| 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 |
|
|
