| 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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|
