Algorithms

Algorithms for Generalized CP Decomposition.

AbstractAlgorithm

Abstract type for GCP algorithms.

Concrete types ConcreteAlgorithm <: AbstractAlgorithm should implement _gcp!(M, X, loss, constraints, algorithm::ConcreteAlgorithm) that modifies the initialization M and returns the modified version.

gcp_objective(M::CPD, X::AbstractArray, loss)

Compute the GCP objective function for the model tensor M, data tensor X, and loss function loss.

gcp_grad_U!(GU, M::CPD, X::AbstractArray, loss)

Compute the GCP gradient with respect to the factor matrices U = (U[1],...,U[N]) for the model tensor M, data tensor X, and loss function loss, and store the result in GU = (GU[1],...,GU[N]).

ALS

Alternating Least Squares. Workhorse algorithm for LeastSquares loss with no constraints.

Algorithm parameters:

• maxiters::Int : max number of iterations (default: 200)

FastALS

Fast Alternating Least Squares.

Efficient ALS algorithm proposed in:

Fast Alternating LS Algorithms for High Order CANDECOMP/PARAFAC Tensor Factorizations. Anh-Huy Phan, Petr Tichavský, Andrzej Cichocki. IEEE Transactions on Signal Processing, 2013. DOI: 10.1109/TSP.2013.2269903

Algorithm parameters:

• maxiters::Int : max number of iterations (default: 200)

LBFGSB

Limited-memory BFGS with Box constraints.

Algorithm parameters:

• m::Int : max number of variable metric corrections (default: 10)
• factr::Float64 : function tolerance in units of machine epsilon (default: 1e7)
• pgtol::Float64 : (projected) gradient tolerance (default: 1e-5)
• maxfun::Int : max number of function evaluations (default: 15000)
• maxiter::Int : max number of iterations (default: 15000)
• iprint::Int : verbosity (default: -1)
  • iprint < 0 means no output
  • iprint = 0 prints only one line at the last iteration
  • 0 < iprint < 99 prints f and |proj g| every iprint iterations
  • iprint = 99 prints details of every iteration except n-vectors
  • iprint = 100 also prints the changes of active set and final x
  • iprint > 100 prints details of every iteration including x and g

Notes:

• this algorithm only supports Float64 numbers

See documentation of LBFGSB.jl for more details.

FastALS_iter!(X, U, λ)

Algorithm for computing MTTKRP sequences is from "Fast Alternating LS Algorithms for High Order CANDECOMP/PARAFAC Tensor Factorizations" by Phan et al., specifically section III-C.

gcp_grad_U!(GU, M::CPD, X::AbstractArray, loss)

Compute the GCP gradient with respect to the factor matrices U = (U[1],...,U[N]) for the model tensor M, data tensor X, and loss function loss, and store the result in GU = (GU[1],...,GU[N]).

gcp_objective(M::CPD, X::AbstractArray, loss)

Compute the GCP objective function for the model tensor M, data tensor X, and loss function loss.