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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 | import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalLabels
namespace OAI
/-! Rectangular matrix multiplication algorithms and their asymptotic exponents. -/
noncomputable section
namespace MatrixMultiplication.RectangularRateLimit
open Filter
open scoped Topology
theorem eventually_two_le_of_positive_log_rate
{a : ℕ → ℕ} {scale : ℕ → ℝ} {A : ℝ}
(scale_positive : ∀ᶠ n in atTop, 0 < scale n)
(rate : Tendsto (fun n => Real.log (a n : ℝ) / scale n) atTop (𝓝 A))
(positive : 0 < A) : ∀ᶠ n in atTop, 2 ≤ a n := by
filter_upwards [rate.eventually_const_lt positive, scale_positive] with n hn hs
have hlog : 0 < Real.log (a n : ℝ) := (div_pos_iff_of_pos_right hs).mp hn
have ha : (1 : ℝ) < a n := (Real.log_pos_iff (Nat.cast_nonneg _)).mp hlog
have : 1 < a n := by exact_mod_cast ha
omega
theorem eventually_aspect_of_rate_gap
{a b : ℕ → ℕ} {scale : ℕ → ℝ} {A B k : ℝ}
(a_positive : ∀ n, 0 < a n) (b_positive : ∀ n, 0 < b n)
(scale_positive : ∀ᶠ n in atTop, 0 < scale n)
(a_rate : Tendsto (fun n => Real.log (a n : ℝ) / scale n) atTop (𝓝 A))
(b_rate : Tendsto (fun n => Real.log (b n : ℝ) / scale n) atTop (𝓝 B))
(gap : k * A < B) : ∀ᶠ n in atTop, (a n : ℝ) ^ k ≤ (b n : ℝ) := by
filter_upwards [(a_rate.const_mul k).eventually_lt b_rate gap, scale_positive]
with n hn hs
have hlog : k * Real.log (a n : ℝ) < Real.log (b n : ℝ) := by
apply (div_lt_div_iff_of_pos_right hs).mp
simpa only [mul_div_assoc] using hn
have ha : (0 : ℝ) < a n := Nat.cast_pos.mpr (a_positive n)
have hb : (0 : ℝ) < b n := Nat.cast_pos.mpr (b_positive n)
apply le_of_lt
apply (Real.log_lt_log_iff (Real.rpow_pos_of_pos ha k) hb).mp
simpa only [Real.log_rpow ha] using hlog
theorem rate_inequality_of_log_bounds
{a multiplicity rank : ℕ → ℕ} {scale : ℕ → ℝ} {A M R exponent : ℝ}
(scale_positive : ∀ᶠ n in atTop, 0 < scale n)
(a_rate : Tendsto (fun n => Real.log (a n : ℝ) / scale n) atTop (𝓝 A))
(multiplicity_rate : Tendsto
(fun n => Real.log (multiplicity n : ℝ) / scale n) atTop (𝓝 M))
(rank_rate : Tendsto (fun n => Real.log (rank n : ℝ) / scale n) atTop (𝓝 R))
(finite_bound : ∀ᶠ n in atTop,
exponent * Real.log (a n : ℝ) ≤
Real.log (rank n : ℝ) - Real.log (multiplicity n : ℝ)) :
exponent * A ≤ R - M := by
apply le_of_tendsto_of_tendsto (a_rate.const_mul exponent)
(rank_rate.sub multiplicity_rate)
filter_upwards [finite_bound, scale_positive] with n hn hs
have h := div_le_div_of_nonneg_right hn hs.le
simpa only [mul_div_assoc, sub_div] using h
theorem exponent_le_of_finite_asi
{a b multiplicity rank : ℕ → ℕ} {scale : ℕ → ℝ}
{A B M R k exponent : ℝ}
(a_positive : ∀ n, 0 < a n) (b_positive : ∀ n, 0 < b n)
(scale_positive : ∀ᶠ n in atTop, 0 < scale n)
(a_rate : Tendsto (fun n => Real.log (a n : ℝ) / scale n) atTop (𝓝 A))
(b_rate : Tendsto (fun n => Real.log (b n : ℝ) / scale n) atTop (𝓝 B))
(multiplicity_rate : Tendsto
(fun n => Real.log (multiplicity n : ℝ) / scale n) atTop (𝓝 M))
(rank_rate : Tendsto (fun n => Real.log (rank n : ℝ) / scale n) atTop (𝓝 R))
(positive : 0 < A) (gap : k * A < B)
(finite_asi : ∀ n, 2 ≤ a n → (a n : ℝ) ^ k ≤ (b n : ℝ) →
exponent * Real.log (a n : ℝ) ≤
Real.log (rank n : ℝ) - Real.log (multiplicity n : ℝ)) :
exponent ≤ (R - M) / A := by
apply (le_div_iff₀ positive).mpr
apply rate_inequality_of_log_bounds scale_positive a_rate multiplicity_rate rank_rate
filter_upwards [eventually_two_le_of_positive_log_rate scale_positive a_rate positive,
eventually_aspect_of_rate_gap a_positive b_positive scale_positive a_rate b_rate gap]
with n ha hab
exact finite_asi n ha hab
end MatrixMultiplication.RectangularRateLimit
end
end OAI
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