File size: 5,754 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Combinatorics.ContingencyTables.Transport.ResidualBoxScale
import OAI.Combinatorics.ContingencyTables.Transport.TabulationPreparationBounds
namespace OAI
/-! # The actual transition-cache boxes in the exact sampler -/
namespace ContingencyTables.SmallGraphProfiles
open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms DensePrograms CompletionCounts Polynomial
open scoped BigOperators Classical
noncomputable section
attribute [local irreducible] unaryMarkers parameterProposalWidth scheduledCacheParameters
residualParameters canonicalCountInput
private lemma proposalBox_eq (z : SamplingParameters) (u v : List ℕ×List ℕ) :
gridStateCount ((scheduledMatrixParameters z,(u,v)).1.2,2)=2^(parameterProposalWidth z) := by
have hm : (scheduledMatrixParameters z,(u,v)).1.2=unaryMarkers (parameterProposalWidth z) := by
unfold scheduledMatrixParameters
rfl
rw [hm,gridStateCount_eq,unaryMarkers_length]
theorem exact_entry_box_bound : ∃ p : Polynomial ℕ,
∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), (∑ i,r i)=∑ j,c j →
∀ u∈profileBoxViews (initialCheck (List.ofFn r,List.ofFn c)),
∀ v∈profileBoxViews (initialCheck (List.ofFn r,List.ofFn c)),
matrixEntryBox (scheduledMatrixParameters (exactSamplingParameters (List.ofFn r,List.ofFn c)),(u,v)) ≤
p.eval (Algorithms.samplingSize n r 0)*exactBoxScale r c := by
obtain ⟨a,ha⟩ := exact_count_input_measure_polynomial
obtain ⟨b,hb⟩ := tabulation_preparation_polynomial
obtain ⟨p,hp⟩ := matrixEntryBox_bound
let A := b.comp a
refine ⟨p.comp A+A+4,?_⟩
intro m n r c htotal u hu v hv
let z := exactSamplingParameters (List.ofFn r,List.ofFn c)
let e : MatrixEntryInput := (scheduledMatrixParameters z,(u,v))
let V := candidateData (initialCheck (List.ofFn r,List.ofFn c),u)
let dim := dimensionAllowance (I := Fin m) (J := Fin n)
let bits := smallBinaryLength r c
let S := Algorithms.samplingSize n r 0
let E := exactBoxScale r c
have hd : 1 ≤ dim := by unfold dim dimensionAllowance; omega
have hbits : 1 ≤ bits := hd.trans (smallBinaryLength_parameters r c).2.1
have h23 : dim^2 ≤ dim^3 := Nat.pow_le_pow_right hd (by decide)
have he : 1 ≤ E := exactBoxScale_pos r c
have hdim : parameterDimension z=dim := by
change marginDimension (List.ofFn r,List.ofFn c)=dim
exact marginDimension_ofFn r c
have hVrow : V.1.2.1=List.ofFn r := rfl
have hVcol : V.1.2.2=List.ofFn c := rfl
have hres : entryResidualParameters e=residualParameters ((V,dim^12),(dim,parameterAccuracy z)) := by
unfold entryResidualParameters
dsimp only [e,entryCacheParameters,sourceData,scheduledMatrixParameters,scheduledCacheParameters,matrixData]
unfold scheduledCacheParameters
dsimp only
rw [show z.1=(List.ofFn r,List.ofFn c) from rfl,marginPadding_ofFn,hdim]
have hnorm := normalized_profile_completion_count r c V hVrow hVcol (parameterAccuracy z)
have hdense := normalized_profile_dense_grid r c V hVrow hVcol (parameterAccuracy z)
have horig : gridStateCount (completionEnumerationData (entryResidualParameters e).1) ≤ E := by
rw [hres]
unfold residualParameters
have hh := arbitrary_profile_completion_count r c V hVrow hVcol
apply hh.trans
apply Nat.pow_le_pow_right (by decide)
have hmul : dim^2*bits ≤ dim^3*bits := Nat.mul_le_mul_right bits h23
change 4*(dim^2*bits) ≤ 40*(dim^3*bits)
omega
have hprop : gridStateCount (e.1.2,2) ≤ E := by
rw [proposalBox_eq z u v,parameterProposalWidth,hdim]
apply Nat.pow_le_pow_right (by decide)
have hc : Nat.clog 2 (32*dim^2) ≤ 32*dim^2 :=
Nat.clog_le_of_le_pow (Nat.le_of_lt Nat.lt_two_pow_self)
have hmul : dim^3 ≤ dim^3*bits := by simpa only [Nat.mul_one] using Nat.mul_le_mul_left (dim^3) hbits
exact hc.trans (by change 32*dim^2 ≤ 40*(dim^3*bits); nlinarith)
have hprepare := hb z
have hnum : parameterDimension z+parameterAccuracy z+parameterCompletionWidth z ≤ A.eval S :=
hprepare.2.1.trans (by simp only [A,eval_comp]; exact polynomial_eval_monotone b (ha m n r c htotal))
have hw : weight e ≤ A.eval S := (hprepare.2.2.2.2 u hu v hv).trans (by
simp only [A,eval_comp]; exact polynomial_eval_monotone b (ha m n r c htotal))
have hpoly : p.eval (weight e) ≤ p.eval (A.eval S) := polynomial_eval_monotone p hw
have hbound := hp e
dsimp only at hbound
have hn : gridStateCount (completionEnumerationData (canonicalCountInput (entryResidualParameters e)).1.1) ≤ E := by
rw [hres]
exact hnorm
have hg : (4*(canonicalCountInput (entryResidualParameters e)).1.2.1^8)^
tabulationDimension (canonicalCountInput (entryResidualParameters e)).1 ≤ E := by
rw [hres]
exact hdense
change matrixEntryBox e ≤ _
have hrs : (entryResidualParameters e).2=(parameterDimension z,parameterAccuracy z) := by
rw [hres]
unfold residualParameters
rw [hdim]
have hcache : (entryCacheParameters e).2=parameterCompletionWidth z := by
unfold e entryCacheParameters scheduledMatrixParameters
unfold scheduledCacheParameters
rfl
have hnum' : (entryResidualParameters e).2.1+(entryResidualParameters e).2.2+
(entryCacheParameters e).2 ≤ A.eval S := by
rw [hrs,hcache]
exact hnum
have hfirst : matrixEntryBox e ≤ 4*E+A.eval S+p.eval (A.eval S) := by
omega
apply hfirst.trans
have hA : A.eval S ≤ A.eval S*E := by simpa only [Nat.mul_one] using Nat.mul_le_mul_left (A.eval S) he
have hpA : p.eval (A.eval S) ≤ p.eval (A.eval S)*E := by
simpa only [Nat.mul_one] using Nat.mul_le_mul_left (p.eval (A.eval S)) he
simp only [eval_add,eval_comp,eval_ofNat]
nlinarith
end
end ContingencyTables.SmallGraphProfiles
end OAI
|