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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 | import Mathlib
import OAI.Probability.Perceptron.Pressure.EnrichedPatternVariance
import OAI.Probability.Perceptron.Cascade.PoissonCountMoments
import OAI.Probability.Perceptron.Variational.CountMixtureVariance
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
def infinitePatternRowsLaw (N : ℕ) : Measure (ℕ → Fin N → ℝ) :=
Measure.infinitePi fun _ : ℕ => Measure.pi fun _ : Fin N => gaussianReal 0 1
instance (N : ℕ) : IsProbabilityMeasure (infinitePatternRowsLaw N) := by
unfold infinitePatternRowsLaw
infer_instance
def patternPrefix (N M : ℕ) (g : ℕ → Fin N → ℝ) : Fin M → Fin N → ℝ :=
fun a => g a.val
lemma patternPrefix_measurable (N M : ℕ) : Measurable (patternPrefix N M) := by
unfold patternPrefix
fun_prop
lemma patternPrefix_measurePreserving (N M : ℕ) : MeasurePreserving (patternPrefix N M)
(infinitePatternRowsLaw N) (finitePatternRowsLaw N M) := by
refine ⟨patternPrefix_measurable N M,?_⟩
change Measure.map (fun g : ℕ → Fin N → ℝ => fun a : Fin M => g a.val)
(Measure.infinitePi fun _ : ℕ => Measure.pi fun _ : Fin N => gaussianReal 0 1)=
Measure.pi fun _ : Fin M => Measure.pi fun _ : Fin N => gaussianReal 0 1
rw [Measure.map_infinitePi_infinitePi_of_inj Fin.val_injective,Measure.infinitePi_eq_pi]
lemma normalizedPatternEnergy_prefix_succ (N M : ℕ) (f : ℝ →ᵇ ℝ)
(g : ℕ → Fin N → ℝ) (x : NormalizedSpin N) :
|normalizedPatternEnergy N (M+1) f (patternPrefix N (M+1) g) x-
normalizedPatternEnergy N M f (patternPrefix N M g) x| ≤ ‖f‖ := by
unfold normalizedPatternEnergy
rw [Fin.sum_univ_castSucc]
simp only [patternPrefix,Fin.val_castSucc,Fin.val_last,add_sub_cancel_left]
simpa only [Real.norm_eq_abs] using f.norm_coe_le_norm (∑ i, g M i*x.val i)
def enrichedPoissonLog (n k : ℕ) (f : ℝ →ᵇ ℝ)
(p d : Fin (n+1) → ℕ) (u : Fin (n+1) → ℝ) (h : Fin (k+1) → ℝ)
(a : ℕ × ((ℕ → Fin (n+1) → ℝ) ×
(EnrichedMark (n+1) (n+1) p × DecoratedCascade (EnrichedMark (n+1) (n+1) p) k))) : ℝ :=
enrichedCascadeLog n a.1 k f (patternPrefix (n+1) a.1 a.2.1) p d u h a.2.2
def enrichedRowsDisorderLaw (n k : ℕ) (p : Fin (n+1) → ℕ) (z : Fin k → ℝ) :=
(infinitePatternRowsLaw (n+1)).prod (enrichedDisorderLaw n k p z)
instance (n k : ℕ) (p : Fin (n+1) → ℕ) (z : Fin k → ℝ) :
IsProbabilityMeasure (enrichedRowsDisorderLaw n k p z) := by
unfold enrichedRowsDisorderLaw
infer_instance
lemma enrichedPoissonLog_measurable (n 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) :
Measurable (enrichedPoissonLog n k f p d u h) := by
apply measurable_from_prod_countable_right
intro M
have hh := (enrichedCascadeLog_parameters_measurable n M k f p d u h z).comp
(((patternPrefix_measurable (n+1) M).comp
(measurable_fst (α := ℕ → Fin (n+1) → ℝ)
(β := EnrichedMark (n+1) (n+1) p × DecoratedCascade (EnrichedMark (n+1) (n+1) p) k))).prodMk measurable_snd)
simpa only [enrichedPoissonLog,Function.comp_def] using hh
lemma enrichedPoissonLog_section_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) :
MemLp (fun a => enrichedPoissonLog n k f p d u h (M,a)) 2 (enrichedRowsDisorderLaw n k p z) ∧
variance (fun a => enrichedPoissonLog n k f p d u h (M,a)) (enrichedRowsDisorderLaw n k p z) ≤
cascadeLogFluctuationConstant k z+(Real.pi^2/8)*
((enrichedFeatureBound (n+1) u:ℝ)*‖enrichedRootMap p d h‖)^2+(M:ℝ)*(2*‖f‖)^2 := by
let E := EnrichedMark (n+1) (n+1) p
let D := E × DecoratedCascade E k
have hp : MeasurePreserving (fun a : (ℕ → Fin (n+1) → ℝ) × D =>
(patternPrefix (n+1) M a.1,a.2))
((infinitePatternRowsLaw (n+1)).prod (enrichedDisorderLaw n k p z))
((finitePatternRowsLaw (n+1) M).prod (enrichedDisorderLaw n k p z)) :=
(patternPrefix_measurePreserving (n+1) M).prod
(MeasurePreserving.id (enrichedDisorderLaw n k p z))
have hs := enrichedFixedCountLog_variance n M k f p d u h z hz hz0 hz1
have hl := hs.1.comp_measurePreserving hp
have hv := (hp.variance_fun_comp hs.1.aestronglyMeasurable.aemeasurable).trans_le hs.2
constructor
· simpa only [Function.comp_def,enrichedFixedCountLog,enrichedPoissonLog,enrichedRowsDisorderLaw] using hl
· simpa only [enrichedFixedCountLog,enrichedPoissonLog,enrichedRowsDisorderLaw] using hv
lemma enrichedPoissonLog_mean_succ (n 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) (M : ℕ) :
|(∫ a, enrichedPoissonLog n k f p d u h (M+1,a) ∂enrichedRowsDisorderLaw n k p z)-
∫ a, enrichedPoissonLog n k f p d u h (M,a) ∂enrichedRowsDisorderLaw n k p z| ≤ ‖f‖ := by
let μ := infinitePatternRowsLaw (n+1)
let G (M : ℕ) := fun g => enrichedExpectedLog n M k f p d u h z (patternPrefix (n+1) M g)
have hG (M : ℕ) : MemLp (G M) 2 μ :=
(enrichedExpectedLog_memLp n M k f p d u h z hz hz0 hz1).comp_measurePreserving
(patternPrefix_measurePreserving (n+1) M)
have he (M : ℕ) : (∫ a, enrichedPoissonLog n k f p d u h (M,a) ∂enrichedRowsDisorderLaw n k p z)=
∫ g, G M g ∂μ :=
integral_prod _ ((enrichedPoissonLog_section_variance n M k f p d u h z hz hz0 hz1).1.integrable (by norm_num))
rw [he,he,← integral_sub ((hG (M+1)).integrable (by norm_num)) ((hG M).integrable (by norm_num))]
have hh (g) : |G (M+1) g-G M g| ≤ ‖f‖ :=
enrichedExpectedLog_pattern_change n (M+1) M k f _ _ p d u h z hz hz0 hz1
(normalizedPatternEnergy_prefix_succ (n+1) M f g)
simpa using norm_integral_le_of_norm_le_const (μ := μ) (f := fun g => G (M+1) g-G M g)
(ae_of_all _ hh)
lemma enrichedPoissonLog_variance (n 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) (r : ℝ≥0) :
let μ := (poissonMeasure r).prod (enrichedRowsDisorderLaw n k p z)
MemLp (enrichedPoissonLog n k f p d u h) 2 μ ∧
variance (enrichedPoissonLog n k f p d u h) μ ≤
cascadeLogFluctuationConstant k z+(Real.pi^2/8)*
((enrichedFeatureBound (n+1) u:ℝ)*‖enrichedRootMap p d h‖)^2+
(r:ℝ)*(2*‖f‖)^2+‖f‖^2*((r:ℝ)+1) := by
let ν := enrichedRowsDisorderLaw n k p z
let F := enrichedPoissonLog n k f p d u h
let G := fun M => ∫ a, F (M,a) ∂ν
let K := cascadeLogFluctuationConstant k z+(Real.pi^2/8)*
((enrichedFeatureBound (n+1) u:ℝ)*‖enrichedRootMap p d h‖)^2
let A := fun M : ℕ => K+(M:ℝ)*(2*‖f‖)^2
have hGdiff := enrichedPoissonLog_mean_succ n k f p d u h z hz hz0 hz1
have hG : MemLp G 2 (poissonMeasure r) := poisson_bounded_increment_memLp r (norm_nonneg f) hGdiff
have hAi : Integrable A (poissonMeasure r) :=
(integrable_const K).add ((poisson_nat_memLp r).integrable (by norm_num) |>.mul_const _)
have hs (M) := enrichedPoissonLog_section_variance n M k f p d u h z hz hz0 hz1
have hp : MemLp F 2 ((poissonMeasure r).prod ν) :=
memLp_prod_of_conditional_square_function (poissonMeasure r) ν
(enrichedPoissonLog_measurable n k f p d u h z) hG hAi (fun M => (hs M).1)
(fun M => by
have hv := (hs M).2
rw [variance_eq_integral (hs M).1.aestronglyMeasurable.aemeasurable] at hv
exact hv)
refine ⟨hp,?_⟩
have hv := variance_prod_le_integral (poissonMeasure r) ν hp (fun M => (hs M).1) hG hAi
(fun M => (hs M).2) (poisson_bounded_increment_variance r (norm_nonneg f) hGdiff)
have he : (∫ M, A M ∂poissonMeasure r)=K+(r:ℝ)*(2*‖f‖)^2 := by
rw [integral_add (integrable_const K) ((poisson_nat_memLp r).integrable (by norm_num) |>.mul_const _)]
simp only [integral_const,integral_mul_const,poisson_nat_mean]
simp
rw [he] at hv
exact hv
end SphericalPerceptronFreeEnergy
end
end OAI
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