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Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 14 (2 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
The border rank of the multiplication of 2 × 2 matrices is seven
 J. Amer. Math. Soc
"... One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, ..."
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Cited by 8 (3 self)
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One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [12] and
New lower bounds for the border rank of matrix multiplication
"... Abstract. The border rank of the matrix multiplication operator for n × n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n 2 − n. Our bounds are better than the previous lower bound (due to Li ..."
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Cited by 7 (2 self)
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Abstract. The border rank of the matrix multiplication operator for n × n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n 2 − n. Our bounds are better than the previous lower bound (due to Lickteig in 1985) of 3 2 n2 + n 2 − 1 for all n ≥ 3. The bounds are obtained by finding new equations that bilinear maps of small border rank must satisfy, i.e., new equations for secant varieties of triple Segre products, that matrix multiplication fails to satisfy.
Explicit Lower Bounds via Geometric Complexity Theory (Extended Abstract)
, 2012
"... We prove the lower bound R(Mm) ≥ 3 2 m² − 2 on the border rank of m × m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense of Mulmuley and Sohoni’s geometric complexity theory (GCT) program. While this bound is weaker than the one recently ob ..."
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We prove the lower bound R(Mm) ≥ 3 2 m² − 2 on the border rank of m × m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense of Mulmuley and Sohoni’s geometric complexity theory (GCT) program. While this bound is weaker than the one recently obtained by Landsberg and Ottaviani, these are the first significant lower bounds obtained within the GCT program. Behind the proof is the new combinatorial concept of obstruction designs, which encode highest weight vectors in Sym d ⊗ 3 (C