Multilinear Singular Value Decomposition (or Tucker's method, see Lathauwer et al.).

The result is a list with the core (head) and rotations so that t == product (hsvd t).

The core and the rotations are truncated to the rank of each mode.

Use hosvd' to get full transformations and rank information about each mode.

Full version of hosvd.

The first element in the result pair is a list with the core (head) and rotations so that t == product (fst (hsvd' t)).

The second element is a list of rank and singular values along each mode, to give some idea about core structure.

Truncate a hosvd decomposition from the desired number of principal components in each dimension.

Experimental implementation of the CP decomposition, based on alternating least squares. We try approximations of increasing rank, until the relative reconstruction error is below a desired percent of Frobenius norm (epsilon).

The approximation of rank k is abandoned if the error does not decrease at least delta% in an iteration.

Practical usage can be based on something like this:

cp finit d e t = cpAuto (finit t) defaultParameters {delta = d, epsilon = e} t

cpS = cp (InitSvd . fst . hosvd')
cpR s = cp (cpInitRandom s)

So we can write

 -- initialization based on hosvd
y = cpS 0.01 1E-6 t

 -- (pseudo)random initialization
z = cpR seed 0.1 0.1 t

Basic CP optimization for a given rank. The result includes the obtained sequence of errors.

For example, a rank 3 approximation can be obtained as follows, where initialization is based on the hosvd:

(y,errs) = cpRank 3 t
     where cpRank r t = cpRun (cpInitSvd (fst $ hosvd' t) r) defaultParameters t

pseudorandom cp initialization from a given seed

cp initialization based on the hosvd

optimization parameters for alternating least squares

nMax = 20, epsilon = 1E-3, delta = 1, post = id, postk = const id, presys = id