# Blind source separation of tensor-valued time series

## TL;DR

This paper extends classic blind source separation methods to tensor-valued time series, providing more effective tools for analyzing high-dimensional data like fMRI and video.

## Contribution

It introduces tensorial generalizations of SOBI, FOBI, and JADE, demonstrating their theoretical properties and superior performance over existing methods.

## Key findings

- Tensorial methods outperform vector-based approaches in various settings.
- The methods are Fisher consistent and orthogonal equivariant.
- Applications to fMRI and video data successfully extract relevant information.

## Abstract

The blind source separation model for multivariate time series generally assumes that the observed series is a linear transformation of an unobserved series with temporally uncorrelated or independent components. Given the observations, the objective is to find a linear transformation that recovers the latent series. Several methods for accomplishing this exist and three particular ones are the classic SOBI and the recently proposed generalized FOBI (gFOBI) and generalized JADE (gJADE), each based on the use of joint lagged moments. In this paper we generalize the methodologies behind these algorithms for tensor-valued time series. We assume that our data consists of a tensor observed at each time point and that the observations are linear transformations of latent tensors we wish to estimate. The tensorial generalizations are shown to have particularly elegant forms and we show that each of them is Fisher consistent and orthogonal equivariant. Comparing the new methods with the original ones in various settings shows that the tensorial extensions are superior to both their vector-valued counterparts and to two existing tensorial dimension reduction methods for i.i.d. data. Finally, applications to fMRI-data and video processing show that the methods are capable of extracting relevant information from noisy high-dimensional data.