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c4cdbbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 | import OAI.Probability.InvariantIsing.Cavity.CavityFrameLogBound
import OAI.Probability.InvariantIsing.Cavity.CavityRotationLogMean
import OAI.Probability.InvariantIsing.Cavity.CavityProjectorPartition
import OAI.Probability.InvariantIsing.Cavity.CavityProjectorHaarMean
namespace OAI
/-! Integrability of the canonical base partition and the random-block
cavity logarithm under the product Haar law. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
namespace InvariantIsing
lemma cavity_canonical_log_partition {N m d depth : ℕ}
(k : Fin m → ℕ) (e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N)
(a₀ : Fin d → Fin m) (V : Orthogonal N) (T : LabeledTree depth)
(lam v : Fin m → ℝ) (u : ℕ → ℝ) (t : ℝ) :
cavityProjectorLogPartition T (fun a => t*lam a+2*perturbationScale N*v a) u
(cavityLabeledProjectorAction V (cavityCanonicalProjectorFrame k e a₀)).1 =
cavityRotationLogMean T
(diagonalPerturbedEigenvalues
(fun i => lam (Sum.elim (fun w => w.1) a₀ (e.symm i)))
(cavityBaseGroup k e a₀) v t) (cavityBaseGroup k e a₀) u V := by
rw [cavityCanonicalProjectorFrame_action_fst]
exact cavity_projector_log_partition _ V T lam v u t
lemma integrable_cavity_canonical_log_partition {N m d depth : ℕ}
(k : Fin m → ℕ) (e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N)
(a₀ : Fin d → Fin m) (ν : Measure (Orthogonal N)) [IsProbabilityMeasure ν]
(T : LabeledTree depth) (lam v : Fin m → ℝ) (u : ℕ → ℝ)
(hu : ∀ j, |u j| ≤ 2) (t : ℝ) :
Integrable (fun V => cavityProjectorLogPartition T
(fun a => t*lam a+2*perturbationScale N*v a) u
(cavityLabeledProjectorAction V (cavityCanonicalProjectorFrame k e a₀)).1) ν := by
simp_rw [cavity_canonical_log_partition k e a₀]
exact integrable_cavityRotationLogMean ν T _ _ u hu
lemma cavity_canonical_capped_log_bound {N n m d depth : ℕ}
(k : Fin m → ℕ) (e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N)
(a₀ : Fin d → Fin m) (V : Orthogonal N) (T : LabeledTree depth)
(lam v : Fin m → ℝ) (u : ℕ → ℝ) (hu : ∀ j, |u j| ≤ 2)
(t cap δ : ℝ) (hcap : 0 ≤ cap) (A : CavityFactorBlocks d n) {D : ℝ}
(hAb : cavityFactorSize A.1 A.2.1 A.2.2 ≤ D) :
|cavityProjectorCappedLog T (fun a => t*lam a+2*perturbationScale N*v a)
u t cap δ A (cavityLabeledProjectorAction V (cavityCanonicalProjectorFrame k e a₀))| ≤
(|t| *D+|δ|)*(1+N) := by
let g : Fin N → Fin m := fun i => Sum.elim (fun w => w.1) a₀ (e.symm i)
let p := cavityLabeledProjectorAction V (cavityCanonicalProjectorFrame k e a₀)
have hp : p.1=fun a => cavitySpectralProjector V (cavitySpectralGroup g a) :=
cavityCanonicalProjectorFrame_action_fst k e a₀ V
have hS : p.2.transpose*p.2=1 := cavity_canonical_action_frame_gram k e a₀ V
exact cavity_projector_capped_log_bound g V T lam v u hu t cap δ hcap A hAb p hp hS
lemma integrable_cavity_canonical_capped_log {Ω : Type*} [MeasurableSpace Ω]
{N n m d depth : ℕ} (P : Measure Ω) [IsProbabilityMeasure P]
(ν : Measure (Orthogonal N)) [IsProbabilityMeasure ν]
(k : Fin m → ℕ) (e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N)
(a₀ : Fin d → Fin m) (T : LabeledTree depth)
(lam v : Fin m → ℝ) (u : ℕ → ℝ) (hu : ∀ j, |u j| ≤ 2)
(t cap δ : ℝ) (hcap : 0 ≤ cap)
(A : Ω → CavityFactorBlocks d n) (hA : Measurable A) {D : ℝ}
(hAb : ∀ ω, cavityFactorSize (A ω).1 (A ω).2.1 (A ω).2.2 ≤ D) :
Integrable (fun q : Ω × Orthogonal N => cavityProjectorCappedLog T
(fun a => t*lam a+2*perturbationScale N*v a) u t cap δ (A q.1)
(cavityLabeledProjectorAction q.2 (cavityCanonicalProjectorFrame k e a₀))) (P.prod ν) := by
have hp : Measurable (fun q : Ω × Orthogonal N =>
cavityLabeledProjectorAction q.2 (cavityCanonicalProjectorFrame k e a₀)) :=
(measurable_cavityCanonicalProjectorAction k e a₀).comp measurable_snd
have hm := measurable_cavityProjectorCappedLog T
(fun a => t*lam a+2*perturbationScale N*v a) u t cap δ
(fun q : Ω × Orthogonal N => A q.1) (hA.comp measurable_fst)
(fun q : Ω × Orthogonal N =>
cavityLabeledProjectorAction q.2 (cavityCanonicalProjectorFrame k e a₀)) hp
apply Integrable.of_bound hm.aestronglyMeasurable ((|t| *D+|δ|)*(1+N))
apply ae_of_all
intro q
rw [Real.norm_eq_abs]
exact cavity_canonical_capped_log_bound k e a₀ q.2 T lam v u hu t cap δ hcap (A q.1) (hAb q.1)
end InvariantIsing
end
end OAI
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