| import Mathlib |
| import OAI.Probability.Perceptron.Variational.TiltMeanCenteredAbsSq |
| import OAI.Probability.Perceptron.Variational.GaussianCoupling |
| import OAI.Probability.Perceptron.Variational.IndexedGaussianCovariance |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace SphericalPerceptronFreeEnergy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
|
|
| lemma sourceEnrichedFeature_add_single (N : ℕ) (p : Fin N → ℕ) (u : Fin N → ℝ) |
| (j : Fin N) (a : ℝ) (x : NormalizedSpin N) (i : EnrichedIndex N N p) : |
| sourceEnrichedFeature N p (u+Pi.single j a) x i = |
| sourceEnrichedFeature N p u x i+ |
| a*perturbationAmplitude N (fun _ => 1) j*sourceGaussianTestFeature N p j x i := by |
| classical |
| cases i with |
| | inl i => simp [sourceEnrichedFeature,sourceGaussianTestFeature,enrichedFeature] |
| | inr i => |
| rcases i with ⟨l,r⟩ |
| by_cases hl : l=j |
| · subst l |
| simp only [sourceEnrichedFeature,sourceGaussianTestFeature,enrichedFeature,perturbationAmplitude, |
| Pi.add_apply,Pi.single_eq_same,mul_one] |
| ring |
| · simp [sourceEnrichedFeature,sourceGaussianTestFeature,enrichedFeature,perturbationAmplitude, |
| Pi.single_apply,hl] |
|
|
| lemma sourceGaussianRow_add_single {N k : ℕ} (p d : Fin N → ℕ) (h : Fin (k+1) → ℝ) |
| (u : Fin N → ℝ) (j : Fin N) (a : ℝ) (i : ℕ) (x : NormalizedSpin N×IndexedLeaf k) : |
| sourceGaussianRow p d h (u+Pi.single j a) i x = |
| sourceGaussianRow p d h u i x+ |
| a*perturbationAmplitude N (fun _ => 1) j*sourceGaussianTestRow p d h j i x := by |
| unfold sourceGaussianRow sourceGaussianTestRow indexedGaussianRow finiteGaussianRow |
| simp_rw [sourceEnrichedFeature_add_single] |
| rw [Finset.mul_sum,← Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro q _ |
| split_ifs <;> ring |
|
|
| lemma enrichedIndexedHamiltonian_add_single (n M k : ℕ) (f : ℝ →ᵇ ℝ) |
| (a : Fin M → Fin (n+1) → ℝ) (p d : Fin (n+1) → ℕ) |
| (h : Fin (k+1) → ℝ) (u : Fin (n+1) → ℝ) (j : Fin (n+1)) (v : ℝ) (g : ℕ → ℝ) |
| (x : NormalizedSpin (n+1)×IndexedLeaf k) : |
| enrichedIndexedHamiltonian n M k f a p d h (u+Pi.single j v) g x = |
| enrichedIndexedHamiltonian n M k f a p d h u g x+ |
| v*perturbationAmplitude (n+1) (fun _ => 1) j* |
| countableGaussianField (sourceGaussianTestRow p d h j) |
| (indexedGaussianRowLength (I := EnrichedIndex (n+1) (n+1) p) k) g x := by |
| unfold enrichedIndexedHamiltonian countableGaussianHamiltonian |
| have hr : sourceGaussianRow p d h (u+Pi.single j v) = fun i x => |
| sourceGaussianRow p d h u i x+ |
| (v*perturbationAmplitude (n+1) (fun _ => 1) j)*sourceGaussianTestRow p d h j i x := by |
| funext i x |
| exact sourceGaussianRow_add_single p d h u j v i x |
| rw [hr,countableGaussianField_add,countableGaussianField_smul] |
| ring |
|
|
| lemma coupling_log_contDiff {S : Type*} [MeasurableSpace S] (μ : Measure S) |
| [IsProbabilityMeasure μ] {H Y : S → ℝ} (hH : Measurable H) (hY : Measurable Y) |
| (he : ∀ a : ℝ, Integrable (fun x => Real.exp (H x+a*Y x)) μ) : |
| ContDiff ℝ 2 (fun b => Real.log (tiltPartition μ (fun x => H x+b*Y x) 1)) := by |
| let ρ := μ.withDensity (fun x => ENNReal.ofReal (Real.exp (H x))) |
| have hf : cgf Y ρ = (fun b => Real.log (tiltPartition μ (fun x => H x+b*Y x) 1)) := by |
| funext b |
| exact congrArg Real.log (coupling_mgf_identity μ hH b) |
| rw [← hf,contDiff_iff_contDiffAt] |
| intro a |
| apply AnalyticAt.contDiffAt |
| apply analyticAt_cgf |
| rw [coupling_expSet μ hH hY he,interior_univ] |
| trivial |
|
|
| lemma coupling_log_convex {S : Type*} [MeasurableSpace S] (μ : Measure S) |
| [IsProbabilityMeasure μ] {H Y : S → ℝ} (hH : Measurable H) (hY : Measurable Y) |
| (he : ∀ a : ℝ, Integrable (fun x => Real.exp (H x+a*Y x)) μ) : |
| ConvexOn ℝ univ (fun b => Real.log (tiltPartition μ (fun x => H x+b*Y x) 1)) := by |
| have hs := coupling_log_contDiff μ hH hY he |
| apply convexOn_of_deriv2_nonneg' convex_univ |
| (hs.differentiable (by norm_num)).differentiableOn |
| ( (ContDiff.deriv' (n := 1) hs).differentiable (by norm_num)).differentiableOn |
| intro a _ |
| rw [← iteratedDeriv_eq_iterate,coupling_log_second_derivative μ hH hY he] |
| exact tiltMean_nonneg μ (fun _ => sq_nonneg _) 1 |
|
|
| lemma convex_derivative_local_bound {F D : ℝ → ℝ} |
| (hc : ConvexOn ℝ univ F) (hd : ∀ x, HasDerivAt F (D x) x) (u v : ℝ) |
| (hv : v ∈ Ioo (u-1) (u+1)) : |
| |D v| ≤ |F (u-2)|+|F (u-1)|+|F (u+1)|+|F (u+2)| := by |
| have hmono : Monotone D := by |
| intro x y hxy |
| simpa only [(hd x).deriv,(hd y).deriv] using |
| hc.monotoneOn_deriv (fun x _ => (hd x).differentiableAt) (mem_univ x) (mem_univ y) hxy |
| have hl := hc.slope_le_of_hasDerivAt (mem_univ (u-2)) (mem_univ (u-1)) |
| (by linarith : u-2 < u-1) (hd (u-1)) |
| have hr := hc.le_slope_of_hasDerivAt (mem_univ (u+1)) (mem_univ (u+2)) |
| (by linarith : u+1 < u+2) (hd (u+1)) |
| have hl' : F (u-1)-F (u-2) ≤ D v := by |
| have hh : (u-1)-(u-2) = (1:ℝ) := by ring |
| simp only [slope,hh,inv_one,one_smul] at hl |
| exact hl.trans (hmono hv.1.le) |
| have hr' : D v ≤ F (u+2)-F (u+1) := by |
| have hh : (u+2)-(u+1) = (1:ℝ) := by ring |
| simp only [slope,hh,inv_one,one_smul] at hr |
| exact (hmono hv.2.le).trans hr |
| apply abs_le.mpr |
| constructor |
| · linarith [neg_abs_le (F (u-1)),le_abs_self (F (u-2)),abs_nonneg (F (u+1)),abs_nonneg (F (u+2))] |
| · linarith [le_abs_self (F (u+2)),neg_abs_le (F (u+1)),abs_nonneg (F (u-1)),abs_nonneg (F (u-2))] |
|
|
| lemma annealed_convex_hasDerivAt {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) |
| {F D : ℝ → Ω → ℝ} (hF : ∀ t, AEStronglyMeasurable (F t) P) |
| (hD : ∀ t, AEStronglyMeasurable (D t) P) (hi : ∀ t, Integrable (F t) P) |
| (hc : ∀ᵐ ω ∂P, ConvexOn ℝ univ (fun t => F t ω) ∧ ∀ t, HasDerivAt (fun t => F t ω) (D t ω) t) |
| (u : ℝ) : |
| Integrable (D u) P ∧ HasDerivAt (fun t => ∫ ω, F t ω ∂P) (∫ ω, D u ω ∂P) u := by |
| let B : Ω → ℝ := fun ω => |F (u-2) ω|+|F (u-1) ω|+|F (u+1) ω|+|F (u+2) ω| |
| apply hasDerivAt_integral_of_dominated_loc_of_deriv_le |
| (Ioo_mem_nhds (by linarith : u-1 < u) (by linarith : u < u+1)) |
| (Eventually.of_forall hF) (hi u) (hD u) |
| · filter_upwards [hc] with ω hω t ht |
| rw [Real.norm_eq_abs] |
| exact convex_derivative_local_bound hω.1 hω.2 u t ht |
| · exact (((hi (u-2)).abs.add (hi (u-1)).abs).add (hi (u+1)).abs).add (hi (u+2)).abs |
| · filter_upwards [hc] with ω hω t _ |
| exact hω.2 t |
|
|
| end SphericalPerceptronFreeEnergy |
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| end |
|
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| end OAI |
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