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# Low-rank matrix approximation using point-wise operators

## Amini, A ; Sharif University of Technology

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- Type of Document: Article
- DOI: 10.1109/TIT.2011.2167714
- Abstract:
- 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 $ {bf A} n imes n be a matrix of rank $kll n$ and assume that the matrix $ {bf B} n imes n is generated by taking the elements of $ {bf A}$ to some real power $p$. In this paper, we show that based on the values of the data matrix $ {bf B}$, one can estimate the value $p$ and, therefore, the underlying low-rank matrix $ {bf A}$; i.e., we are reducing the dimensionality of $ {bf B}$ by using point-wise operators. Moreover, the estimation algorithm does not need to know the rank of $ {bf A}$. We also provide bounds on the quality of the approximation and validate the stability of the proposed algorithm with simulations in noisy environments
- Keywords:
- Low-rank matrix ; Data matrices ; Dimensional subspace ; Dimensionality reduction ; Estimation algorithm ; High dimensional data ; Highly nonlinear ; Linear methods ; Linear projections ; Low-dimensional structures ; Low-rank matrices ; Matrix ; Noisy environment ; Point-wise operator ; Real power ; SIMPLE algorithm ; Statistical pattern recognition ; Approximation algorithms ; Data compression ; Estimation ; Pattern recognition ; Wireless sensor networks ; Matrix algebra
- Source: IEEE Transactions on Information Theory ; Volume 58, Issue 1 , September , 2012 , Pages 302-310 ; 00189448 (ISSN)
- URL: http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6017118&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6017118