| 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 |
|
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| end |
|
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| end OAI |
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