| 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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