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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 | import Mathlib
import OAI.Probability.Perceptron.Variational.VarianceProdMemLp
import OAI.Probability.Perceptron.Variational.EnrichedMoments
import OAI.Probability.Perceptron.Cascade.CascadeParameters
import OAI.Probability.Perceptron.Variational.GaussianLinearStability
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
lemma enrichedTerminal_parameters_measurable (n M : ℕ) (f : ℝ →ᵇ ℝ)
(p : Fin (n+1) → ℕ) (u : Fin (n+1) → ℝ) (htop : ℝ) :
Measurable (fun a : (Fin M → Fin (n+1) → ℝ) × EnrichedMark (n+1) (n+1) p =>
enrichedTerminal n M f a.1 p u htop a.2) := by
let V := fun x : NormalizedSpin (n+1) => innerSL ℝ (sourceEnrichedFeature (n+1) p u x)
have hV : Continuous V := (innerSL ℝ).continuous.comp (enrichedFeature_continuous _ _ _ _ _)
have hW := normalizedPatternEnergy_continuous (n+1) M f
have hm : Measurable (fun a : ((Fin M → Fin (n+1) → ℝ) × EnrichedMark (n+1) (n+1) p) × NormalizedSpin (n+1) =>
Real.exp (normalizedPatternEnergy (n+1) M f a.1.1 a.2+V a.2 a.1.2)) :=
((hW.comp ((continuous_fst.comp continuous_fst).prodMk continuous_snd)).add
((hV.comp continuous_snd).clm_apply (continuous_snd.comp continuous_fst))).rexp.measurable
exact hm.stronglyMeasurable.integral_prod_right'.measurable.log.sub_const _
lemma enrichedCascadeLog_parameters_measurable (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) :
Measurable (fun a : (Fin M → Fin (n+1) → ℝ) ×
(EnrichedMark (n+1) (n+1) p × DecoratedCascade (EnrichedMark (n+1) (n+1) p) k) =>
enrichedCascadeLog n M k f a.1 p d u h a.2) := by
have hh := gaussianLinearCascadeLog_parameter_measurable
(P := Fin M → Fin (n+1) → ℝ) (E := EnrichedMark (n+1) (n+1) p)
(enrichedIncrementMap p d h) (enrichedRootMap p d h)
(H := fun g => enrichedTerminal n M f g p u (h (Fin.last k)))
(enrichedTerminal_parameters_measurable n M f p u (h (Fin.last k))) k z
simpa only [enrichedCascadeLog,gaussianLinearCascadeLog] using hh
def enrichedExpectedLog (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)
(g : Fin M → Fin (n+1) → ℝ) : ℝ :=
∫ a, enrichedCascadeLog n M k f g p d u h a ∂enrichedDisorderLaw n k p z
lemma enrichedExpectedLog_measurable (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) :
Measurable (enrichedExpectedLog n M k f p d u h z) :=
(enrichedCascadeLog_parameters_measurable n M k f p d u h z).stronglyMeasurable.integral_prod_right'.measurable
lemma enrichedTerminal_pattern_change (n M M' : ℕ) (f : ℝ →ᵇ ℝ)
(g : Fin M → Fin (n+1) → ℝ) (g' : Fin M' → Fin (n+1) → ℝ)
(p : Fin (n+1) → ℕ) (u : Fin (n+1) → ℝ) (htop : ℝ) {C : ℝ}
(hC : ∀ x, |normalizedPatternEnergy (n+1) M f g x-
normalizedPatternEnergy (n+1) M' f g' x| ≤ C)
(a : EnrichedMark (n+1) (n+1) p) :
|enrichedTerminal n M f g p u htop a-enrichedTerminal n M' f g' p u htop a| ≤ C := by
have hW (M : ℕ) (g : Fin M → Fin (n+1) → ℝ) :
Continuous (normalizedPatternEnergy (n+1) M f g) :=
(normalizedPatternEnergy_continuous (n+1) M f).comp (continuous_const.prodMk continuous_id)
have hV : Continuous (fun x : NormalizedSpin (n+1) => innerSL ℝ (sourceEnrichedFeature (n+1) p u x)) :=
(innerSL ℝ).continuous.comp (enrichedFeature_continuous _ _ _ _ _)
simp only [enrichedTerminal,sub_sub_sub_cancel_right]
exact vectorLogPartition_data_bound (unitSphereLaw (n+1)) (hW M g).measurable (hW M' g').measurable
hV.measurable (normalizedPatternEnergy_bound _ _ _ _) (normalizedPatternEnergy_bound _ _ _ _)
(C := enrichedFeatureBound (n+1) u)
(fun x => by rw [innerSL_apply_norm,sourceEnrichedFeature_norm]) hC a
lemma enrichedExpectedLog_pattern_change (n M M' k : ℕ) (f : ℝ →ᵇ ℝ)
(g : Fin M → Fin (n+1) → 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) {C : ℝ}
(hC : ∀ x, |normalizedPatternEnergy (n+1) M f g x-
normalizedPatternEnergy (n+1) M' f g' x| ≤ C) :
|enrichedExpectedLog n M k f p d u h z g-enrichedExpectedLog n M' k f p d u h z g'| ≤ C :=
gaussianLinearCascadeLog_mean_difference (enrichedIncrementMap p d h) (enrichedRootMap p d h)
(enrichedTerminal_lipschitz n M f g p u _) (enrichedTerminal_lipschitz n M' f g' p u _)
(enrichedTerminal_pattern_change n M M' f g g' p u _ hC) k z hz hz0 hz1
lemma normalizedPatternEnergy_update (N M : ℕ) (f : ℝ →ᵇ ℝ)
(g : Fin M → Fin N → ℝ) (a : Fin M) (v : Fin N → ℝ) (x : NormalizedSpin N) :
|normalizedPatternEnergy N M f (Function.update g a v) x-normalizedPatternEnergy N M f g x| ≤ 2*‖f‖ := by
classical
unfold normalizedPatternEnergy
rw [← Finset.sum_sub_distrib]
have he : (∑ b : Fin M, (f (∑ i, (Function.update g a v) b i*x.val i)-f (∑ i, g b i*x.val i)))=
f (∑ i, v i*x.val i)-f (∑ i, g a i*x.val i) := by
rw [Finset.sum_eq_single a]
· simp
· intro b _ hb
simp [Function.update_of_ne hb]
· simp
rw [he]
exact (abs_sub _ _).trans (by
have hb := f.norm_coe_le_norm (∑ i, v i*x.val i)
have ha := f.norm_coe_le_norm (∑ i, g a i*x.val i)
simp only [Real.norm_eq_abs] at hb ha
linarith)
lemma enrichedExpectedLog_update (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)
(g : Fin M → Fin (n+1) → ℝ) (a : Fin M) (v : Fin (n+1) → ℝ) :
|enrichedExpectedLog n M k f p d u h z (Function.update g a v)-
enrichedExpectedLog n M k f p d u h z g| ≤ 2*‖f‖ :=
enrichedExpectedLog_pattern_change n M M k f _ _ p d u h z hz hz0 hz1
(normalizedPatternEnergy_update (n+1) M f g a v)
lemma enrichedExpectedLog_bounded (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) :
∃ C : ℝ, ∀ g, |enrichedExpectedLog n M k f p d u h z g| ≤ C := by
refine ⟨|enrichedExpectedLog n M k f p d u h z 0|+2*(M:ℝ)*‖f‖,fun g => ?_⟩
have hh := enrichedExpectedLog_pattern_change n M M k f g 0 p d u h z hz hz0 hz1
(C := 2*(M:ℝ)*‖f‖) (fun x => (abs_sub _ _).trans (by
have h1 := normalizedPatternEnergy_bound (n+1) M f g x
have h0 := normalizedPatternEnergy_bound (n+1) M f 0 x
linarith))
have ht := abs_sub_le (enrichedExpectedLog n M k f p d u h z g)
(enrichedExpectedLog n M k f p d u h z 0) 0
simp only [sub_zero] at ht
linarith
end SphericalPerceptronFreeEnergy
end
end OAI
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