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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 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.Source
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldSourceRates
open AllFieldSource Filter
open scoped Topology
private theorem tendsto_log_linear_div_nat (a : ℝ) (ha : 0 < a) :
Tendsto (fun N : ℕ => Real.log (a * N + 1) / (N : ℝ)) atTop (𝓝 0) := by
have hnat : Tendsto (fun N : ℕ => (N : ℝ)) atTop atTop :=
tendsto_natCast_atTop_atTop
have harg : Tendsto (fun N : ℕ => a * N + 1) atTop atTop :=
(hnat.const_mul_atTop ha).atTop_add_nonneg (fun _ => zero_le_one)
have hlog : Tendsto (fun N : ℕ => Real.log (a * N + 1) / (a * N + 1))
atTop (𝓝 0) := by
simpa only [Function.comp_def, id_eq] using
Real.isLittleO_log_id_atTop.tendsto_div_nhds_zero.comp harg
have hinv : Tendsto (fun N : ℕ => (1 : ℝ) / N) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop hnat
have hratio : Tendsto (fun N : ℕ => (a * N + 1) / (N : ℝ)) atTop (𝓝 a) := by
have hsum : Tendsto (fun N : ℕ => a + 1 / (N : ℝ)) atTop (𝓝 a) := by
simpa only [add_zero] using hinv.const_add a
apply hsum.congr'
filter_upwards [eventually_ne_atTop (0 : ℕ)] with N hN
have hN' : (N : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hN
field_simp [hN']
have hprod : Tendsto (fun N : ℕ =>
(Real.log (a * N + 1) / (a * N + 1)) * ((a * N + 1) / (N : ℝ)))
atTop (𝓝 0) := by
simpa only [zero_mul] using hlog.mul hratio
apply hprod.congr
intro N
have harg' : a * (N : ℝ) + 1 ≠ 0 := by positivity
exact div_mul_div_cancel₀ harg'
theorem tendsto_log_polynomial_overhead (K : ℕ) (hK : 0 < K) :
Tendsto (fun N : ℕ =>
Real.log (((3 * (8 * K * N) + 1) ^ 2 : ℕ) : ℝ) / ((K : ℝ) * N))
atTop (𝓝 0) := by
have hK' : (0 : ℝ) < K := Nat.cast_pos.mpr hK
have hbase := tendsto_log_linear_div_nat (3 * (8 * (K : ℝ))) (by positivity)
have hscaled : Tendsto (fun N : ℕ =>
(2 * (Real.log (3 * (8 * (K : ℝ)) * N + 1) / (N : ℝ))) / (K : ℝ))
atTop (𝓝 0) := by
simpa only [mul_zero, zero_div] using (hbase.const_mul 2).div_const (K : ℝ)
apply hscaled.congr
intro N
have harg : ((3 * (8 * K * N) + 1 : ℕ) : ℝ) =
3 * (8 * (K : ℝ)) * N + 1 := by push_cast; ring
rw [Nat.cast_pow, Real.log_pow, harg]
norm_num only [Nat.cast_ofNat]
ring
theorem tendsto_log_exponential_source (K : ℕ) (hK : 0 < K) :
Tendsto (fun N : ℕ =>
Real.log ((7 ^ (8 * K * N) : ℕ) : ℝ) / ((K : ℝ) * N))
atTop (𝓝 (8 * Real.log 7)) := by
have hK' : (K : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hK)
apply tendsto_const_nhds.congr'
filter_upwards [eventually_ne_atTop (0 : ℕ)] with N hN
have hN' : (N : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hN
rw [Nat.cast_pow, Real.log_pow]
push_cast
field_simp [hK', hN']
theorem log_rankBudget (Copies : Type*) [Fintype Copies] (K N : ℕ)
(hcopies : 0 < Fintype.card Copies) :
Real.log (rankBudget Copies K N : ℝ) =
Real.log (Fintype.card Copies : ℝ) +
Real.log (((3 * (8 * K * N) + 1) ^ 2 : ℕ) : ℝ) +
Real.log ((7 ^ (8 * K * N) : ℕ) : ℝ) := by
have hc : (Fintype.card Copies : ℝ) ≠ 0 :=
Nat.cast_ne_zero.mpr (Nat.ne_of_gt hcopies)
have hp : (((3 * (8 * K * N) + 1) ^ 2 : ℕ) : ℝ) ≠ 0 := by positivity
have he : ((7 ^ (8 * K * N) : ℕ) : ℝ) ≠ 0 := by positivity
simp only [rankBudget, Nat.cast_mul]
rw [Real.log_mul hc (mul_ne_zero hp he), Real.log_mul hp he, ← add_assoc]
theorem tendsto_log_rankBudget (Copies : ℕ → Type*) [∀ N, Fintype (Copies N)]
(K : ℕ) (hK : 0 < K)
(hcopies : ∀ N, 0 < Fintype.card (Copies N))
(hcopyRate : Tendsto (fun N : ℕ =>
Real.log (Fintype.card (Copies N) : ℝ) / (N : ℝ)) atTop (𝓝 0)) :
Tendsto (fun N : ℕ =>
Real.log (rankBudget (Copies N) K N : ℝ) / ((K : ℝ) * N))
atTop (𝓝 (8 * Real.log 7)) := by
have hc : Tendsto (fun N : ℕ =>
Real.log (Fintype.card (Copies N) : ℝ) / ((K : ℝ) * N)) atTop (𝓝 0) := by
have h : Tendsto (fun N : ℕ =>
(Real.log (Fintype.card (Copies N) : ℝ) / (N : ℝ)) / (K : ℝ))
atTop (𝓝 0) := by
simpa only [zero_div] using hcopyRate.div_const (K : ℝ)
apply h.congr
intro N
ring
have hsum := (hc.add (tendsto_log_polynomial_overhead K hK)).add
(tendsto_log_exponential_source K hK)
simp only [zero_add] at hsum
apply hsum.congr
intro N
rw [log_rankBudget (Copies N) K N (hcopies N), add_div, add_div]
end MatrixMultiplication.AllFieldSourceRates
end
end OAI
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