Codes of implemented CANDECOMP methods.

Helps to maintain the pool of reusable arrays of different sizes and reduce the burden on garbage collection.

Canonical polyadic N-mode tensor decomposition (CANDECOMP/PARAFAC).

CUR (columns-U-rows) 3-tensor decomposition.

Parallel Factor Analysis (PARAFAC) decomposition

Helper object for spnntucker().

State of sparse (semi-)nonnegative Tucker decomposition

Sparse N-mode array.

vals contains all (L) non-zero elements. pos is NxL matrix of non-zero elements indices. dims is NTuple of array dimensions.

Convert dense N-dimensional array into N-dimensional a SparseArray.

Abstract decomposition of N-mode tensor.

Tucker decomposition of a N-mode tensor.

Returns CANDECOMP rank.

Computes CANDECOMP by ALS (Alternating Least Squares) method.

Computes CANDECOMP by SGSD (Simultaneous Generalized Schur Decomposition) method.

Stub for non-implemented CANDECOMP algorithms.

Checks the validity of the core tensor dimensions, where core tensor is r^N hypercube.

Checks the validity of the core tensor dimensions.

Unfolds the tensor into matrix such that the specified mode becomes matrix column.

Generates random factor matrices for Tucker/CANDECOMP decompositions if core tensor is r^N hypercube.

Returns:

• a vector of N (orig[n], r)-sized matrices

Generates random factor matrices for Tucker/CANDECOMP etc decompositions.

• orig_dims original tensor dimensions
• core_dims core tensor dimensions

Returns:

• a vector of N (orig[n], core[n])-sized matrices

Unfolds the tensor into matrix such that the specified mode becomes matrix row.

Unfolds the tensor into matrix, such that the specified group of modes becomes matrix rows and the other one becomes columns.

• row_modes vector of modes to be unfolded as rows
• col_modes vector of modes to be unfolded as columns

Gets an array of specific size from the pool. The returned array should be returned back to the pool using release!().

Calculates canonical polyadic tensor decomposition (CANDECOMP/PARAFAC).

Returns: CANDECOMP object

Re-composes the tensor from CANDECOMP decomposition.

Composes a full tensor from Tucker decomposition.

Returns the core tensor of Tucker decomposition.

Returns the factor matrices of Tucker decomposition.

High-order singular value decomposition (HO-SVD).

khatrirao!(dest::AbstractMatrix{T}, mtxs::NTuple{N, <:AbstractMatrix{T}})

In-place Khatri-Rao matrices product (column-wise Kronecker product) calculation.

khatrirao(mtxs::NTuple{N, <:AbstractMatrix{T}})

Calculates Khatri-Rao product of a sequence of matrices (column-wise Kronecker product).

Non-negative CANDECOMP tensor decomposition.

PARAFAC2 model

Returns relative tensor decomposition error, NaN if not available.

Returns relative error between the re-composed and the original tensors.

Returns relative error between re-composed and the original tensor.

Releases an array returned by acquire!() back into the pool.

Scale the factors and core of the initial decomposition. Each decompositon component is scaled proportional to the number of its elements. After scaling, |decomp|=s

Sparse (semi-)nonnegative Tucker decomposition

Decomposes nonnegative tensor tnsr into optionally nonnegative core tensor and sparse nonnegative factor matrices factors.

• tnsr nonnegative N-mode tensor to decompose
• core_dims size of a core densor
• core_nonneg if true, the output core tensor is nonnegative
• tol the target error of decomposition relative to the Frobenius norm of tnsr
• max_iter maximum number of iterations if error stays above tol
• max_time max running time
• lambdasN+1 vector of non-negative sparsity regularizer coefficients for the factor matrices and the core tensor
• Lmin lower bound for Lipschitz constant for the gradients of the residual error eqn{l(Z,U) = fnorm(tnsr - ttl(Z, U))byZand eachU`
• rw controls the extrapolation weight
• boundsN+1 vector of the maximal absolute values allows for the elements of core tensor and factor matrices (effective only if the regularization is disabled)
• ini_decomp initial decomposition, if equals to :hosvd, hosvd() is used to generate the starting decomposition, if nothing, a random decomposition is used
• verbose more output algorithm progress

Returns:

• Tucker decomposition object with additional properties:

* `:converged` method convergence indicator
* `:rel_residue` the Frobenius norm of the residual error `l(Z,U)` plus regularization penalty (if any)
* `:niter` number of iterations
* `:nredo` number of times `core` and `factor` were recalculated to avoid the increase in objective function
* `:iter_diag` convergence info for each iteration, see `SPNNTuckerState`

The function uses the alternating proximal gradient method to solve the following optimization problem: deqn{min 0.5 |tnsr - Z times1 U1 ldots timesK UK |{F^2} + sum{n=1}^{K} lambdan |Un|1 + lambda{K+1} |Z|1, ;text{where; Z geq 0, Ui geq 0.} If core_nonneg is FALSE, core tensor Z is allowed to have negative elements and eqn{z{i,j}=max(0,z{i,j}-lambda{K+1}/L{K+1}}) rule is replaced by eqn{z{i,j}=sign(z{i,j})max(0,|z{i,j}|-lambda{K+1}/L_{K+1})}. The method stops if either the relative improvement of the error is below the tolerance tol for 3 consequitive iterations or both the relative error improvement and relative error (wrt the tnsr norm) are below the tolerance. Otherwise it stops if the maximal number of iterations or the time limit were reached.

The implementation is based on ntds_fapg() MATLAB code by Yangyang Xu and Wotao Yin.

See Y. Xu, "Alternating proximal gradient method for sparse nonnegative Tucker decomposition", Math. Prog. Comp., 7, 39-70, 2015. See http://www.caam.rice.edu/~optimization/bcu/`

SS-HOPM (Shifted Symmetric Higher-order Power Method) for computing tensor eigenpars.

Contract N-mode tensor and M matrices.

• tensor tensor to contract
• matrices matrices to contract
• modes corresponding modes of matrices to contract
• transpose if true, matrices are contracted along their columns

Contract N-mode tensor and M matrices.

• dest array to hold the result
• src source tensor to contract
• matrices matrices to contract
• modes corresponding modes of matrices to contract
• transpose if true, matrices are contracted along their columns

Calculates CUR decomposition for 3-mode tensors.