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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.CW75ReorderedBalanced
import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.CW75ReorderedFiniteRates
import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalRateBridge
import OAI.LinearAlgebra.MatrixMultiplication.Rectangular.Witness
namespace OAI
/-! Explicit complex square and rectangular matrix multiplication bounds. -/
noncomputable section
namespace MatrixMultiplication.CW75ReorderedBalancedRates
open MatrixMultiplication.Foundation Filter CW75LabelHierarchy
open ConditionalLabels ComplexWitness CW75ReorderedBalanced
open scoped BigOperators Topology Classical
variable (counts : Scalar → ℕ) (hD : 0 < ∑ a, counts a)
(p : FiniteLaw Scalar)
(mass : ∀ a, p.mass a = (counts a : ℝ) / (∑ b, counts b : ℕ))
include hD mass
theorem tendsto_log_rows :
Tendsto (fun t => Real.log (CW75ReorderedFinite.rows counts t : ℝ) /
((t : ℝ) * (∑ a, counts a : ℕ))) atTop
(𝓝 (lawPairingRate p curveLabels 5 curveReaderPair .xz)) := by
have h := tendsto_log_stagePairingSize_law counts hD p mass
curveLabels 5 curveReaderPair .xz
apply h.congr
intro t
rw [(CW75ReorderedFinite.dimensions_eq_pairingSize counts t).1]
theorem tendsto_log_inner :
Tendsto (fun t => Real.log (CW75ReorderedFinite.inner counts t : ℝ) /
((t : ℝ) * (∑ a, counts a : ℕ))) atTop
(𝓝 (lawPairingRate p curveLabels 5 curveReaderPair .xy)) := by
have h := tendsto_log_stagePairingSize_law counts hD p mass
curveLabels 5 curveReaderPair .xy
apply h.congr
intro t
rw [(CW75ReorderedFinite.dimensions_eq_pairingSize counts t).2.1]
theorem tendsto_log_columns :
Tendsto (fun t => Real.log (CW75ReorderedFinite.columns counts t : ℝ) /
((t : ℝ) * (∑ a, counts a : ℕ))) atTop
(𝓝 (lawPairingRate p curveLabels 5 curveReaderPair .yz)) := by
have h := tendsto_log_stagePairingSize_law counts hD p mass
curveLabels 5 curveReaderPair .yz
apply h.congr
intro t
rw [(CW75ReorderedFinite.dimensions_eq_pairingSize counts t).2.2]
theorem tendsto_log_witness_outer :
Tendsto (fun t => Real.log ((witness counts t).outer : ℝ) /
((t : ℝ) * (∑ a, counts a : ℕ))) atTop
(𝓝 (lawPairingRate p curveLabels 5 curveReaderPair .xy +
lawPairingRate p curveLabels 5 curveReaderPair .xz)) := by
have h := (tendsto_log_rows counts hD p mass).add
(tendsto_log_inner counts hD p mass)
rw [add_comm (lawPairingRate p curveLabels 5 curveReaderPair .xz)] at h
apply h.congr
intro t
rw [witness_outer, Nat.cast_mul, Real.log_mul
(Nat.cast_ne_zero.mpr (Nat.ne_of_gt (CW75ReorderedFinite.rows_pos counts t)))
(Nat.cast_ne_zero.mpr (Nat.ne_of_gt (CW75ReorderedFinite.inner_pos counts t))), add_div]
theorem tendsto_log_witness_inner :
Tendsto (fun t => Real.log ((witness counts t).inner : ℝ) /
((t : ℝ) * (∑ a, counts a : ℕ))) atTop
(𝓝 (2 * lawPairingRate p curveLabels 5 curveReaderPair .yz)) := by
simpa only [witness_inner, Nat.cast_pow, Real.log_pow, Nat.cast_ofNat,
mul_div_assoc] using (tendsto_log_columns counts hD p mass).const_mul 2
theorem tendsto_log_witness_multiplicity :
Tendsto (fun t => Real.log ((witness counts t).multiplicity : ℝ) /
((t : ℝ) * (∑ a, counts a : ℕ))) atTop
(𝓝 (2 * finiteEntropy p.mass)) := by
simpa only [witness_multiplicity, Nat.cast_pow, Real.log_pow, Nat.cast_ofNat,
mul_div_assoc] using
(CW75ReorderedFiniteRates.tendsto_log_copies counts hD p mass).const_mul 2
theorem tendsto_log_witness_rank :
Tendsto (fun t => Real.log ((witness counts t).rank : ℝ) /
((t : ℝ) * (∑ a, counts a : ℕ))) atTop
(𝓝 (6 * Real.log 3 + 2 * finiteEntropy p.mass)) := by
have h := (CW75ReorderedFiniteRates.tendsto_log_rankBudget counts hD p mass).const_mul 2
have hcost : 2 * (3 * Real.log 3 + finiteEntropy p.mass) =
6 * Real.log 3 + 2 * finiteEntropy p.mass := by ring
simpa only [witness_rank, Nat.cast_pow, Real.log_pow, Nat.cast_ofNat,
mul_div_assoc, hcost] using h
omit mass
theorem scale_eventually_pos :
∀ᶠ t : ℕ in atTop, 0 < (t : ℝ) * (∑ a, counts a : ℕ) := by
filter_upwards [eventually_ge_atTop 1] with t ht
exact mul_pos (Nat.cast_pos.mpr (by omega)) (Nat.cast_pos.mpr hD)
include mass
theorem omega_lt_of_fixed_count_law (k target : ℝ)
(k_nonneg : 0 ≤ k)
(incident_positive : 0 < lawPairingRate p curveLabels 5 curveReaderPair .xy +
lawPairingRate p curveLabels 5 curveReaderPair .xz)
(aspect_gap : k * (lawPairingRate p curveLabels 5 curveReaderPair .xy +
lawPairingRate p curveLabels 5 curveReaderPair .xz) <
2 * lawPairingRate p curveLabels 5 curveReaderPair .yz)
(cost_gap : 6 * Real.log 3 < target *
(lawPairingRate p curveLabels 5 curveReaderPair .xy +
lawPairingRate p curveLabels 5 curveReaderPair .xz)) :
Arithmetic.rectangularOmega ℂ k < target :=
RectangularWitness.omega_lt_of_balanced_family (witness counts)
(fun t => (t : ℝ) * (∑ a, counts a : ℕ)) k target
(lawPairingRate p curveLabels 5 curveReaderPair .yz)
(lawPairingRate p curveLabels 5 curveReaderPair .xy +
lawPairingRate p curveLabels 5 curveReaderPair .xz)
(finiteEntropy p.mass) k_nonneg incident_positive aspect_gap cost_gap
(scale_eventually_pos counts hD)
(tendsto_log_witness_outer counts hD p mass)
(tendsto_log_witness_inner counts hD p mass)
(tendsto_log_witness_multiplicity counts hD p mass)
(tendsto_log_witness_rank counts hD p mass)
end MatrixMultiplication.CW75ReorderedBalancedRates
end
end OAI
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