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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 | import OAI.Probability.InvariantIsing.Fields.FieldPairingError
import OAI.Probability.InvariantIsing.Fields.FieldBounds
namespace OAI
/-! Approximate self-consistency makes the actual scalar field values
approach the entropy supremum. Uniform height bounds avoid needing a limit
of the fields themselves. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped Topology
namespace InvariantIsing
lemma field_trial_upper (p : OverlapPath) (h : FieldStep) :
fieldValue h 0 + fieldPairing p h / 2 ≤ h.height (Fin.last h.depth) / 2 := by
have hp : fieldPairing p h ≤ h.height (Fin.last h.depth) := by
have hh := integral_mono (integrable_path_mul_field p h) (integrable_const (h.height (Fin.last h.depth)))
(fun s => (mul_le_of_le_one_left (fieldFunction_nonneg h s) (p.le_one s)).trans
(fieldFunction_le_last h s))
simpa only [fieldPairing, integral_const, probReal_univ, one_smul] using hh
linarith [fieldValue_nonpos h]
theorem field_entropy_sequence_of_support (p : OverlapPath) (h : ℕ → FieldStep)
{C : ℝ} (hC : ∀ n, (h n).height (Fin.last (h n).depth) ≤ C)
(hsupport : ∀ n k, fieldValue k 0 ≤ fieldValue (h n) 0 -
(∫ s, fieldMagnetizationPath (h n) s * (fieldFunction k s - fieldFunction (h n) s)
∂pathMeasure) / 2)
(hself : Tendsto (fun n => ∫ s, |fieldMagnetizationPath (h n) s - p s| ∂pathMeasure)
atTop (𝓝 0)) :
entropyFunctional p ≠ ⊤ ∧
Tendsto (fun n => fieldValue (h n) 0 + fieldPairing p (h n) / 2)
atTop (𝓝 (entropyFunctional p).toReal) := by
let V := fun n => fieldValue (h n) 0 + fieldPairing p (h n) / 2
let e := fun n => ∫ s, |fieldMagnetizationPath (h n) s - p s| ∂pathMeasure
have he0 n : 0 ≤ e n := integral_nonneg (fun _ => abs_nonneg _)
have hV n : V n ≤ C / 2 := (field_trial_upper p (h n)).trans (by linarith [hC n])
have happrox (n : ℕ) (k : FieldStep) :
fieldValue k 0 + fieldPairing p k / 2 ≤ V n +
((k.height (Fin.last k.depth) + C) / 2) * e n := by
have hb := field_trial_le_of_support p (h n) k (hsupport n k)
change fieldValue k 0 + fieldPairing p k / 2 ≤ V n +
((k.height (Fin.last k.depth) + (h n).height (Fin.last (h n).depth)) / 2) * e n at hb
have hc : (k.height (Fin.last k.depth) + (h n).height (Fin.last (h n).depth)) / 2 ≤
(k.height (Fin.last k.depth) + C) / 2 := by linarith [hC n]
exact hb.trans (add_le_add le_rfl (mul_le_mul_of_nonneg_right hc (he0 n)))
have hall (k : FieldStep) : fieldValue k 0 + fieldPairing p k / 2 ≤ C / 2 := by
have ht : Tendsto (fun n => C / 2 +
((k.height (Fin.last k.depth) + C) / 2) * e n) atTop (𝓝 (C / 2)) := by
simpa only [mul_zero, add_zero] using
(hself.const_mul ((k.height (Fin.last k.depth) + C) / 2)).const_add (C / 2)
apply ge_of_tendsto ht
exact Eventually.of_forall (fun n => (happrox n k).trans (add_le_add (hV n) le_rfl))
have hE : entropyFunctional p ≤ ((C / 2 : ℝ) : EReal) := by
apply iSup_le
intro k
exact_mod_cast hall k
have htop : entropyFunctional p ≠ ⊤ := ne_top_of_le_ne_top (EReal.coe_ne_top _) hE
have hbot : entropyFunctional p ≠ ⊥ :=
ne_bot_of_le_ne_bot (by simp) (entropyFunctional_nonneg p)
let S := (entropyFunctional p).toReal
have hS : entropyFunctional p = (S : EReal) := (EReal.coe_toReal htop hbot).symm
have hVS n : V n ≤ S := by
have hh := le_iSup (fun k : FieldStep =>
((fieldValue k 0 + fieldPairing p k / 2 : ℝ) : EReal)) (h n)
change (V n : EReal) ≤ entropyFunctional p at hh
rw [hS] at hh
exact_mod_cast hh
refine ⟨htop, Metric.tendsto_nhds.mpr ?_⟩
intro ε hε
have hlt : ((S - ε / 2 : ℝ) : EReal) < entropyFunctional p := by
rw [hS]
exact_mod_cast (show S - ε / 2 < S by linarith)
obtain ⟨k, hk⟩ := lt_iSup_iff.mp hlt
have hk' : S - ε / 2 < fieldValue k 0 + fieldPairing p k / 2 := by exact_mod_cast hk
have hsmall := (hself.const_mul ((k.height (Fin.last k.depth) + C) / 2)).eventually
(gt_mem_nhds (show ((k.height (Fin.last k.depth) + C) / 2) * 0 < ε / 2 by
simpa using half_pos hε))
filter_upwards [hsmall] with n hn
rw [Real.dist_eq, abs_of_nonpos (sub_nonpos.mpr (hVS n))]
have hb := happrox n k
change ((k.height (Fin.last k.depth) + C) / 2) * e n < ε / 2 at hn
change -(V n - S) < ε
linarith
end InvariantIsing
end
end OAI
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