| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Probability.AdaptiveLaw |
| import OAI.Combinatorics.MatchingCount.Graphs.LabelOrder |
| import OAI.Combinatorics.MatchingCount.Sampling.TableMarking |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.Variation |
| lemma imageLaw_instances {A B : Type*} (F₁ F₂ : Fintype A) (f : A → B) : |
| @imageLaw A B F₁ f=@imageLaw A B F₂ f := by |
| cases Subsingleton.elim F₁ F₂ |
| rfl |
| lemma tv_instances {A : Type*} (F₁ F₂ : Fintype A) (p q : A → ℝ) : |
| @tv A F₁ p q=@tv A F₂ p q := by |
| cases Subsingleton.elim F₁ F₂ |
| rfl |
| end MatchingFPRAS.Variation |
| namespace MatchingFPRAS.RationalSource |
| open Finset Transport Variation Enlargement |
|
|
| variable {V : Type*} [Fintype V] [LinearOrder V] [Nontrivial V] |
| attribute [local instance] NumberedHierarchy.labelOrder sym2Order bagEdgeOrder bagEdgeFintype |
| attribute [local instance] UnitRefresh.Computable.allowedDec prodOrder sumOrder |
| noncomputable local instance (w : V → V → ℚ) (K : ℕ) : DecidableEq (BagPM (bags w K)) := Classical.decEq _ |
| noncomputable local instance (w : V → V → ℚ) (K : ℕ) : Fintype (BagPM (bags w K)) := Fintype.ofFinite _ |
|
|
| def centralEdge (w : V → V → ℚ) (K : ℕ) (e : LogicalEdge V) : BagEdge (bags w K) := |
| coloredReal w K (e,⟨2*(2*K+2),by omega⟩) |
| def probePortion (w : V → V → ℚ) (K D : ℕ) (e : BagEdge (bags w K)) : ℚ := |
| if ∃ q,centralEdge w K q=e then 1/(D:ℚ) else 0 |
| def portions (w : V → V → ℚ) (K D : ℕ) (e : BagEdge (bags w K)) : Fin 3 → ℚ := |
| ![realPortion w K e,bagHeight w K e.color/D-probePortion w K D e,probePortion w K D e] |
| lemma portions_sum (w : V → V → ℚ) (K D : ℕ) (e : BagEdge (bags w K)) : |
| ∑ i,portions w K D e i=bagWeight w K D e := by |
| simp [portions,Fin.sum_univ_succ,bagWeight] |
| lemma portions_nonneg (w : V → V → ℚ) (hw : ∀ x y,0<w x y) (K D : ℕ) |
| (e : BagEdge (bags w K)) (i : Fin 3) : 0≤portions w K D e i := by |
| fin_cases i |
| · exact realPortion_nonneg w hw K e |
| · change 0≤bagHeight w K e.color/D-probePortion w K D e |
| apply sub_nonneg.mpr |
| unfold probePortion |
| split_ifs |
| · exact div_le_div_of_nonneg_right (bagHeight_one w K _) (Nat.cast_nonneg D) |
| · exact div_nonneg (by dsimp [bagHeight]; positivity) (Nat.cast_nonneg D) |
| · change 0≤probePortion w K D e |
| unfold probePortion |
| split_ifs <;> positivity |
| lemma bagWeight_pos (w : V → V → ℚ) (hw : ∀ x y,0<w x y) (K D : ℕ) (hD : 0<D) |
| (e : BagEdge (bags w K)) : 0<bagWeight w K D e := by |
| apply add_pos_of_nonneg_of_pos (realPortion_nonneg w hw K e) |
| exact div_pos (by dsimp [bagHeight]; positivity) (by exact_mod_cast hD) |
|
|
| def markingTable (w : V → V → ℚ) (hw : ∀ x y,0<w x y) (K D : ℕ) (hD : 0<D) |
| (e : BagEdge (bags w K)) : WeightedOptions (Fin 3) where |
| size:=3 |
| out:=id |
| mass i:=portions w K D e i/bagWeight w K D e |
| nonneg i:=div_nonneg (portions_nonneg w hw K D e i) (bagWeight_pos w hw K D hD e).le |
| total:=by rw [←sum_div,portions_sum,div_self (bagWeight_pos w hw K D hD e).ne'] |
| lemma markingTable_law (w : V → V → ℚ) (hw : ∀ x y,0<w x y) (K D : ℕ) (hD : 0<D) |
| (e : BagEdge (bags w K)) (i : Fin 3) : |
| (markingTable w hw K D hD e).law i=portions w K D e i/bagWeight w K D e := by |
| change (∑ j : Fin 3,if j=i then portions w K D e j/bagWeight w K D e else 0)=_ |
| simp only [sum_ite_eq',mem_univ,ite_true] |
|
|
| abbrev Marked (w : V → V → ℚ) (K : ℕ) := TableMarking.Marked (I:=Fin 3) (BagEdge.Incident (bags:=bags w K)) |
| noncomputable local instance sourceMarkedFintype (w : V → V → ℚ) (K : ℕ) : |
| Fintype (Marked w K) := Fintype.ofFinite _ |
| noncomputable def markedLaw (w : V → V → ℚ) (K D : ℕ) : Marked w K → ℝ := |
| sigmaLift (gibbs (fun e=>(bagWeight w K D e:ℝ))) |
| (fun M=>product (fun e : M.val=>fun i=>(portions w K D e.val i/bagWeight w K D e.val:ℚ))) |
|
|
| def markedSampler (w : V → V → ℚ) (hw : ∀ x y,0<w x y) (K D m : ℕ) |
| (hD : 0<D) (hN : 0<Fintype.card (SourceVertex V K)) |
| (h : ∀ x y,x≠y → BottleneckAlgorithm.edge w x y≤(4:ℚ)^K) |
| (hq : 0<Transport.ladderQ (Fintype.card (SourceVertex V K)) D) (M : BagPM (bags w K)) (η : ℚ) := |
| TableMarking.draw BagEdge.Incident (topSampler w K D m hN h hq M η) |
| (markingTable w hw K D hD) |
| (bitPrecision (6*Fintype.card (BagEdge (bags w K))) η) |
|
|
| noncomputable local instance sourceMarkingBitsFintype {A : Type*} [Fintype A] |
| (w : V → V → ℚ) (K k : ℕ) : |
| Fintype (A × (BagEdge (bags w K) → Fin k → Bool)) := Fintype.ofFinite _ |
|
|
| lemma marking_tv {A : Type*} [Fintype A] [Nonempty A] |
| (w : V → V → ℚ) (hw : ∀ x y,0<w x y) (K D : ℕ) (hD : 0<D) |
| (f : A → BagPM (bags w K)) (k : ℕ) : |
| tv (imageLaw (TableMarking.draw BagEdge.Incident f (markingTable w hw K D hD) k)) |
| (markedLaw w K D)≤tv (imageLaw f) (gibbs (fun e=>(bagWeight w K D e:ℝ)))+ |
| 6*Fintype.card (BagEdge (bags w K))/(2:ℝ)^k := by |
| have he : TableMarking.law BagEdge.Incident (fun e=>(bagWeight w K D e:ℝ)) |
| (markingTable w hw K D hD)=markedLaw w K D := by |
| unfold TableMarking.law TableMarking.portion markedLaw |
| simp only [markingTable_law] |
| have hh:=TableMarking.draw_tv_size BagEdge.Incident f |
| (fun e=>(bagWeight w K D e:ℝ)) (markingTable w hw K D hD) k 3 (fun _=>rfl) |
| rw [he] at hh |
| norm_num only [Nat.cast_ofNat] at hh |
| rw [imageLaw_instances _ (sourceMarkingBitsFintype w K k), |
| tv_instances _ (sourceMarkedFintype w K)] at hh |
| exact hh |
|
|
| noncomputable def markedSamplerLaw (w : V → V → ℚ) (hw : ∀ x y,0<w x y) (K D m : ℕ) |
| (hD : 0<D) (hN : 0<Fintype.card (SourceVertex V K)) |
| (h : ∀ x y,x≠y → BottleneckAlgorithm.edge w x y≤(4:ℚ)^K) |
| (hq : 0<Transport.ladderQ (Fintype.card (SourceVertex V K)) D) (M : BagPM (bags w K)) (η : ℚ) : |
| Marked w K → ℝ := imageLaw (markedSampler w hw K D m hD hN h hq M η) |
|
|
| theorem markedSampler_tv (w : V → V → ℚ) (hw : ∀ x y,0<w x y) (K D m : ℕ) (hD : 1≤D) |
| (hN : 2≤Fintype.card (SourceVertex V K)) (hm : 1 < m) |
| (h : ∀ x y,x≠y → BottleneckAlgorithm.edge w x y≤(4:ℚ)^K) |
| (hq : 0<Transport.ladderQ (Fintype.card (SourceVertex V K)) D) (M : BagPM (bags w K)) (η : ℚ) (hη : 0<η) : |
| tv (markedSamplerLaw w hw K D m (lt_of_lt_of_le Nat.zero_lt_one hD) (lt_of_lt_of_le Nat.zero_lt_two hN) h hq M η) |
| (markedLaw w K D)≤2*(η:ℝ)+(m:ℝ)⁻¹ := by |
| have hh:=marking_tv w hw K D (lt_of_lt_of_le Nat.zero_lt_one hD) |
| (topSampler w K D m (lt_of_lt_of_le Nat.zero_lt_two hN) h hq M η) |
| (bitPrecision (6*Fintype.card (BagEdge (bags w K))) η) |
| have hs:=topSampler_tv w hw K D m hD hN hm h hq M η hη |
| have hb:=bitPrecision_error (6*Fintype.card (BagEdge (bags w K))) η hη |
| push_cast at hb |
| change tv (markedSamplerLaw w hw K D m (lt_of_lt_of_le Nat.zero_lt_one hD) |
| (lt_of_lt_of_le Nat.zero_lt_two hN) h hq M η) _≤_ at hh |
| linarith |
| end MatchingFPRAS.RationalSource |
|
|
| end OAI |
|
|
