| import Mathlib |
| import OAI.Probability.Perceptron.Interpolation.ArrayGeometry |
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|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
| variable {A S K : Type*} [MeasurableSpace A] [MeasurableSpace S] |
| variable [TopologicalSpace K] [MeasurableSpace K] [BorelSpace K] |
| [SecondCountableTopology K] [CompactSpace K] |
| variable (κ : Kernel A S) [IsMarkovKernel κ] (P : Measure A) [IsProbabilityMeasure P] |
| variable (H : A → S → ℝ) (hH : Measurable (Function.uncurry H)) |
| variable (Q : S → S → K) (hQ : Measurable (Function.uncurry Q)) |
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|
| def overlapArray (x : ℕ → S) : CompactArray K := fun i j => Q (x i) (x j) |
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|
| include hQ in |
| omit [TopologicalSpace K] [BorelSpace K] [SecondCountableTopology K] [CompactSpace K] in |
| lemma overlapArray_measurable : Measurable (overlapArray Q) := by |
| unfold overlapArray |
| exact Measurable.of_eval fun row => Measurable.of_eval fun column => |
| hQ.comp (show Measurable (fun sequence : ℕ → S => (sequence row,sequence column)) from |
| (measurable_pi_apply row).prodMk (measurable_pi_apply column)) |
|
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| def gibbsOverlapArrayLaw : ProbabilityMeasure (CompactArray K) := |
| ⟨(annealedInfiniteReplicaMeasure κ P H hH).map (fun a => overlapArray Q a.2), |
| (Measure.isProbabilityMeasure_map_iff |
| ((overlapArray_measurable Q hQ).comp measurable_snd).aemeasurable).2 inferInstance⟩ |
|
|
| omit [TopologicalSpace K] [BorelSpace K] [SecondCountableTopology K] [CompactSpace K] in |
| lemma gibbsOverlapArray_integral {F : CompactArray K → ℝ} (hF : Measurable F) : |
| (∫ R, F R ∂gibbsOverlapArrayLaw κ P H hH Q hQ) = |
| ∫ a, F (overlapArray Q a.2) ∂annealedInfiniteReplicaMeasure κ P H hH := |
| integral_map ((overlapArray_measurable Q hQ).comp measurable_snd).aemeasurable hF.aestronglyMeasurable |
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|
| omit [CompactSpace K] |
|
|
| lemma gibbsOverlapArray_block (hexp : ∀ᵐ a ∂P, |
| Integrable (fun x => Real.exp (H a x)) (κ a)) (r : ℕ) |
| (F : CompactBlock K r →ᵇ ℝ) : |
| (∫ R, F (compactBlock r R) ∂gibbsOverlapArrayLaw κ P H hH Q hQ) = |
| ∫ a, gibbsReplicaMean (κ a) (H a) r |
| (fun x => F (fun i j => Q (x i) (x j))) ∂P := by |
| rw [gibbsOverlapArray_integral κ P H hH Q hQ |
| (F:=fun R => F (compactBlock r R)) (F.continuous.measurable.comp (compactBlock_continuous r).measurable)] |
| exact annealedInfiniteReplica_gibbs_prefix κ P H hH hexp r |
| (F.continuous.measurable.comp (Measurable.of_eval fun row => Measurable.of_eval fun column => |
| hQ.comp ((measurable_pi_apply row).prodMk (measurable_pi_apply column)))) |
| (fun x => by |
| change |F (fun i j => Q (x i) (x j))|≤‖F‖ |
| simpa only [Real.norm_eq_abs] using F.norm_coe_le_norm (fun i j => Q (x i) (x j))) |
|
|
| lemma compactGGDefect_gibbsOverlapArray (hexp : ∀ᵐ a ∂P, |
| Integrable (fun x => Real.exp (H a x)) (κ a)) |
| (r : ℕ) (i : Fin r) (f : CompactBlock K r →ᵇ ℝ) (g : K →ᵇ ℝ) |
| (diag : ℝ) (hdiag : ∀ x, g (Q x x)=diag) (hsymm : ∀ x y, g (Q x y)=g (Q y x)) : |
| compactGGDefect (gibbsOverlapArrayLaw κ P H hH Q hQ) r i f g = |
| -kernelGGDefect κ P H (fun x y => g (Q x y)) diag r i |
| (fun x => f (fun j l => Q (x j) (x l))) := by |
| classical |
| let μ := gibbsOverlapArrayLaw κ P H hH Q hQ |
| let F : CompactArray K →ᵇ ℝ := f.compContinuous ⟨compactBlock r,compactBlock_continuous r⟩ |
| let E (a b : ℕ) : CompactArray K →ᵇ ℝ := g.compContinuous ⟨fun R => R a b,by fun_prop⟩ |
| let Es (a b : Fin r) : CompactBlock K r →ᵇ ℝ := g.compContinuous ⟨fun R => R a b,by fun_prop⟩ |
| have hmean := gibbsOverlapArray_block κ P H hH Q hQ hexp r f |
| have hsum := gibbsOverlapArray_block κ P H hH Q hQ hexp r (f * ∑ l, Es i l) |
| have hp := gibbsOverlapArray_block κ P H hH Q hQ hexp 2 |
| (g.compContinuous ⟨fun R => R 0 1,by fun_prop⟩) |
| have hpair : (∫ R : CompactArray K, g (R 0 1) ∂μ) = |
| ∫ a, gibbsReplicaMean (κ a) (H a) 2 (fun x => g (Q (x 1) (x 0))) ∂P := by |
| change (∫ R : CompactArray K, g (R 0 1) ∂μ) = |
| ∫ a, gibbsReplicaMean (κ a) (H a) 2 (fun x => g (Q (x 0) (x 1))) ∂P at hp |
| rw [hp] |
| apply integral_congr_ae |
| exact Eventually.of_forall fun a => congrArg (gibbsReplicaMean (κ a) (H a) 2) |
| (funext fun x => hsymm (x 0) (x 1)) |
| let fnew : CompactBlock K (r+1) →ᵇ ℝ := |
| (f.compContinuous ⟨fun R => fun j l => R j.castSucc l.castSucc,by fun_prop⟩) * |
| (g.compContinuous ⟨fun R => R i.castSucc (Fin.last r),by fun_prop⟩) |
| have hn := gibbsOverlapArray_block κ P H hH Q hQ hexp (r+1) fnew |
| have hnew : (∫ R, f (compactBlock r R)*g (R i r) ∂μ) = |
| ∫ a, gibbsReplicaMean (κ a) (H a) (r+1) |
| (fun x => f (fun j l => Q (x j.succ) (x l.succ))*g (Q (x i.succ) (x 0))) ∂P := by |
| change (∫ R, fnew (compactBlock (r+1) R) ∂μ) = _ |
| rw [hn] |
| apply integral_congr_ae |
| filter_upwards [hexp] with a ha |
| symm |
| have hG : Measurable (Function.uncurry (fun (x : Fin r → S) (y : S) => |
| f (fun j l => Q (x j) (x l))*g (Q (x i) y))) := by |
| apply Measurable.mul |
| · exact f.continuous.measurable.comp (Measurable.of_eval fun row => |
| Measurable.of_eval fun column => hQ.comp |
| (show Measurable (fun replica : (Fin r → S)×S => |
| (replica.1 row,replica.1 column)) from by fun_prop)) |
| · exact g.continuous.measurable.comp (hQ.comp |
| (show Measurable (fun a : (Fin r → S)×S => (a.1 i,a.2)) from by fun_prop)) |
| simpa only [fnew,BoundedContinuousFunction.mul_apply,BoundedContinuousFunction.compContinuous_apply, |
| ContinuousMap.coe_mk] using gibbsReplicaMean_fresh_last (κ a) (hH.of_uncurry_left) ha r hG |
| have hdiagint : (∫ R, f (compactBlock r R)*g (R i i) ∂μ) = |
| diag*(∫ R, f (compactBlock r R) ∂μ) := by |
| change (∫ R, (F*E i i) R ∂gibbsOverlapArrayLaw κ P H hH Q hQ) = |
| diag*(∫ R, F R ∂gibbsOverlapArrayLaw κ P H hH Q hQ) |
| rw [gibbsOverlapArray_integral κ P H hH Q hQ (F * E i i).continuous.measurable, |
| gibbsOverlapArray_integral κ P H hH Q hQ F.continuous.measurable] |
| simp only [F,E,BoundedContinuousFunction.mul_apply,BoundedContinuousFunction.compContinuous_apply, |
| ContinuousMap.coe_mk,overlapArray,hdiag] |
| rw [integral_mul_const,mul_comm] |
| have hsumint : (∫ R, f (compactBlock r R)*∑ l : Fin r, g (R i l) ∂μ) = |
| (∑ l ∈ Finset.univ.erase i, ∫ R, f (compactBlock r R)*g (R i l) ∂μ)+ |
| diag*(∫ R, f (compactBlock r R) ∂μ) := by |
| simp_rw [Finset.mul_sum] |
| rw [integral_finsetSum (f:=fun (l : Fin r) (R : CompactArray K) => f (compactBlock r R)*g (R i l)) |
| Finset.univ (fun l _ => by exact (F*E i l).integrable μ)] |
| rw [← Finset.sum_erase_add _ _ (Finset.mem_univ i),hdiagint] |
| simp only [BoundedContinuousFunction.mul_apply,BoundedContinuousFunction.sum_apply, |
| Es,BoundedContinuousFunction.compContinuous_apply,ContinuousMap.coe_mk] at hsum |
| change (∫ R, f (compactBlock r R)*∑ l : Fin r, g (R i l) ∂μ) = _ at hsum |
| unfold compactGGDefect kernelGGDefect |
| change _ = - ((_)-diag*(_)+(_)*(_)-r*(_)) |
| rw [← hmean,← hpair,← hnew,← hsum,hsumint] |
| ring |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
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| end OAI |
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