File size: 3,245 Bytes
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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.PairMarginals
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldPairConditionalEntropy
open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory
open AllFieldNativeCapacity AllFieldPairMarginals
open scoped BigOperators
theorem entropy_sub_eq_conditional_sum
{A B C : Type*} [Fintype A] [Fintype B] [Fintype C]
(p : FiniteLaw A) (q : FiniteLaw B) (m : FiniteLaw C)
(f : A → C) (g : B → C) (hp : p.map f = m) (hq : q.map g = m) :
finiteEntropy p.mass - finiteEntropy q.mass =
∑ k, m.mass k * (finiteEntropy (p.conditional f k).mass -
finiteEntropy (q.conditional g k).mass) := by
rw [p.entropy_eq_map_add_conditional f, q.entropy_eq_map_add_conditional g, hp, hq]
simp only [mul_sub, Finset.sum_sub_distrib]
ring
def stageASplitWeight (g : Shape) (hg : g ∈ positiveInitial) (side : Fin 3) :
Fin (below g).length → Fin 17 :=
fun j => ⟨(below g).get j side,
Nat.lt_succ_of_le ((below_le (List.get_mem _ _) side).trans
((positiveInitial_shape_spec g hg).1 side))⟩
def stageBSplitWeight (t : Shape) (ht : t ∈ positiveSecond) (side : Fin 3) :
Fin (below t).length → Fin 17 :=
fun j => ⟨(below t).get j side,
Nat.lt_succ_of_le ((below_le (List.get_mem _ _) side).trans
((positiveSecond_shape_spec t ht).1 side))⟩
theorem stageASplitLaw_ownWeight_marginal (g : Shape) (hg : g ∈ positiveInitial)
(side : Fin 3) :
(stageASplitLaw g hg).map (stageASplitWeight g hg side) =
stageAMarginalLaw g hg side := rfl
theorem stageBSplitLaw_ownWeight_marginal (t : Shape) (ht : t ∈ positiveSecond)
(side : Fin 3) :
(stageBSplitLaw t ht).map (stageBSplitWeight t ht side) =
stageBMarginalLaw t ht side := rfl
theorem stageA_entropy_sub_pair_eq_conditional_sum
(g : Shape) (hg : g ∈ positiveInitial) (side : Fin 3) :
finiteEntropy (stageASplitLaw g hg).mass -
finiteEntropy (stageAPairLaw g hg side).mass =
∑ k, (stageAMarginalLaw g hg side).mass k *
(finiteEntropy ((stageASplitLaw g hg).conditional
(stageASplitWeight g hg side) k).mass -
finiteEntropy ((stageAPairLaw g hg side).conditional
(fun ij => halfWeightCode ij.1) k).mass) :=
entropy_sub_eq_conditional_sum _ _ _ _ _
(stageASplitLaw_ownWeight_marginal g hg side)
(stageAPairLaw_leftWeight_marginal g hg side)
theorem stageB_entropy_sub_pair_eq_conditional_sum
(t : Shape) (ht : t ∈ positiveSecond) (side : Fin 3) :
finiteEntropy (stageBSplitLaw t ht).mass -
finiteEntropy (stageBPairLaw t ht side).mass =
∑ k, (stageBMarginalLaw t ht side).mass k *
(finiteEntropy ((stageBSplitLaw t ht).conditional
(stageBSplitWeight t ht side) k).mass -
finiteEntropy ((stageBPairLaw t ht side).conditional
(fun ij => statisticWeightCode ij.1) k).mass) :=
entropy_sub_eq_conditional_sum _ _ _ _ _
(stageBSplitLaw_ownWeight_marginal t ht side)
(stageBPairLaw_leftWeight_marginal t ht side)
end MatrixMultiplication.AllFieldPairConditionalEntropy
end
end OAI
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