File size: 3,424 Bytes
b317718 | 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 | import OAI.Probability.FreeIsing.Model
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace Problem367.SpinRecovery
def spinSign : IsingSpin → ℝ
| .neg => -1
| .pos => 1
def decode (x : ℕ → ℝ) : IsingSpin := by
classical
exact if Tendsto (fun n : ℕ => x n / (n : ℝ)) atTop (𝓝 (-1 : ℝ))
then .neg else .pos
theorem measurable_decode : Measurable decode := by
unfold decode
apply Measurable.ite
· exact measurableSet_tendsto (𝓝 (-1 : ℝ))
(f := fun (n : ℕ) (x : ℕ → ℝ) => x n / (n : ℝ))
(by intro n; fun_prop)
· exact measurable_const
· exact measurable_const
theorem decode_of_tendsto {x : ℕ → ℝ} {s : IsingSpin}
(h : Tendsto (fun n : ℕ => x n / (n : ℝ)) atTop (𝓝 (spinSign s))) :
decode x = s := by
cases s with
| neg =>
simp only [spinSign] at h
simp [decode, h]
| pos =>
have hn : ¬ Tendsto (fun n : ℕ => x n / (n : ℝ)) atTop (𝓝 (-1 : ℝ)) := by
intro hm
have he : (-1 : ℝ) = 1 := tendsto_nhds_unique hm h
norm_num at he
simp [decode, hn]
theorem decode_signal_add_noise (s : IsingSpin) {z : ℕ → ℝ}
(hz : Tendsto (fun n : ℕ => z n / (n : ℝ)) atTop (𝓝 (0 : ℝ))) :
decode (fun n : ℕ => (n : ℝ) * spinSign s + z n) = s := by
apply decode_of_tendsto
have he : (fun n : ℕ => ((n : ℝ) * spinSign s + z n) / (n : ℝ)) =ᶠ[atTop]
(fun n : ℕ => spinSign s + z n / (n : ℝ)) := by
filter_upwards [eventually_ge_atTop 1] with n hn
have hn' : (n : ℝ) ≠ 0 := by exact_mod_cast (by omega : n ≠ 0)
field_simp
apply (tendsto_congr' he).2
simpa using tendsto_const_nhds.add hz
def decodeArray {V : Type*} (x : V → ℕ → ℝ) : V → IsingSpin :=
fun v => decode (x v)
theorem measurable_decodeArray {V : Type*} :
Measurable (decodeArray (V := V)) := by
exact Measurable.of_eval fun vertex =>
measurable_decode.comp (measurable_pi_apply vertex)
theorem decodeArray_comp {V W : Type*} (x : V → ℕ → ℝ) (f : W → V) :
decodeArray (x ∘ f) = decodeArray x ∘ f := rfl
theorem decodeArray_relabel {d : ℕ} (g : TreeAut d)
(x : TreeVertex d → ℕ → ℝ) :
decodeArray (relabel g x) = relabel g (decodeArray x) := rfl
theorem decodeArray_of_tendsto {V : Type*} {x : V → ℕ → ℝ}
{s : V → IsingSpin}
(h : ∀ v, Tendsto (fun n : ℕ => x v n / (n : ℝ)) atTop (𝓝 (spinSign (s v)))) :
decodeArray x = s := by
funext v
exact decode_of_tendsto (h v)
theorem decodeArray_ae_eq {Ω V : Type*} [MeasurableSpace Ω] [Countable V]
{μ : Measure Ω} {x : Ω → V → ℕ → ℝ} {s : Ω → V → IsingSpin}
(h : ∀ v, ∀ᵐ ω ∂μ,
Tendsto (fun n : ℕ => x ω v n / (n : ℝ)) atTop (𝓝 (spinSign (s ω v)))) :
(fun ω => decodeArray (x ω)) =ᵐ[μ] s := by
filter_upwards [ae_all_iff.2 h] with ω hω
exact decodeArray_of_tendsto hω
theorem map_decodeArray_eq {Ω V : Type*} [MeasurableSpace Ω] [Countable V]
{μ : Measure Ω} {x : Ω → V → ℕ → ℝ} {s : Ω → V → IsingSpin}
(h : ∀ v, ∀ᵐ ω ∂μ,
Tendsto (fun n : ℕ => x ω v n / (n : ℝ)) atTop (𝓝 (spinSign (s ω v)))) :
Measure.map (fun ω => decodeArray (x ω)) μ = Measure.map s μ :=
Measure.map_congr (decodeArray_ae_eq h)
end Problem367.SpinRecovery
end
end OAI
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