| import Mathlib |
| import OAI.Probability.Perceptron.Pressure.EnrichedConcentration |
| import OAI.Probability.Perceptron.Pressure.EnrichedPatternMean |
| import OAI.Probability.Perceptron.Variational.MeasurableVariance |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| def finitePatternRowsLaw (N M : ℕ) : Measure (Fin M → Fin N → ℝ) := |
| Measure.pi fun _ : Fin M => Measure.pi fun _ : Fin N => gaussianReal 0 1 |
| |
| instance (N M : ℕ) : IsProbabilityMeasure (finitePatternRowsLaw N M) := by |
| unfold finitePatternRowsLaw |
| infer_instance |
| |
| lemma enrichedExpectedLog_memLp (n M k : ℕ) (f : ℝ →ᵇ 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 (enrichedExpectedLog n M k f p d u h z) 2 (finitePatternRowsLaw (n+1) M) := by |
| obtain ⟨C,hC⟩ := enrichedExpectedLog_bounded n M k f p d u h z hz hz0 hz1 |
| exact MemLp.of_bound (enrichedExpectedLog_measurable n M k f p d u h z).aestronglyMeasurable |
| C (ae_of_all _ hC) |
| |
| lemma enrichedExpectedLog_variance (n M k : ℕ) (f : ℝ →ᵇ 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) : |
| variance (enrichedExpectedLog n M k f p d u h z) (finitePatternRowsLaw (n+1) M) ≤ |
| (M:ℝ)*(2*‖f‖)^2 := by |
| obtain ⟨C,hC⟩ := enrichedExpectedLog_bounded n M k f p d u h z hz hz0 hz1 |
| exact variance_pi_le_of_measurable_bounded_differences |
| (Measure.pi fun _ : Fin (n+1) => gaussianReal 0 1) M |
| (enrichedExpectedLog_measurable n M k f p d u h z) hC |
| (enrichedExpectedLog_update n M k f p d u h z hz hz0 hz1) |
| |
| def enrichedFixedCountLog (n M k : ℕ) (f : ℝ →ᵇ ℝ) |
| (p d : Fin (n+1) → ℕ) (u : Fin (n+1) → ℝ) (h : Fin (k+1) → ℝ) |
| (a : (Fin M → Fin (n+1) → ℝ) × |
| (EnrichedMark (n+1) (n+1) p × DecoratedCascade (EnrichedMark (n+1) (n+1) p) k)) : ℝ := |
| enrichedCascadeLog n M k f a.1 p d u h a.2 |
| |
| lemma enrichedFixedCountLog_variance (n M k : ℕ) (f : ℝ →ᵇ 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) : |
| let μ := (finitePatternRowsLaw (n+1) M).prod (enrichedDisorderLaw n k p z) |
| MemLp (enrichedFixedCountLog n M k f p d u h) 2 μ ∧ |
| variance (enrichedFixedCountLog n M k f p d u h) μ ≤ |
| cascadeLogFluctuationConstant k z+(Real.pi^2/8)* |
| ((enrichedFeatureBound (n+1) u:ℝ)*‖enrichedRootMap p d h‖)^2+(M:ℝ)*(2*‖f‖)^2 := by |
| let μ := finitePatternRowsLaw (n+1) M |
| let ν := enrichedDisorderLaw n k p z |
| let F := enrichedFixedCountLog n M k f p d u h |
| let G := enrichedExpectedLog n M k f p d u h z |
| let K := cascadeLogFluctuationConstant k z+(Real.pi^2/8)* |
| ((enrichedFeatureBound (n+1) u:ℝ)*‖enrichedRootMap p d h‖)^2 |
| have hG : MemLp G 2 μ := enrichedExpectedLog_memLp n M k f p d u h z hz hz0 hz1 |
| have hs (g : Fin M → Fin (n+1) → ℝ) : |
| MemLp (fun a => F (g,a)) 2 ν ∧ variance (fun a => F (g,a)) ν ≤ K := |
| enrichedCascadeLog_fixed_patterns_variance n M k f g p d u h z hz hz0 hz1 |
| have hp : MemLp F 2 (μ.prod ν) := memLp_prod_of_conditional_square μ ν |
| (enrichedCascadeLog_parameters_measurable n M k f p d u h z) hG K (fun g => (hs g).1) |
| (fun g => by |
| have hv := (hs g).2 |
| rw [variance_eq_integral (hs g).1.aestronglyMeasurable.aemeasurable] at hv |
| exact hv) |
| refine ⟨hp,?_⟩ |
| exact variance_prod_le_memLp μ ν hp (fun g => (hs g).1) hG (fun g => (hs g).2) |
| (enrichedExpectedLog_variance n M k f p d u h z hz hz0 hz1) |
| |
| end SphericalPerceptronFreeEnergy |
| end |
| |
| end OAI |
| |
