A mathematical model is called sparse if it is a combination of only a few non-zero terms. In sparse interpolation, the aim is to determine both the support of the sparse linear combination and the scalar coefficients in the representation, from a small or minimal amount of data samples. Sparse techniques solve the problem statement from a number of samples proportional to the number of terms in the representation rather than the number of available data points or available generating elements. Sparse representations reduce the complexity in several ways: data collection, algorithmic complexity, model complexity.
In this tutorial, we introduce sparse interpolation. We indicate the connections between sparse interpolation, generalized eigenvalue computation, exponential analysis, rational approximation, and tensor decomposition. In the past few years, insight gained from the computer algebra community combined with methods developed by the numerical analysis community, has led to significant progress in several very practical and real-life signal processing applications. We make use of tools such as the singular value decomposition and various convergence results for Padé approximants to regularize an otherwise inverse problem. Classical resolution limitations in signal processing with respect to frequency and decay rates, are overcome.
We particularly focus on multi-exponential models. These models appear, for instance, in transient detection, motor fault diagnosis, electrophysiology, magnetic resonance and infrared spectroscopy, vibration analysis, fluorescence lifetime imaging, music signal processing, direction of arrival estimation in wireless communication systems, dynamic spectrum management such as in cognitive radio, and so on. The connection with tensor decomposition leads to new possibilities to exploit sparsity in analyzing tensor-structured datasets.