      ## Welcome!

Higher-order tensor methods are intensively studied in many disciplines nowadays. The developments gradually allow us to move from classical vector and matrix based methods in applied mathematics and mathematical engineering to methods that involve tensors of arbitrary order. This step from linear transformations, quadratic and bilinear forms to polynomials and multilinear forms is pivotal for many applications in diverse disciplines. Furthermore, tensor methods have firm roots in multilinear algebra, algebraic geometry, numerical mathematics and optimization.

The uniqueness of canonical polyadic decomposition, under mild conditions which do not have a matrix counterpart, makes it a powerful tool for signal separation and data analysis. Multilinear singular value decomposition and low multilinear rank approximation are key in multi-way extensions of principal component analysis. Coupled decompositions provide a unifying framework for tackling intricate multimodal data fusion and inference tasks by breaking them in simpler pieces. Tensor trains and hierarchical Tucker decompositions allow one to break the curse of dimensionality in a numerically reliable manner and show promise for big data analytics in combination with compressed sensing.

The sessions consist of about 2 hours of theory and 1.5 hours of computer exercises each. For the exercises we will make use of Tensorlab.

List of topics:

• Tucker decomposition, multilinear singular value decomposition and low multilinear rank approximation
• Canonical polyadic decomposition and low rank approximation
• Constrained decompositions
• Tensor trains, hierarchical Tucker decomposition and cross-approximation
• Algebraic algorithms, alternating least squares, numerical optimization-based computation, randomized algorithms
• Multilinear principal component analysis
• Deterministic signal separation and independent component analysis
• Coupled decompositions and data fusion
• Mixture modeling, topic modeling, factorization machines
• Big data and curse of dimensionality