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92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | import Mathlib.LinearAlgebra.Basis.Defs
import Mathlib.LinearAlgebra.Matrix.Rank
import Mathlib.Tactic.Ring
namespace OAI
section
namespace UniqueGamesTheorem.Quadratic
open Module
open scoped BigOperators Matrix
noncomputable section
variable {R F V ι n E : Type*}
variable [Field R] [Field F] [Algebra R F]
variable [AddCommGroup V] [Module R V]
variable [Fintype ι] [Fintype n]
/-- The `|n| * |ι|` alignment columns for a fixed parametrization and input set.
The input set is represented by any map `point : E → V`; no injectivity is
needed for the exact algebraic equivalence. -/
def alignmentMatrix (b : Basis ι R V) (point : E → V) (z : V → n → F) :
Matrix E (n × ι) F :=
fun e ij => algebraMap R F (b.repr (point e) ij.2) * z (point e) ij.1
/-- Basis values of a linear lift give the coefficients of the alignment
column combination. The identity holds for every lift, without selecting
one in advance of the parametrization. -/
theorem dot_lift_eq_alignment_mulVec (b : Basis ι R V)
(point : E → V) (z : V → n → F) (g : V →ₗ[R] (n → F)) (e : E) :
(∑ i, z (point e) i * g (point e) i) =
(alignmentMatrix b point z *ᵥ (fun ij => g (b ij.2) ij.1)) e := by
have hg : g (point e) = ∑ j, b.repr (point e) j • g (b j) := by
calc
g (point e) = g (∑ j, b.repr (point e) j • b j) :=
congrArg g (b.sum_repr (point e)).symm
_ = ∑ j, b.repr (point e) j • g (b j) := by simp only [map_sum, map_smul]
rw [hg]
simp only [Matrix.mulVec, dotProduct, alignmentMatrix,
Finset.sum_apply, Algebra.smul_def, Pi.mul_apply,
Pi.algebraMap_apply, Finset.mul_sum,
Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro i hi
apply Finset.sum_congr rfl
intro j hj
ring
/-- Existence of *any* linear lift aligning on the chosen inputs is one
column-membership event. In particular there is no field-size-dependent
union bound over the possible lifts. -/
theorem exists_lift_alignment_iff_mem_range (b : Basis ι R V)
(point : E → V) (z : V → n → F) (c : E → F) :
(∃ g : V →ₗ[R] (n → F), ∀ e,
(∑ i, z (point e) i * g (point e) i) = c e) ↔
c ∈ LinearMap.range (alignmentMatrix b point z).mulVecLin := by
constructor
· rintro ⟨g, hg⟩
refine ⟨fun ij => g (b ij.2) ij.1, ?_⟩
ext e
change (alignmentMatrix b point z *ᵥ (fun ij => g (b ij.2) ij.1)) e = c e
rw [← dot_lift_eq_alignment_mulVec]
exact hg e
· rintro ⟨u, hu⟩
let g : V →ₗ[R] (n → F) := b.constr R (fun j i => u (i, j))
refine ⟨g, ?_⟩
intro e
rw [dot_lift_eq_alignment_mulVec b point z g e]
have hg : (fun ij : n × ι => g (b ij.2) ij.1) = u := by
funext ij
exact congrFun (b.constr_basis R (fun j i => u (i, j)) ij.2) ij.1
rw [hg]
exact congrFun hu e
/-- The column-space dimension is at most the number of free basis values.
For three field coordinates and a binary basis of size `r`, this is `3r`. -/
theorem finrank_alignment_range_le (b : Basis ι R V)
(point : E → V) (z : V → n → F) [Fintype E] :
Module.finrank F (LinearMap.range (alignmentMatrix b point z).mulVecLin) ≤
Fintype.card n * Fintype.card ι := by
simpa only [Module.finrank_fintype_fun_eq_card, Fintype.card_prod] using
(LinearMap.finrank_range_le (alignmentMatrix b point z).mulVecLin)
end
end UniqueGamesTheorem.Quadratic
end
end OAI
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