| import OAI.LinearAlgebra.MatrixMultiplication.Completion.FiniteRealization |
| import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.SquareWitness |
|
|
| namespace OAI |
|
|
| /-! Readable tensor completion and its finite arithmetic realization. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.CompletionSquareRates |
|
|
| open MatrixMultiplication.Foundation CompletionHierarchyWords CompletionFiniteRealization |
| open Filter StageHierarchyResources |
| open scoped BigOperators Topology Classical |
|
|
| variable {A X Y Z : Type} [Fintype A] [Fintype X] [Fintype Y] [Fintype Z] |
| variable (H : ReadableHierarchy A X Y Z) (counts : A → ℕ) |
| variable (hD : 0 < ∑ a, counts a) |
| variable (complete : Function.Injective (labelRecordOf H.labels H.depth)) |
| variable {T : Tensor ℂ X Y Z} {factors d D : ℕ} |
| variable (source : Tensor.PolynomialApproximation T (3 ^ factors) d D) |
| variable (supported : ∀ x y z, T x y z ≠ 0 → |
| ∃ a, H.x a = x ∧ H.y a = y ∧ H.z a = z) |
| variable (unit : ∀ a, T (H.x a) (H.y a) (H.z a) = 1) |
| variable (factors_pos : 0 < factors) |
|
|
| def positiveRealization (n : ℕ) : FiniteMMRealization := |
| realization H counts (n + 1) complete source supported unit factors_pos |
| (Nat.mul_pos (Nat.succ_pos n) hD) |
|
|
| def countMass (a : A) : ℝ := (counts a : ℝ) / (∑ a, counts a : ℕ) |
|
|
| private theorem normalize_log (K t population factors : ℕ) |
| (_ : 0 < t) (_ : 0 < population) (_ : 0 < factors) : |
| Real.log (K : ℝ) / ((factors * (t * population) : ℕ) : ℝ) = |
| (Real.log (K : ℝ) / ((t : ℝ) * (population : ℝ))) / (factors : ℝ) := by |
| simp only [Nat.cast_mul, div_div] |
| congr 1 |
| ring |
|
|
| theorem multiplicity_rate : |
| Tendsto (fun n => |
| Real.log ((positiveRealization H counts hD complete source supported unit factors_pos n).multiplicity : ℝ) / |
| ((positiveRealization H counts hD complete source supported unit factors_pos n).occurrences : ℝ)) |
| atTop (𝓝 (finiteEntropy (countMass counts) / factors)) := by |
| have hpop := (tendsto_log_exactWords_card_mul counts hD).comp (tendsto_add_atTop_nat 1) |
| have hnormalized := hpop.div_const (factors : ℝ) |
| change Tendsto _ atTop (𝓝 (finiteEntropy (countMass counts) / factors)) at hnormalized |
| apply hnormalized.congr |
| intro n |
| symm |
| change Real.log (copies H counts (n + 1) : ℝ) / |
| ((factors * blocks counts (n + 1) : ℕ) : ℝ) = _ |
| simp only [copies, blocks, CompletionExecution.realize_label_card H counts (n + 1) complete] |
| exact normalize_log _ (n + 1) (∑ a, counts a) factors (Nat.succ_pos n) hD factors_pos |
|
|
| theorem auxiliary_rate : |
| Tendsto (fun n => |
| Real.log ((positiveRealization H counts hD complete source supported unit factors_pos n).auxiliaryRank : ℝ) / |
| ((positiveRealization H counts hD complete source supported unit factors_pos n).occurrences : ℝ)) |
| atTop (𝓝 (finiteEntropy (countMass counts) / factors)) := by |
| have hrank := (tendsto_log_stageAuxiliaryBudget counts hD H.labels H.depth complete).comp |
| (tendsto_add_atTop_nat 1) |
| have hnormalized := hrank.div_const (factors : ℝ) |
| change Tendsto _ atTop (𝓝 (finiteEntropy (countMass counts) / factors)) at hnormalized |
| apply hnormalized.congr |
| intro n |
| symm |
| change Real.log ((CompletionExecution.realize H counts (n + 1)).execution.rankBound : ℝ) / |
| ((factors * blocks counts (n + 1) : ℕ) : ℝ) = _ |
| simp only [CompletionExecution.realize_rankBound, blocks] |
| exact normalize_log _ (n + 1) (∑ a, counts a) factors (Nat.succ_pos n) hD factors_pos |
|
|
| theorem volume_rate : |
| Tendsto (fun n => |
| Real.log ((positiveRealization H counts hD complete source supported unit factors_pos n).rates.volume : ℝ) / |
| ((positiveRealization H counts hD complete source supported unit factors_pos n).occurrences : ℝ)) |
| atTop (𝓝 (finiteEntropy (countMass counts) / factors)) := by |
| have hpop := (tendsto_log_exactWords_card_mul counts hD).comp (tendsto_add_atTop_nat 1) |
| have hnormalized := hpop.div_const (factors : ℝ) |
| change Tendsto _ atTop (𝓝 (finiteEntropy (countMass counts) / factors)) at hnormalized |
| apply hnormalized.congr |
| intro n |
| symm |
| change Real.log ((rows H counts (n + 1) * inner H counts (n + 1) * |
| columns H counts (n + 1) : ℕ) : ℝ) / |
| ((factors * blocks counts (n + 1) : ℕ) : ℝ) = _ |
| rw [volume_eq_copies H counts (n + 1) complete] |
| simp only [copies, blocks, CompletionExecution.realize_label_card H counts (n + 1) complete] |
| exact normalize_log _ (n + 1) (∑ a, counts a) factors (Nat.succ_pos n) hD factors_pos |
|
|
| theorem score_limit (β : ℝ) : |
| Tendsto (fun n => |
| (positiveRealization H counts hD complete source supported unit factors_pos n).rates.score β) |
| atTop (𝓝 (β / (3 * factors) * finiteEntropy (countMass counts))) := by |
| have hm := multiplicity_rate H counts hD complete source supported unit factors_pos |
| have ha := auxiliary_rate H counts hD complete source supported unit factors_pos |
| have hv := volume_rate H counts hD complete source supported unit factors_pos |
| have hl := (hm.sub ha).add (hv.const_mul (β / 3)) |
| have he : finiteEntropy (countMass counts) / factors - |
| finiteEntropy (countMass counts) / factors + |
| β / 3 * (finiteEntropy (countMass counts) / factors) = |
| β / (3 * factors) * finiteEntropy (countMass counts) := by ring |
| rw [he] at hl |
| apply hl.congr |
| intro n |
| simp only [FiniteConstructionRates.score, FiniteMMRealization.rates] |
| ring |
|
|
| include H hD complete source supported unit factors_pos in |
| theorem omega_lt_of_entropy_gap (β : ℝ) |
| (gap : Real.log 3 < β / (3 * factors) * finiteEntropy (countMass counts)) : |
| MatrixMultiplication.Foundation.Arithmetic.omega < β := |
| ComplexWitness.omega_lt_of_score_limit |
| (positiveRealization H counts hD complete source supported unit factors_pos) |
| (score_limit H counts hD complete source supported unit factors_pos β) gap |
|
|
| end MatrixMultiplication.CompletionSquareRates |
|
|
| end |
|
|
| end OAI |
|
|
