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a2ee19a | 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 | 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
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