| import Mathlib | |
| import OAI.Probability.Perceptron.Variational.GaussianLinearVariance | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction | |
| namespace SphericalPerceptronFreeEnergy | |
| lemma enrichedTerminal_C1 (n M : ℕ) (f : ℝ →ᵇ ℝ) | |
| (g : Fin M → Fin (n+1) → ℝ) (p : Fin (n+1) → ℕ) (u : Fin (n+1) → ℝ) | |
| (htop : ℝ) : | |
| ∃ D : EnrichedMark (n+1) (n+1) p → StrongDual ℝ (EnrichedMark (n+1) (n+1) p), | |
| Continuous D ∧ ∀ x, HasFDerivAt (enrichedTerminal n M f g p u htop) (D x) x := by | |
| let W := normalizedPatternEnergy (n+1) M f g | |
| let V := fun x : NormalizedSpin (n+1) => innerSL ℝ (sourceEnrichedFeature (n+1) p u x) | |
| have hW : Continuous W := (normalizedPatternEnergy_continuous (n+1) M f).comp | |
| (continuous_const.prodMk continuous_id) | |
| have hV : Continuous V := (innerSL ℝ).continuous.comp (enrichedFeature_continuous _ _ _ _ _) | |
| have hWB : ∀ x, |W x| ≤ (M:ℝ)*‖f‖ := normalizedPatternEnergy_bound _ _ _ _ | |
| have hVB : ∀ x, ‖V x‖ ≤ enrichedFeatureBound (n+1) u := fun x => by | |
| change ‖innerSL ℝ (sourceEnrichedFeature (n+1) p u x)‖ ≤ _ | |
| rw [innerSL_apply_norm,sourceEnrichedFeature_norm] | |
| refine ⟨vectorLogDerivative (unitSphereLaw (n+1)) W V, | |
| vectorLogDerivative_continuous (unitSphereLaw (n+1)) hW.measurable hV.measurable | |
| (enrichedFeatureBound (n+1) u).coe_nonneg hWB hVB,?_⟩ | |
| intro x | |
| exact (vectorLogPartition_hasFDerivAt (unitSphereLaw (n+1)) hW.measurable hV.measurable | |
| (enrichedFeatureBound (n+1) u).coe_nonneg hWB hVB x).sub_const _ | |
| lemma enrichedCascadeLog_fixed_patterns_variance (n M k : ℕ) (f : ℝ →ᵇ ℝ) | |
| (g : Fin M → Fin (n+1) → ℝ) (p d : Fin (n+1) → ℕ) (u : Fin (n+1) → ℝ) | |
| (h : Fin (k+1) → ℝ) (z : Fin k → ℝ) | |
| (hz : StrictMono z) (hz0 : ∀ i, 0<z i) (hz1 : ∀ i, z i<1) : | |
| MemLp (enrichedCascadeLog n M k f g p d u h) 2 (enrichedDisorderLaw n k p z) ∧ | |
| variance (enrichedCascadeLog n M k f g p d u h) (enrichedDisorderLaw n k p z) ≤ | |
| cascadeLogFluctuationConstant k z+(Real.pi^2/8)* | |
| ((enrichedFeatureBound (n+1) u:ℝ)*‖enrichedRootMap p d h‖)^2 := by | |
| obtain ⟨D,hD,hd⟩ := enrichedTerminal_C1 n M f g p u (h (Fin.last k)) | |
| exact gaussianLinearCascadeLog_variance (enrichedIncrementMap p d h) (enrichedRootMap p d h) | |
| (enrichedTerminal_lipschitz n M f g p u _) hd hD k z hz hz0 hz1 | |
| end SphericalPerceptronFreeEnergy | |
| end | |
| end OAI | |