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 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.
low-rank matrix point-wise operator dimensionality reduction