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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 | import OAI.Probability.InvariantIsing.Core.FiniteGaussianMaximum
namespace OAI
/-! The finite Gaussian maximum comparison used for rectangular spectral edges. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators
namespace InvariantIsing
variable {X : Type*} [Fintype X] [Nonempty X]
omit [Fintype X] [Nonempty X] in
lemma gaussianCross_scale_both {d : ℕ} (A C : X → Fin d → ℝ) (β : ℝ) (x y : X) :
gaussianCross (fun x i => β*A x i) (fun x i => β*C x i) x y =
β^2*gaussianCross A C x y := by
unfold gaussianCross
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
ring
theorem finite_gaussian_maximum_comparison {d : ℕ} (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, finiteGaussianMaximum C g ∂Measure.pi (fun _ : Fin (d+1) => gaussianReal 0 1)) ≤
∫ g, finiteGaussianMaximum A g ∂Measure.pi (fun _ : Fin (d+1) => gaussianReal 0 1) := by
let μ := Measure.pi (fun _ : Fin (d+1) => gaussianReal 0 1)
let MC := ∫ g, finiteGaussianMaximum C g ∂μ
let MA := ∫ g, finiteGaussianMaximum A g ∂μ
let K := Real.log (Fintype.card X)
have hw : GibbsReference (fun _ : X => (1 : ℝ)) :=
⟨fun _ => zero_le_one,⟨Classical.choice ‹Nonempty X›,zero_lt_one⟩⟩
have hbound (β : ℝ) (hβ : 0 < β) : β*MC ≤ β*MA+K := by
let LC := fun g : Fin (d+1) → ℝ =>
Real.log (finitePartition (fun _ : X => 1) (fun x => β*linearGaussian C g x))
let LA := fun g : Fin (d+1) → ℝ =>
Real.log (finitePartition (fun _ : X => 1) (fun x => β*linearGaussian A g x))
have hiC : Integrable LC μ := by
simpa only [zero_add] using integrable_log_finitePartition hw (fun _ => 0) C β
have hiA : Integrable LA μ := by
simpa only [zero_add] using integrable_log_finitePartition hw (fun _ => 0) A β
have hlow : β*MC ≤ ∫ g, LC g ∂μ := by
rw [← integral_const_mul]
exact integral_mono ((finiteGaussianMaximum_integrable C).const_mul β) hiC
(fun g => (finiteGaussianMaximum_log_bounds C g β hβ).1)
have hupp : (∫ g, LA g ∂μ) ≤ β*MA+K := by
have hconst : (∫ _ : Fin (d+1) → ℝ, K ∂μ) = K := by simp
rw [← integral_const_mul,← hconst,← integral_add]
· exact integral_mono hiA (((finiteGaussianMaximum_integrable A).const_mul β).add (integrable_const K))
(fun g => (finiteGaussianMaximum_log_bounds A g β hβ).2)
· exact (finiteGaussianMaximum_integrable A).const_mul β
· exact integrable_const K
have hmid : (∫ g, LC g ∂μ) ≤ ∫ g, LA g ∂μ := by
have h := finite_gaussian_log_comparison hw (fun _ => 0)
(fun x i => β*C x i) (fun x i => β*A x i)
(fun x y => by rw [gaussianCross_scale_both,hCA x y,mul_zero])
(fun x => by rw [gaussianCross_scale_both,gaussianCross_scale_both,hdiag x])
(fun x y => by
rw [gaussianCross_scale_both,gaussianCross_scale_both]
exact mul_le_mul_of_nonneg_left (hoff x y) (sq_nonneg β))
simpa only [linearGaussian_scale,zero_add] using h
exact hlow.trans (hmid.trans hupp)
change MC ≤ MA
by_contra h
have hpos : 0 < MC-MA := sub_pos.mpr (lt_of_not_ge h)
let β := (|K|+1)/(MC-MA)
have hβ : 0 < β := div_pos (by positivity) hpos
have he : β*(MC-MA) = |K|+1 := div_mul_cancel₀ _ hpos.ne'
have hh := hbound β hβ
have hk := le_abs_self K
nlinarith
end InvariantIsing
end
end OAI
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