hTensor-0.7.0: Multidimensional arrays and simple tensor computations.

Stability provisional Alberto Ruiz

Numeric.LinearAlgebra.Array.Decomposition

Contents

Description

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

Synopsis

# HOSVD

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.

# CP

Arguments

 :: (Int -> [Array Double]) Initialization function for each rank -> ALSParam None Double optimization parameters -> Array Double input array -> [Array Double] factors

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
```

Arguments

 :: [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
```

Arguments

 :: Int seed -> NArray i t target array to decompose -> Int rank -> [NArray None Double] random starting point

pseudorandom cp initialization from a given seed

Arguments

 :: [NArray None Double] hosvd decomposition of the target array -> Int rank -> [NArray None Double] starting point

cp initialization based on the hosvd

# Utilities

data ALSParam i t Source

optimization parameters for alternating least squares

Constructors

 ALSParam FieldsnMax :: Intmaximum number of iterations delta :: Doubleminimum relative improvement in the optimization (percent, e.g. 0.1) epsilon :: Doublemaximum 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 tpost-processing function for the k-th argument (e.g. `const` `id`) presys :: Matrix t -> Matrix tpreprocessing function for the linear systems (eg. `id`, or `infoRank`)

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