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7d9f027 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import Mathlib
import OAI.Probability.SKGap.Terminal.TerminalProbability
import OAI.Probability.SKGap.Terminal.TerminalHighProbability
import OAI.Probability.SKGap.Posterior.ChannelTranslation
namespace OAI
section
noncomputable section
namespace SKGap
open MeasureTheory ProbabilityTheory Real Set
open scoped BigOperators ENNReal
variable {n : ℕ}
lemma channel_field_shift {T : ℝ} (hT : 0 ≤ T) (x : Spin n) (z : Fin n→ℝ) :
(fun i=>sqrt T*(z i+sqrt T*spinValue (x i)))=observationField T x z := by
funext i
simp only [observationField]
calc
_ = sqrt T*z i+(sqrt T)^2*spinValue (x i) := by ring
_ = _ := by rw [sq_sqrt hT];ring
lemma gaussianChannel_terminal_bad (g : Disorder n) {T c : ℝ} (hT : 0 ≤ T)
(hh : ∀ x : Spin n,(channelReference n).real {z | ¬∀ f : Spin n→ℝ,
variance g (observationField T x z) f ≤ 2*dirichlet g (observationField T x z) f} ≤ exp (-c*(n:ℝ))) :
(gaussianChannelLaw g T).real {z | ¬∀ f : Spin n→ℝ,
variance g (fun i=>sqrt T*z i) f ≤ 2*dirichlet g (fun i=>sqrt T*z i) f} ≤ exp (-c*(n:ℝ)) := by
let B : Set (Fin n→ℝ) := {z | ¬∀ f : Spin n→ℝ,
variance g (fun i=>sqrt T*z i) f ≤ 2*dirichlet g (fun i=>sqrt T*z i) f}
have hB : MeasurableSet B :=
((terminal_goodSet_closed g 2).measurableSet.compl).preimage (by fun_prop)
have he := gaussianChannel_event g hT B hB
have hx (x : Spin n) : channelReference n
((fun z : Fin n→ℝ=>fun i=>z i+sqrt T*spinValue (x i)) ⁻¹' B) ≤ ENNReal.ofReal (exp (-c*(n:ℝ))) := by
have hb : ((fun z : Fin n→ℝ=>fun i=>z i+sqrt T*spinValue (x i)) ⁻¹' B)=
{z | ¬∀ f : Spin n→ℝ,variance g (observationField T x z) f ≤
2*dirichlet g (observationField T x z) f} := by
ext z
simp only [Set.mem_preimage,B,Set.mem_ofPred_eq,channel_field_shift hT]
rw [hb]
exact (ENNReal.toReal_le_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mp
(by simpa only [measureReal_def,ENNReal.toReal_ofReal (exp_pos _).le] using hh x)
have hs : (∑ x : Spin n,ENNReal.ofReal (mass g 0 x))=1 := by
rw [← ENNReal.ofReal_sum_of_nonneg (fun x _=>mass_nonneg g 0 x),sum_mass]
exact ENNReal.ofReal_one
have hb : gaussianChannelLaw g T B ≤ ENNReal.ofReal (exp (-c*(n:ℝ))) := by
rw [he]
calc
_ ≤ ∑ x,ENNReal.ofReal (mass g 0 x)*ENNReal.ofReal (exp (-c*(n:ℝ))) :=
Finset.sum_le_sum (fun x _=>by gcongr;exact hx x)
_ = _ := by rw [← Finset.sum_mul,hs,one_mul]
have hr := ENNReal.toReal_mono ENNReal.ofReal_ne_top hb
simpa only [measureReal_def,ENNReal.toReal_ofReal (exp_pos _).le] using hr
end SKGap
end
end
section
noncomputable section
namespace SKGap
open MeasureTheory ProbabilityTheory Real Set Filter
open scoped BigOperators Topology Matrix.Norms.L2Operator
variable {n : ℕ}
lemma terminalMatrixBound_nonneg {j : ℝ} (hj : 0 ≤ j) : 0 ≤ terminalMatrixBound j := by
unfold terminalMatrixBound terminalRowBound
have hlog : 0 ≤ log (2:ℝ) := log_nonneg (by norm_num)
positivity
lemma terminal_endpoint_from_data {M α ε T : ℝ} (hM : 0 ≤ M) (hT : 0 < T) (hε : 0 ≤ ε)
(hsmall : M^2 ≤ (T/4)^2*α/2)
(hnum : 400001*(ε+M*(2*sqrt (1/cosh (T/4)^2)+1/cosh (T/4)^2)) ≤ 1/4)
(hp : exp (-T/8)*(exp 1-1) ≤ α/4)
(hn₁ : terminalEntryScale n ≤ 1/1000) (hn₂ : 800000*(M*terminalEntryScale n^2) ≤ 1/4)
(g : Disorder n) (hg : terminalMatrixData M α ε g) :
(gaussianChannelLaw g T).real {z | ¬∀ f : Spin n→ℝ,
variance g (fun i=>sqrt T*z i) f ≤ 2*dirichlet g (fun i=>sqrt T*z i) f} ≤ exp (-(α/4)*(n:ℝ)) := by
apply gaussianChannel_terminal_bad g hT.le
intro x
have hc (i k : Fin n) : |tanh (coupling g i k)| ≤ 1/1000 :=
(abs_tanh_le_abs _).trans ((hg.1 i k).trans hn₁)
have hh := terminal_probability_fixedSpin g hM hT hε hg.2.1 hg.2.2.1 hg.2.2.2.1 hc hg.2.2.2.2.1
(terminalMatrixData_forms g hg) hsmall (by linarith only [hnum,hn₂]) hp x
convert hh using 1
congr 1
ring
lemma terminal_numeric_eventually (M : ℝ) : ∀ᶠ n : ℕ in atTop,
terminalEntryScale n ≤ 1/1000 ∧ 800000*(M*terminalEntryScale n^2) ≤ 1/4 := by
have ht := (terminalEntryScale_tendsto.pow 2).const_mul M |>.const_mul 800000
simp only [zero_pow (by decide : (2:ℕ) ≠ 0),mul_zero] at ht
exact (terminalEntryScale_tendsto.eventually (ge_mem_nhds (by norm_num : (0:ℝ) < 1/1000))).and
(ht.eventually (ge_mem_nhds (by norm_num : (0:ℝ) < 1/4)))
theorem terminal_gap_high_probability {β : ℝ} (hβ : 0 < β) :
∃ T a : ℝ,0 < T ∧ 0 < a ∧ ∃ Ω : (n : ℕ)→Set (Disorder n),
(∀ n,MeasurableSet (Ω n)) ∧
Tendsto (fun n=>(disorderLaw β n).real (Ω n)) atTop (𝓝 1) ∧
∀ (n : ℕ) (g : Disorder n),g ∈ Ω n →
(gaussianChannelLaw g T).real {z | ¬∀ f : Spin n→ℝ,
variance g (fun i=>sqrt T*z i) f ≤ 2*dirichlet g (fun i=>sqrt T*z i) f} ≤ exp (-a*(n:ℝ)) := by
let ε : ℝ := 1/(16*400001)
have hε : 0 < ε := by norm_num [ε]
obtain ⟨α,l,hα,hαhalf,hl,hedge,h₅,h₆⟩ := sparse_parameters (sq_pos_of_pos hβ) hε
let M := terminalMatrixBound (β^2)
have hM : 0 ≤ M := terminalMatrixBound_nonneg (sq_nonneg β)
obtain ⟨T,hT,hsmall,hnum,hp⟩ := terminalTime_exists hM hα (show 400001*ε < 1/4 by norm_num [ε])
obtain ⟨N,hN⟩ := eventually_atTop.mp (terminal_numeric_eventually M)
let Ω : (n : ℕ)→Set (Disorder n) := fun n=>if N ≤ n then {g | terminalMatrixData M α ε g} else ∅
have hΩ (n : ℕ) : MeasurableSet (Ω n) := by
dsimp [Ω]
split_ifs
· exact terminalMatrixData_measurable M α ε
· exact MeasurableSet.empty
refine ⟨T,α/4,hT,(by positivity),Ω,hΩ,?_,?_⟩
· apply Tendsto.congr' _ (terminalMatrixData_actual_whp hβ hα.le hε hl hedge h₅ h₆)
filter_upwards [eventually_ge_atTop N] with n hn
simp only [Ω,ite_eq_left hn]
rfl
· intro n g hg
have hn : N ≤ n := by
by_contra hn
simp only [Ω,ite_eq_right hn,Set.mem_empty_iff_false] at hg
have hd : terminalMatrixData M α ε g := by simpa only [Ω,ite_eq_left hn,Set.mem_ofPred_eq] using hg
exact terminal_endpoint_from_data hM hT hε.le hsmall hnum hp (hN n hn).1 (hN n hn).2 g hd
end SKGap
end
end
end OAI
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