| import OAI.LinearAlgebra.MatrixMultiplication.Duality.FiniteRealization |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalRateBridge |
|
|
| namespace OAI |
|
|
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.DualFiniteRealization |
|
|
| open MatrixMultiplication.Foundation CompletionHierarchyWords CompletionExecution |
| open CompletionFiniteRealization StageHierarchyResources ConditionalLabels Filter |
| 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 (p : FiniteLaw A) |
| variable (mass : ∀ a, p.mass a = (counts a : ℝ) / (∑ b, counts b : ℕ)) |
|
|
| def pairRate (pair : ReaderPair) : ℝ := |
| lawPairingRate p H.labels H.depth (fun n => H.pair n.val) pair |
|
|
| include hD complete mass in |
| omit [Fintype X] [Fintype Y] [Fintype Z] in |
| theorem copies_limit : |
| Tendsto (fun t : ℕ => Real.log (copies H counts t : ℝ) / |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop (𝓝 (finiteEntropy p.mass)) := by |
| have hmass : (fun a => (counts a : ℝ) / (∑ b, counts b : ℕ)) = p.mass := |
| (funext mass).symm |
| simpa only [copies, realize_label_card H counts _ complete, hmass] using |
| tendsto_log_exactWords_card_mul counts hD |
|
|
| include hD complete mass in |
| omit [Fintype X] [Fintype Y] [Fintype Z] in |
| theorem auxiliary_limit : |
| Tendsto (fun t : ℕ => Real.log ((realize H counts t).execution.rankBound : ℝ) / |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop (𝓝 (finiteEntropy p.mass)) := by |
| have hmass : (fun a => (counts a : ℝ) / (∑ b, counts b : ℕ)) = p.mass := |
| (funext mass).symm |
| simpa only [realize_rankBound, hmass] using |
| tendsto_log_stageAuxiliaryBudget counts hD H.labels H.depth complete |
|
|
| include hD in |
| theorem source_rank_limit (R : ℕ) : |
| Tendsto (fun t : ℕ => |
| Real.log ((R ^ CompletionFiniteRealization.blocks counts t : ℕ) : ℝ) / |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop (𝓝 (Real.log R)) := by |
| have hD' : ((∑ a, counts a : ℕ) : ℝ) ≠ 0 := |
| Nat.cast_ne_zero.mpr (Nat.ne_of_gt hD) |
| apply tendsto_const_nhds.congr' |
| filter_upwards [eventually_ne_atTop (0 : ℕ)] with t ht |
| have htD : (t : ℝ) * (∑ a, counts a : ℕ) ≠ 0 := |
| mul_ne_zero (Nat.cast_ne_zero.mpr ht) hD' |
| rw [Nat.cast_pow, Real.log_pow, CompletionFiniteRealization.blocks, Nat.cast_mul] |
| exact (mul_div_cancel_left₀ (Real.log R) htD).symm |
| |
| include hD complete mass in |
| omit [Fintype X] [Fintype Y] [Fintype Z] in |
| theorem rank_budget_limit {R : ℕ} (hR : 0 < R) : |
| Tendsto (fun t : ℕ => Real.log (rankBudget H counts t R : ℝ) / |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop |
| (𝓝 (Real.log R + finiteEntropy p.mass)) := by |
| apply ((source_rank_limit counts hD R).add |
| (auxiliary_limit H counts hD complete p mass)).congr |
| intro t |
| have hs : ((R ^ CompletionFiniteRealization.blocks counts t : ℕ) : ℝ) ≠ 0 := |
| Nat.cast_ne_zero.mpr (pow_ne_zero _ (Nat.ne_of_gt hR)) |
| have ha : ((realize H counts t).execution.rankBound : ℝ) ≠ 0 := |
| Nat.cast_ne_zero.mpr (Nat.ne_of_gt (auxiliaryRank_pos H counts t)) |
| rw [rankBudget, Nat.cast_mul, Real.log_mul hs ha, add_div] |
| |
| variable {T : Tensor ℂ X Y Z} {R d D : ℕ} |
| variable (source : Tensor.PolynomialApproximation T R d D) (hR : 0 < R) |
| 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) |
| |
| include hD mass in |
| theorem witness_outer_limit : |
| Tendsto (fun t : ℕ => |
| Real.log ((witness H counts t complete source hR supported unit).outer : ℝ) / |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop |
| (𝓝 (pairRate H p .xy + pairRate H p .xz)) := by |
| have hi := tendsto_log_stagePairingSize_law counts hD p mass H.labels H.depth |
| (fun n => H.pair n.val) .xy |
| have hr := tendsto_log_stagePairingSize_law counts hD p mass H.labels H.depth |
| (fun n => H.pair n.val) .xz |
| apply (hi.add hr).congr |
| intro t |
| rw [(witness_dimensions H counts t complete source hR supported unit).1, |
| Nat.cast_mul, Real.log_mul] |
| · ring |
| · exact Nat.cast_ne_zero.mpr (Nat.ne_of_gt |
| (pairingSize_pos _ _ _ (fun n => stageRefinementCount_pos counts H.labels n.val t))) |
| · exact Nat.cast_ne_zero.mpr (Nat.ne_of_gt |
| (pairingSize_pos _ _ _ (fun n => stageRefinementCount_pos counts H.labels n.val t))) |
| |
| include hD mass in |
| theorem witness_inner_limit : |
| Tendsto (fun t : ℕ => |
| Real.log ((witness H counts t complete source hR supported unit).inner : ℝ) / |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop (𝓝 (2 * pairRate H p .yz)) := by |
| apply ((tendsto_log_stagePairingSize_law counts hD p mass H.labels H.depth |
| (fun n => H.pair n.val) .yz).const_mul 2).congr |
| intro t |
| rw [(witness_dimensions H counts t complete source hR supported unit).2, |
| Nat.cast_pow, Real.log_pow] |
| norm_num |
| ring |
| |
| include hD mass in |
| theorem witness_multiplicity_limit : |
| Tendsto (fun t : ℕ => |
| Real.log ((witness H counts t complete source hR supported unit).multiplicity : ℝ) / |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop (𝓝 (2 * finiteEntropy p.mass)) := by |
| apply ((copies_limit H counts hD complete p mass).const_mul 2).congr |
| intro t |
| rw [witness_multiplicity, Nat.cast_pow, Real.log_pow] |
| norm_num |
| ring |
| |
| include hD mass in |
| theorem witness_rank_limit : |
| Tendsto (fun t : ℕ => |
| Real.log ((witness H counts t complete source hR supported unit).rank : ℝ) / |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop |
| (𝓝 (2 * (Real.log R + finiteEntropy p.mass))) := by |
| apply ((rank_budget_limit H counts hD complete p mass hR).const_mul 2).congr |
| intro t |
| rw [witness_rank, Nat.cast_pow, Real.log_pow] |
| simp only [rankBudget, Nat.cast_mul, Nat.cast_pow] |
| norm_num |
| ring |
| |
| end MatrixMultiplication.DualFiniteRealization |
| |
| end |
| |
| end OAI |
| |
