Final Report Summary - DECODA (Tensor Decomposition for Data Analysis with Applications to Health and Environment)
The problem addressed in the DECODA project is related to the approximation of a function of several variables by a sum of functions with separate variables. For instance, if a two-variate function is discretized, the problem consists of approximating a matrix by another of lower rank, an old problem in linear algebra. The latter has been solved long ago by resorting to the SVD and now permits, for instance, to complete low-rank data matrices when entries are missing. However, a matrix cannot be approximated in a unique way by a sum of rank-one terms. In addition, things are much less obvious for functions in more than two variables, which is the framework of the project.
When discretized, such arrays of numbers have more than two ways, and are seen as tensors. At that point, one can emphasize striking differences with matrices, including the following: (i) the rank is often larger than the dimensions, (ii) the best low-rank approximation may not exist, (iii) when it exists, its approximate decomposition into a sum of rank-one terms is generally unique.
Property (ii) has been extensively studied in the project. In particular, when data are real and nonnegative, the problem becomes well-posed if all terms in the decomposition are also assumed nonnegative real, which makes sense in many application fields (e.g. chemical concentrations, light spectra, graphs, probabilities, social networks, etc). Property (iii) has been proved during the course of the project; it is very important because it restores identifiability of rank-one contributions, which was lost with matrices, and opens the door to a large realm of applications.
Beside the above mentioned mathematical contributions, several applications have been explored: (a) localization of brain sources via non-invasive EEG, useful to heal epilepsy or Alzheimer diseases, (b) antenna array processing, e.g. localization of seismic sources during icequakes, (c) unsupervised detection of toxic molecules in liquids at low concentrations by fluorescence spectroscopy, (d) observation of the earth surface at a sub-pixel level by hyperspectral imaging, or (e) detection and classification of toxic or odorous molecules in the ambiant air. The latter application is currently the subject of further studies.
When discretized, such arrays of numbers have more than two ways, and are seen as tensors. At that point, one can emphasize striking differences with matrices, including the following: (i) the rank is often larger than the dimensions, (ii) the best low-rank approximation may not exist, (iii) when it exists, its approximate decomposition into a sum of rank-one terms is generally unique.
Property (ii) has been extensively studied in the project. In particular, when data are real and nonnegative, the problem becomes well-posed if all terms in the decomposition are also assumed nonnegative real, which makes sense in many application fields (e.g. chemical concentrations, light spectra, graphs, probabilities, social networks, etc). Property (iii) has been proved during the course of the project; it is very important because it restores identifiability of rank-one contributions, which was lost with matrices, and opens the door to a large realm of applications.
Beside the above mentioned mathematical contributions, several applications have been explored: (a) localization of brain sources via non-invasive EEG, useful to heal epilepsy or Alzheimer diseases, (b) antenna array processing, e.g. localization of seismic sources during icequakes, (c) unsupervised detection of toxic molecules in liquids at low concentrations by fluorescence spectroscopy, (d) observation of the earth surface at a sub-pixel level by hyperspectral imaging, or (e) detection and classification of toxic or odorous molecules in the ambiant air. The latter application is currently the subject of further studies.