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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 | import OAI.LinearAlgebra.MatrixMultiplication.Recovery.RecoveryGrowth
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Fintype.BigOperators
namespace OAI
/-! Finite entropy, rate estimates and ordered asymptotic limits. -/
noncomputable section
namespace MatrixMultiplication.FloorExponentialRates
open Filter
open scoped BigOperators Topology Classical
def count (rate : ℝ) (m : ℕ) : ℕ := ⌊Real.exp ((m : ℝ) * rate)⌋₊
theorem count_pos {rate : ℝ} (hr : 0 ≤ rate) (m : ℕ) : 0 < count rate m := by
have he : (1 : ℝ) ≤ Real.exp ((m : ℝ) * rate) :=
Real.one_le_exp (mul_nonneg (Nat.cast_nonneg _) hr)
exact lt_of_lt_of_le Nat.zero_lt_one (Nat.le_floor (by simpa only [Nat.cast_one] using he))
theorem log_count_bounds {rate : ℝ} (hr : 0 ≤ rate) (m : ℕ) :
(m : ℝ) * rate - Real.log 2 ≤ Real.log (count rate m : ℝ) ∧
Real.log (count rate m : ℝ) ≤ (m : ℝ) * rate := by
have hp : (0 : ℝ) < count rate m := Nat.cast_pos.mpr (count_pos hr m)
have hone : (1 : ℝ) ≤ count rate m := by exact_mod_cast count_pos hr m
have hround := Nat.lt_floor_add_one (Real.exp ((m : ℝ) * rate))
have hhalf : Real.exp ((m : ℝ) * rate) / 2 ≤ (count rate m : ℝ) := by
change Real.exp ((m : ℝ) * rate) < (count rate m : ℝ) + 1 at hround
linarith
constructor
· have h := Real.log_le_log (by positivity : 0 < Real.exp ((m : ℝ) * rate) / 2) hhalf
simpa only [Real.log_div (Real.exp_ne_zero _) (by norm_num : (2 : ℝ) ≠ 0),
Real.log_exp] using h
· have h := Real.log_le_log hp (Nat.floor_le (Real.exp_pos _).le)
simpa only [Real.log_exp] using h
theorem tendsto_log_count_div_nat {rate : ℝ} (hr : 0 ≤ rate) :
Tendsto (fun m : ℕ => Real.log (count rate m : ℝ) / (m : ℝ))
atTop (𝓝 rate) := by
have hinv : Tendsto (fun m : ℕ => Real.log 2 / (m : ℝ)) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop
have hlo : Tendsto (fun m : ℕ => rate - Real.log 2 / (m : ℝ)) atTop (𝓝 rate) := by
simpa only [sub_zero] using tendsto_const_nhds.sub hinv
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' hlo tendsto_const_nhds
· filter_upwards [eventually_gt_atTop (0 : ℕ)] with m hm
have hm' : (0 : ℝ) < m := Nat.cast_pos.mpr hm
have h := div_le_div_of_nonneg_right (log_count_bounds hr m).1 hm'.le
simpa only [sub_div, mul_div_cancel_left₀ rate hm'.ne'] using h
· filter_upwards [eventually_gt_atTop (0 : ℕ)] with m hm
have hm' : (0 : ℝ) < m := Nat.cast_pos.mpr hm
have h := div_le_div_of_nonneg_right (log_count_bounds hr m).2 hm'.le
simpa only [mul_div_cancel_left₀ rate hm'.ne'] using h
theorem tendsto_log_prod_count_div_nat {I : Type*} [Fintype I] (rate : I → ℝ)
(hr : ∀ i, 0 ≤ rate i) :
Tendsto (fun m : ℕ => Real.log ((∏ i, count (rate i) m : ℕ) : ℝ) / (m : ℝ))
atTop (𝓝 (∑ i, rate i)) := by
have h := tendsto_finsetSum Finset.univ (fun i _ => tendsto_log_count_div_nat (hr i))
apply h.congr
intro m
rw [Nat.cast_prod, Real.log_prod (s := Finset.univ)
(f := fun i => (count (rate i) m : ℝ))
(fun i _ => Nat.cast_ne_zero.mpr (Nat.ne_of_gt (count_pos (hr i) m)))]
simp only [div_eq_mul_inv, Finset.sum_mul]
abbrev Labels {I : Type*} (rate : I → ℝ) (m : ℕ) := ∀ i, Fin (count (rate i) m)
theorem card_labels {I : Type*} [Fintype I] (rate : I → ℝ) (m : ℕ) :
Fintype.card (Labels rate m) = ∏ i, count (rate i) m := by
simp only [Labels, Fintype.card_pi, Fintype.card_fin]
theorem card_labels_pos {I : Type*} [Fintype I] (rate : I → ℝ)
(hr : ∀ i, 0 ≤ rate i) (m : ℕ) : 0 < Fintype.card (Labels rate m) := by
rw [card_labels]
exact Finset.prod_pos fun i _ => count_pos (hr i) m
theorem tendsto_log_card_labels_div_nat {I : Type*} [Fintype I] (rate : I → ℝ)
(hr : ∀ i, 0 ≤ rate i) :
Tendsto (fun m : ℕ => Real.log (Fintype.card (Labels rate m) : ℝ) / (m : ℝ))
atTop (𝓝 (∑ i, rate i)) := by
simpa only [card_labels] using tendsto_log_prod_count_div_nat rate hr
end MatrixMultiplication.FloorExponentialRates
end
end OAI
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