File size: 3,533 Bytes
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 | import OAI.LinearAlgebra.MatrixMultiplication.Arithmetic.Exponent
import OAI.LinearAlgebra.MatrixMultiplication.Numerical.ComplexCertificatesDual
import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexWitness
import OAI.LinearAlgebra.MatrixMultiplication.Rectangular.ASI
import OAI.LinearAlgebra.MatrixMultiplication.Rectangular.RateLimit
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
namespace MatrixMultiplication.DualExponentBound
open Filter
open scoped Topology
def fixedAspect : ℝ := 4651 / 10000
theorem target_lt_fixedAspect : (93 : ℝ) / 200 < fixedAspect := by
norm_num [fixedAspect]
theorem fixedAspect_lt_certificate : fixedAspect < (1548637 : ℝ) / 3327360 := by
norm_num [fixedAspect]
theorem fixedAspect_lt_dual_aspect : fixedAspect < DualWitness.aspect :=
fixedAspect_lt_certificate.trans ComplexCertificates.dual_aspect_lower
theorem fixedAspect_mem_unit : fixedAspect ∈ Set.Icc (0 : ℝ) 1 := by
norm_num [fixedAspect]
structure ActualFamily (k budget : ℝ) where
witness : ℕ → ComplexWitness.FiniteRectangularWitness
scale : ℕ → ℝ
outerRate : ℝ
innerRate : ℝ
multiplicityRate : ℝ
rankRate : ℝ
scale_positive : ∀ᶠ n in atTop, 0 < scale n
outer_positive : 0 < outerRate
outer_limit : Tendsto
(fun n => Real.log ((witness n).outer : ℝ) / scale n) atTop (𝓝 outerRate)
inner_limit : Tendsto
(fun n => Real.log ((witness n).inner : ℝ) / scale n) atTop (𝓝 innerRate)
multiplicity_limit : Tendsto
(fun n => Real.log ((witness n).multiplicity : ℝ) / scale n) atTop (𝓝 multiplicityRate)
rank_limit : Tendsto
(fun n => Real.log ((witness n).rank : ℝ) / scale n) atTop (𝓝 rankRate)
aspect_gap : k * outerRate < innerRate
cost_bound : rankRate - multiplicityRate ≤ budget * outerRate
namespace ActualFamily
theorem omega_le {k budget : ℝ} (P : ActualFamily k budget) (hk : 0 ≤ k) :
Arithmetic.rectangularOmega ℂ k ≤ budget := by
have hbound := RectangularRateLimit.exponent_le_of_finite_asi
(fun n => (P.witness n).outer_pos) (fun n => (P.witness n).inner_pos)
P.scale_positive P.outer_limit P.inner_limit P.multiplicity_limit P.rank_limit
P.outer_positive P.aspect_gap
(fun n ha hab => Arithmetic.equalRectangular_log_inequality_of_polynomialApproximation
ha (P.witness n).multiplicity_pos hk hab (P.witness n).approximation)
exact hbound.trans ((div_le_iff₀ P.outer_positive).2 P.cost_bound)
end ActualFamily
theorem omega_eq_two_of_arbitrarily_close_families {k : ℝ} (hk : 0 ≤ k)
(families : ∀ ε : ℝ, 0 < ε → Nonempty (ActualFamily k (2 + ε))) :
Arithmetic.rectangularOmega ℂ k = 2 := by
apply le_antisymm _ (Arithmetic.rectangularOmega_two_le k)
by_contra h
have hω : 2 < Arithmetic.rectangularOmega ℂ k := lt_of_not_ge h
have hε : 0 < (Arithmetic.rectangularOmega ℂ k - 2) / 2 := by linarith
obtain ⟨P⟩ := families ((Arithmetic.rectangularOmega ℂ k - 2) / 2) hε
have hbound := P.omega_le hk
linarith
theorem alpha_gt_target_of_arbitrarily_close_families
(families : ∀ ε : ℝ, 0 < ε → Nonempty (ActualFamily fixedAspect (2 + ε))) :
(93 : ℝ) / 200 < Arithmetic.complexAlpha := by
have hω := omega_eq_two_of_arbitrarily_close_families fixedAspect_mem_unit.1 families
exact target_lt_fixedAspect.trans_le (Arithmetic.le_complexAlpha fixedAspect_mem_unit hω)
end MatrixMultiplication.DualExponentBound
end
end OAI
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