| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.CenteringVelocityPicard |
|
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| namespace OAI |
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| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory |
| open scoped Classical BigOperators NNReal |
|
|
| lemma productPointEquiv_lipschitz_one (d e : ℕ) : |
| LipschitzWith 1 (productPointEquiv d e) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro y z |
| simp only [NNReal.coe_one,one_mul,dist_eq_norm,←map_sub] |
| rw [Prod.norm_def] |
| exact max_le (productPointEquiv_fst_norm_le _) (productPointEquiv_snd_norm_le _) |
|
|
| lemma productPointEquiv_symm_norm_le (d e : ℕ) (p : Point d × Point e) : |
| ‖(productPointEquiv d e).symm p‖≤2*‖p‖ := by |
| have h := productPointEquiv_norm_sq ((productPointEquiv d e).symm p) |
| simp only [ContinuousLinearEquiv.apply_symm_apply] at h |
| have h1 : ‖p.1‖≤‖p‖ := norm_fst_le p |
| have h2 : ‖p.2‖≤‖p‖ := norm_snd_le p |
| have h1' := pow_le_pow_left₀ (norm_nonneg p.1) h1 2 |
| have h2' := pow_le_pow_left₀ (norm_nonneg p.2) h2 2 |
| nlinarith [norm_nonneg p,norm_nonneg ((productPointEquiv d e).symm p)] |
|
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| lemma productPointEquiv_symm_lipschitz_two (d e : ℕ) : |
| LipschitzWith 2 (productPointEquiv d e).symm := by |
| apply LipschitzWith.of_dist_le_mul |
| intro y z |
| simpa only [NNReal.coe_ofNat,dist_eq_norm,←map_sub] using |
| productPointEquiv_symm_norm_le d e (y-z) |
|
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| lemma centering_joint_product_law {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T : ℝ} |
| (hr : 0<r) (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) : |
| (gibbs (centeringPotential F x r T)).map (productPointEquiv d d)= |
| (interpolationLaw F x r T).prod (stdGaussian (Point d)) := by |
| rw [centeringPotential,productPotential_gibbs |
| (interpolationPotential_smooth hF x hr.le hl hT0 hT1).continuous |
| (gaussianPotential_polySmooth d).smooth.continuous,gaussianPotential_gibbs] |
| rw [interpolationLaw_eq_gibbs hF x hr.le (by linarith only [hl]) |
| (by nlinarith [probability_time hT0 hT1])] |
| end LogConcaveSampling |
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| end |
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| end |
|
|
| section |
|
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| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory Quadrature TensorEnergy RMSIntegral |
| open scoped Classical BigOperators NNReal |
|
|
| def centeringVelocityJoint {d : ℕ} (F : Point d → ℝ) (x : Point d) |
| (r T h ψ s : ℝ) (n m N : ℕ) (e : ProbabilityNode T h (n+1)) |
| (y : Point (d+d)) : Point (d+d) := |
| (productPointEquiv d d).symm |
| (centeringVelocityPicard F x r T h ψ s n m N e (productPointEquiv d d y)) |
|
|
| lemma actual_centering_joint_field {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0<r) (hlam : 0<lam) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) (s : ℝ) (y : Point (d+d)) : |
| productPointEquiv d d (skewLieField (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) y)= |
| skewCenteringField (centeringKernel hF x hr hl hT0 hT1) |
| (fun z => conditionalFieldMean F x r T z-primitiveExpectedField F x r) s |
| (productPointEquiv d d y) := by |
| apply jointSkew_field ((interpolationPotential_smooth hF x hr.le hl hT0 hT1).differentiable (by simp)) |
| · intro i j |
| exact (contDiff_const.inner ℝ |
| ((centeringKernel_smooth hF x hr hlam hl hT0 hT1).clm_apply contDiff_const)).differentiable (by simp) |
| · exact centeringKernel_strong hF x hr hlam hl hT0 hT1 |
|
|
| lemma centeringVelocityJoint_error_sq {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T h ψ s : ℝ} (hr : 0<r) (hlam : 0<lam) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) |
| (n m N : ℕ) (e : ProbabilityNode T h (n+1)) (y : Point (d+d)) : |
| ‖centeringVelocityJoint F x r T h ψ s n m N e y- |
| skewLieField (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) y‖^2= |
| (r*s)⁻¹^2 * ‖kernelQuadraturePicard F x r T h ψ (n+1) m N e |
| (terminalQuadratureWeight T h n) (productPointEquiv d d y)- |
| r • (centeringKernel hF x hr hl hT0 hT1 (productPointEquiv d d y).1).adjoint |
| (productPointEquiv d d y).2‖^2 := by |
| rw [productPointEquiv_norm_sq,map_sub,actual_centering_joint_field hF x hr hlam hl hT0 hT1] |
| simp only [centeringVelocityJoint,ContinuousLinearEquiv.apply_symm_apply, |
| centeringVelocityPicard,skewCenteringField,Prod.fst_sub,Prod.snd_sub,sub_self,norm_zero] |
| have hscale : -(r*s)⁻¹*r= -s⁻¹ := by field_simp |
| rw [←hscale,←smul_smul,←smul_sub,norm_smul,mul_pow] |
| simp only [Real.norm_eq_abs,sq_abs,neg_sq] |
| ring |
|
|
| lemma centeringVelocityJoint_rms_of_kernel {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T h ψ s : ℝ} (hr : 0<r) (hlam : 0<lam) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) |
| (n m N : ℕ) (e : ProbabilityNode T h (n+1)) {B : ℝ} |
| (hK : Continuous (kernelQuadraturePicard F x r T h ψ (n+1) m N e (terminalQuadratureWeight T h n))) |
| (hi : Integrable (fun p => ‖kernelQuadraturePicard F x r T h ψ (n+1) m N e |
| (terminalQuadratureWeight T h n) p-r • |
| (centeringKernel hF x hr hl hT0 hT1 p.1).adjoint p.2‖^2) |
| ((interpolationLaw F x r T).prod (stdGaussian (Point d)))) |
| (hb : (∫p,‖kernelQuadraturePicard F x r T h ψ (n+1) m N e (terminalQuadratureWeight T h n) p- |
| r • (centeringKernel hF x hr hl hT0 hT1 p.1).adjoint p.2‖^2 |
| ∂((interpolationLaw F x r T).prod (stdGaussian (Point d))))≤B) : |
| let err := fun y => centeringVelocityJoint F x r T h ψ s n m N e y- |
| skewLieField (centeringPotential F x r T) (jointSkew (centeringKernel hF x hr hl hT0 hT1) s) y |
| Integrable (fun y => ‖err y‖^2) (gibbs (centeringPotential F x r T)) ∧ |
| (∫y,‖err y‖^2 ∂gibbs (centeringPotential F x r T))≤(r*s)⁻¹^2*B := by |
| intro err |
| let f := fun p : Point d × Point d => ‖kernelQuadraturePicard F x r T h ψ (n+1) m N e |
| (terminalQuadratureWeight T h n) p-r •(centeringKernel hF x hr hl hT0 hT1 p.1).adjoint p.2‖^2 |
| have hc : Continuous f := by |
| apply Continuous.pow |
| apply Continuous.norm |
| apply hK.sub |
| exact (continuous_const_smul r).comp |
| ((ContinuousLinearMap.adjoint.continuous.comp |
| ((centeringKernel_smooth hF x hr hlam hl hT0 hT1).continuous.comp continuous_fst)).clm_apply continuous_snd) |
| have he := centering_joint_product_law hF x hr hl hT0 hT1 |
| have hp : Integrable (fun y => f (productPointEquiv d d y)) (gibbs (centeringPotential F x r T)) := by |
| apply (integrable_map_measure hc.measurable.aestronglyMeasurable |
| (productPointEquiv d d).continuous.measurable.aemeasurable).mp |
| rw [he] |
| exact hi |
| have hb' : (∫y,f (productPointEquiv d d y) ∂gibbs (centeringPotential F x r T))≤B := by |
| rw [←integral_map (productPointEquiv d d).continuous.measurable.aemeasurable hc.measurable.aestronglyMeasurable,he] |
| exact hb |
| dsimp only [err] |
| simp_rw [centeringVelocityJoint_error_sq hF x hr hlam hl hT0 hT1] |
| refine ⟨hp.const_mul _,?_⟩ |
| rw [integral_const_mul] |
| exact mul_le_mul_of_nonneg_left hb' (sq_nonneg _) |
| end LogConcaveSampling |
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| end |
|
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| end |
|
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| end OAI |
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