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cdec422 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.MaskedAssignment
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.OrdinarySelection
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointMaskedRates
open Filter JointCoarseHashing JointExtractionRates JointMaskedSelection
open JointOrdinarySelection (hashLevel hashSet)
open scoped Topology
attribute [local instance] Classical.propDecidable
def errorBound (degree : ℕ → ℕ) (B p : ℕ) (H : ℝ) (N : ℕ) : ℝ :=
(degree N : ℝ) / (hashModulus H N : ℝ) *
(1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * (N : ℝ))
theorem polynomialFactor_le (B p N : ℕ) :
1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * (N : ℝ) ≤
((B : ℝ) + 1) * ((N : ℝ) + 1) ^ (p + 1) := by
have hN : (0 : ℝ) ≤ N := Nat.cast_nonneg _
have hx : (0 : ℝ) ≤ (N : ℝ) + 1 := by linarith
have hpower : (1 : ℝ) ≤ ((N : ℝ) + 1) ^ (p + 1) :=
one_le_pow₀ (by linarith : (1 : ℝ) ≤ (N : ℝ) + 1)
calc
_ ≤ 1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * ((N : ℝ) + 1) :=
add_le_add le_rfl (mul_le_mul_of_nonneg_left (by linarith)
(mul_nonneg (Nat.cast_nonneg _) (pow_nonneg hx _)))
_ = 1 + (B : ℝ) * ((N : ℝ) + 1) ^ (p + 1) := by rw [pow_succ]; ring
_ ≤ ((N : ℝ) + 1) ^ (p + 1) + (B : ℝ) * ((N : ℝ) + 1) ^ (p + 1) :=
add_le_add hpower le_rfl
_ = _ := by ring
theorem eventually_errorBound_le_half (degree : ℕ → ℕ) (B p : ℕ)
{D H : ℝ} (hH : 0 ≤ H) (hgap : D < H)
(hdegreeRate : ∀ η : ℝ, 0 < η →
∀ᶠ N in atTop, Real.log (degree N : ℝ) / (N : ℝ) ≤ D + η) :
∀ᶠ N in atTop, errorBound degree B p H N ≤ 1 / 2 := by
let c : ℝ := (H - D) / 2
have hc : 0 < c := by dsimp [c]; linarith
have hclt : c < H - D := by dsimp [c]; linarith
have hratio := eventually_degree_div_hashModulus_le_exp hH hdegreeRate hclt
apply eventually_error_le_half ((B : ℝ) + 1) (p + 1) hc
filter_upwards [hratio] with N hN
have hx : (0 : ℝ) ≤ (N : ℝ) + 1 := add_nonneg (Nat.cast_nonneg N) zero_le_one
have hfactor : 0 ≤ 1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * (N : ℝ) :=
add_nonneg zero_le_one
(mul_nonneg (mul_nonneg (Nat.cast_nonneg _) (pow_nonneg hx _)) (Nat.cast_nonneg _))
calc
errorBound degree B p H N ≤ Real.exp (-c * (N : ℝ)) *
(1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * (N : ℝ)) :=
mul_le_mul_of_nonneg_right hN hfactor
_ ≤ Real.exp (-c * (N : ℝ)) * (((B : ℝ) + 1) * ((N : ℝ) + 1) ^ (p + 1)) :=
mul_le_mul_of_nonneg_left (polynomialFactor_le B p N) (Real.exp_pos _).le
_ = ((B : ℝ) + 1) * (((N : ℝ) + 1) ^ (p + 1) * Real.exp (-c * (N : ℝ))) := by ring
theorem count_error_le_envelope {P E O V : Type*} (d : OrbitData P E O V)
(e : E) (degree B p N M : ℕ)
(heligibility : (d.eligibilityCompetitors e).card ≤ degree)
(horbits : (d.orbits e).card ≤ B * (N + 1) ^ p) :
((d.eligibilityCompetitors e).card : ℝ) / (M : ℝ) +
((d.orbits e).card : ℝ) * (N : ℝ) * ((degree : ℝ) / (M : ℝ)) ≤
(degree : ℝ) / (M : ℝ) * (1 + (B : ℝ) * ((N : ℝ) + 1) ^ p * (N : ℝ)) := by
have hdegreeR : ((d.eligibilityCompetitors e).card : ℝ) ≤ (degree : ℝ) := by
exact_mod_cast heligibility
have horbitsR : ((d.orbits e).card : ℝ) ≤ (B : ℝ) * ((N : ℝ) + 1) ^ p := by
exact_mod_cast horbits
have hratio : (0 : ℝ) ≤ (degree : ℝ) / (M : ℝ) :=
div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
calc
_ ≤ (degree : ℝ) / (M : ℝ) +
((B : ℝ) * ((N : ℝ) + 1) ^ p) * (N : ℝ) * ((degree : ℝ) / (M : ℝ)) :=
add_le_add (div_le_div_of_nonneg_right hdegreeR (Nat.cast_nonneg _))
(mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_right horbitsR (Nat.cast_nonneg _)) hratio)
_ = _ := by ring
theorem eventually_exists_selection
{P E O V : ℕ → Type*} [∀ N, Fintype (P N)] [∀ N, Fintype (E N)]
(d : ∀ N, OrbitData (P N) (E N) (O N) (V N))
(hinj : ∀ N, Function.Injective (d N).coarse)
(degree : ℕ → ℕ) (B p : ℕ)
(heligibility : ∀ N e, ((d N).eligibilityCompetitors e).card ≤ degree N)
(hdegree : ∀ N e, ∀ o ∈ (d N).orbits e, ∀ v ∈ (d N).passing e o,
((d N).competitors e o v).card ≤ degree N)
(horbits : ∀ N e, ((d N).orbits e).card ≤ B * (N + 1) ^ p)
{A D H C₀ : ℝ} (hH : 0 ≤ H) (hgap : D < H)
(hdegreeRate : ∀ η : ℝ, 0 < η →
∀ᶠ N in atTop, Real.log (degree N : ℝ) / (N : ℝ) ≤ D + η)
(hdet : ∀ᶠ N in atTop, ∀ e, ∀ o ∈ (d N).orbits e,
JointLossCounts.deterministicLossFraction ((d N).full e o) ((d N).passing e o) ≤
C₀ / (N : ℝ))
(hpos : ∀ᶠ N in atTop, 0 < Fintype.card (E N))
(hE : Tendsto (fun N => Real.log (Fintype.card (E N) : ℝ) / (N : ℝ)) atTop (𝓝 A))
{ε : ℝ} (hε : 0 < ε) :
∀ᶠ N in atTop, ∃ (s : Sample (P N) (hashLevel H N)) (G : Finset (E N)),
G.card = ⌊Real.exp ((N : ℝ) * (A - H - ε))⌋₊ ∧
(∀ e ∈ G, (d N).Good (hashLevel H N) (hashSet H N) (N : ℝ) e s) ∧
(∀ e ∈ G, ∀ f ∈ G, e ≠ f → ((d N).coarse e).1 ≠ ((d N).coarse f).1) ∧
∀ e ∈ G, ∀ o ∈ (d N).orbits e,
JointLossCounts.missingFraction ((d N).full e o) ((d N).passing e o)
((d N).variableBad (hashLevel H N) (hashSet H N) e o) s ≤ (C₀ + 1) / (N : ℝ) := by
classical
have huniform := eventually_errorBound_le_half degree B p hH hgap hdegreeRate
have hsmall : ∀ᶠ N in atTop, ∀ e,
(((d N).eligibilityCompetitors e).card : ℝ) / (hashModulus H N : ℝ) +
(((d N).orbits e).card : ℝ) * (N : ℝ) *
((degree N : ℝ) / (hashModulus H N : ℝ)) ≤ 1 / 2 := by
filter_upwards [huniform] with N hN e
exact (count_error_le_envelope (d N) e (degree N) B p N (hashModulus H N)
(heligibility N e) (horbits N e)).trans hN
have hpositive : ∀ᶠ N : ℕ in atTop, (0 : ℝ) < N := by
filter_upwards [eventually_gt_atTop (0 : ℕ)] with N hN
exact_mod_cast hN
let good := fun N (s : Sample (P N) (hashLevel H N)) (e : E N) =>
(d N).Good (hashLevel H N) (hashSet H N) (N : ℝ) e s
have hinc : ∀ᶠ N in atTop, ∀ e,
(((JointAPFree.modularSet (hashModulus H N)).card : ℝ) /
(hashModulus H N : ℝ) ^ 2 / 2) *
(Fintype.card (Sample (P N) (hashLevel H N)) : ℝ) ≤
(goodIncidence (good N) e : ℝ) := by
filter_upwards [hsmall, hpositive] with N hN hNpos e
rw [show goodIncidence (good N) e =
((d N).goodEvent (hashLevel H N) (hashSet H N) (N : ℝ) e).card from
(d N).goodIncidence_eq _ _ _ _]
exact (d N).goodEvent_card_lower (hashLevel H N) (hashSet H N) (N : ℝ) hNpos e
(degree N) (hdegree N e) (hN e)
have hyield := eventually_exists_many_good_explicit good hH hpos hinc hE hε
have hselection := eventually_exists_good_subfamily good
(fun N => ⌊Real.exp ((N : ℝ) * (A - H - ε))⌋₊) hyield
filter_upwards [hselection, hdet] with N hN hdetN
obtain ⟨s, G, hcard, hgood⟩ := hN
refine ⟨s, G, hcard, hgood, ?_, ?_⟩
· intro e he f hf hne
exact (d N).good_no_shared_x (hashLevel H N) (hashSet H N) (N : ℝ) f e s
(hgood f hf) (hgood e he) (fun h => hne ((hinj N) h))
· intro e he o ho
exact (d N).good_missingFraction_le (hashLevel H N) (hashSet H N) C₀ (N : ℝ) e s
(hgood e he) (hdetN e) o ho
end MatrixMultiplication.JointMaskedRates
end
end OAI
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