| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalLabels |
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| namespace OAI |
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| /-! Rectangular matrix multiplication algorithms and their asymptotic exponents. -/ |
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| noncomputable section |
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| namespace MatrixMultiplication.RectangularRateLimit |
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| open Filter |
| open scoped Topology |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| end MatrixMultiplication.RectangularRateLimit |
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| end |
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| end OAI |
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