| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.MatrixCovarianceAlgebra |
| import OAI.Probability.SKBarriers.Hierarchy.BlockCovarianceComparison |
| |
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
|
|
| def matrixInner (d : ℕ) (Q R : Fin d → Fin d → ℝ) : ℝ := ∑ a, ∑ b, Q a b*R a b |
| |
| def matrixSquare (d : ℕ) (Q : Fin d → Fin d → ℝ) : ℝ := ∑ a, ∑ b, (Q a b)^2 |
| |
| theorem matrixSquare_sub (d : ℕ) (R Q : Fin d → Fin d → ℝ) : |
| matrixSquare d (fun a b => R a b-Q a b)=matrixSquare d R-2*matrixInner d Q R+matrixSquare d Q := by |
| simp only [matrixSquare,matrixInner,Finset.mul_sum,← Finset.sum_sub_distrib,← Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro a _ |
| apply Finset.sum_congr rfl |
| intro b _ |
| ring |
|
|
| theorem matrixSquare_nonneg (d : ℕ) (R : Fin d → Fin d → ℝ) : 0 ≤ matrixSquare d R := |
| Finset.sum_nonneg (fun _ _ => Finset.sum_nonneg (fun _ _ => sq_nonneg _)) |
|
|
| theorem matrixCovariance_lower (d : ℕ) (R Q : Fin d → Fin d → ℝ) : |
| -matrixSquare d Q ≤ matrixSquare d R-2*matrixInner d Q R := by |
| have H := matrixSquare_nonneg d (fun a b => R a b-Q a b) |
| rw [matrixSquare_sub] at H |
| linarith |
|
|
| def factorPath (d r k : ℕ) (B : Fin (k+1) → Fin r → Fin d → ℝ) |
| (j : Fin (k+1)) (a b : Fin d) : ℝ := |
| ∑ l, if l ≤ j then factorCovariance d r (B l) a b else 0 |
|
|
| theorem factorPath_last (d r k : ℕ) (B : Fin (k+1) → Fin r → Fin d → ℝ) (a b : Fin d) : |
| factorPath d r k B (Fin.last k) a b=∑ l, factorCovariance d r (B l) a b := by |
| simp only [factorPath,Fin.le_last,ite_true] |
|
|
| theorem factorPath_inner (d r k : ℕ) (B : Fin (k+1) → Fin r → Fin d → ℝ) |
| (j : Fin (k+1)) (R : Fin d → Fin d → ℝ) : |
| matrixInner d (factorPath d r k B j) R= |
| ∑ l, if l ≤ j then matrixInner d (factorCovariance d r (B l)) R else 0 := by |
| simp only [matrixInner,factorPath,Finset.sum_mul,ite_mul,zero_mul] |
| calc |
| _ = ∑ a, ∑ l, ∑ b, if l ≤ j then factorCovariance d r (B l) a b*R a b else 0 := by |
| apply Finset.sum_congr rfl |
| intro a _ |
| exact Finset.sum_comm |
| _ = ∑ l, ∑ a, ∑ b, if l ≤ j then factorCovariance d r (B l) a b*R a b else 0 := Finset.sum_comm |
| _ = _ := by |
| apply Finset.sum_congr rfl |
| intro l _ |
| split_ifs < |
|
|
| section |
| variable {S : Type} [Fintype S] [Nonempty S] |
|
|
| def matrixHamiltonianObservables (N d : ℕ) (β : ℝ) (X : S → Fin N → Fin d → ℝ) : |
| Fin (N*N) → S → ℝ := fun u s => matrixPairFeature N d β (X s) (finProdFinEquiv.symm u) |
|
|
| def matrixFieldObservables (N d r k : ℕ) (β : ℝ) (B : Fin (k+1) → Fin r → Fin d → ℝ) |
| (X : S → Fin N → Fin d → ℝ) : Fin (k+1) → Fin (N*r) → S → ℝ := |
| fun l u s => matrixFieldFeature N d r β (B l) (X s) (finProdFinEquiv.symm u) |
|
|
| omit [Fintype S] [Nonempty S] in |
| theorem matrixHamiltonianObservables_covariance {N d : ℕ} (hN : 0<N) (β : ℝ) |
| (X : S → Fin N → Fin d → ℝ) (s t : S) : |
| (∑ u, matrixHamiltonianObservables N d β X u s*matrixHamiltonianObservables N d β X u t)= |
| ((N:ℝ)*β^2/2)*matrixSquare d (replicaCrossOverlap N d (X s) (X t)) := by |
| unfold matrixHamiltonianObservables |
| rw [← finProdFinEquiv.sum_comp] |
| simpa only [Equiv.symm_apply_apply,matrixSquare] using matrixPairFeature_covariance hN β (X s) (X t) |
|
|
| omit [Fintype S] [Nonempty S] in |
| theorem matrixFieldObservables_covariance {N d r k : ℕ} (β : ℝ) |
| (B : Fin (k+1) → Fin r → Fin d → ℝ) (X : S → Fin N → Fin d → ℝ) |
| (l : Fin (k+1)) (s t : S) : |
| (∑ u, matrixFieldObservables N d r k β B X l u s*matrixFieldObservables N d r k β B X l u t)= |
| ((N:ℝ)*β^2)*matrixInner d (factorCovariance d r (B l)) (replicaCrossOverlap N d (X s) (X t)) := by |
| unfold matrixFieldObservables |
| rw [← finProdFinEquiv.sum_comp] |
| simpa only [Equiv.symm_apply_apply,matrixInner] using matrixFieldFeature_covariance β (B l) (X s) (X t) |
|
|
| omit [Fintype S] [Nonempty S] in |
| theorem matrixFieldObservables_prefix {N d r k : ℕ} (β : ℝ) |
| (B : Fin (k+1) → Fin r → Fin d → ℝ) (X : S → Fin N → Fin d → ℝ) |
| (j : Fin (k+1)) (s t : S) : |
| (∑ l, if l ≤ j then ∑ u, matrixFieldObservables N d r k β B X l u s* |
| matrixFieldObservables N d r k β B X l u t else 0)= |
| ((N:ℝ)*β^2)*matrixInner d (factorPath d r k B j) (replicaCrossOverlap N d (X s) (X t)) := by |
| simp_rw [matrixFieldObservables_covariance,factorPath_inner,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro l _ |
| split_ifs < |
|
|
| theorem matrixFiniteInterpolation {N d r k : ℕ} (hN : 0<N) (hr : 0<r) (β : ℝ) |
| (B : Fin (k+1) → Fin r → Fin d → ℝ) (X : S → Fin N → Fin d → ℝ) |
| (D : Fin d → Fin d → ℝ) (hself : ∀ s, replicaCrossOverlap N d (X s) (X s)=D) |
| (c : S → ℝ) (w : Fin (k+1) → ℝ) (hw : ∀ b, 0 ≤ w b) (hs : ∑ b, w b=1) : |
| let n:=blockDimension (N*N) (N*r) k |
| let H:=matrixHamiltonianObservables N d β X |
| let F:=matrixFieldObservables N d r k β B X |
| let m:=weightedBlockMass (N*N) (N*r) k w |
| let I:=blockInterpolationChoice (N*N) (N*r) k |
| hierarchyPressure n m (affineLogPartition c (observableExponent n (blockObservable H F) |
| (interpolationCoefficient n I 1))) 0 ≤ |
| hierarchyPressure n m (affineLogPartition c (observableExponent n (blockObservable H F) |
| (interpolationCoefficient n I 0))) 0+ |
| ((N:ℝ)*β^2/4)*(matrixSquare d D-2*matrixInner d (factorPath d r k B (Fin.last k)) D+ |
| ∑ j, w j*matrixSquare d (factorPath d r k B j)) := by |
| let K : ℝ := (N:ℝ)*β^2/4 |
| have hK : 0 ≤ K := by positivity |
| have H := blockObservable_compare (Nat.mul_pos hN hr) |
| (matrixHamiltonianObservables N d β X) (matrixFieldObservables N d r k β B X) c w hw hs |
| (K*(matrixSquare d D-2*matrixInner d (factorPath d r k B (Fin.last k)) D)) |
| (fun j => K*matrixSquare d (factorPath d r k B j)) |
| (fun s => by |
| simp only [pow_two] |
| rw [matrixHamiltonianObservables_covariance hN] |
| have HF := matrixFieldObservables_prefix β B X (Fin.last k) s s |
| simp only [Fin.le_last,ite_true] at HF |
| rw [HF,hself s] |
| dsimp only [K] |
| ring_nf |
| exact le_rfl) |
| (fun j s t => by |
| rw [matrixHamiltonianObservables_covariance hN,matrixFieldObservables_prefix] |
| have HK := mul_le_mul_of_nonneg_left (matrixCovariance_lower d (replicaCrossOverlap N d (X s) (X t)) |
| (factorPath d r k B j)) hK |
| dsimp only [K] at HK ⊢ |
| nlinarith) |
| have he : (∑ j, w j*(K*matrixSquare d (factorPath d r k B j)))= |
| K*∑ j, w j*matrixSquare d (factorPath d r k B j) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| rw [he] at H |
| dsimp only |
| convert H using 1 |
| dsimp only [K] |
| ring |
|
|
| end |
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
