.aemeasurable, |
| map_withDensity_of_measurePreserving (shear T hT) (shear_measurePreserving hT) (by unfold gibbsDensity; fun_prop)] |
| congr 2 |
| funext p |
| change ENNReal.ofReal (Real.exp (-H p.1))* |
| ENNReal.ofReal (Real.exp (-‖p.2-T p.1‖^2/(2*σ^2))) = _ |
| rw [← ENNReal.ofReal_mul (Real.exp_pos _).le,← Real.exp_add] |
| congr 2 |
| ring |
|
|
| end EulerDensity |
| end LogConcaveSampling |
|
|
| end UpperProof |
| end |
| end |
| end |
|
|
| end OAI |
|
|
