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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 | import Mathlib
import OAI.LinearAlgebra.MatrixFields.Extraction.ConditionalLabels
namespace OAI
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
noncomputable section
namespace MatrixMultiplication.ConditionalLabels
open MatrixMultiplication.Foundation Filter
open scoped BigOperators Topology Classical
universe u v w
variable {A : Type u} [Fintype A] {Label : ℕ → Type v}
[∀ n, Fintype (Label n)]
theorem refinementEntropy_eq_prefix_difference (p : FiniteLaw A)
(labels : ∀ n, A → Label n) (depth n : ℕ) (hn : n < depth) :
(FiniteLabelHierarchy.ofLawLabels p labels depth).refinementEntropy n =
finiteEntropy (p.map (labelRecordOf labels (n + 1))).mass -
finiteEntropy (p.map (labelRecordOf labels n)).mass := by
have h := (FiniteLabelHierarchy.ofLawLabels p labels depth).prefixLaw_entropy_succ
n (Nat.succ_le_of_lt hn)
rw [FiniteLabelHierarchy.ofLawLabels_prefixLaw_mass,
FiniteLabelHierarchy.ofLawLabels_prefixLaw_mass] at h
linarith
theorem tendsto_map_entropy {J : Type w} {filter : Filter J}
{p : J → FiniteLaw A} {q : FiniteLaw A}
(mass : ∀ a, Tendsto (fun j => (p j).mass a) filter (𝓝 (q.mass a)))
{B : Type*} [Fintype B] (label : A → B) :
Tendsto (fun j => finiteEntropy ((p j).map label).mass) filter
(𝓝 (finiteEntropy (q.map label).mass)) := by
apply FiniteLaw.entropy_tendsto
intro b
simp only [FiniteLaw.map_mass]
apply tendsto_finsetSum
intro a _
by_cases h : label a = b
· simpa only [h, ite_true] using mass a
· simp only [h, ite_false]
exact tendsto_const_nhds
theorem tendsto_lawPairingRate {J : Type w} {filter : Filter J}
{p : J → FiniteLaw A} {q : FiniteLaw A}
(mass : ∀ a, Tendsto (fun j => (p j).mass a) filter (𝓝 (q.mass a)))
(labels : ∀ n, A → Label n) (depth : ℕ)
(reader : Fin depth → ReaderPair) (pair : ReaderPair) :
Tendsto (fun j => lawPairingRate (p j) labels depth reader pair) filter
(𝓝 (lawPairingRate q labels depth reader pair)) := by
unfold lawPairingRate pairingRate
apply tendsto_finsetSum
intro n _
by_cases h : reader n = pair
· simp only [h, ite_true,
refinementEntropy_eq_prefix_difference _ labels depth n.val n.isLt]
exact (tendsto_map_entropy mass (labelRecordOf labels (n.val + 1))).sub
(tendsto_map_entropy mass (labelRecordOf labels n.val))
· simp only [h, ite_false]
exact tendsto_const_nhds
theorem exists_rational_law_sequence (p : FiniteLaw A) :
∃ q : ℕ → RationalLaw A,
∀ a, Tendsto (fun n => (q n).toFiniteLaw.mass a) atTop (𝓝 (p.mass a)) := by
have hexists (n : ℕ) := p.exists_rational_approximation
(1 / ((n : ℝ) + 1)) (by positivity)
choose q _support close using hexists
refine ⟨q, fun a => ?_⟩
have hdifference : Tendsto
(fun n => (q n).toFiniteLaw.mass a - p.mass a) atTop (𝓝 0) := by
apply squeeze_zero_norm _ tendsto_one_div_add_atTop_nhds_zero_nat
intro n
simpa only [Real.norm_eq_abs, RationalLaw.toFiniteLaw_mass] using (close n a).le
simpa only [sub_add_cancel, zero_add] using hdifference.add_const (p.mass a)
theorem exists_rational_with_strict_margins (p : FiniteLaw A)
(labels : ∀ n, A → Label n) (depth : ℕ) (reader : Fin depth → ReaderPair)
{k C target : ℝ}
(positive : 0 < lawPairingRate p labels depth reader .xy +
lawPairingRate p labels depth reader .xz)
(aspect : k * (lawPairingRate p labels depth reader .xy +
lawPairingRate p labels depth reader .xz) <
2 * lawPairingRate p labels depth reader .yz)
(cost : 2 * C < target * (lawPairingRate p labels depth reader .xy +
lawPairingRate p labels depth reader .xz)) :
∃ q : RationalLaw A,
0 < lawPairingRate q.toFiniteLaw labels depth reader .xy +
lawPairingRate q.toFiniteLaw labels depth reader .xz ∧
k * (lawPairingRate q.toFiniteLaw labels depth reader .xy +
lawPairingRate q.toFiniteLaw labels depth reader .xz) <
2 * lawPairingRate q.toFiniteLaw labels depth reader .yz ∧
2 * C < target * (lawPairingRate q.toFiniteLaw labels depth reader .xy +
lawPairingRate q.toFiniteLaw labels depth reader .xz) := by
obtain ⟨q, hq⟩ := exists_rational_law_sequence p
have hincident := (tendsto_lawPairingRate hq labels depth reader .xy).add
(tendsto_lawPairingRate hq labels depth reader .xz)
have hhidden := tendsto_lawPairingRate hq labels depth reader .yz
have hpos := hincident.eventually_const_lt positive
have haspect := (hincident.const_mul k).eventually_lt (hhidden.const_mul 2) aspect
have hcost := (hincident.const_mul target).eventually_const_lt cost
obtain ⟨n, hnpos, hnaspect, hncost⟩ := (hpos.and (haspect.and hcost)).exists
exact ⟨q n, hnpos, hnaspect, hncost⟩
end MatrixMultiplication.ConditionalLabels
end
end MatrixAllFields
end OAI
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