| import Mathlib |
| import OAI.Probability.Perceptron.Variational.TensorFeatures |
| import OAI.Probability.Perceptron.Variational.NonlinearDirectionIBP |
|
|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma spinOverlap_self {N : ℕ} (x : NormalizedSpin N) : spinOverlap x x = 1 := by |
| have hx : ‖x.val‖ = 1 := by simp |
| simp [spinOverlap,hx] |
|
|
| lemma spinOverlap_sum {N : ℕ} (x y : NormalizedSpin N) : |
| spinOverlap x y = ∑ i, x.val i*y.val i := by |
| simp [spinOverlap,PiLp.inner_apply,mul_comm] |
|
|
| lemma normalizedSpin_coord_bound {N : ℕ} (x : NormalizedSpin N) (i : Fin N) : |x.val i| ≤ 1 := by |
| have hx : ‖x.val‖ = 1 := by simp |
| simpa only [Real.norm_eq_abs,hx] using PiLp.norm_apply_le x.val i |
|
|
| lemma spherical_pattern_direction_ibp {n : ℕ} |
| (μ : Measure (NormalizedSpin (n+1))) [IsProbabilityMeasure μ] |
| (f : Jet3) (G : ℝ →ᵇ ℝ) {h : NormalizedSpin (n+1) → ℝ} |
| (hh : Measurable h) {A : ℝ} (hA : 0≤A) (hhA : ∀ x, |h x|≤A) : |
| let b := gaussianField (n+1) (fun i (x : NormalizedSpin (n+1)) => x.val i) |
| let H := fun g x => h x+f.f (b g x) |
| (∫ g, gibbsReplicaMean μ (H g) 2 (fun x => |
| G (spinOverlap (x 0) (x 1))*f.d1 (b g (x 0))* |
| (b g (x 1)-spinOverlap (x 0) (x 1)*b g (x 0))) |
| ∂Measure.pi (fun _ => gaussianReal 0 1)) = |
| ∫ g, gibbsReplicaMean μ (H g) 2 (fun x => |
| G (spinOverlap (x 0) (x 1))*f.d1 (b g (x 0))*f.d1 (b g (x 1))* |
| (1-spinOverlap (x 0) (x 1)^2)) - |
| 2*gibbsReplicaMean μ (H g) 3 (fun x => |
| G (spinOverlap (x 1) (x 2))*f.d1 (b g (x 1))*f.d1 (b g (x 0))* |
| (spinOverlap (x 2) (x 0)-spinOverlap (x 1) (x 2)*spinOverlap (x 1) (x 0))) |
| ∂Measure.pi (fun _ => gaussianReal 0 1) := by |
| dsimp only |
| let v := fun i (x : NormalizedSpin (n+1)) => x.val i |
| let w := fun i (x : Fin 2→NormalizedSpin (n+1)) => |
| G (spinOverlap (x 0) (x 1))*(v i (x 1)-spinOverlap (x 0) (x 1)*v i (x 0)) |
| have hv : ∀ i, Measurable (v i) := fun i => by dsimp [v]; fun_prop |
| have hw : ∀ i, Measurable (w i) := fun i => by dsimp [w,v,spinOverlap]; fun_prop |
| have hwB (i x) : |w i x| ≤ ‖G‖*2 := by |
| dsimp [w]; rw [abs_mul] |
| apply mul_le_mul (G.norm_coe_le_norm _) _ (abs_nonneg _) (norm_nonneg _) |
| apply (abs_sub _ _).trans |
| have h1 := normalizedSpin_coord_bound (x 1) i |
| have h0 := normalizedSpin_coord_bound (x 0) i |
| have hr := spinOverlap_abs_le (x 0) (x 1) |
| dsimp [v]; rw [abs_mul] |
| have := mul_le_mul hr h0 (abs_nonneg _) (by norm_num : (0:ℝ)≤1) |
| linarith |
| have hK (x : Fin 2→NormalizedSpin (n+1)) (y : NormalizedSpin (n+1)) : |
| (∑ i, w i x*v i y) = G (spinOverlap (x 0) (x 1))* |
| (spinOverlap (x 1) y-spinOverlap (x 0) (x 1)*spinOverlap (x 0) y) := by |
| simp only [w,v,spinOverlap_sum,Finset.mul_sum,Finset.sum_sub_distrib,sub_mul,mul_sub] |
| congr 1 <;> (apply Finset.sum_congr rfl; intros; ring) |
| have hb (g : Fin (n+1)→ℝ) (x : Fin 2→NormalizedSpin (n+1)) : |
| (∑ i, g i*w i x) = G (spinOverlap (x 0) (x 1))* |
| (gaussianField (n+1) v g (x 1)-spinOverlap (x 0) (x 1)*gaussianField (n+1) v g (x 0)) := by |
| simp only [w,gaussianField,Finset.mul_sum,Finset.sum_sub_distrib,mul_sub] |
| congr 1 <;> (apply Finset.sum_congr rfl; intros; ring) |
| have he := nonlinear_pattern_direction_ibp μ (0:Fin 2) f hv hh hw hA |
| (by norm_num : (0:ℝ)≤1) (by positivity : 0≤‖G‖*2) hhA |
| (fun i x => normalizedSpin_coord_bound x i) hwB |
| dsimp only at he |
| simp_rw [hK,hb] at he |
| convert he using 1 |
| · congr 1; funext g; congr 1; funext x; ring |
| · congr 1; funext g |
| congr 1 |
| · congr 1; funext x |
| simp only [Fin.sum_univ_two,spinOverlap_self] |
| rw [show spinOverlap (x 1) (x 0) = spinOverlap (x 0) (x 1) from real_inner_comm _ _] |
| ring |
| · congr 1; congr 1; funext x |
| simp only [Fin.succ_zero_eq_one,Fin.succ_one_eq_two] |
| ring |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
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| end OAI |
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