hTensor- Multidimensional arrays and simple tensor computations.

MaintainerAlberto Ruiz <>




Common multidimensional array decompositions. See the paper by Kolda & Balder.



hosvd :: Array Double -> [Array Double]Source

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.

hosvd' :: Array Double -> ([Array Double], [(Int, Vector Double)])Source

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.

truncateFactors :: [Int] -> [Array Double] -> [Array Double]Source

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




:: (Int -> [Array Double])

Initialization function for each rank

-> ALSParam None Double

optimization parameters

-> Array Double

input array

-> [Array Double]


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



:: [Array Double]

starting point

-> ALSParam None Double

optimization parameters

-> Array Double

input array

-> ([Array Double], [Double])

factors and error history

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



:: Int


-> NArray i t

target array to decompose

-> Int


-> [NArray None Double]

random starting point

pseudorandom cp initialization from a given seed



:: [NArray None Double]

hosvd decomposition of the target array

-> Int


-> [NArray None Double]

starting point

cp initialization based on the hosvd


data ALSParam i t Source

optimization parameters for alternating least squares




nMax :: Int

maximum number of iterations

delta :: Double

minimum relative improvement in the optimization (percent, e.g. 0.1)

epsilon :: Double

maximum relative error. For nonhomogeneous problems it is the reconstruction error in percent (e.g. 1E-3), and for homogeneous problems is the frobenius norm of the expected zero structure in the right hand side.

post :: [NArray i t] -> [NArray i t]

post-processing function after each full iteration (e.g. id)

postk :: Int -> NArray i t -> NArray i t

post-processing function for the k-th argument (e.g. const id)

presys :: Matrix t -> Matrix t

preprocessing function for the linear systems (eg. id, or infoRank)

defaultParameters :: ALSParam i tSource

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