250 research outputs found
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Detecting the community structure and activity patterns of temporal networks: a non-negative tensor factorization approach
The increasing availability of temporal network data is calling for more
research on extracting and characterizing mesoscopic structures in temporal
networks and on relating such structure to specific functions or properties of
the system. An outstanding challenge is the extension of the results achieved
for static networks to time-varying networks, where the topological structure
of the system and the temporal activity patterns of its components are
intertwined. Here we investigate the use of a latent factor decomposition
technique, non-negative tensor factorization, to extract the community-activity
structure of temporal networks. The method is intrinsically temporal and allows
to simultaneously identify communities and to track their activity over time.
We represent the time-varying adjacency matrix of a temporal network as a
three-way tensor and approximate this tensor as a sum of terms that can be
interpreted as communities of nodes with an associated activity time series. We
summarize known computational techniques for tensor decomposition and discuss
some quality metrics that can be used to tune the complexity of the factorized
representation. We subsequently apply tensor factorization to a temporal
network for which a ground truth is available for both the community structure
and the temporal activity patterns. The data we use describe the social
interactions of students in a school, the associations between students and
school classes, and the spatio-temporal trajectories of students over time. We
show that non-negative tensor factorization is capable of recovering the class
structure with high accuracy. In particular, the extracted tensor components
can be validated either as known school classes, or in terms of correlated
activity patterns, i.e., of spatial and temporal coincidences that are
determined by the known school activity schedule
A new penalized nonnegative third order tensor decomposition using a block coordinate proximal gradient approach: application to 3D fluorescence spectroscopy
International audienceIn this article, we address the problem of tensor factorization subject to certain constraints. We focus on the Canonical Polyadic Decomposition (CPD) also known as Parafac. The interest of this multi-linear decomposition coupled with 3D fluorescence spectroscopy is now well established in the fields of environmental data analysis, biochemistry and chemistry. When real experimental data (possibly corrupted by noise) are processed, the actual rank of the " observed " tensor is generally unknown. Moreover, when the amount of data is very large, this inverse problem may become numerically ill-posed and consequently hard to solve. The use of proper constraints reflecting some a priori knowledge about the latent (or hidden) tracked variables and/or additional information through the addition of penalty functions can prove very helpful in estimating more relevant components rather than totally arbitrary ones. The counterpart is that the cost functions that have to be considered can be non convex and sometimes even non differentiable making their optimization more difficult, leading to a higher computing time and a slower convergence speed. Block alternating proximal approaches offer a rigorous and flexible framework to properly address that problem since they are applicable to a large class of cost functions while remaining quite easy to implement. Here, we suggest a new block coordinate variable metric forward-backward method which can be seen as a special case of Majorize-Minimize (MM) approaches to derive a new penalized nonnegative third order CPD algorithm. Its interest, efficiency, robustness and flexibility are illustrated thanks to computer simulations carried out on both simulated and real experimental 3D fluorescence spectroscopy data
Accelerating jackknife resampling for the Canonical Polyadic Decomposition
The Canonical Polyadic (CP) tensor decomposition is frequently used as a
model in applications in a variety of different fields. Using jackknife
resampling to estimate parameter uncertainties is often desirable but results
in an increase of the already high computational cost. Upon observation that
the resampled tensors, though different, are nearly identical, we show that it
is possible to extend the recently proposed Concurrent ALS (CALS) technique to
a jackknife resampling scenario. This extension gives access to the
computational efficiency advantage of CALS for the price of a modest increase
(typically a few percent) in the number of floating point operations. Numerical
experiments on both synthetic and real-world datasets demonstrate that the new
workflow based on a CALS extension can be several times faster than a
straightforward workflow where the jackknife submodels are processed
individually
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