File size: 3,351 Bytes
d5b7741 | 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 | import OAI.Combinatorics.ContingencyTables.Transport.TranslatedCompletionChain
namespace OAI
/-! # Exact edge flow of the completion-translation chain -/
namespace ContingencyTables.CompletionTranslation
open scoped BigOperators Classical
open CommonBasesFPRAS
variable {X I J : Type*} [Fintype X] [Nonempty X] [Fintype I] [Fintype J]
omit [Nonempty X] in
lemma completionMove_flow (R : X → Option I → ℕ) (P : X → Option J → ℕ)
(htotal : ∀ x,(∑ i,R x i)=∑ j,P x j)
(Adj : X → X → Prop) (β : ℝ) (x y : X) :
((blockCount R P x:ℝ)/MetropolisChain.normalizer (fun z => (blockCount R P z:ℝ)))*
completionMove R P Adj β x y =
if Adj x y then β*(acceptedCount R P x y:ℝ)/
MetropolisChain.normalizer (fun z => (blockCount R P z:ℝ)) else 0 := by
have hx : (blockCount R P x:ℝ)≠0 := ne_of_gt (by exact_mod_cast blockCount_pos R P htotal x)
unfold completionMove
split_ifs
· exact div_mul_div_cancel₀' hx _ _
· exact mul_zero _
/-- The flow formula includes the holding terms: those contribute zero
energy. The numerator counts literal acceptable completion translations. -/
theorem completionChain_energy (R : X → Option I → ℕ) (P : X → Option J → ℕ)
(htotal : ∀ x,(∑ i,R x i)=∑ j,P x j)
(Adj : X → X → Prop) (hsym : ∀ x y,Adj x y ↔ Adj y x)
(β : ℝ) (hβ : 0 ≤ β) (D : ℕ)
(hdegree : ∀ x,(Finset.univ.filter (Adj x)).card ≤ D)
(hproposal : (D:ℝ)*β ≤ 1/2) (H : X → ℝ) :
(completionChain R P htotal Adj hsym β hβ D hdegree hproposal).energy H=
(∑ x,∑ y,if Adj x y then
β*(acceptedCount R P x y:ℝ)*(H x-H y)^2 else 0)/
(2*MetropolisChain.normalizer (fun z => (blockCount R P z:ℝ))) := by
unfold completionChain
rw [FiniteChain.ofMoves_energy]
simp_rw [completionMove_flow R P htotal Adj β]
have hterm (x y : X) :
(if Adj x y then β*(acceptedCount R P x y:ℝ)/
MetropolisChain.normalizer (fun z => (blockCount R P z:ℝ)) else 0)*(H x-H y)^2 =
(if Adj x y then β*(acceptedCount R P x y:ℝ)*(H x-H y)^2 else 0)/
MetropolisChain.normalizer (fun z => (blockCount R P z:ℝ)) := by
split_ifs <;> ring
simp_rw [hterm,← Finset.sum_div]
ring
omit [Fintype X] [Nonempty X] in
/-- A half-acceptance estimate yields the min-weight comparison on every
actual edge, with no request for completion-count evaluation. -/
lemma completionMove_lower (R : X → Option I → ℕ) (P : X → Option J → ℕ)
(htotal : ∀ x,(∑ i,R x i)=∑ j,P x j)
(Adj : X → X → Prop) (β : ℝ) (hβ : 0 ≤ β) (x y : X)
(hxy : Adj x y)
(haccept : (blockCount R P x:ℝ)/2 ≤ acceptedCount R P x y) :
β/2*min 1 ((blockCount R P y:ℝ)/blockCount R P x) ≤ completionMove R P Adj β x y := by
have hx : (0:ℝ)<blockCount R P x := by exact_mod_cast blockCount_pos R P htotal x
have hm : β/2*min 1 ((blockCount R P y:ℝ)/blockCount R P x) ≤ β/2 := by
simpa only [mul_one] using mul_le_mul_of_nonneg_left
(min_le_left (1:ℝ) ((blockCount R P y:ℝ)/blockCount R P x)) (by positivity : 0 ≤ β/2)
apply hm.trans
unfold completionMove
rw [ite_eq_left hxy]
apply (le_div_iff₀ hx).mpr
nlinarith [mul_nonneg hβ (sub_nonneg.mpr haccept)]
end ContingencyTables.CompletionTranslation
end OAI
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