| import OAI.LinearAlgebra.MatrixMultiplication.FieldParameters.Weights |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexFiniteEntropy |
|
|
| namespace OAI |
|
|
| /-! Fixed rational distributions and their entropy and capacity formulas. -/ |
|
|
| namespace MatrixMultiplication.AllFieldParameters |
|
|
| open MatrixMultiplication.Foundation |
| open scoped BigOperators |
|
|
| noncomputable section |
|
|
| def homogeneousEntropy {A : Type*} [Fintype A] (p : A → ℝ) : ℝ := |
| (∑ a, p a) * Real.log (∑ a, p a) + finiteEntropy p |
|
|
| theorem homogeneousEntropy_normalized {A : Type*} [Fintype A] |
| (p : A → ℝ) (hp : ∑ a, p a = 1) : |
| homogeneousEntropy p = finiteEntropy p := by |
| simp [homogeneousEntropy, hp] |
|
|
| theorem entropyTerm_divide (x m : ℝ) : |
| entropyTerm (x / m) = entropyTerm x / m + (x / m) * Real.log m := by |
| by_cases hx : x = 0 |
| · simp [hx] |
| by_cases hm : m = 0 |
| · simp [hm] |
| rw [entropyTerm, Real.log_div hx hm] |
| unfold entropyTerm |
| ring |
|
|
| theorem homogeneousEntropy_eq_mass_mul {A : Type*} [Fintype A] |
| (p : A → ℝ) (hm : ∑ a, p a ≠ 0) : |
| homogeneousEntropy p = |
| (∑ a, p a) * finiteEntropy (fun a => p a / ∑ b, p b) := by |
| simp only [finiteEntropy, entropyTerm_divide, Finset.sum_add_distrib, |
| ← Finset.sum_div, ← Finset.sum_mul] |
| unfold homogeneousEntropy finiteEntropy |
| rw [div_self hm] |
| field_simp |
| ring |
|
|
| theorem homogeneousEntropy_zero {A : Type*} [Fintype A] : |
| homogeneousEntropy (fun _ : A => 0) = 0 := by |
| simp [homogeneousEntropy, finiteEntropy] |
|
|
| theorem homogeneousEntropy_expansion {A : Type*} [Fintype A] (p : A → ℝ) : |
| homogeneousEntropy p = |
| (∑ a, p a) * Real.log (∑ a, p a) - ∑ a, p a * Real.log (p a) := by |
| simp only [homogeneousEntropy, finiteEntropy, entropyTerm, neg_mul, |
| Finset.sum_neg_distrib, sub_eq_add_neg] |
|
|
| end |
|
|
| end MatrixMultiplication.AllFieldParameters |
|
|
| end OAI |
|
|
