openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Entropy /Optimization.lean
| import Mathlib.Algebra.BigOperators.Field | |
| import Mathlib.Analysis.SpecialFunctions.BinaryEntropy | |
| import Mathlib.Analysis.SpecialFunctions.Pow.Real | |
| import Mathlib.Tactic.FieldSimp | |
| import Mathlib.Tactic.Ring | |
| namespace OAI | |
| /-! | |
| # Entropy optimization for finitely many sectors | |
| The normalized weights `q = A / (A + B)` and `1 - q = B / (A + B)` | |
| make the entropy-weighted geometric mean equal to `A + B`. This is the | |
| scalar identity used in Section 5.1 of the auxiliary-separation argument. | |
| The same calculation also applies to an arbitrary nonempty finite family. | |
| -/ | |
| open scoped BigOperators | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| /-- The logarithmic form of the entropy identity at the normalized weights. -/ | |
| theorem binary_entropy_log_partition {A B : ℝ} (hA : 0 < A) (hB : 0 < B) : | |
| Real.binEntropy (A / (A + B)) + | |
| Real.log A * (A / (A + B)) + | |
| Real.log B * (1 - A / (A + B)) = Real.log (A + B) := by | |
| have hsum : A + B ≠ 0 := ne_of_gt (add_pos hA hB) | |
| have hcomplement : 1 - A / (A + B) = B / (A + B) := by | |
| field_simp | |
| ring | |
| rw [Real.binEntropy, hcomplement, Real.log_inv, Real.log_inv, | |
| Real.log_div hA.ne' hsum, Real.log_div hB.ne' hsum] | |
| field_simp | |
| ring | |
| /-- Entropy compensates the weighted geometric mean at the normalized weights. -/ | |
| theorem binary_entropy_variational_identity {A B : ℝ} (hA : 0 < A) (hB : 0 < B) : | |
| Real.exp (Real.binEntropy (A / (A + B))) * | |
| A ^ (A / (A + B)) * B ^ (1 - A / (A + B)) = A + B := by | |
| rw [Real.rpow_def_of_pos hA, Real.rpow_def_of_pos hB, | |
| ← Real.exp_add, ← Real.exp_add, binary_entropy_log_partition hA hB, | |
| Real.exp_log (add_pos hA hB)] | |
| /-- The identity with an explicitly named optimizing proportion. -/ | |
| theorem binary_entropy_variational_identity_of_ratio {A B q : ℝ} | |
| (hA : 0 < A) (hB : 0 < B) (hq : q = A / (A + B)) : | |
| Real.exp (-q * Real.log q - (1 - q) * Real.log (1 - q)) * | |
| A ^ q * B ^ (1 - q) = A + B := by | |
| subst q | |
| have hentropy : | |
| -(A / (A + B)) * Real.log (A / (A + B)) - | |
| (1 - A / (A + B)) * Real.log (1 - A / (A + B)) = | |
| Real.binEntropy (A / (A + B)) := by | |
| simp only [Real.binEntropy, Real.log_inv] | |
| ring | |
| rw [hentropy] | |
| exact binary_entropy_variational_identity hA hB | |
| /-- The logarithmic entropy identity for positive weights on a finite set. -/ | |
| theorem finite_entropy_log_partition {ι : Type*} (s : Finset ι) (A : ι → ℝ) | |
| (hs : s.Nonempty) (hA : ∀ i ∈ s, 0 < A i) : | |
| -(∑ i ∈ s, (A i / (∑ j ∈ s, A j)) * Real.log (A i / (∑ j ∈ s, A j))) + | |
| (∑ i ∈ s, Real.log (A i) * (A i / (∑ j ∈ s, A j))) = | |
| Real.log (∑ j ∈ s, A j) := by | |
| have hsum : 0 < ∑ j ∈ s, A j := Finset.sum_pos hA hs | |
| calc | |
| _ = ∑ i ∈ s, (A i / (∑ j ∈ s, A j)) * Real.log (∑ j ∈ s, A j) := by | |
| rw [← Finset.sum_neg_distrib, ← Finset.sum_add_distrib] | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| rw [Real.log_div (hA i hi).ne' hsum.ne'] | |
| ring | |
| _ = (∑ i ∈ s, A i / (∑ j ∈ s, A j)) * Real.log (∑ j ∈ s, A j) := by | |
| rw [Finset.sum_mul] | |
| _ = _ := by rw [← Finset.sum_div, div_self hsum.ne', one_mul] | |
| /-- Normalizing any positive finite family attains its sum in the entropy formula. -/ | |
| theorem finite_entropy_variational_identity {ι : Type*} (s : Finset ι) (A : ι → ℝ) | |
| (hs : s.Nonempty) (hA : ∀ i ∈ s, 0 < A i) : | |
| Real.exp (-(∑ i ∈ s, (A i / (∑ j ∈ s, A j)) * | |
| Real.log (A i / (∑ j ∈ s, A j)))) * | |
| (∏ i ∈ s, (A i) ^ (A i / (∑ j ∈ s, A j))) = ∑ j ∈ s, A j := by | |
| have hsum : 0 < ∑ j ∈ s, A j := Finset.sum_pos hA hs | |
| have hprod : (∏ i ∈ s, (A i) ^ (A i / (∑ j ∈ s, A j))) = | |
| Real.exp (∑ i ∈ s, Real.log (A i) * (A i / (∑ j ∈ s, A j))) := by | |
| rw [Real.exp_sum] | |
| apply Finset.prod_congr rfl | |
| intro i hi | |
| exact Real.rpow_def_of_pos (hA i hi) _ | |
| rw [hprod, ← Real.exp_add, finite_entropy_log_partition s A hs hA, | |
| Real.exp_log hsum] | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |