TensorBox

TENSORBOX is a Matlab package containing state-of-the-art algorithms for decompositions of multiway array data into rank-1 tensors such as

• CANDECOMP/PARAFAC
• Tucker decomposition
• Generalized Kronecker tensor decomposition
• Tensor deconvolution
• Tensor train decomposition
• Best rank-1 tensor approximation
• Tensor decomposition with given error bound (or the denoising problem)
• Application for BSS, harmonic retrieval, image denoising, image completion, dictionary learning, ...

Algorithms for CANDECOMP/PARAFAC decomposition (CPD)

• FastALS : fast ALS algorithm employs the fast method to compute CP gradients. Other alternating and all-at-once algorithms for CPD can be accelerated using the similar method. See HALS, QALS, MLS, fLM and subfunction of FastALS: cp_gradient.

• HALS hierarchical ALS algorithm for nonnegative CPD (NCPD)

• QALS recursive ALS algorithm for nonnegative CPD (NCPD)

• MLS multiplicative algorithm for NCPD

• fLM fast damped Gauss-Newton or Levenberg-Marquardt algorithm for CPD and NCPD (see cp_fLMa and cpx_fLMa, ncp_fLM)

• FCP fast algorithm for higher order CPD through tensor unfolding

• XGRAM extended generalized rank annihilation method (GRAM) and direct trilinear decomposition (DLTD) to higher order CPD

• CPO-ALS1 ALS algorithm for CPD with column-wise orthogonal factor

• CPO-ALS2 ALS algorithm for CPD with column-wise orthogonal factor

• CRIB Cramer-Rao Induced Bound for CPD.

Algorithms for Tensor Deflation and Rank-1 tensor extraction

• ASU Alternating Subspace update. The algorithm extracts a rank-1 tensor from a rank-R tensor, i.e., deflation. It can be used to sequentially decompose a rank-R tensor over R rank-1 tensor extraction.

• CRB for the tensor deflation

Algorithms for Tucker decomposition (TD)

• HALS hierarchical ALS algorithm for nonnegative TD.
• LM Levenberg-Marquardt algorithm with log-barrier function.
• LMPU A simplified version of the LM algorithm which updates only one factor matrix and a core tensor at a time.
• O2LB A simplified version of the LM algorithm which updates only one factor matrix or a core tensor at a time.
• CrNc Crank-Nicholson algorithm for orthogonal Tucker decomposition.

Algorithms for Error Preserving Correction method

which solves the optimization problem min sum_r |norm_of_rank-1_tensor_r-th|_F^2 s.t. |Y - X|_F^2 <= error_bound

• ANC alternating correction algorithm
• SQP sequential QP algorithm
• ITP Iterior point algorithm

Algorithms for CPD with bound constraint

which solves the optimization problem min |Y - X|_F^2 s.t. sum_r |norm_of_rank-1_tensor_r-th|_F^2 <= error_bound

• ANC alternating correction algorithm
• SQP sequential QP algorithm
• ITP Iterior point algorithm

Algorithms for best rank-1 tensor approximation

• Closed form expression to find best rank-1 for tensor 2x2x2

• Closed form expression to find best rank-1 for tensor 2x2x2x2

• LM iterative algorithm with optimal damping parameter

• RORO rotational rank-1 tensor approximation

Algorithms for rank-(L o M) block term decompositions

• bcdLoR_als

Algorithms for tensor denoising

i.e. min rank(C) s.t. |Y - X|_F^2 <= error_bound

Algorithms for Tensor train decomposition

• ASCU alternating single core update
• ADCU alternating double core update
• A3CU alternating trible core update

Algorithms for Tensor to CPD conversion

• Exact conversion between CPS and TT tensors
• Iterative algorithms for fiting a CP tensor to a TT-tensor

Algorithms for BSS based on Tensor network decomposition

• Exact conversion between CPS and TT tensors
• Iterative algorithms for fiting a CP tensor to a TT-tensor

Algorithms for Rank-1 Tensor Deconvolution

Algorithms for generalized Kronecker Tensor decomposition

with low-rank constraint and sparsity constraints

Examples for image denoising and completion

Algorithm for Quadratic Programming over sphere (QPS)