Low-Rank Matrix Approximation Using Point-Wise Operators

The problem of extracting low-dimensional structure from high-dimensional data arises in many applications such as machine learning, statistical pattern recognition, wireless sensor networks, and data compression. If the data is restricted to a lower dimensional subspace, then simple algorithms usin...

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Bibliographic Details
Published in:IEEE transactions on information theory 2012-01, Vol.58 (1), p.302-310
Main Authors: Amini, Arash, Karbasi, A., Marvasti, F.
Format: Article
Language:English
Subjects:
Algorithms
Applied sciences
Approximation
Approximation methods
Artificial intelligence
Coding, codes
Dimensionality reduction
Estimates
Estimating techniques
Exact sciences and technology
Information theory
Information, signal and communications theory
low-rank matrix
Mathematical analysis
Matrix decomposition
Nonlinearity
Operators
Pattern recognition
point-wise operator
Polynomials
Services and terminals of telecommunications
Signal and communications theory
Signal processing
Simulation
Subspaces
Systems, networks and services of telecommunications
Taylor series
Telecommunications
Telecommunications and information theory
Telemetry. Remote supervision. Telewarning. Remote control
Tin
Tomography
Ultrasonic imaging
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Summary:The problem of extracting low-dimensional structure from high-dimensional data arises in many applications such as machine learning, statistical pattern recognition, wireless sensor networks, and data compression. If the data is restricted to a lower dimensional subspace, then simple algorithms using linear projections can find the subspace and consequently estimate its dimensionality. However, if the data lies on a low-dimensional but nonlinear space (e.g., manifolds), then its structure may be highly nonlinear and, hence, linear methods are doomed to fail. In this paper, we introduce a new technique for dimensionality reduction based on point-wise operators. More precisely, let be a matrix of rank and assume that the matrix is generated by taking the elements of to some real power . In this paper, we show that based on the values of the data matrix , one can estimate the value and, therefore, the underlying low-rank matrix ; i.e., we are reducing the dimensionality of by using point-wise operators. Moreover, the estimation algorithm does not need to know the rank of . We also provide bounds on the quality of the approximation and validate the stability of the proposed algorithm with simulations in noisy environments.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2011.2167714