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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 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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | import OAI.Probability.InvariantIsing.Gaussian.GordonValueBound
namespace OAI
/-! Finite minimum--maximum values and their two-level log-sum-exp bounds. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators
namespace InvariantIsing
variable {U V : Type*} [Fintype U] [Nonempty U] [Fintype V] [Nonempty V]
def finiteRowMax (H : U × V → ℝ) (v : V) : ℝ :=
Finset.univ.sup' Finset.univ_nonempty (fun u => H (u,v))
def finiteMinmax (H : U × V → ℝ) : ℝ :=
Finset.univ.inf' Finset.univ_nonempty (finiteRowMax H)
lemma finiteMinmax_le_row (H : U × V → ℝ) (v : V) : finiteMinmax H ≤ finiteRowMax H v :=
Finset.inf'_le _ (Finset.mem_univ v)
lemma finite_log_sum_exp_bounds {X : Type*} [Fintype X] [Nonempty X] (H : X → ℝ) :
Finset.univ.sup' Finset.univ_nonempty H ≤ Real.log (finitePartition (fun _ => 1) H) ∧
Real.log (finitePartition (fun _ => 1) H) ≤
Finset.univ.sup' Finset.univ_nonempty H+Real.log (Fintype.card X) := by
constructor
· obtain ⟨x,hx,he⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty H
rw [he]
have hs : Real.exp (H x) ≤ finitePartition (fun _ => 1) H := by
simpa only [finitePartition,one_mul] using
Finset.single_le_sum (fun y _ => (Real.exp_pos (H y)).le) hx
simpa only [Real.log_exp] using Real.log_le_log (Real.exp_pos _) hs
· have he := log_finitePartition_le_add (gordon_reference (X := X))
(fun x => show H x ≤ 0+Finset.univ.sup' Finset.univ_nonempty H by
simpa only [zero_add] using Finset.le_sup' H (Finset.mem_univ x))
simpa only [finitePartition,Real.exp_zero,mul_one,Finset.sum_const,Finset.card_univ,
nsmul_eq_mul,mul_one,add_comm] using he
lemma gordonValue_finiteMinmax_bounds (H : U × V → ℝ) (β : ℝ) (hβ : 0 < β) :
β*finiteMinmax H-Real.log (Fintype.card V) ≤ gordonValue (fun x => β*H x) ∧
gordonValue (fun x => β*H x) ≤ β*finiteMinmax H+Real.log (Fintype.card U) := by
have hr (v : V) : β*finiteRowMax H v ≤ gordonRow (fun x => β*H x) v ∧
gordonRow (fun x => β*H x) v ≤ β*finiteRowMax H v+Real.log (Fintype.card U) := by
have hs : Finset.univ.sup' Finset.univ_nonempty (fun u => β*H (u,v)) =
β*finiteRowMax H v := by
obtain ⟨u,hu,he⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (fun u => H (u,v))
apply le_antisymm
· apply Finset.sup'_le
intro a _
exact mul_le_mul_of_nonneg_left
(Finset.le_sup' (fun u => H (u,v)) (Finset.mem_univ a)) hβ.le
· change β*(Finset.univ.sup' Finset.univ_nonempty (fun u => H (u,v))) ≤ _
rw [he]
exact Finset.le_sup' (fun u => β*H (u,v)) hu
simpa only [hs,gordonRow] using finite_log_sum_exp_bounds (fun u => β*H (u,v))
constructor
· have hb (v : V) : -gordonRow (fun x => β*H x) v ≤ 0+(-β*finiteMinmax H) := by
have hm := mul_le_mul_of_nonneg_left (finiteMinmax_le_row H v) hβ.le
linarith [(hr v).1]
have he := log_finitePartition_le_add (gordon_reference (X := V)) hb
have hz : finitePartition (fun _ : V => (1 : ℝ)) (fun _ => 0) = Fintype.card V := by
simp [finitePartition]
rw [hz] at he
change β*finiteMinmax H-Real.log (Fintype.card V) ≤ -Real.log _
linarith
· obtain ⟨v,hv,he⟩ := Finset.exists_mem_eq_inf' Finset.univ_nonempty (finiteRowMax H)
have hl : -gordonRow (fun x => β*H x) v ≤ Real.log
(finitePartition (fun _ : V => 1) (fun v => -gordonRow (fun x => β*H x) v)) := by
exact (Finset.le_sup' (fun v => -gordonRow (fun x => β*H x) v) hv).trans
(finite_log_sum_exp_bounds _).1
have hmin : finiteMinmax H = finiteRowMax H v := he
have hh := (hr v).2
rw [← hmin] at hh
change -Real.log _ ≤ _
linarith
lemma finiteMinmax_linear_integrable {d : ℕ} (C : U × V → Fin d → ℝ) :
Integrable (fun g => finiteMinmax (fun x => linearGaussian C g x))
(Measure.pi (fun _ : Fin d => gaussianReal 0 1)) := by
have hm : Measurable (fun g => finiteMinmax (fun x => linearGaussian C g x)) := by
have hr (v : V) : Continuous (fun g => finiteRowMax (fun x => linearGaussian C g x) v) :=
Continuous.finset_sup'_apply Finset.univ_nonempty (fun u _ => by
unfold linearGaussian; fun_prop)
exact (Continuous.finset_inf'_apply Finset.univ_nonempty (fun v _ => hr v)).measurable
apply (integrable_finsetSum Finset.univ (fun x _ => (integrable_linearGaussian C x).abs)).mono'
hm.aestronglyMeasurable
apply ae_of_all
intro g
obtain ⟨v,_,hv⟩ := Finset.exists_mem_eq_inf' Finset.univ_nonempty
(finiteRowMax (fun x => linearGaussian C g x))
obtain ⟨u,_,hu⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty
(fun u => linearGaussian C g (u,v))
change ‖Finset.univ.inf' Finset.univ_nonempty _‖ ≤ _
rw [hv]
change ‖Finset.univ.sup' Finset.univ_nonempty _‖ ≤ _
rw [hu,Real.norm_eq_abs]
exact Finset.single_le_sum (fun y _ => abs_nonneg (linearGaussian C g y)) (Finset.mem_univ (u,v))
end InvariantIsing
end
end OAI
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