45,318 research outputs found
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
Scalable Task-Based Algorithm for Multiplication of Block-Rank-Sparse Matrices
A task-based formulation of Scalable Universal Matrix Multiplication
Algorithm (SUMMA), a popular algorithm for matrix multiplication (MM), is
applied to the multiplication of hierarchy-free, rank-structured matrices that
appear in the domain of quantum chemistry (QC). The novel features of our
formulation are: (1) concurrent scheduling of multiple SUMMA iterations, and
(2) fine-grained task-based composition. These features make it tolerant of the
load imbalance due to the irregular matrix structure and eliminate all
artifactual sources of global synchronization.Scalability of iterative
computation of square-root inverse of block-rank-sparse QC matrices is
demonstrated; for full-rank (dense) matrices the performance of our SUMMA
formulation usually exceeds that of the state-of-the-art dense MM
implementations (ScaLAPACK and Cyclops Tensor Framework).Comment: 8 pages, 6 figures, accepted to IA3 2015. arXiv admin note: text
overlap with arXiv:1504.0504
Quantum Groups
These notes correspond rather accurately to the translation of the lectures
given at the Fifth Mexican School of Particles and Fields, held in Guanajuato,
Gto., in December~1992. They constitute a brief and elementary introduction to
quantum symmetries from a physical point of view, along the lines of the
forthcoming book by C. G\'omez, G. Sierra and myself.Comment: 37 pages, plain.te
Information Recovery from Pairwise Measurements
A variety of information processing tasks in practice involve recovering
objects from single-shot graph-based measurements, particularly those taken
over the edges of some measurement graph . This paper concerns the
situation where each object takes value over a group of different values,
and where one is interested to recover all these values based on observations
of certain pairwise relations over . The imperfection of
measurements presents two major challenges for information recovery: 1)
: a (dominant) portion of measurements are
corrupted; 2) : a significant fraction of pairs are
unobservable, i.e. can be highly sparse.
Under a natural random outlier model, we characterize the , that is, the critical threshold of non-corruption rate
below which exact information recovery is infeasible. This accommodates a very
general class of pairwise relations. For various homogeneous random graph
models (e.g. Erdos Renyi random graphs, random geometric graphs, small world
graphs), the minimax recovery rate depends almost exclusively on the edge
sparsity of the measurement graph irrespective of other graphical
metrics. This fundamental limit decays with the group size at a square root
rate before entering a connectivity-limited regime. Under the Erdos Renyi
random graph, a tractable combinatorial algorithm is proposed to approach the
limit for large (), while order-optimal recovery is
enabled by semidefinite programs in the small regime.
The extended (and most updated) version of this work can be found at
(http://arxiv.org/abs/1504.01369).Comment: This version is no longer updated -- please find the latest version
at (arXiv:1504.01369
Algebraic conversions
An examination of the pure algebraic properties of computational type conversion leads to a new
generalizations of the concept of a homomorphism for which the term conversion seems appropriate.
While an homomorphism is a mapping that respects the value of all terms, a conversion is a mapping
that respects the value of all sufficiently small terms. Such a mapping has practical value, as well
as theoretical interest that stems from conversions forming a category. This paper gives a precise
definition of the concept and demonstrates an application to formal computer science based on
work completed by the author in his PhD thesis
Algebraic {}-Integration and Fourier Theory on Quantum and Braided Spaces
We introduce an algebraic theory of integration on quantum planes and other
braided spaces. In the one dimensional case we obtain a novel picture of the
Jackson -integral as indefinite integration on the braided group of
functions in one variable . Here is treated with braid statistics
rather than the usual bosonic or Grassmann ones. We show that the definite
integral can also be evaluated algebraically as multiples of the
integral of a -Gaussian, with remaining as a bosonic scaling variable
associated with the -deformation. Further composing our algebraic
integration with a representation then leads to ordinary numbers for the
integral. We also use our integration to develop a full theory of -Fourier
transformation . We use the braided addition and braided-antipode to define a convolution product, and prove a
convolution theorem. We prove also that . We prove the analogous results
on any braided group, including integration and Fourier transformation on
quantum planes associated to general R-matrices, including -Euclidean and
-Minkowski spaces.Comment: 50 pages. Minor changes, added 3 reference
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