| import OAI.Probability.InvariantIsing.Cavity.CavityGaussianCurveMean |
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| namespace OAI |
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| /-! 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] |
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| 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] |
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| 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 |
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| end InvariantIsing |
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| end |
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| end OAI |
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