Tucker decomposition

The Tucker decomposition factors a tensor into a small core tensor multiplied along each mode by a factor matrix. For order 3, T = G x_1 U x_2 V x_3 W, where G is the core and U, V, W are matrices acting on the first, second, and third modes respectively (the x_n notation means multiply the core by that matrix along mode n). It is the multilinear analog of the SVD's T = U*Sigma*V^T, but with a full core in place of a diagonal.

Think of the factor matrices as choosing a better, lower-dimensional basis for each mode, and the core as the tensor expressed in those new bases. If U, V, W have r1, r2, r3 columns, the core is only r1-by-r2-by-r3, far smaller than the original. The list (r1, r2, r3) is the multilinear rank — the ranks of the three matrix unfoldings — and unlike CP rank it is easy to compute.

The standard recipe is the higher-order SVD (HOSVD): take an ordinary SVD of each mode's unfolding, keep the top singular vectors as the factor matrices, and form the core by projecting. Truncating those singular vectors gives an excellent compression that, while not the provably optimal Tucker approximation, comes with a clean quasi-optimality bound.

How does it relate to CP? CP forces the core to be diagonal (only superdiagonal entries nonzero), making it parsimonious and essentially unique but rigid and hard. Tucker allows a full, dense core, making it flexible, stable, and easy to compute, but NOT unique (you can rotate factor matrices and counter-rotate the core). Use CP to find interpretable factors; use Tucker to compress and denoise.