| import OAI.LinearAlgebra.MatrixMultiplication.Completion.Laws |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexHierarchyEntropy |
|
|
| namespace OAI |
|
|
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.DualEntropyHelpers |
|
|
| open MatrixMultiplication.Foundation |
| open scoped BigOperators |
|
|
| variable {A B C X : Type*} [Fintype A] [Fintype B] [Fintype C] [Fintype X] |
|
|
| theorem map_comp_mass (p : FiniteLaw A) (f : A → B) (g : B → C) : |
| ((p.map f).map g).mass = (p.map (fun a => g (f a))).mass := by |
| classical |
| funext c |
| simp only [FiniteLaw.map_mass] |
| calc |
| (∑ b, if g b = c then ∑ a, if f a = b then p.mass a else 0 else 0) = |
| ∑ b, ∑ a, if f a = b then (if g b = c then p.mass a else 0) else 0 := by |
| apply Finset.sum_congr rfl |
| intro b _ |
| by_cases h : g b = c <;> simp [h] |
| _ = ∑ a, ∑ b, if f a = b then (if g b = c then p.mass a else 0) else 0 := |
| Finset.sum_comm |
| _ = ∑ a, if g (f a) = c then p.mass a else 0 := by |
| apply Finset.sum_congr rfl |
| intro a _ |
| simp |
|
|
| theorem deterministic_pair_entropy (p : FiniteLaw A) (f : A → B) (g : B → C) : |
| finiteEntropy (p.map (fun a => (f a, g (f a)))).mass = |
| finiteEntropy (p.map f).mass := by |
| rw [← map_comp_mass p f (fun b => (b, g b))] |
| exact (p.map f).map_entropy_of_injective _ (fun _ _ h => congrArg Prod.fst h) |
|
|
| theorem pair_entropy (p : FiniteLaw A) (f : A → B) (g : A → C) : |
| finiteEntropy (p.map (fun a => (f a, g a))).mass = |
| finiteEntropy (p.map f).mass + |
| ∑ b, (p.map f).mass b * finiteEntropy ((p.conditional f b).map g).mass := by |
| have hm : ((p.map f).joint (fun b => (p.conditional f b).map g)).mass = |
| (p.map (fun a => (f a, g a))).mass := by |
| funext bc |
| exact p.map_mass_mul_conditional_map f g bc.1 bc.2 |
| rw [← hm, FiniteLaw.joint_entropy] |
|
|
| theorem conditional_independence_entropy (p : FiniteLaw A) |
| (f : A → B) (g : A → C) (x : A → X) |
| (independent : ∀ b, 0 < (p.map f).mass b → ∀ c xx, |
| ((p.conditional f b).map (fun a => (g a, x a))).mass (c, xx) = |
| ((p.conditional f b).map g).mass c * |
| ((p.conditional f b).map x).mass xx) : |
| finiteEntropy (p.map (fun a => ((f a, g a), x a))).mass - |
| finiteEntropy (p.map (fun a => (f a, g a))).mass = |
| finiteEntropy (p.map (fun a => (f a, x a))).mass - |
| finiteEntropy (p.map f).mass := by |
| classical |
| have hassoc : finiteEntropy (p.map (fun a => ((f a, g a), x a))).mass = |
| finiteEntropy (p.map (fun a => (f a, (g a, x a)))).mass := by |
| have h := (p.map (fun a => ((f a, g a), x a))).map_entropy_of_injective |
| (Equiv.prodAssoc B C X) (Equiv.prodAssoc B C X).injective |
| rw [map_comp_mass] at h |
| exact h.symm |
| have hcond (b : B) : |
| (p.map f).mass b * |
| finiteEntropy ((p.conditional f b).map (fun a => (g a, x a))).mass = |
| (p.map f).mass b * |
| (finiteEntropy ((p.conditional f b).map g).mass + |
| finiteEntropy ((p.conditional f b).map x).mass) := by |
| by_cases hb : (p.map f).mass b = 0 |
| · simp only [hb, zero_mul] |
| · have hpos : 0 < (p.map f).mass b := |
| lt_of_le_of_ne ((p.map f).nonneg b) (Ne.symm hb) |
| have hm : ((p.conditional f b).map (fun a => (g a, x a))).mass = |
| (((p.conditional f b).map g).joint |
| (fun _ => (p.conditional f b).map x)).mass := by |
| funext cx |
| exact independent b hpos cx.1 cx.2 |
| rw [hm, CompletionLaws.independent_entropy] |
| rw [hassoc, pair_entropy p f (fun a => (g a, x a)), |
| pair_entropy p f g, pair_entropy p f x] |
| simp_rw [hcond] |
| simp only [mul_add, Finset.sum_add_distrib] |
| ring |
|
|
| end MatrixMultiplication.DualEntropyHelpers |
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|
| end |
|
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| end OAI |
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|
