openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Cavity /CavityCanonicalLogIntegrability.lean
| 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 | |