Last year, I have read JM Landsberg's beautiful book Tensors: Geometry and Applications, especially because I was eager to learn about algebro-geometric settings for the exponent of matrix multiplication. This led to JM pairing me with his former student Luke Oeding to work on a contribution for the workshop Questions in geometry arising in the sciences. Luke and I met for a few days at Fraunhofer IAIS last summer and identified the following problem to be of mutual interest:
Matrices are at the heart of many computational techniques we are using in multimedia pattern recognition. In audio signal processing we have spectrogram matrices and spectral dictionaries as two examples. Typically, these matrices have more structure than just coming from a complete matrix group. As there is only a limited number of factors or sources contributing, these matrices should be of low rank, although this information may be hidden in noise. Thus, a mathematical question arising is how we can characterise varieties of matrices of the form low rank plus noise. Can we detect this structure efficiently?