| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.BlockStein |
| import OAI.Probability.SKBarriers.Replicas.ReplicaOverlap |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| open Matrix |
| section ReplicaVarianceSplit |
| variable {S : Type} [Fintype S] |
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|
| theorem finiteReplicaMoment_const (p : S → ℝ) (hp : ∑ s, p s = 1) (c : ℝ) : |
| finiteReplicaMoment p (fun _ _ => c) = c := by |
| simp only [finiteReplicaMoment,← Finset.mul_sum,← Finset.sum_mul,hp,mul_one,one_mul] |
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| theorem finiteReplicaMoment_add (p : S → ℝ) (g h : S → S → ℝ) : |
| finiteReplicaMoment p (fun s t => g s t+h s t) = finiteReplicaMoment p g+finiteReplicaMoment p h := by |
| simp only [finiteReplicaMoment,mul_add,Finset.sum_add_distrib] |
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| theorem finiteReplicaMoment_const_mul (p : S → ℝ) (g : S → S → ℝ) (c : ℝ) : |
| finiteReplicaMoment p (fun s t => c*g s t) = c*finiteReplicaMoment p g := by |
| simp only [finiteReplicaMoment,Finset.mul_sum] |
| apply Finset.sum_congr rfl; intro s _ |
| apply Finset.sum_congr rfl; intro t _ |
| ring |
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| theorem finiteReplicaMoment_sub_square (p : S → ℝ) (hp : ∑ s, p s = 1) (g : S → S → ℝ) (r : ℝ) : |
| finiteReplicaMoment p (fun s t => (g s t-r)^2) = |
| finiteReplicaMoment p (fun s t => (g s t)^2)-2*r*finiteReplicaMoment p g+r^2 := by |
| have he : (fun s t => (g s t-r)^2) = fun s t => (g s t)^2+(-2*r)*g s t+r^2 := by |
| funext s t; ring |
| rw [he,finiteReplicaMoment_add,finiteReplicaMoment_add,finiteReplicaMoment_const_mul,finiteReplicaMoment_const p hp] |
| ring |
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| theorem finiteReplicaMoment_variance_split (p : S → ℝ) (hp : ∑ s, p s = 1) (g : S → S → ℝ) (r : ℝ) : |
| finiteReplicaMoment p (fun s t => (g s t-r)^2) = |
| finiteReplicaMoment p (fun s t => (g s t-finiteReplicaMoment p g)^2)+(finiteReplicaMoment p g-r)^2 := by |
| rw [finiteReplicaMoment_sub_square p hp,finiteReplicaMoment_sub_square p hp] |
| ring |
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|
| theorem finiteReplicaMoment_normalizedDot {N : ℕ} (p : S → ℝ) (v : S → Fin N → ℝ) : |
| finiteReplicaMoment p (fun s t => normalizedDot (v s) (v t)) = normalizedSquare (finiteMean p v) := by |
| have he : (fun s t => normalizedDot (v s) (v t)) = fun s t => (N : ℝ)⁻¹*(v s ⬝ᵥ v t) := by |
| funext s t; simp only [normalizedDot,dotProduct,div_eq_mul_inv]; ring |
| rw [he,finiteReplicaMoment_const_mul,finiteReplicaMoment_dot] |
| simp only [normalizedSquare,dotProduct,pow_two,div_eq_mul_inv,mul_comm] |
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| theorem finiteCovariance_trace_square_nonneg {I : Type} [Fintype I] |
| (p : S → ℝ) (v : S → I → ℝ) (hp : ∀ s, 0 ≤ p s) (hs : ∑ s, p s = 1) : |
| 0 ≤ ((finiteCovariance p v)*(finiteCovariance p v)).trace := by |
| have h := (finiteCovariance_posSemidef p v hp hs).isHermitian |
| simpa only [h.eq] using (Matrix.posSemidef_conjTranspose_mul_self (finiteCovariance p v)).trace_nonneg |
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| theorem finiteReplica_normalized_variance_le {N : ℕ} (hN : 0 < N) |
| (p : S → ℝ) (v : S → Fin N → ℝ) (hp : ∀ s, 0 ≤ p s) (hs : ∑ s, p s = 1) |
| (hv : ∀ s i, |v s i| ≤ 1) : |
| finiteReplicaMoment p (fun s t => (normalizedDot (v s) (v t)-normalizedSquare (finiteMean p v))^2) ≤ |
| ((finiteCovariance p v)*(finiteCovariance p v)).trace/(N : ℝ)^2+ |
| 2*Real.sqrt (((finiteCovariance p v)*(finiteCovariance p v)).trace/(N : ℝ)^2) := by |
| let T := ((finiteCovariance p v)*(finiteCovariance p v)).trace |
| let m := finiteMean p v |
| have hn : (0 : ℝ) < N := by exact_mod_cast hN |
| have hT : 0 ≤ T := finiteCovariance_trace_square_nonneg p v hp hs |
| have he : (fun s t => (normalizedDot (v s) (v t)-normalizedSquare m)^2) = |
| fun s t => ((N : ℝ)^2)⁻¹*(v s ⬝ᵥ v t-m ⬝ᵥ m)^2 := by |
| funext s t |
| simp only [normalizedDot,normalizedSquare,dotProduct,pow_two,div_eq_mul_inv] |
| ring |
| rw [he,finiteReplicaMoment_const_mul] |
| have H := mul_le_mul_of_nonneg_left (finiteReplica_variance_le p v hp hs hv) |
| (inv_nonneg.2 (sq_nonneg (N : ℝ))) |
| apply H.trans_eq |
| simp only [Fintype.card_fin] |
| rw [Real.sqrt_div hT,Real.sqrt_sq hn.le] |
| change ((N : ℝ)^2)⁻¹*(T+2*(N : ℝ)*Real.sqrt T) = T/(N : ℝ)^2+2*(Real.sqrt T/(N : ℝ)) |
| field_simp [ne_of_gt hn] |
| end ReplicaVarianceSplit |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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