File size: 8,459 Bytes
97c3405 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 | import OAI.Combinatorics.ContingencyTables.Counting.CompanionMultiplierUpdate
import OAI.Combinatorics.ContingencyTables.Transport.ObservableTrajectory
namespace OAI
/-! # Concentration of the literal companion phase observables -/
namespace ContingencyTables.CompanionConcentration
open scoped BigOperators Classical
open CommonBasesFPRAS CompanionFactorialApplication CompanionAnnealing CompanionMultiplierUpdate
open TwoLiftStates BinaryContraction DegreeCompletion ObservableBounds
variable {I J : Type*} [Fintype I] [Fintype J]
private lemma observable_constant_nonnegative (p q : ℝ) (hp : 0 ≤ p) (hq : 0 ≤ q) :
0 ≤ observableConstant p q := by
unfold observableConstant
positivity
private lemma bounded_variance_factor (p K v N : ℝ) (hK : 0 ≤ K)
(hN : 0 ≤ N) (hv : v ≤ 1) :
2*(10*p^2)*K*v/N ≤ 20*p^2*K/N := by
apply div_le_div_of_nonneg_right _ hN
nlinarith [mul_le_mul_of_nonneg_left hv (show 0 ≤ 2*(10*p^2)*K by positivity)]
lemma pairs_ne_nil (A : Instance I J) (hp : 0< Fintype.card (Labels A)) :
(pairs : List ((Labels A×Bool)×(Labels A×Bool)))≠[] := by
apply List.ne_nil_of_length_pos
simpa only [PairedLabels.length] using hp
lemma actual_degree (A : Instance I J) (j : ℕ) :
DegreePositive (weight A j) (pairs : List ((Labels A×Bool)×(Labels A×Bool))).length ∧
OmissionProperty (weight A j) (pairs : List ((Labels A×Bool)×(Labels A×Bool))).length := by
have hs := first_properties A ((j:ℝ)/scale A) (div_nonneg (Nat.cast_nonneg j) (scale_positive A).le)
simpa only [weight,PairedLabels.length] using hs
lemma square_width_one_le (A : Instance I J) : (1:ℝ)≤(A.W:ℝ)^2 := by
have hW : (2:ℝ)≤ A.W := by exact_mod_cast A.hW
nlinarith
/-- Every actual phase observable has the source's mean-square bound
20 p² Kobs/N under a 10 p²-warm start of its original labelled chain. -/
theorem bounded_MSE (A : Instance I J) (j : ℕ) (hp : 0< Fintype.card (Labels A))
(w : ((Labels A×Bool)×(Labels A×Bool)) → ((Labels A×Bool)×(Labels A×Bool)) → ℝ)
(hw : GoodMultipliers pairs (weight A j) w) (o : PhaseObservables (pairs : List ((Labels A×Bool)×(Labels A×Bool))))
(nu : ChainIndex (pairs : List ((Labels A×Bool)×(Labels A×Bool))) → ℝ)
(hwarm : ∀ S,nu S≤(10*(Fintype.card (Labels A):ℝ)^2)*
chainWeight pairs w (weight A j) S.val/chainTotal pairs w (weight A j))
(N : ℕ) (hN : 0< N) :
let C := metropolisChain pairs PairedLabels.fresh (pairs_ne_nil A hp) PairedLabels.cover w
(completeDegree (weight A j) (pairs : List ((Labels A×Bool)×(Labels A×Bool))).length)
hw.positive (completeDegree_positive _ _ (actual_degree A j).1)
let G := fun S : ChainIndex (pairs : List ((Labels A×Bool)×(Labels A×Bool))) =>
phaseObservable pairs (ratio A j) o S.val
(∑ S,nu S*C.squareMoment (fun S => G S-C.mean G) N S)/(N:ℝ)^2≤
20*(Fintype.card (Labels A):ℝ)^2*
ObservableBounds.observableConstant (Fintype.card (Labels A)) ((A.W:ℝ)^2)/N := by
let gs : List ((Labels A×Bool)×(Labels A×Bool)) := pairs
let F := completeDegree (weight A j) gs.length
have hF := completeDegree_positive (weight A j) gs.length (actual_degree A j).1
let C := metropolisChain gs PairedLabels.fresh (pairs_ne_nil A hp) PairedLabels.cover w F hw.positive hF
let G : Set (Labels A×Bool) → ℝ := phaseObservable gs (ratio A j) o
have hG : ∀ S,0≤ G S ∧ G S≤1 := by
intro S
exact phaseObservable_bounded gs (ratio A j)
(fun T _ => ⟨by linarith [(ratio_bounds A j T).1],(ratio_bounds A j T).2⟩) o S
have h := ObservableTrajectory.omission_chain_timeAverage gs PairedLabels.fresh (pairs_ne_nil A hp)
PairedLabels.cover w (weight A j) G hw.positive (actual_degree A j).1 (actual_degree A j).2
((A.W:ℝ)^2) (square_width_one_le A) (phaseObservable_class gs (ratio A j) o)
hw.lower hw.upper (10*(Fintype.card (Labels A):ℝ)^2) (by positivity) nu hwarm N hN
have hv := C.variance_le_one (fun S => G S.val) (fun S => hG S.val)
change C.variance (fun S => G S.val)≤1 at hv
rw [metropolisChain_variance,chainVariance_completion gs (weight A j) G w
PairedLabels.fresh PairedLabels.cover] at hv
have hm : C.mean (fun S => G S.val)=chainMean gs w (weight A j) G := by
rw [metropolisChain_mean,chainMean_completion gs (weight A j) G w
PairedLabels.fresh PairedLabels.cover]
change (∑ S,nu S*C.squareMoment (fun S => G S.val-C.mean (fun T => G T.val)) N S)/(N:ℝ)^2≤ _
rw [hm]
calc
_ ≤ 2*(10*(Fintype.card (Labels A):ℝ)^2)*ObservableBounds.observableConstant gs.length ((A.W:ℝ)^2)*
chainVariance gs w (weight A j) G/N := h
_ ≤ 20*(Fintype.card (Labels A):ℝ)^2*observableConstant gs.length ((A.W:ℝ)^2)/N :=
bounded_variance_factor _ _ _ _
(observable_constant_nonnegative _ _ (Nat.cast_nonneg _) (sq_nonneg _)) (Nat.cast_nonneg _) hv
_ = _ := by simp only [gs,PairedLabels.length]
/-- The source's finite union bound for all actual phase observables.
The warm-start hypothesis is the restart endpoint condition, rather than
an independent phase-concentration assumption. -/
theorem phase_failure (A : Instance I J) (j : ℕ) (hp : 0< Fintype.card (Labels A))
(w : ((Labels A×Bool)×(Labels A×Bool)) → ((Labels A×Bool)×(Labels A×Bool)) → ℝ)
(hw : GoodMultipliers pairs (weight A j) w)
(nu : ChainIndex (pairs : List ((Labels A×Bool)×(Labels A×Bool))) → ℝ)
(hnu : ∀ S,0≤ nu S)
(hwarm : ∀ S,nu S≤(10*(Fintype.card (Labels A):ℝ)^2)*
chainWeight pairs w (weight A j) S.val/chainTotal pairs w (weight A j))
(N : ℕ) (hN : 0< N) (eta : ℝ) (heta : 0< eta) :
let C := metropolisChain pairs PairedLabels.fresh (pairs_ne_nil A hp) PairedLabels.cover w
(completeDegree (weight A j) (pairs : List ((Labels A×Bool)×(Labels A×Bool))).length)
hw.positive (completeDegree_positive _ _ (actual_degree A j).1)
C.pathProbability nu (N-1) (fun l => ∃ o : PhaseObservables (pairs : List ((Labels A×Bool)×(Labels A×Bool))),
eta*C.mean (fun S => phaseObservable pairs (ratio A j) o S.val)<
|FiniteChain.pathSum (fun S => phaseObservable pairs (ratio A j) o S.val) l/(N:ℝ)-
C.mean (fun S => phaseObservable pairs (ratio A j) o S.val)|)≤
16000*(Fintype.card (Labels A):ℝ)^8*
ObservableBounds.observableConstant (Fintype.card (Labels A)) ((A.W:ℝ)^2)/((N:ℝ)*eta^2) := by
let gs : List ((Labels A×Bool)×(Labels A×Bool)) := pairs
let F := completeDegree (weight A j) gs.length
have hF := completeDegree_positive (weight A j) gs.length (actual_degree A j).1
let C := metropolisChain gs PairedLabels.fresh (pairs_ne_nil A hp) PairedLabels.cover w F hw.positive hF
let p : ℝ := Fintype.card (Labels A)
let K := ObservableBounds.observableConstant (Fintype.card (Labels A)) ((A.W:ℝ)^2)
have hpR : 0< p := by dsimp only [p]; exact_mod_cast hp
have hK : 0≤ K := observable_constant_nonnegative _ _ (Nat.cast_nonneg _) (sq_nonneg _)
have hNR : (0:ℝ)< N := by exact_mod_cast hN
have hmean (o : PhaseObservables gs) :
1/(20*p^2)≤ C.mean (fun S => phaseObservable gs (ratio A j) o S.val) := by
rw [metropolisChain_mean,chainMean_completion gs (weight A j)
(phaseObservable gs (ratio A j) o) w PairedLabels.fresh PairedLabels.cover]
have hh := phaseObservable_mean_lower gs PairedLabels.fresh (pairs_ne_nil A hp) PairedLabels.cover
w (weight A j) (ratio A j) hw.positive (weight_positive A j)
(fun S _ => (ratio_bounds A j S).1) hw.lower hw.upper o
simpa only [p,gs,PairedLabels.length] using hh
have hb := C.relative_union_bound nu hnu
(fun o S => phaseObservable gs (ratio A j) o S.val)
(1/(20*p^2)) eta (20*p^2*K/N) (by positivity) heta (by positivity) hmean N hN
(fun o => bounded_MSE A j hp w hw o nu hwarm N hN)
have hJ : (Fintype.card (PhaseObservables gs):ℝ)≤2*p^2 := by
have hc := phaseObservables_card gs PairedLabels.fresh (pairs_ne_nil A hp)
simp only [gs,PairedLabels.length] at hc
dsimp only [p,gs]
exact_mod_cast hc
apply hb.trans
calc
_ ≤ (2*p^2)*(20*p^2*K/N)/(eta*(1/(20*p^2)))^2 :=
div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_right hJ (by positivity)) (sq_nonneg _)
_ = 16000*p^8*K/((N:ℝ)*eta^2) := by
field_simp [hpR.ne',heta.ne',hNR.ne']
ring
end ContingencyTables.CompanionConcentration
end OAI
|