Tucker decomposition

The Tucker decomposition is a generalization of the matrix singular value decomposition (SVD) to higher-order tensors, introduced by Albert W. Tucker in the 1960s. While the SVD decomposes a matrix into a sum of rank-one matrices, the Tucker decomposition expresses a tensor as a core tensor multiplied by factor matrices along each mode. The factor matrices capture the latent structure of each dimension; the core tensor captures the interactions among dimensions. It is the foundational operation of multilinear algebra and the basis for tensor-based methods in machine learning, signal processing, and neuroscience.

The Tucker decomposition reveals that the complexity of high-dimensional data is not merely a matter of dimensionality but of multilinear rank — a measure of the information content that generalizes matrix rank to arbitrary dimensions. A tensor may have low multilinear rank even when its total number of elements is enormous, which means that the apparent complexity of the data is a surface effect and the underlying structure is far simpler. This insight is the basis for tensor completion, tensor regression, and tensor-based compression in data-intensive fields.

The Tucker decomposition is not merely a mathematical convenience. It is a claim about the ontology of structured data: that high-dimensional objects have a skeleton, and that the skeleton can be extracted. The claim is true for some data and false for others, and the field has not yet developed a theory of which data types have low multilinear rank and which do not. Until such a theory exists, tensor methods will remain an engineering practice rather than a science.