File size: 3,658 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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactSamplingParameters
import OAI.Combinatorics.ContingencyTables.Sampling.ExactMatrixTrial
import OAI.Combinatorics.ContingencyTables.Machines.ScheduledResidualLaw
namespace OAI
/-! # Computing the complete input to the exact matrix rejection loop -/
namespace ContingencyTables.ProfilePrograms
open MatchingFPRAS MatchingFPRAS.TreeTyped DensePrograms Polynomial
attribute [local irreducible] exactResidualInput scheduledResidualCounts originalCandidates
def exactRareMeasure (z : CompletionMargins) : ℕ :=
exactParameterMeasure z+scheduledResidualMeasure (exactResidualInput z)+
originalEnumerationMeasure z+weight z
def exactRareWeights (z : CompletionMargins) : List ℕ :=
scheduledResidualCounts (exactResidualInput z)
def exactRareWeightsRealizer : Realizer exactRareWeights :=
composition (f := exactResidualInput) (g := scheduledResidualCounts)
exactResidualInputRealizer scheduledResidualCountsRealizer
lemma work_exactRareWeights : WorkBound exactRareWeightsRealizer exactRareMeasure := by
unfold exactRareWeightsRealizer
have hp : WorkBound exactResidualInputRealizer exactRareMeasure :=
work_remeasure work_exactResidualInput X (by
intro z
simp only [eval_X,exactRareMeasure]
omega)
apply work_composition hp work_scheduledResidualCounts X
intro z
simp only [eval_X,exactRareMeasure]
omega
attribute [local irreducible] exactRareWeights
def exactMatrixInput (z : CompletionMargins) : ExactMatrixTrial.Input :=
((originalCandidates z,GreedyFeasibleTable.listMatrix z.1 z.2),exactRareWeights z)
def exactMatrixInputRealizer : Realizer exactMatrixInput :=
pair (pair originalCandidatesRealizer GreedyFeasibleTable.tableRealizer) exactRareWeightsRealizer
lemma work_exactMatrixInput : WorkBound exactMatrixInputRealizer exactRareMeasure := by
have hi (z : CompletionMargins) : weight z ≤ X.eval (exactRareMeasure z) := by
simp only [eval_X,exactRareMeasure]
omega
have he : WorkBound originalCandidatesRealizer exactRareMeasure :=
work_remeasure work_originalCandidates X (by
intro z
simp only [eval_X,exactRareMeasure]
omega)
exact work_pair (work_pair he (work_of_polynomial GreedyFeasibleTable.polynomial_table X hi) X hi)
work_exactRareWeights X hi
end ContingencyTables.ProfilePrograms
namespace ContingencyTables.SmallGraphProfiles
open ProfilePrograms DensePrograms OriginalTableBox ResidualMixture FirstSuccess CompletionCounts
open scoped BigOperators Classical
noncomputable section
variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
local notation "d" => dimensionAllowance (I := Fin m) (J := Fin n)
local notation "D" => correctionPrecision r c
local notation "q" => largeTailCount r (d^20)*largeTailCount c (d^20)
lemma exactRareWeights_pos
(hPL : PublishedInputs.FiniteBoxPrekopaLeindler q)
(hCheeger : PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*d^8)))
(htotal : (∑ i,r i)=∑ j,c j) :
0<(exactRareWeights (List.ofFn r,List.ofFn c)).sum := by
rw [exactRareWeights,exactResidualInput_ofFn]
exact scheduledResidualCounts_sum r c hPL hCheeger htotal
lemma exactMatrixInput_code (htotal : (∑ i,r i)=∑ j,c j) :
exactMatrixInput (List.ofFn r,List.ofFn c)=
((originalCandidates (List.ofFn r,List.ofFn c),matrixCode (GreedyFeasibleTable.table r c htotal).val),
scheduledResidualCounts (((List.ofFn r,List.ofFn c),2*D),2^D)) := by
unfold exactMatrixInput exactRareWeights
rw [exactResidualInput_ofFn,GreedyFeasibleTable.listMatrix_ofFn]
rfl
end
end ContingencyTables.SmallGraphProfiles
end OAI
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