2,783 research outputs found
Tensor Networks for Big Data Analytics and Large-Scale Optimization Problems
In this paper we review basic and emerging models and associated algorithms
for large-scale tensor networks, especially Tensor Train (TT) decompositions
using novel mathematical and graphical representations. We discus the concept
of tensorization (i.e., creating very high-order tensors from lower-order
original data) and super compression of data achieved via quantized tensor
train (QTT) networks. The purpose of a tensorization and quantization is to
achieve, via low-rank tensor approximations "super" compression, and
meaningful, compact representation of structured data. The main objective of
this paper is to show how tensor networks can be used to solve a wide class of
big data optimization problems (that are far from tractable by classical
numerical methods) by applying tensorization and performing all operations
using relatively small size matrices and tensors and applying iteratively
optimized and approximative tensor contractions.
Keywords: Tensor networks, tensor train (TT) decompositions, matrix product
states (MPS), matrix product operators (MPO), basic tensor operations,
tensorization, distributed representation od data optimization problems for
very large-scale problems: generalized eigenvalue decomposition (GEVD),
PCA/SVD, canonical correlation analysis (CCA).Comment: arXiv admin note: text overlap with arXiv:1403.204
Very Large-Scale Singular Value Decomposition Using Tensor Train Networks
We propose new algorithms for singular value decomposition (SVD) of very
large-scale matrices based on a low-rank tensor approximation technique called
the tensor train (TT) format. The proposed algorithms can compute several
dominant singular values and corresponding singular vectors for large-scale
structured matrices given in a TT format. The computational complexity of the
proposed methods scales logarithmically with the matrix size under the
assumption that both the matrix and the singular vectors admit low-rank TT
decompositions. The proposed methods, which are called the alternating least
squares for SVD (ALS-SVD) and modified alternating least squares for SVD
(MALS-SVD), compute the left and right singular vectors approximately through
block TT decompositions. The very large-scale optimization problem is reduced
to sequential small-scale optimization problems, and each core tensor of the
block TT decompositions can be updated by applying any standard optimization
methods. The optimal ranks of the block TT decompositions are determined
adaptively during iteration process, so that we can achieve high approximation
accuracy. Extensive numerical simulations are conducted for several types of
TT-structured matrices such as Hilbert matrix, Toeplitz matrix, random matrix
with prescribed singular values, and tridiagonal matrix. The simulation results
demonstrate the effectiveness of the proposed methods compared with standard
SVD algorithms and TT-based algorithms developed for symmetric eigenvalue
decomposition
Steady state particle distribution of a dilute sedimenting suspension
Sedimentation of a non-Brownian suspension of hard particles is studied. It
is shown that in the low concentration limit a two-particle distribution
function ensuring finite particle correlation length can be found and
explicitly calculated. The sedimentation coefficient is computed. Results are
compared with experiment.Comment: 4 pages, 2 figure
Stable, Robust and Super Fast Reconstruction of Tensors Using Multi-Way Projections
In the framework of multidimensional Compressed Sensing (CS), we introduce an
analytical reconstruction formula that allows one to recover an th-order
data tensor
from a reduced set of multi-way compressive measurements by exploiting its low
multilinear-rank structure. Moreover, we show that, an interesting property of
multi-way measurements allows us to build the reconstruction based on
compressive linear measurements taken only in two selected modes, independently
of the tensor order . In addition, it is proved that, in the matrix case and
in a particular case with rd-order tensors where the same 2D sensor operator
is applied to all mode-3 slices, the proposed reconstruction
is stable in the sense that the approximation
error is comparable to the one provided by the best low-multilinear-rank
approximation, where is a threshold parameter that controls the
approximation error. Through the analysis of the upper bound of the
approximation error we show that, in the 2D case, an optimal value for the
threshold parameter exists, which is confirmed by our
simulation results. On the other hand, our experiments on 3D datasets show that
very good reconstructions are obtained using , which means that this
parameter does not need to be tuned. Our extensive simulation results
demonstrate the stability and robustness of the method when it is applied to
real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity
based CS methods specialized for multidimensional signals is also included. A
very attractive characteristic of the proposed method is that it provides a
direct computation, i.e. it is non-iterative in contrast to all existing
sparsity based CS algorithms, thus providing super fast computations, even for
large datasets.Comment: Submitted to IEEE Transactions on Signal Processin
Multi-tensor Completion for Estimating Missing Values in Video Data
Many tensor-based data completion methods aim to solve image and video
in-painting problems. But, all methods were only developed for a single
dataset. In most of real applications, we can usually obtain more than one
dataset to reflect one phenomenon, and all the datasets are mutually related in
some sense. Thus one question raised whether such the relationship can improve
the performance of data completion or not? In the paper, we proposed a novel
and efficient method by exploiting the relationship among datasets for
multi-video data completion. Numerical results show that the proposed method
significantly improve the performance of video in-painting, particularly in the
case of very high missing percentage
Employed Unemployed? On Shadow Employment During Transition
Unregistered employment poses two types of challenges to the researchers: (i) reliably evaluating the wage differential between formally and informally employed and (ii) accounting for the push and pull factors in general and the effects of business cycle in particular. We address the former with the use of propensity score matching and analyse the evolution of the estimated average compensations and the differentials with reference to GDP and unemployment fluctuations. Using 13 years of quarterly labour force survey data from Poland on de iure unemployed but de facto employed individuals we find, that in-the-shadow compensations tend to be higher and procyclical. We also find considerable distributional heterogeneity.undeclared employment, propensity score matching, transition
Shadow Employment In Transition - A Matter of Choice or No Choice?
Shadow employment may follow from two main labour market failures. In the first, official market labour taxation distortions make it ineffective for some agents to engage in registered employment due to a tax wedge, which makes the revenues from unofficial employment higher than the corresponding official ones (tax evasion hypothesis). The alternative explanation draws to labour market tightness - for workers regular employment may be unattainable, which results in seeking earning opportunities beyond the boundaries of the official labour market (market segmentation hypothesis). We use a unique data set from a survey on undeclared employment. Using propensity score matching and decomposition techniques we demonstrate that workers of the shadow economy are characterised by slightly higher endowments, while their revenues are considerably lower than among the matched official economy counterparts. Although unobservable heterogeneity is considerable, results are robust and point to social exclusion and the market segmentation hypothesis.shadow employment, propensity score matching, Oaxaca-Blinder decomposition, transition, Poland
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