| import Mathlib |
| import OAI.Probability.LogConcave.JetEstimates.NormalizedLaplace |
| import OAI.Probability.LogConcave.Sampling.PrimitiveField |
|
|
| 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 |
|
|
| namespace LogConcaveSampling |
| open MeasureTheory |
| open scoped RealInnerProductSpace |
|
|
| lemma integral_gibbs_add_const {d : ℕ} {H : Point d → ℝ} (hc : Continuous H) |
| (hi : Integrable (fun z => Real.exp (-H z))) (c : ℝ) (f : Point d → ℝ) : |
| (∫ z,f z ∂gibbs (fun z => H z+c))=∫ z,f z ∂gibbs H := by |
| have he : (fun z => Real.exp (-(H z+c)))=fun z => Real.exp (-c)*Real.exp (-H z) := by |
| funext z |
| rw [← Real.exp_add] |
| congr 1 |
| ring |
| have hi' : Integrable (fun z => Real.exp (-(H z+c))) := by rw [he]; exact hi.const_mul _ |
| rw [integral_gibbs (hc.add_const c) hi', integral_gibbs hc hi] |
| simp_rw [congrFun he,mul_assoc] |
| rw [integral_const_mul,integral_const_mul,mul_div_mul_left _ _ (Real.exp_pos _).ne'] |
|
|
| lemma HasGaussianLowerTail.integrable_exp {d : ℕ} {H : Point d → ℝ} |
| (ht : HasGaussianLowerTail H) (hc : Continuous H) : |
| Integrable (fun z => Real.exp (-H z)) := by |
| simpa only [inner_zero_left,zero_sub,smul_eq_mul,mul_one] using |
| integrable_tilted H hc continuous_const ht (growth_const (1:ℝ)) 0 |
|
|
| lemma HasGaussianLowerTail.comp_inverse_affine {d : ℕ} {H : Point d → ℝ} |
| (ht : HasGaussianLowerTail H) (v : Point d) {s : ℝ} (hs : 0<s) : |
| HasGaussianLowerTail (fun z => H (s⁻¹ • (z-v))) := by |
| obtain ⟨m,hm,C,hb⟩ := ht |
| refine ⟨m/(2*s^2),by positivity,C+(m/s^2)*‖v‖^2,fun z => ?_⟩ |
| have hh := hb (s⁻¹ • (z-v)) |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hs),mul_pow,inv_pow] at hh |
| have hn := norm_add_le (z-v) v |
| rw [sub_add_cancel] at hn |
| have hk := pow_le_pow_left₀ (norm_nonneg z) hn 2 |
| have hsq : ‖z‖^2 ≤ 2*‖z-v‖^2+2*‖v‖^2 := by |
| nlinarith [sq_nonneg (‖z-v‖-‖v‖)] |
| have hd : m/(2*s^2)*2=m/s^2 := by field_simp |
| have hmul := mul_le_mul_of_nonneg_left hsq (by positivity : 0 ≤ m / (2*s^2)) |
| simp only [mul_add,← mul_assoc,hd] at hmul |
| have he : m*(s^2)⁻¹=m/s^2 := by ring |
| rw [← mul_assoc,he] at hh |
| linarith |
|
|
| lemma normalizedLaplace_gibbs {d : ℕ} {H : Point d → ℝ} (hc : Continuous H) |
| (hi : Integrable (fun z => Real.exp (-H z))) (θ : Point d) |
| (hit : Integrable (fun z => Real.exp (inner ℝ θ z-H z))) (f : Point d → ℝ) : |
| Appell.normalizedLaplace (gibbs H) f θ = |
| ∫ z,f z ∂gibbs (fun z => H z-inner ℝ θ z) := by |
| have he : ∀ z,Real.exp (-H z)*Real.exp (inner ℝ θ z)=Real.exp (inner ℝ θ z-H z) := by |
| intro z |
| rw [← Real.exp_add] |
| congr 1 |
| ring |
| have hit' : Integrable (fun z => Real.exp (-(H z-inner ℝ θ z))) := by |
| simpa only [neg_sub] using hit |
| rw [Appell.normalizedLaplace,Appell.laplace,Appell.laplace] |
| have hct : Continuous (fun z => H z-inner ℝ θ z) := by fun_prop |
| rw [integral_gibbs hc hi,integral_gibbs hc hi,integral_gibbs hct hit'] |
| simp only [smul_eq_mul,mul_one,← mul_assoc,he,neg_sub] |
| have hA : (∫ z,Real.exp (-H z))≠0 := (integral_exp_pos hi).ne' |
| have hB : (∫ z,Real.exp (inner ℝ θ z-H z))≠0 := (integral_exp_pos hit).ne' |
| field_simp |
|
|
| end LogConcaveSampling |
|
|
| namespace LogConcaveSampling |
| open MeasureTheory |
| open scoped RealInnerProductSpace |
|
|
| lemma conditionalPotential_continuous {d : ℕ} {F : Point d → ℝ} (hc : Continuous F) |
| (x : Point d) (r ρ : ℝ) (y : Point d) : Continuous (conditionalPotential F x r ρ y) := by |
| unfold conditionalPotential |
| fun_prop |
|
|
| lemma conditional_hasGaussianLowerTail {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0≤ρ) (hρ1 : ρ<1) (y : Point d) : |
| HasGaussianLowerTail (conditionalPotential F x r ρ y) := by |
| obtain ⟨ha,ha1⟩ := probability_time hρ0 hρ1 |
| have hs := conditional_smallness hl ha ha1 |
| have ht := (hF.hasGaussianLowerTail (x+r • (ρ • y)) |
| (mul_nonneg hr (Real.sqrt_nonneg _)) (by linarith : |
| (lam:ℝ)*(r*Real.sqrt (1-ρ^2))^2<1)).comp_inverse_affine |
| (ρ • y) (Real.sqrt_pos.mpr ha) |
| simpa only [conditionalPotential_pullback F x r ρ y _ ha] using ht |
|
|
| lemma conditionalPotential_shift {d : ℕ} (F : Point d → ℝ) (x : Point d) |
| (r ρ : ℝ) (y h z : Point d) : |
| conditionalPotential F x r ρ (y+h) z = |
| conditionalPotential F x r ρ y z-inner ℝ ((ρ/(1-ρ^2)) • h) z+ |
| (ρ^2/(2*(1-ρ^2)))*(‖y+h‖^2-‖y‖^2) := by |
| simp only [conditionalPotential,norm_sub_sq_real,inner_smul_right,norm_smul, |
| Real.norm_eq_abs,mul_pow,sq_abs,inner_add_right,inner_smul_left,starRingEnd_apply,star_trivial] |
| rw [real_inner_comm h z] |
| simp only [div_eq_mul_inv,mul_inv_rev] |
| ring |
|
|
| def conditionalMeanScalar {d : ℕ} (F : Point d → ℝ) (x : Point d) (r ρ : ℝ) |
| (f : Point d → ℝ) (y : Point d) : ℝ := ∫ z,f z ∂gibbs (conditionalPotential F x r ρ y) |
|
|
| theorem conditionalMeanScalar_shift {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0≤ρ) (hρ1 : ρ<1) |
| (f : Point d → ℝ) (y h : Point d) : |
| conditionalMeanScalar F x r ρ f (y+h)= |
| Appell.normalizedLaplace (gibbs (conditionalPotential F x r ρ y)) f |
| ((ρ/(1-ρ^2)) • h) := by |
| let H := conditionalPotential F x r ρ y |
| let θ := (ρ/(1-ρ^2)) • h |
| have hH : Continuous H := conditionalPotential_continuous hF.smooth.continuous x r ρ y |
| have ht : HasGaussianLowerTail H := conditional_hasGaussianLowerTail hF x hr hl hρ0 hρ1 y |
| have hi := ht.integrable_exp hH |
| have hit : Integrable (fun z => Real.exp (inner ℝ θ z-H z)) := by |
| simpa only [smul_eq_mul,mul_one] using |
| integrable_tilted H hH continuous_const ht (growth_const (1:ℝ)) θ |
| rw [normalizedLaplace_gibbs hH hi θ hit] |
| unfold conditionalMeanScalar |
| have hp : conditionalPotential F x r ρ (y+h) = fun z => |
| (H z-inner ℝ θ z)+(ρ^2/(2*(1-ρ^2)))*(‖y+h‖^2-‖y‖^2) := by |
| funext z |
| exact conditionalPotential_shift F x r ρ y h z |
| rw [hp] |
| apply integral_gibbs_add_const (hH.sub (continuous_const.inner continuous_id)) |
| simpa only [Pi.sub_apply,id_eq,neg_sub] using hit |
|
|
| theorem conditionalMeanScalar_smooth {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0≤ρ) (hρ1 : ρ<1) |
| {f : Point d → ℝ} (hf : Continuous f) (hg : Appell.HasGrowth f) : |
| ContDiff ℝ (⊤ : ℕ∞) (conditionalMeanScalar F x r ρ f) := by |
| let μ := gibbs (conditionalPotential F x r ρ 0) |
| let := conditionalLaw_probability hF x hr hl hρ0 hρ1 (0 : Point d) |
| have hμ := conditionalLaw_hasExpMoments hF x hr hl hρ0 hρ1 (0 : Point d) |
| have he : conditionalMeanScalar F x r ρ f = fun y => |
| Appell.normalizedLaplace μ f ((ρ/(1-ρ^2)) • y) := by |
| funext y |
| simpa only [zero_add] using conditionalMeanScalar_shift hF x hr hl hρ0 hρ1 f 0 y |
| rw [he] |
| exact (Appell.smooth_normalizedLaplace hμ hf hg).comp (by fun_prop) |
|
|
| end LogConcaveSampling |
|
|
| end UpperProof |
| end |
| end |
| end |
|
|
| end OAI |
|
|
