AdaptiveCenterRGO: mathematical reading route

Construct global kernels by conditional kernels, measurable comap/product/map, then integrate existing section recovery over measurable joint-event sections.

Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.

theorem adaptive_center_recovery (μ : Measure E) [IsProbabilityMeasure μ] (b a : ℝ)
    (hb : 0 ≤ b) (ha : 0 < a) :
    ∃ (T H : Kernel E E) (B : Kernel (E × E) (E × E)),
      IsMarkovKernel T ∧ IsMarkovKernel H ∧ IsMarkovKernel B ∧
      (∀ u, T u = μ.tilted (fun x => -(b/2)*‖x-u‖^2)) ∧
      (∀ u, H u = GaussianSmoothing.gaussianSmoothing (T u) (Real.sqrt a)) ∧
      (∀ u y, B (u,y) = Measure.map (Prod.mk u)
        ((T u).tilted (fun x => -‖x-y‖^2/(2*a)))) ∧
      (∀ u y, B (u,y) = Measure.map (Prod.mk u)
        (μ.tilted (fun x => -((b+a⁻¹)/2)*
          ‖x-(b+a⁻¹)⁻¹ • (b • u+a⁻¹ • y)‖^2))) ∧
      ∀ (ν : Measure E), IsProbabilityMeasure ν → B ∘ₘ (ν ⊗ₘ H) = ν ⊗ₘ T := by
  have precisionKernel (μ : Measure E) [IsProbabilityMeasure μ] (q : ℝ) (hq : 0 < q) :
      ∃ T : Kernel E E, IsMarkovKernel T ∧
        ∀ u, T u = μ.tilted (fun x => -(q/2)*‖x-u‖^2) := by
    obtain ⟨T,hT,hfiber,_⟩ := GaussianConditionalKernel.exists_tilted_isCondKernel μ (inv_pos.mpr hq)
    refine ⟨T,hT,?_⟩
    intro u
    rw [hfiber]
    congr 1
    funext x
    field_simp
  
  have targetKernel (μ : Measure E) [IsProbabilityMeasure μ] (b : ℝ) (hb : 0 ≤ b) :
      ∃ T : Kernel E E, IsMarkovKernel T ∧
        ∀ u, T u = μ.tilted (fun x => -(b/2)*‖x-u‖^2) := by
    by_cases hb0 : b = 0
    · subst b
      refine ⟨Kernel.const E μ,inferInstance,?_⟩
      intro u
      simp
    · exact precisionKernel μ b (lt_of_le_of_ne hb (Ne.symm hb0))
  
  have globalKernels (μ : Measure E) [IsProbabilityMeasure μ] (b a : ℝ)
      (hb : 0 ≤ b) (ha : 0 < a) :
      ∃ (T H : Kernel E E) (K : Kernel (E × E) E) (B : Kernel (E × E) (E × E)),
        IsMarkovKernel T ∧ IsMarkovKernel H ∧ IsMarkovKernel K ∧ IsMarkovKernel B ∧
        (∀ u, T u = μ.tilted (fun x => -(b/2)*‖x-u‖^2)) ∧
        (∀ u, H u = GaussianSmoothing.gaussianSmoothing (T u) (Real.sqrt a)) ∧
        (∀ u y, K (u,y) = (T u).tilted (fun x => -‖x-y‖^2/(2*a))) ∧
        (∀ u y, K (u,y) = μ.tilted (fun x => -((b+a⁻¹)/2)*
          ‖x-(b+a⁻¹)⁻¹ • (b • u+a⁻¹ • y)‖^2)) ∧
        ∀ u y, B (u,y) = Measure.map (Prod.mk u) (K (u,y)) := by
    obtain ⟨T,hT,hTf⟩ := targetKernel μ b hb
    let := hT
    obtain ⟨R,hR,hRf⟩ := precisionKernel μ (b+a⁻¹) (add_pos_of_nonneg_of_pos hb (inv_pos.mpr ha))
    let := hR
    let c : E × E → E := fun p => (b+a⁻¹)⁻¹ • (b • p.1+a⁻¹ • p.2)
    have hc : Measurable c := by fun_prop
    let K := R.comap c hc
    have hK : IsMarkovKernel K := inferInstance
    let H := (T ×ₖ Kernel.const E (GaussianSmoothing.scaledStdGaussian (E := E) (Real.sqrt a))).map
      (fun p : E × E => p.1+p.2)
    have hH : IsMarkovKernel H := by
      dsimp only [H]
      exact Kernel.IsMarkovKernel.map _ (by fun_prop)
    let B := (Kernel.deterministic (Prod.fst : E × E → E) measurable_fst) ×ₖ K
    have hB : IsMarkovKernel B := inferInstance
    have hKf (u y : E) : K (u,y) = μ.tilted (fun x => -((b+a⁻¹)/2)*
        ‖x-(b+a⁻¹)⁻¹ • (b • u+a⁻¹ • y)‖^2) := hRf (c (u,y))
    refine ⟨T,H,K,B,hT,hH,hK,hB,hTf,?_,?_,hKf,?_⟩
    · intro u
      dsimp only [H]
      rw [Kernel.map_apply _ (by fun_prop), Kernel.prod_apply]
      rfl
    · intro u y
      rw [hKf,hTf]
      have heq : (fun x : E => -‖x-y‖^2/(2*a)) = (fun x => -(a⁻¹/2)*‖x-y‖^2) := by
        funext x
        field_simp
      rw [heq]
      exact (RGOClosure.quadratic_tilt_tilt μ hb (inv_pos.mpr ha) u y).symm
    · intro u y
      dsimp only [B]
      rw [Kernel.prod_apply, Kernel.deterministic_apply, Measure.dirac_prod]
  
  obtain ⟨T,H,K,B,hT,hH,hK,hB,hTf,hHf,hKf,hKupdated,hBf⟩ := globalKernels μ b a hb ha
  let := hT
  let := hH
  let := hK
  let := hB
  have hrec (u : E) : (K.comap (Prod.mk u) measurable_prodMk_left) ∘ₘ H u = T u := by
    obtain ⟨_,R,hR,hRf,_,hrecover,_⟩ := RGOBackward.rgo_backward_recovery μ b a hb ha u
    have heq : K.comap (Prod.mk u) measurable_prodMk_left = R := by
      ext y : 1
      change K (u,y) = R y
      rw [hKf,hTf,hRf]
    rw [heq,hHf,hTf]
    exact hrecover
  refine ⟨T,H,B,hT,hH,hB,hTf,hHf,?_,?_,?_⟩
  · intro u y
    rw [hBf,hKf]
  · intro u y
    rw [hBf,hKupdated]
  · intro ν hν
    let := hν
    ext s hs
    rw [Measure.bind_apply hs B.aemeasurable, Measure.lintegral_compProd (B.measurable_coe hs),
      Measure.compProd_apply hs]
    apply lintegral_congr
    intro u
    have hs' := hs.preimage (measurable_prodMk_left (x := u))
    have heq := congrArg (fun m : Measure E => m ((Prod.mk u) ⁻¹' s)) (hrec u)
    rw [Measure.bind_apply hs' (K.comap (Prod.mk u) measurable_prodMk_left).aemeasurable] at heq
    rw [← heq]
    apply lintegral_congr
    intro y
    rw [hBf,Measure.map_apply measurable_prodMk_left hs]
    rfl

end AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.AdaptiveCenterRGO

Exact module and namespace context