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a89d282 | 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 | import OAI.Probability.InvariantIsing.Cavity.CavityGaussianCurveMean
namespace OAI
/-! Covariance order gives the finite Gaussian log-partition comparison. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators
namespace InvariantIsing
variable {X : Type*} [Fintype X]
lemma gaussian_gibbs_linear_mean_nonneg {d : ℕ} {w : X → ℝ} (hw : GibbsReference w)
(H : X → ℝ) (A C : X → Fin (d+1) → ℝ)
(hK : ∀ x y, gaussianCross A C x y ≤ gaussianCross A C x x) :
0 ≤ ∫ g, ∑ x, gaussianGibbs w H C g x * linearGaussian A g x
∂Measure.pi (fun _ : Fin (d+1) => gaussianReal 0 1) := by
have hi x : Integrable (fun g => gaussianGibbs w H C g x * linearGaussian A g x)
(Measure.pi (fun _ : Fin (d+1) => gaussianReal 0 1)) := by
simpa only [mul_comm] using integrable_linearGaussian_mul_gibbs hw H A C x x
rw [integral_finsetSum _ (fun x _ => hi x)]
apply Finset.sum_nonneg
intro x _
have he := gaussian_gibbs_insertion hw H A C x
simp only [mul_comm (gaussianGibbs w H C _ x)]
rw [he]
apply integral_nonneg
intro g
apply mul_nonneg (finiteGibbs_nonneg hw _ x)
apply sub_nonneg.mpr
calc
_ ≤ ∑ y, gaussianGibbs w H C g y * gaussianCross A C x x := by
apply Finset.sum_le_sum
intro y _
exact mul_le_mul_of_nonneg_left (hK x y) (finiteGibbs_nonneg hw _ y)
_ = gaussianCross A C x x := by
rw [← Finset.sum_mul]
change (∑ y, finiteGibbs w _ y) * _ = _
rw [finiteGibbs_sum hw,one_mul]
theorem finite_gaussian_log_comparison {d : ℕ} {w : X → ℝ} (hw : GibbsReference w)
(H : X → ℝ) (C A : X → Fin (d+1) → ℝ)
(hCA : ∀ x y, gaussianCross C A x y = 0)
(hdiag : ∀ x, gaussianCross A A x x = gaussianCross C C x x)
(hoff : ∀ x y, gaussianCross A A x y ≤ gaussianCross C C x y) :
(∫ g, Real.log (finitePartition w (fun x => H x+linearGaussian C g x))
∂Measure.pi (fun _ : Fin (d+1) => gaussianReal 0 1)) ≤
∫ g, Real.log (finitePartition w (fun x => H x+linearGaussian A g x))
∂Measure.pi (fun _ : Fin (d+1) => gaussianReal 0 1) := by
have hd (s : ℝ) := cavityGaussianCurveMean_hasDerivAt hw H (fun _ => 0) 0 C A s
have hs (s : ℝ) (hs : s ∈ Set.Icc (0 : ℝ) (Real.pi/2)) :
0 ≤ cavityGaussianCurveSlope w H (fun _ => 0) 0 C A s := by
have hsc : 0 ≤ Real.sin s * Real.cos s := mul_nonneg
(Real.sin_nonneg_of_nonneg_of_le_pi hs.1 (by linarith [hs.2,Real.pi_pos]))
(Real.cos_nonneg_of_mem_Icc ⟨by linarith [hs.1,Real.pi_pos],hs.2⟩)
have hcross : ∀ x y,
gaussianCross (cavityGaussianCurveDerivative C A s) (cavityGaussianCurve C A s) x y ≤
gaussianCross (cavityGaussianCurveDerivative C A s) (cavityGaussianCurve C A s) x x := by
intro x y
rw [cavityGaussianCurve_cross C A s x y hCA,cavityGaussianCurve_cross C A s x x hCA,
hdiag x,sub_self,mul_zero]
exact mul_nonpos_of_nonneg_of_nonpos hsc (sub_nonpos.mpr (hoff x y))
have h := gaussian_gibbs_linear_mean_nonneg hw H (cavityGaussianCurveDerivative C A s)
(cavityGaussianCurve C A s) hcross
have hp : cavityPenaltyCurve H (fun _ => 0) 0 s = H := by
funext x
simp only [cavityPenaltyCurve,mul_zero,zero_mul,sub_zero]
simpa only [cavityGaussianCurveSlope,hp,zero_mul,mul_zero,zero_add] using h
have hm : MonotoneOn (cavityGaussianCurveMean w H (fun _ => 0) 0 C A)
(Set.Icc (0 : ℝ) (Real.pi/2)) := by
apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _)
(continuous_iff_continuousAt.mpr (fun s => (hd s).continuousAt)).continuousOn
(fun s _ => (hd s).hasDerivWithinAt)
intro s hs'
exact hs s (interior_subset hs')
have hp : (0 : ℝ) ≤ Real.pi/2 := by positivity
have he := hm ⟨le_rfl,hp⟩ ⟨hp,le_rfl⟩ hp
simpa only [cavityGaussianCurveMean,cavityPenaltyCurve,linearGaussian,cavityGaussianCurve,
Real.cos_zero,Real.sin_zero,Real.cos_pi_div_two,Real.sin_pi_div_two,
one_mul,zero_mul,mul_zero,sub_zero,zero_add,add_zero] using he
end InvariantIsing
end
end OAI
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