| import OAI.Probability.InvariantIsing.Gaussian.GaussianComparisonSign |
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| namespace OAI |
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| /-! 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] |
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| 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) |
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| 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 |
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| 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 |
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| end InvariantIsing |
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| end |
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| end OAI |
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