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f18e7de | 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 | import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexWitness
import OAI.LinearAlgebra.MatrixMultiplication.Numerical.ComplexCertificates
import OAI.LinearAlgebra.MatrixMultiplication.Rectangular.ASI
import OAI.LinearAlgebra.MatrixMultiplication.Rectangular.RateLimit
namespace OAI
/-! Rectangular matrix multiplication algorithms and their asymptotic exponents. -/
namespace MatrixMultiplication.RectangularWitness
open Filter
open scoped Topology
theorem aspect_lower {g h : ℝ}
(hg : (27961 : ℝ) / 25000 < g)
(hh₀ : 0 < h) (hh : h < (15761 : ℝ) / 5000) :
(55922 : ℝ) / 78805 < 2 * g / h := by
apply (lt_div_iff₀ hh₀).2
nlinarith
theorem target_aspect_lt {g h : ℝ}
(hg : (27961 : ℝ) / 25000 < g)
(hh₀ : 0 < h) (hh : h < (15761 : ℝ) / 5000) :
(709 : ℝ) / 1000 < 2 * g / h := by
exact (by norm_num : (709 : ℝ) / 1000 < 55922 / 78805).trans
(aspect_lower hg hh₀ hh)
theorem aspect_lt_one {g h : ℝ}
(hg : g < (6991 : ℝ) / 6250)
(hh : (31519 : ℝ) / 10000 < h) : 2 * g / h < 1 := by
have hh₀ : 0 < h := by linarith
apply (div_lt_iff₀ hh₀).2
linarith
theorem exponent_upper {c h : ℝ}
(hc : c < (10987 : ℝ) / 10000)
(hh : (31519 : ℝ) / 10000 < h) :
6 * c / h < (65922 : ℝ) / 31519 := by
have hh₀ : 0 < h := by linarith
apply (div_lt_iff₀ hh₀).2
nlinarith
theorem exponent_lt_target {c h : ℝ}
(hc : c < (10987 : ℝ) / 10000)
(hh : (31519 : ℝ) / 10000 < h) :
6 * c / h < (523 : ℝ) / 250 := by
exact (exponent_upper hc hh).trans
(by norm_num : (65922 : ℝ) / 31519 < 523 / 250)
theorem omega_lt_of_balanced_family
(W : ℕ → ComplexWitness.FiniteRectangularWitness)
(scale : ℕ → ℝ) (k target g h H : ℝ)
(k_nonneg : 0 ≤ k) (h_positive : 0 < h)
(aspect_gap : k * h < 2 * g) (cost_gap : 6 * Real.log 3 < target * h)
(scale_positive : ∀ᶠ n in atTop, 0 < scale n)
(outer_rate : Tendsto
(fun n => Real.log ((W n).outer : ℝ) / scale n) atTop (𝓝 h))
(inner_rate : Tendsto
(fun n => Real.log ((W n).inner : ℝ) / scale n) atTop (𝓝 (2 * g)))
(multiplicity_rate : Tendsto
(fun n => Real.log ((W n).multiplicity : ℝ) / scale n) atTop (𝓝 (2 * H)))
(rank_rate : Tendsto
(fun n => Real.log ((W n).rank : ℝ) / scale n) atTop
(𝓝 (6 * Real.log 3 + 2 * H))) :
Arithmetic.rectangularOmega ℂ k < target := by
have hbound := RectangularRateLimit.exponent_le_of_finite_asi
(fun n => (W n).outer_pos) (fun n => (W n).inner_pos)
scale_positive outer_rate inner_rate multiplicity_rate rank_rate h_positive aspect_gap
(fun n ha hab => Arithmetic.equalRectangular_log_inequality_of_polynomialApproximation
ha (W n).multiplicity_pos k_nonneg hab (W n).approximation)
have hcancel : (6 * Real.log 3 + 2 * H - 2 * H) / h = 6 * Real.log 3 / h := by ring
rw [hcancel] at hbound
exact hbound.trans_lt ((div_lt_iff₀ h_positive).2 cost_gap)
theorem omega_lt_target_of_nearby_balanced_family
(W : ℕ → ComplexWitness.FiniteRectangularWitness)
(scale : ℕ → ℝ) (g h H : ℝ)
(g_lower : (27961 : ℝ) / 25000 < g)
(h_lower : (31519 : ℝ) / 10000 < h)
(h_upper : h < (15761 : ℝ) / 5000)
(scale_positive : ∀ᶠ n in atTop, 0 < scale n)
(outer_rate : Tendsto
(fun n => Real.log ((W n).outer : ℝ) / scale n) atTop
(𝓝 h))
(inner_rate : Tendsto
(fun n => Real.log ((W n).inner : ℝ) / scale n) atTop
(𝓝 (2 * g)))
(multiplicity_rate : Tendsto
(fun n => Real.log ((W n).multiplicity : ℝ) / scale n) atTop (𝓝 (2 * H)))
(rank_rate : Tendsto
(fun n => Real.log ((W n).rank : ℝ) / scale n) atTop
(𝓝 (6 * Real.log 3 + 2 * H))) :
Arithmetic.rectangularOmega ℂ ((709 : ℝ) / 1000) < (523 : ℝ) / 250 := by
have hh₀ : 0 < h := by linarith
have hgap : (709 : ℝ) / 1000 * h < 2 * g :=
(lt_div_iff₀ hh₀).1 (target_aspect_lt g_lower hh₀ h_upper)
apply omega_lt_of_balanced_family W scale (709 / 1000) (523 / 250) g h H
(by norm_num) hh₀ hgap _ scale_positive outer_rate inner_rate multiplicity_rate rank_rate
exact (div_lt_iff₀ hh₀).1
(exponent_lt_target ComplexCertificates.log_three_upper h_lower)
theorem omega_lt_target_of_balanced_family
(W : ℕ → ComplexWitness.FiniteRectangularWitness)
(scale : ℕ → ℝ) (H : ℝ)
(scale_positive : ∀ᶠ n in atTop, 0 < scale n)
(outer_rate : Tendsto
(fun n => Real.log ((W n).outer : ℝ) / scale n) atTop
(𝓝 ComplexCertificates.curveH))
(inner_rate : Tendsto
(fun n => Real.log ((W n).inner : ℝ) / scale n) atTop
(𝓝 (2 * ComplexCertificates.curveG)))
(multiplicity_rate : Tendsto
(fun n => Real.log ((W n).multiplicity : ℝ) / scale n) atTop (𝓝 (2 * H)))
(rank_rate : Tendsto
(fun n => Real.log ((W n).rank : ℝ) / scale n) atTop
(𝓝 (6 * Real.log 3 + 2 * H))) :
Arithmetic.rectangularOmega ℂ ((709 : ℝ) / 1000) < (523 : ℝ) / 250 :=
omega_lt_target_of_nearby_balanced_family W scale
ComplexCertificates.curveG ComplexCertificates.curveH H
ComplexCertificates.curveG_bounds.1 ComplexCertificates.curveH_bounds.1
ComplexCertificates.curveH_bounds.2 scale_positive outer_rate inner_rate
multiplicity_rate rank_rate
end MatrixMultiplication.RectangularWitness
end OAI
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