29,020 research outputs found
New lower bounds for the rank of matrix multiplication
The rank of the matrix multiplication operator for nxn matrices is one of the
most studied quantities in algebraic complexity theory. I prove that the rank
is at least n^2-o(n^2). More precisely, for any integer p\leq n -1, the rank is
at least (3- 1/(p+1))n^2-(1+2p\binom{2p}{p-1})n. The previous lower bound, due
to Blaser, was 5n^2/2-3n (the case p=1).
The new bounds improve Blaser's bound for all n>84. I also prove lower bounds
for rectangular matrices significantly better than the the previous bound.Comment: Completely rewritten, mistake in error term in previous version
corrected. To appear in SICOM
On Matrix Multiplication and Polynomial Identity Testing
We show that lower bounds on the border rank of matrix multiplication can be
used to non-trivially derandomize polynomial identity testing for small
algebraic circuits. Letting denote the border rank of matrix multiplication, we construct a hitting set generator
with seed length that hits
-variate circuits of multiplicative complexity . If the matrix
multiplication exponent is not 2, our generator has seed length
and hits circuits of size for
sufficiently small . Surprisingly, the fact that
already yields new, non-trivial hitting set
generators for circuits of sublinear multiplicative complexity
Discreteness of asymptotic tensor ranks
Tensor parameters that are amortized or regularized over large tensor powers,
often called "asymptotic" tensor parameters, play a central role in several
areas including algebraic complexity theory (constructing fast matrix
multiplication algorithms), quantum information (entanglement cost and
distillable entanglement), and additive combinatorics (bounds on cap sets,
sunflower-free sets, etc.). Examples are the asymptotic tensor rank, asymptotic
slice rank and asymptotic subrank. Recent works (Costa-Dalai,
Blatter-Draisma-Rupniewski, Christandl-Gesmundo-Zuiddam) have investigated
notions of discreteness (no accumulation points) or "gaps" in the values of
such tensor parameters.
We prove a general discreteness theorem for asymptotic tensor parameters of
order-three tensors and use this to prove that (1) over any finite field, the
asymptotic subrank and the asymptotic slice rank have no accumulation points,
and (2) over the complex numbers, the asymptotic slice rank has no accumulation
points.
Central to our approach are two new general lower bounds on the asymptotic
subrank of tensors, which measures how much a tensor can be diagonalized. The
first lower bound says that the asymptotic subrank of any concise three-tensor
is at least the cube-root of the smallest dimension. The second lower bound
says that any three-tensor that is "narrow enough" (has one dimension much
smaller than the other two) has maximal asymptotic subrank.
Our proofs rely on new lower bounds on the maximum rank in matrix subspaces
that are obtained by slicing a three-tensor in the three different directions.
We prove that for any concise tensor the product of any two such maximum ranks
must be large, and as a consequence there are always two distinct directions
with large max-rank
Discreteness of asymptotic tensor ranks
Tensor parameters that are amortized or regularized over large tensor powers, often called "asymptotic" tensor parameters, play a central role in several areas including algebraic complexity theory (constructing fast matrix multiplication algorithms), quantum information (entanglement cost and distillable entanglement), and additive combinatorics (bounds on cap sets, sunflower-free sets, etc.). Examples are the asymptotic tensor rank, asymptotic slice rank and asymptotic subrank. Recent works (Costa-Dalai, Blatter-Draisma-Rupniewski, Christandl-Gesmundo-Zuiddam) have investigated notions of discreteness (no accumulation points) or "gaps" in the values of such tensor parameters.
We prove a general discreteness theorem for asymptotic tensor parameters of order-three tensors and use this to prove that (1) over any finite field (and in fact any finite set of coefficients in any field), the asymptotic subrank and the asymptotic slice rank have no accumulation points, and (2) over the complex numbers, the asymptotic slice rank has no accumulation points.
Central to our approach are two new general lower bounds on the asymptotic subrank of tensors, which measures how much a tensor can be diagonalized. The first lower bound says that the asymptotic subrank of any concise three-tensor is at least the cube-root of the smallest dimension. The second lower bound says that any concise three-tensor that is "narrow enough" (has one dimension much smaller than the other two) has maximal asymptotic subrank.
Our proofs rely on new lower bounds on the maximum rank in matrix subspaces that are obtained by slicing a three-tensor in the three different directions. We prove that for any concise tensor, the product of any two such maximum ranks must be large, and as a consequence there are always two distinct directions with large max-rank
Limits on the Universal Method for Matrix Multiplication
In this work, we prove limitations on the known methods for designing matrix multiplication algorithms. Alman and Vassilevska Williams [Alman and Williams, 2018] recently defined the Universal Method, which substantially generalizes all the known approaches including Strassen\u27s Laser Method [V. Strassen, 1987] and Cohn and Umans\u27 Group Theoretic Method [Cohn and Umans, 2003]. We prove concrete lower bounds on the algorithms one can design by applying the Universal Method to many different tensors. Our proofs use new tools for upper bounding the asymptotic slice rank of a wide range of tensors. Our main result is that the Universal method applied to any Coppersmith-Winograd tensor CW_q cannot yield a bound on omega, the exponent of matrix multiplication, better than 2.16805. By comparison, it was previously only known that the weaker "Galactic Method" applied to CW_q could not achieve an exponent of 2.
We also study the Laser Method (which is, in principle, a highly special case of the Universal Method) and prove that it is "complete" for matrix multiplication algorithms: when it applies to a tensor T, it achieves omega = 2 if and only if it is possible for the Universal method applied to T to achieve omega = 2. Hence, the Laser Method, which was originally used as an algorithmic tool, can also be seen as a lower bounding tool. For example, in their landmark paper, Coppersmith and Winograd [Coppersmith and Winograd, 1990] achieved a bound of omega <= 2.376, by applying the Laser Method to CW_q. By our result, the fact that they did not achieve omega=2 implies a lower bound on the Universal Method applied to CW_q. Indeed, if it were possible for the Universal Method applied to CW_q to achieve omega=2, then Coppersmith and Winograd\u27s application of the Laser Method would have achieved omega=2
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