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c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Gaussian.GaussianComparisonSign
namespace OAI
/-! Finite Gaussian maxima and their log-sum-exp approximation. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators
namespace InvariantIsing
variable {X : Type*} [Fintype X] [Nonempty X]
def finiteGaussianMaximum {d : ℕ} (C : X → Fin d → ℝ) (g : Fin d → ℝ) : ℝ :=
Finset.univ.sup' Finset.univ_nonempty (fun x => linearGaussian C g x)
lemma finiteGaussianMaximum_integrable {d : ℕ} (C : X → Fin d → ℝ) :
Integrable (finiteGaussianMaximum C) (Measure.pi (fun _ : Fin d => gaussianReal 0 1)) := by
have hm : Measurable (finiteGaussianMaximum C) := by
have hc := Finset.measurable_sup' (s := (Finset.univ : Finset X)) Finset.univ_nonempty
(fun x _ => show Measurable (fun g => linearGaussian C g x) from by unfold linearGaussian; fun_prop)
convert hc using 1
funext g
rw [Finset.sup'_apply]
rfl
apply (integrable_finsetSum Finset.univ (fun x _ => (integrable_linearGaussian C x).abs)).mono'
hm.aestronglyMeasurable
apply ae_of_all
intro g
obtain ⟨x,hx,he⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (fun x => linearGaussian C g x)
change ‖Finset.univ.sup' Finset.univ_nonempty _‖ ≤ _
rw [he,Real.norm_eq_abs]
exact Finset.single_le_sum (fun y _ => abs_nonneg (linearGaussian C g y)) hx
lemma finiteGaussianMaximum_log_bounds {d : ℕ} (C : X → Fin d → ℝ)
(g : Fin d → ℝ) (β : ℝ) (hβ : 0 < β) :
β * finiteGaussianMaximum C g ≤
Real.log (finitePartition (fun _ : X => 1) (fun x => β*linearGaussian C g x)) ∧
Real.log (finitePartition (fun _ : X => 1) (fun x => β*linearGaussian C g x)) ≤
β * finiteGaussianMaximum C g + Real.log (Fintype.card X) := by
have hw : GibbsReference (fun _ : X => (1 : ℝ)) :=
⟨fun _ => zero_le_one,⟨Classical.choice ‹Nonempty X›,zero_lt_one⟩⟩
constructor
· obtain ⟨x,hx,he⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (fun x => linearGaussian C g x)
have hs : Real.exp (β * finiteGaussianMaximum C g) ≤
finitePartition (fun _ : X => 1) (fun x => β*linearGaussian C g x) := by
unfold finiteGaussianMaximum
rw [he]
simpa only [finitePartition,one_mul] using
Finset.single_le_sum (fun y _ => (Real.exp_pos (β*linearGaussian C g y)).le) hx
simpa only [Real.log_exp] using Real.log_le_log (Real.exp_pos _) hs
· have hb (x : X) : β*linearGaussian C g x ≤ 0+β*finiteGaussianMaximum C g := by
simpa only [zero_add,finiteGaussianMaximum] using mul_le_mul_of_nonneg_left
(Finset.le_sup' (fun x => linearGaussian C g x) (Finset.mem_univ x)) hβ.le
have he := log_finitePartition_le_add hw hb
simpa only [finitePartition,Real.exp_zero,mul_one,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,
mul_one,add_comm] using he
end InvariantIsing
end
end OAI
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