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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 | import Mathlib
import OAI.Probability.Perceptron.Variational.TensorFeatures
import OAI.Probability.Perceptron.Variational.NonlinearDirectionIBP
namespace OAI
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
end OAI
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