On the Geometry of Border Rank Algorithms for n × 2 by 2 × 2 Matrix Multiplication
نویسندگان
چکیده
منابع مشابه
On the Geometry of Border Rank Algorithms for n × 2 by 2 × 2 Matrix Multiplication
We make an in-depth study of the known border rank (i.e. approximate) algorithms for the matrix multiplication tensor M⟨n,2,2⟩ ∈ C⊗C⊗C encoding the multiplication of an n × 2 matrix by a 2 × 2 matrix.
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Let M ⟨n⟩ ∈ C n 2 ⊗C n 2 ⊗C n 2 denote the matrix multiplication tensor for n × n matrices. We use the border substitution method [2, 3, 6] combined with Koszul flattenings [8] to prove the border rank lower bound R(M ⟨n,n,w⟩) ≥ 2n 2 − ⌈log 2 (n)⌉ − 1.
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We establish basic information about border rank algorithms for the matrix multiplication tensor and other tensors with symmetry. We prove that border rank algorithms for tensors with symmetry (such as matrix multiplication and the determinant polynomial) come in families that include representatives with normal forms. These normal forms will be useful both to develop new efficient algorithms a...
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One of the leading problems of algebraic complexity theory is matrix multiplication. The näıve multiplication of two n× n matrices uses n multiplications. In 1969 Strassen [19] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [11] and Winograd [20] proved independently that there is no algorithm for multiply...
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A notion of a Frobenius manifold with a nice real structure was introduced by Hertling. It is called CDV structure (Cecotti-Dubrovin-Vafa structure). In this paper, we introduce a “positivity condition” on CDV structures and show that any Frobenius manifold of rank two with real spectrum can be equipped with a positive CDV structure. We extend naturally the symmetries of Frobenius structures gi...
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ژورنال
عنوان ژورنال: Experimental Mathematics
سال: 2016
ISSN: 1058-6458,1944-950X
DOI: 10.1080/10586458.2016.1162230