openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Duality /FiniteRealization.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.FiniteRealization | |
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.MatrixBalancing | |
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexWitness | |
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.BaseTensorBudgets | |
| namespace OAI | |
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.DualFiniteRealization | |
| open MatrixMultiplication.Foundation RecursiveCompletion | |
| open CompletionHierarchyWords CompletionExecution CompletionFiniteRealization | |
| open scoped BigOperators Classical | |
| variable {A X Y Z : Type} [Fintype A] | |
| variable (H : ReadableHierarchy A X Y Z) (counts : A → ℕ) (t : ℕ) | |
| variable (complete : Function.Injective (labelRecordOf H.labels H.depth)) | |
| variable [Fintype X] [Fintype Y] [Fintype Z] | |
| def rankBudget (R : ℕ) : ℕ := | |
| R ^ CompletionFiniteRealization.blocks counts t * (realize H counts t).execution.rankBound | |
| def sourcePower {T : Tensor ℂ X Y Z} {R d D : ℕ} | |
| (source : Tensor.PolynomialApproximation T R d D) : | |
| Tensor.PolynomialApproximation | |
| (Tensor.power T (CompletionFiniteRealization.blocks counts t)) | |
| (R ^ CompletionFiniteRealization.blocks counts t) | |
| (d * CompletionFiniteRealization.blocks counts t) | |
| (D * CompletionFiniteRealization.blocks counts t) := | |
| source.power (CompletionFiniteRealization.blocks counts t) | |
| def approximation {T : Tensor ℂ X Y Z} {R d D : ℕ} | |
| (source : Tensor.PolynomialApproximation T R d D) | |
| (supported : ∀ x y z, T x y z ≠ 0 → ∃ a, H.x a = x ∧ H.y a = y ∧ H.z a = z) | |
| (unit : ∀ a, T (H.x a) (H.y a) (H.z a) = 1) : | |
| Tensor.PolynomialApproximation | |
| (Tensor.directSum (fun _ : Fin (copies H counts t) => | |
| Tensor.matrixCoefficients (K := ℂ) | |
| (Fin (rows H counts t)) (Fin (inner H counts t)) (Fin (columns H counts t)))) | |
| (rankBudget H counts t R) (order H counts t d) (degree H counts t D) := by | |
| have hs : ∀ x y z, ¬ support H counts t x y z → | |
| Tensor.power T (CompletionFiniteRealization.blocks counts t) x y z = 0 := by | |
| intro x y z outside | |
| by_contra nonzero | |
| apply outside | |
| intro i | |
| exact supported _ _ _ ((Finset.prod_ne_zero_iff.mp nonzero) i (Finset.mem_univ i)) | |
| have full := ExecutionApproximation.approximation (realize H counts t).execution | |
| (sourcePower counts t source) hs | |
| have selected := full.pullback (leftSelect H counts t complete) | |
| (middleSelect H counts t complete) (rightSelect H counts t complete) | |
| rw [selected_output H counts t complete T unit] at selected | |
| exact selected | |
| omit [Fintype X] [Fintype Y] [Fintype Z] in | |
| theorem rankBudget_pos {R : ℕ} (hR : 0 < R) : 0 < rankBudget H counts t R := | |
| Nat.mul_pos (pow_pos hR _) (auxiliaryRank_pos H counts t) | |
| def witness {T : Tensor ℂ X Y Z} {R d D : ℕ} | |
| (source : Tensor.PolynomialApproximation T R d D) (hR : 0 < R) | |
| (supported : ∀ x y z, T x y z ≠ 0 → ∃ a, H.x a = x ∧ H.y a = y ∧ H.z a = z) | |
| (unit : ∀ a, T (H.x a) (H.y a) (H.z a) = 1) : | |
| ComplexWitness.FiniteRectangularWitness where | |
| outer := rows H counts t * inner H counts t | |
| inner := columns H counts t ^ 2 | |
| multiplicity := copies H counts t ^ 2 | |
| rank := rankBudget H counts t R ^ 2 | |
| outer_pos := Nat.mul_pos (dimensions_positive H counts t).1 | |
| (dimensions_positive H counts t).2.1 | |
| inner_pos := pow_pos (dimensions_positive H counts t).2.2 2 | |
| multiplicity_pos := pow_pos (copies_pos H counts t complete) 2 | |
| rank_pos := pow_pos (rankBudget_pos H counts t hR) 2 | |
| order := 2 * order H counts t d | |
| degree := 2 * degree H counts t D | |
| approximation := MatrixBalancing.finiteBalanced | |
| (approximation H counts t complete source supported unit) | |
| section Projections | |
| 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) | |
| @[simp] theorem witness_outer : | |
| (witness H counts t complete source hR supported unit).outer = | |
| rows H counts t * inner H counts t := rfl | |
| @[simp] theorem witness_inner : | |
| (witness H counts t complete source hR supported unit).inner = columns H counts t ^ 2 := rfl | |
| @[simp] theorem witness_multiplicity : | |
| (witness H counts t complete source hR supported unit).multiplicity = copies H counts t ^ 2 := | |
| rfl | |
| @[simp] theorem witness_rank : | |
| (witness H counts t complete source hR supported unit).rank = | |
| (R ^ CompletionFiniteRealization.blocks counts t * | |
| (realize H counts t).execution.rankBound) ^ 2 := rfl | |
| theorem witness_dimensions : | |
| (witness H counts t complete source hR supported unit).outer = | |
| ConditionalLabels.stagePairingSize counts H.labels H.depth | |
| (fun n => H.pair n.val) .xz t * | |
| ConditionalLabels.stagePairingSize counts H.labels H.depth | |
| (fun n => H.pair n.val) .xy t ∧ | |
| (witness H counts t complete source hR supported unit).inner = | |
| ConditionalLabels.stagePairingSize counts H.labels H.depth | |
| (fun n => H.pair n.val) .yz t ^ 2 := by | |
| rcases realize_dimensions H counts t with ⟨hr, hi, hc⟩ | |
| change (realize H counts t).shape.rows * (realize H counts t).shape.inner = _ ∧ | |
| (realize H counts t).shape.columns ^ 2 = _ | |
| simp only [hr, hi, hc, and_self] | |
| theorem witness_rank_stageBudget : | |
| (witness H counts t complete source hR supported unit).rank = | |
| (R ^ CompletionFiniteRealization.blocks counts t * | |
| StageHierarchyResources.stageAuxiliaryBudget counts H.labels H.depth t) ^ 2 := by | |
| rw [witness_rank, realize_rankBound] | |
| end Projections | |
| omit [Fintype X] [Fintype Y] [Fintype Z] in | |
| theorem rankBudget_base (base factors : ℕ) : | |
| rankBudget H counts t (base ^ factors) = | |
| base ^ (factors * CompletionFiniteRealization.blocks counts t) * | |
| StageHierarchyResources.stageAuxiliaryBudget counts H.labels H.depth t := by | |
| simp only [rankBudget, ← pow_mul, realize_rankBound] | |
| def dualSource : | |
| Tensor.PolynomialApproximation (Script.tensor BaseTwo.flagged Script.dual).tensor | |
| (2 ^ 288) (Script.dual.order 1) (Script.dual.degree 3) := by | |
| simpa only [Script.dual_factors] using | |
| Script.approximation BaseTwo.flagged BaseTwo.approximation Script.dual | |
| end MatrixMultiplication.DualFiniteRealization | |
| end | |
| end OAI | |