File size: 5,515 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 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 | import OAI.Combinatorics.ContingencyTables.Sampling.AllSamplingAccuracy
import OAI.Combinatorics.ContingencyTables.Sampling.ProfileRuntimeBound
import OAI.Combinatorics.ContingencyTables.Machines.FiniteWordTableLaw
namespace OAI
/-! # The bounded public sampler on every margin pair -/
namespace ContingencyTables.ProfilePrograms
open MatchingFPRAS MatchingFPRAS.TreeTyped
noncomputable def allSamplingMachine : RandomMachine :=
FiniteWordExecution.machine publicWordCountRealizer publicSamplingAllRealizer
noncomputable def allSamplingMachineTime (as : List MatchingFPRAS.Symbol) : ℕ :=
FiniteWordExecution.machineTime publicWordCountRealizer publicSamplingAllRealizer as
lemma publicSamplingAll_alphabet (as : List MatchingFPRAS.Symbol) (bits : List Bool) :
∀ a∈publicSamplingAll (as,bits),a≠6 ∧ a≠7 := printMatrix_alphabet _
lemma allSamplingMachineTime_polynomial :
∃ p : Polynomial ℕ,∀ as,allSamplingMachineTime as≤p.eval
(FiniteWordExecution.wordMeasure publicWordCount as) :=
FiniteWordExecution.machineTime_polynomial polynomial_publicWordCount polynomial_publicSamplingAll
end ContingencyTables.ProfilePrograms
namespace ContingencyTables.SmallGraphProfiles
open ProfilePrograms DensePrograms CompletionCounts FirstPaperPhysicalMarginal
open MatchingFPRAS MatchingFPRAS.PhysicalStream MatchingFPRAS.TreeTyped Turing Algorithms Polynomial
open ResidualMixture FirstSuccess FairBits
open scoped BigOperators Classical
noncomputable section
lemma allSamplingMachineTime_public_bound :
∃ C d : ℕ,0<C ∧ ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),
(∑ i,r i)=∑ j,c j → ∀ k,
allSamplingMachineTime (encodeSamplingInput r c k)≤C*(samplingSize n r k)^d := by
obtain ⟨p,hp⟩ := allSamplingMachineTime_polynomial
refine ⟨p.eval (344^644+91)+1,1288*p.natDegree,by omega,?_⟩
intro m n r c htotal k
have hS : 1 ≤ samplingSize n r k := by unfold samplingSize; omega
calc
_ ≤ p.eval (FiniteWordExecution.wordMeasure publicWordCount (encodeSamplingInput r c k)) := hp _
_ ≤ p.eval ((344^644+91)*(samplingSize n r k)^1288) :=
polynomial_eval_monotone p (public_wordMeasure_bound r c htotal k)
_ ≤ p.eval (344^644+91)*(samplingSize n r k)^(1288*p.natDegree) :=
polynomial_scaled_power p (344^644+91) (samplingSize n r k) 1288 hS
_ ≤ _ := Nat.mul_le_mul_right _ (Nat.le_succ _)
variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
lemma allSamplingMachine_outputs (htotal : (∑ i,r i)=∑ j,c j) (k t : ℕ)
(ht : allSamplingMachineTime (encodeSamplingInput r c k)≤t) :
∀ bits : Bits t,∃ X : Table r c,
OutputsTable allSamplingMachine (encodeSamplingInput r c k) bits X.val := by
let as := encodeSamplingInput r c k
let P : Tape MatchingFPRAS.Symbol → Prop := fun tape => ∃ X : Table r c,
tape.right₀=ListBlank.mk (encodeMatrix X.val)
have hf : ∀ bits : Bits (publicWordCount as),P (Tape.mk₁ (publicSamplingAll (as,List.ofFn bits))) := by
dsimp only [as]
rw [publicWordCount_spec]
intro bits
refine ⟨scheduledTableDraw r c htotal k bits,?_⟩
rw [publicSamplingAll_spec r c htotal]
simp only [Tape.mk₁,Tape.mk₂,Tape.mk'_right₀]
have hP := FiniteWordExecution.machine_all publicWordCountRealizer publicSamplingAllRealizer as
(samplingInput_nonzero r c k) (publicSamplingAll_alphabet as) P hf t ht
have hhalt := (FiniteWordExecution.machine_law publicWordCountRealizer publicSamplingAllRealizer as
(samplingInput_nonzero r c k) (publicSamplingAll_alphabet as) t ht).1
intro bits
obtain ⟨X,hX⟩ := hP bits
exact ⟨X,hhalt bits,hX⟩
lemma allSamplingMachine_mass (htotal : (∑ i,r i)=∑ j,c j) (k t : ℕ)
(ht : allSamplingMachineTime (encodeSamplingInput r c k)≤t) (X : Table r c) :
tableMass allSamplingMachine (encodeSamplingInput r c k) t X.val=
((mapLaw (uniformLaw (α := Bits (finiteSamplerBits r c k)))
(scheduledTableDraw r c htotal k)).mass X:ℝ) :=
FiniteWordExecution.machine_tableMass publicWordCountRealizer publicSamplingAllRealizer
(encodeSamplingInput r c k) (samplingInput_nonzero r c k)
(publicSamplingAll_alphabet _) (publicWordCount_spec r c k) (scheduledTableDraw r c htotal k)
(publicSamplingAll_spec r c htotal k) t ht X
/-- Paper 200-01, main theorem (i), conditional only on Prékopa (1973)
Theorem 3/(2.3) on finite boxes and Levin--Peres (2017) Theorem 13.10,
(13.6), Lemma 13.7 on the completion grids. -/
theorem boundedSampling_conditional
(hPL : ∀ q,PublishedInputs.FiniteBoxPrekopaLeindler q)
(hCheeger : ∀ q d,PublishedInputs.FiniteCheeger
(GridBoundary.Grid (I := Fin q) (4*d^8))) : BoundedSamplingStatement := by
obtain ⟨C,D,hC,hbound⟩ := allSamplingMachineTime_public_bound
refine ⟨allSamplingMachine,C,D,hC,?_⟩
intro m n r c htotal k hk
dsimp only
have ht := hbound m n r c htotal k
refine ⟨allSamplingMachine_outputs r c htotal k _ ht,?_⟩
let : Nonempty (Table r c) := ⟨GreedyFeasibleTable.table r c htotal⟩
have hv := scheduledTableDraw_accuracy r c (hPL _)
(hCheeger _ (dimensionAllowance (I := Fin m) (J := Fin n))) htotal k hk
dsimp only at hv
simp_rw [allSamplingMachine_mass r c htotal k _ ht]
simpa only [variation,Rat.cast_div,Rat.cast_sum,Rat.cast_abs,Rat.cast_sub,
Rat.cast_ofNat,Rat.cast_inv,uniformLaw,uniformMass,Rat.cast_one,Rat.cast_natCast,one_div] using hv
end
end ContingencyTables.SmallGraphProfiles
end OAI
|