20,810 research outputs found
Towards an Efficient Use of the BLAS Library for Multilinear Tensor Contractions
Mathematical operators whose transformation rules constitute the building
blocks of a multi-linear algebra are widely used in physics and engineering
applications where they are very often represented as tensors. In the last
century, thanks to the advances in tensor calculus, it was possible to uncover
new research fields and make remarkable progress in the existing ones, from
electromagnetism to the dynamics of fluids and from the mechanics of rigid
bodies to quantum mechanics of many atoms. By now, the formal mathematical and
geometrical properties of tensors are well defined and understood; conversely,
in the context of scientific and high-performance computing, many tensor-
related problems are still open. In this paper, we address the problem of
efficiently computing contractions among two tensors of arbitrary dimension by
using kernels from the highly optimized BLAS library. In particular, we
establish precise conditions to determine if and when GEMM, the kernel for
matrix products, can be used. Such conditions take into consideration both the
nature of the operation and the storage scheme of the tensors, and induce a
classification of the contractions into three groups. For each group, we
provide a recipe to guide the users towards the most effective use of BLAS.Comment: 27 Pages, 7 figures and additional tikz generated diagrams. Submitted
to Applied Mathematics and Computatio
Topology of Networks in Generalized Musical Spaces
The abstraction of musical structures (notes, melodies, chords, harmonic or
rhythmic progressions, etc.) as mathematical objects in a geometrical space is
one of the great accomplishments of contemporary music theory. Building on this
foundation, I generalize the concept of musical spaces as networks and derive
functional principles of compositional design by the direct analysis of the
network topology. This approach provides a novel framework for the analysis and
quantification of similarity of musical objects and structures, and suggests a
way to relate such measures to the human perception of different musical
entities. Finally, the analysis of a single work or a corpus of compositions as
complex networks provides alternative ways of interpreting the compositional
process of a composer by quantifying emergent behaviors with well-established
statistical mechanics techniques. Interpreting the latter as probabilistic
randomness in the network, I develop novel compositional design frameworks that
are central to my own artistic research
Measuring Information Leakage in Website Fingerprinting Attacks and Defenses
Tor provides low-latency anonymous and uncensored network access against a
local or network adversary. Due to the design choice to minimize traffic
overhead (and increase the pool of potential users) Tor allows some information
about the client's connections to leak. Attacks using (features extracted from)
this information to infer the website a user visits are called Website
Fingerprinting (WF) attacks. We develop a methodology and tools to measure the
amount of leaked information about a website. We apply this tool to a
comprehensive set of features extracted from a large set of websites and WF
defense mechanisms, allowing us to make more fine-grained observations about WF
attacks and defenses.Comment: In Proceedings of the 2018 ACM SIGSAC Conference on Computer and
Communications Security (CCS '18
Products, coproducts and singular value decomposition
Products and coproducts may be recognized as morphisms in a monoidal tensor
category of vector spaces. To gain invariant data of these morphisms, we can
use singular value decomposition which attaches singular values, ie generalized
eigenvalues, to these maps. We show, for the case of Grassmann and Clifford
products, that twist maps significantly alter these data reducing degeneracies.
Since non group like coproducts give rise to non classical behavior of the
algebra of functions, ie make them noncommutative, we hope to be able to learn
more about such geometries. Remarkably the coproduct for positive singular
values of eigenvectors in yields directly corresponding eigenvectors in
A\otimes A.Comment: 17 pages, three eps-figure
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