2,001 research outputs found
Stochastic variance reduced multiplicative update for nonnegative matrix factorization
Nonnegative matrix factorization (NMF), a dimensionality reduction and factor
analysis method, is a special case in which factor matrices have low-rank
nonnegative constraints. Considering the stochastic learning in NMF, we
specifically address the multiplicative update (MU) rule, which is the most
popular, but which has slow convergence property. This present paper introduces
on the stochastic MU rule a variance-reduced technique of stochastic gradient.
Numerical comparisons suggest that our proposed algorithms robustly outperform
state-of-the-art algorithms across different synthetic and real-world datasets.Comment: IEEE International Conference on Acoustics, Speech and Signal
Processing (ICASSP2018
Accelerated Parallel and Distributed Algorithm using Limited Internal Memory for Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) is a powerful technique for dimension
reduction, extracting latent factors and learning part-based representation.
For large datasets, NMF performance depends on some major issues: fast
algorithms, fully parallel distributed feasibility and limited internal memory.
This research aims to design a fast fully parallel and distributed algorithm
using limited internal memory to reach high NMF performance for large datasets.
In particular, we propose a flexible accelerated algorithm for NMF with all its
regularized variants based on full decomposition, which is a
combination of an anti-lopsided algorithm and a fast block coordinate descent
algorithm. The proposed algorithm takes advantages of both these algorithms to
achieve a linear convergence rate of in
optimizing each factor matrix when fixing the other factor one in the sub-space
of passive variables, where is the number of latent components; where
. In addition, the algorithm can exploit the data
sparseness to run on large datasets with limited internal memory of machines.
Furthermore, our experimental results are highly competitive with 7
state-of-the-art methods about three significant aspects of convergence,
optimality and average of the iteration number. Therefore, the proposed
algorithm is superior to fast block coordinate descent methods and accelerated
methods
Algorithmic Acceleration of Parallel ALS for Collaborative Filtering: Speeding up Distributed Big Data Recommendation in Spark
Collaborative filtering algorithms are important building blocks in many
practical recommendation systems. For example, many large-scale data processing
environments include collaborative filtering models for which the Alternating
Least Squares (ALS) algorithm is used to compute latent factor matrix
decompositions. In this paper, we propose an approach to accelerate the
convergence of parallel ALS-based optimization methods for collaborative
filtering using a nonlinear conjugate gradient (NCG) wrapper around the ALS
iterations. We also provide a parallel implementation of the accelerated
ALS-NCG algorithm in the Apache Spark distributed data processing environment,
and an efficient line search technique as part of the ALS-NCG implementation
that requires only one pass over the data on distributed datasets. In serial
numerical experiments on a linux workstation and parallel numerical experiments
on a 16 node cluster with 256 computing cores, we demonstrate that the combined
ALS-NCG method requires many fewer iterations and less time than standalone ALS
to reach movie rankings with high accuracy on the MovieLens 20M dataset. In
parallel, ALS-NCG can achieve an acceleration factor of 4 or greater in clock
time when an accurate solution is desired; furthermore, the acceleration factor
increases as greater numerical precision is required in the solution. In
addition, the NCG acceleration mechanism is efficient in parallel and scales
linearly with problem size on synthetic datasets with up to nearly 1 billion
ratings. The acceleration mechanism is general and may also be applicable to
other optimization methods for collaborative filtering.Comment: Proceedings of ICPADS 2015, Melbourne, AU. 10 pages; 6 figures; 4
table
Novel Algorithms based on Majorization Minimization for Nonnegative Matrix Factorization
Matrix decomposition is ubiquitous and has applications in various fields
like speech processing, data mining and image processing to name a few. Under
matrix decomposition, nonnegative matrix factorization is used to decompose a
nonnegative matrix into a product of two nonnegative matrices which gives some
meaningful interpretation of the data. Thus, nonnegative matrix factorization
has an edge over the other decomposition techniques. In this paper, we propose
two novel iterative algorithms based on Majorization Minimization (MM)-in which
we formulate a novel upper bound and minimize it to get a closed form solution
at every iteration. Since the algorithms are based on MM, it is ensured that
the proposed methods will be monotonic. The proposed algorithms differ in the
updating approach of the two nonnegative matrices. The first
algorithm-Iterative Nonnegative Matrix Factorization (INOM) sequentially
updates the two nonnegative matrices while the second algorithm-Parallel
Iterative Nonnegative Matrix Factorization (PARINOM) parallely updates them. We
also prove that the proposed algorithms converge to the stationary point of the
problem. Simulations were conducted to compare the proposed methods with the
existing ones and was found that the proposed algorithms performs better than
the existing ones in terms of computational speed and convergence.
KeyWords: Nonnegative matrix factorization, Majorization Minimization, Big
Data, Parallel, Multiplicative Updat
Sparse Deep Nonnegative Matrix Factorization
Nonnegative matrix factorization is a powerful technique to realize dimension
reduction and pattern recognition through single-layer data representation
learning. Deep learning, however, with its carefully designed hierarchical
structure, is able to combine hidden features to form more representative
features for pattern recognition. In this paper, we proposed sparse deep
nonnegative matrix factorization models to analyze complex data for more
accurate classification and better feature interpretation. Such models are
designed to learn localized features or generate more discriminative
representations for samples in distinct classes by imposing -norm penalty
on the columns of certain factors. By extending one-layer model into
multi-layer one with sparsity, we provided a hierarchical way to analyze big
data and extract hidden features intuitively due to nonnegativity. We adopted
the Nesterov's accelerated gradient algorithm to accelerate the computing
process with the convergence rate of after steps iteration. We
also analyzed the computing complexity of our framework to demonstrate their
efficiency. To improve the performance of dealing with linearly inseparable
data, we also considered to incorporate popular nonlinear functions into this
framework and explored their performance. We applied our models onto two
benchmarking image datasets, demonstrating our models can achieve competitive
or better classification performance and produce intuitive interpretations
compared with the typical NMF and competing multi-layer models.Comment: 13 pages, 8 figure
Tensor Completion Algorithms in Big Data Analytics
Tensor completion is a problem of filling the missing or unobserved entries
of partially observed tensors. Due to the multidimensional character of tensors
in describing complex datasets, tensor completion algorithms and their
applications have received wide attention and achievement in areas like data
mining, computer vision, signal processing, and neuroscience. In this survey,
we provide a modern overview of recent advances in tensor completion algorithms
from the perspective of big data analytics characterized by diverse variety,
large volume, and high velocity. We characterize these advances from four
perspectives: general tensor completion algorithms, tensor completion with
auxiliary information (variety), scalable tensor completion algorithms
(volume), and dynamic tensor completion algorithms (velocity). Further, we
identify several tensor completion applications on real-world data-driven
problems and present some common experimental frameworks popularized in the
literature. Our goal is to summarize these popular methods and introduce them
to researchers and practitioners for promoting future research and
applications. We conclude with a discussion of key challenges and promising
research directions in this community for future exploration
Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset
Recent research on problem formulations based on decomposition into low-rank
plus sparse matrices shows a suitable framework to separate moving objects from
the background. The most representative problem formulation is the Robust
Principal Component Analysis (RPCA) solved via Principal Component Pursuit
(PCP) which decomposes a data matrix in a low-rank matrix and a sparse matrix.
However, similar robust implicit or explicit decompositions can be made in the
following problem formulations: Robust Non-negative Matrix Factorization
(RNMF), Robust Matrix Completion (RMC), Robust Subspace Recovery (RSR), Robust
Subspace Tracking (RST) and Robust Low-Rank Minimization (RLRM). The main goal
of these similar problem formulations is to obtain explicitly or implicitly a
decomposition into low-rank matrix plus additive matrices. In this context,
this work aims to initiate a rigorous and comprehensive review of the similar
problem formulations in robust subspace learning and tracking based on
decomposition into low-rank plus additive matrices for testing and ranking
existing algorithms for background/foreground separation. For this, we first
provide a preliminary review of the recent developments in the different
problem formulations which allows us to define a unified view that we called
Decomposition into Low-rank plus Additive Matrices (DLAM). Then, we examine
carefully each method in each robust subspace learning/tracking frameworks with
their decomposition, their loss functions, their optimization problem and their
solvers. Furthermore, we investigate if incremental algorithms and real-time
implementations can be achieved for background/foreground separation. Finally,
experimental results on a large-scale dataset called Background Models
Challenge (BMC 2012) show the comparative performance of 32 different robust
subspace learning/tracking methods.Comment: 121 pages, 5 figures, submitted to Computer Science Review. arXiv
admin note: text overlap with arXiv:1312.7167, arXiv:1109.6297,
arXiv:1207.3438, arXiv:1105.2126, arXiv:1404.7592, arXiv:1210.0805,
arXiv:1403.8067 by other authors, Computer Science Review, November 201
Harnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation
Low-rank modeling plays a pivotal role in signal processing and machine
learning, with applications ranging from collaborative filtering, video
surveillance, medical imaging, to dimensionality reduction and adaptive
filtering. Many modern high-dimensional data and interactions thereof can be
modeled as lying approximately in a low-dimensional subspace or manifold,
possibly with additional structures, and its proper exploitations lead to
significant reduction of costs in sensing, computation and storage. In recent
years, there is a plethora of progress in understanding how to exploit low-rank
structures using computationally efficient procedures in a provable manner,
including both convex and nonconvex approaches. On one side, convex relaxations
such as nuclear norm minimization often lead to statistically optimal
procedures for estimating low-rank matrices, where first-order methods are
developed to address the computational challenges; on the other side, there is
emerging evidence that properly designed nonconvex procedures, such as
projected gradient descent, often provide globally optimal solutions with a
much lower computational cost in many problems. This survey article will
provide a unified overview of these recent advances on low-rank matrix
estimation from incomplete measurements. Attention is paid to rigorous
characterization of the performance of these algorithms, and to problems where
the low-rank matrix have additional structural properties that require new
algorithmic designs and theoretical analysis.Comment: To appear in IEEE Signal Processing Magazin
A Flexible and Efficient Algorithmic Framework for Constrained Matrix and Tensor Factorization
We propose a general algorithmic framework for constrained matrix and tensor
factorization, which is widely used in signal processing and machine learning.
The new framework is a hybrid between alternating optimization (AO) and the
alternating direction method of multipliers (ADMM): each matrix factor is
updated in turn, using ADMM, hence the name AO-ADMM. This combination can
naturally accommodate a great variety of constraints on the factor matrices,
and almost all possible loss measures for the fitting. Computation caching and
warm start strategies are used to ensure that each update is evaluated
efficiently, while the outer AO framework exploits recent developments in block
coordinate descent (BCD)-type methods which help ensure that every limit point
is a stationary point, as well as faster and more robust convergence in
practice. Three special cases are studied in detail: non-negative matrix/tensor
factorization, constrained matrix/tensor completion, and dictionary learning.
Extensive simulations and experiments with real data are used to showcase the
effectiveness and broad applicability of the proposed framework
A Survey on Matrix Completion: Perspective of Signal Processing
Matrix completion (MC) is a promising technique which is able to recover an
intact matrix with low-rank property from sub-sampled/incomplete data. Its
application varies from computer vision, signal processing to wireless network,
and thereby receives much attention in the past several years. There are plenty
of works addressing the behaviors and applications of MC methodologies. This
work provides a comprehensive review for MC approaches from the perspective of
signal processing. In particular, the MC problem is first grouped into six
optimization problems to help readers understand MC algorithms. Next, four
representative types of optimization algorithms solving the MC problem are
reviewed. Ultimately, three different application fields of MC are described
and evaluated.Comment: 12 pages, 9 figure
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