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import OAI.Probability.LogConcave.Numerics.CenteringVelocityPicard
namespace OAI
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)]
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)
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
end
end
section
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
end
end
end OAI
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