32 research outputs found
A note on the gap between rank and border rank
We study the tensor rank of the tensor corresponding to the algebra of
n-variate complex polynomials modulo the dth power of each variable. As a
result we find a sequence of tensors with a large gap between rank and border
rank, and thus a counterexample to a conjecture of Rhodes. At the same time we
obtain a new lower bound on the tensor rank of tensor powers of the generalised
W-state tensor. In addition, we exactly determine the tensor rank of the tensor
cube of the three-party W-state tensor, thus answering a question of Chen et
al.Comment: To appear in Linear Algebra and its Application
The asymptotic spectrum of graphs and the Shannon capacity
We introduce the asymptotic spectrum of graphs and apply the theory of
asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new
dual characterisation of the Shannon capacity of graphs. Elements in the
asymptotic spectrum of graphs include the Lov\'asz theta number, the fractional
clique cover number, the complement of the fractional orthogonal rank and the
fractional Haemers bounds
On the orthogonal rank of Cayley graphs and impossibility of quantum round elimination
After Bob sends Alice a bit, she responds with a lengthy reply. At the cost
of a factor of two in the total communication, Alice could just as well have
given the two possible replies without listening and have Bob select which
applies to him. Motivated by a conjecture stating that this form of "round
elimination" is impossible in exact quantum communication complexity, we study
the orthogonal rank and a symmetric variant thereof for a certain family of
Cayley graphs. The orthogonal rank of a graph is the smallest number for
which one can label each vertex with a nonzero -dimensional complex vector
such that adjacent vertices receive orthogonal vectors.
We show an exp lower bound on the orthogonal rank of the graph on
in which two strings are adjacent if they have Hamming distance at
least . In combination with previous work, this implies an affirmative
answer to the above conjecture.Comment: 13 page
Asymptotic tensor rank of graph tensors: beyond matrix multiplication
We present an upper bound on the exponent of the asymptotic behaviour of the
tensor rank of a family of tensors defined by the complete graph on
vertices. For , we show that the exponent per edge is at most 0.77,
outperforming the best known upper bound on the exponent per edge for matrix
multiplication (), which is approximately 0.79. We raise the question
whether for some the exponent per edge can be below , i.e. can
outperform matrix multiplication even if the matrix multiplication exponent
equals 2. In order to obtain our results, we generalise to higher order tensors
a result by Strassen on the asymptotic subrank of tight tensors and a result by
Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our
results have applications in entanglement theory and communication complexity
The border support rank of two-by-two matrix multiplication is seven
We show that the border support rank of the tensor corresponding to
two-by-two matrix multiplication is seven over the complex numbers. We do this
by constructing two polynomials that vanish on all complex tensors with format
four-by-four-by-four and border rank at most six, but that do not vanish
simultaneously on any tensor with the same support as the two-by-two matrix
multiplication tensor. This extends the work of Hauenstein, Ikenmeyer, and
Landsberg. We also give two proofs that the support rank of the two-by-two
matrix multiplication tensor is seven over any field: one proof using a result
of De Groote saying that the decomposition of this tensor is unique up to
sandwiching, and another proof via the substitution method. These results
answer a question asked by Cohn and Umans. Studying the border support rank of
the matrix multiplication tensor is relevant for the design of matrix
multiplication algorithms, because upper bounds on the border support rank of
the matrix multiplication tensor lead to upper bounds on the computational
complexity of matrix multiplication, via a construction of Cohn and Umans.
Moreover, support rank has applications in quantum communication complexity
The asymptotic induced matching number of hypergraphs: balanced binary strings
We compute the asymptotic induced matching number of the -partite
-uniform hypergraphs whose edges are the -bit strings of Hamming weight
, for any large enough even number . Our lower bound relies on the
higher-order extension of the well-known Coppersmith-Winograd method from
algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam.
Our result is motivated by the study of the power of this method as well as of
the power of the Strassen support functionals (which provide upper bounds on
the asymptotic induced matching number), and the connections to questions in
tensor theory, quantum information theory and theoretical computer science.
Phrased in the language of tensors, as a direct consequence of our result, we
determine the asymptotic subrank of any tensor with support given by the
aforementioned hypergraphs. In the context of quantum information theory, our
result amounts to an asymptotically optimal -party stochastic local
operations and classical communication (slocc) protocol for the problem of
distilling GHZ-type entanglement from a subfamily of Dicke-type entanglement
Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication
We study nondeterministic multiparty quantum communication with a quantum
generalization of broadcasts. We show that, with number-in-hand classical
inputs, the communication complexity of a Boolean function in this
communication model equals the logarithm of the support rank of the
corresponding tensor, whereas the approximation complexity in this model equals
the logarithm of the border support rank. This characterisation allows us to
prove a log-rank conjecture posed by Villagra et al. for nondeterministic
multiparty quantum communication with message-passing.
The support rank characterization of the communication model connects quantum
communication complexity intimately to the theory of asymptotic entanglement
transformation and algebraic complexity theory. In this context, we introduce
the graphwise equality problem. For a cycle graph, the complexity of this
communication problem is closely related to the complexity of the computational
problem of multiplying matrices, or more precisely, it equals the logarithm of
the asymptotic support rank of the iterated matrix multiplication tensor. We
employ Strassen's laser method to show that asymptotically there exist
nontrivial protocols for every odd-player cyclic equality problem. We exhibit
an efficient protocol for the 5-player problem for small inputs, and we show
how Young flattenings yield nontrivial complexity lower bounds
A note on the gap between rank and border rank
We study the tensor rank of a certain algebra. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a counterexample to a conjecture of Rhodes. We also obtain a new lower bound on the tensor rank of powers of the generalized W-state
Tensor rank is not multiplicative under the tensor product
The tensor rank of a tensor t is the smallest number r such that t can be
decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an
l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is
sub-multiplicative under the tensor product. We revisit the connection between
restrictions and degenerations. A result of our study is that tensor rank is
not in general multiplicative under the tensor product. This answers a question
of Draisma and Saptharishi. Specifically, if a tensor t has border rank
strictly smaller than its rank, then the tensor rank of t is not multiplicative
under taking a sufficiently hight tensor product power. The "tensor Kronecker
product" from algebraic complexity theory is related to our tensor product but
different, namely it multiplies two k-tensors to get a k-tensor.
Nonmultiplicativity of the tensor Kronecker product has been known since the
work of Strassen.
It remains an open question whether border rank and asymptotic rank are
multiplicative under the tensor product. Interestingly, lower bounds on border
rank obtained from generalised flattenings (including Young flattenings)
multiply under the tensor product