57 research outputs found
Matrix multiplication over word-size modular rings using approximate formulae
International audienceBini–Capovani–Lotti–Romani approximate formula (or border rank) for matrix multiplication achieves abetter complexity than Strassen’s matrix multiplication formula. In this paper, we show a novel way touse the approximate formula in the special case where the ring is Z/pZ. Besides, we show an implementation à la FFLAS–FFPACK, where p is a word-size modulo, that improves on state-of-the-art Z/pZ matrix multiplication implementations
Nonnegative approximations of nonnegative tensors
We study the decomposition of a nonnegative tensor into a minimal sum of
outer product of nonnegative vectors and the associated parsimonious naive
Bayes probabilistic model. We show that the corresponding approximation
problem, which is central to nonnegative PARAFAC, will always have optimal
solutions. The result holds for any choice of norms and, under a mild
assumption, even Bregman divergences.Comment: 14 page
Fault-Tolerant Strassen-Like Matrix Multiplication
In this study, we propose a simple method for fault-tolerant Strassen-like
matrix multiplications. The proposed method is based on using two distinct
Strassen-like algorithms instead of replicating a given one. We have realized
that using two different algorithms, new check relations arise resulting in
more local computations. These local computations are found using computer
aided search. To improve performance, special parity (extra) sub-matrix
multiplications (PSMMs) are generated (two of them) at the expense of
increasing communication/computation cost of the system. Our preliminary
results demonstrate that the proposed method outperforms a Strassen-like
algorithm with two copies and secures a very close performance to three copy
version using only 2 PSMMs, reducing the total number of compute nodes by
around 24\% i.e., from 21 to 16.Comment: 6 pages, 2 figure
Cryptography from tensor problems
We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler
Using 1-Factorization from Graph Theory for Quantum Speedups on Clique Problems
The clique problems, including -CLIQUE and Triangle Finding, form an
important class of computational problems; the former is an NP-complete
problem, while the latter directly gives lower bounds for Matrix
Multiplication. A number of previous efforts have approached these problems
with Quantum Computing methods, such as Amplitude Amplification. In this paper,
we provide new Quantum oracle designs based on the 1-factorization of complete
graphs, all of which have depth instead of the presented in
previous studies. Also, we discuss the usage of one of these oracles in
bringing the Triangle Finding time complexity down to , compared to the classical record. Finally, we benchmark the
number of required Amplitude Amplification iterations for another presented
oracle, for solving -CLIQUE.Comment: 14 pages, 8 figure
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