*Discovering latent variables in high dimensional high-order data.*

# Tensor Factorization Models

- Tucker Decomposition

- Unconstrained Tucker decomposition has rotational ambiguity.
- By imposing proper constraints on all/partial factors, we are able to extract multiple sets of unique components with specific physical meaning, which is also referred to as MBSS [PDF].
- As most existing nonnegative Tucker decomposition algorithms are very slow, we developed a quite fast one by incorporating dimensionality reduction techniques. [PDF][MATLAB code]

- CP Decomposition

- Under mild conditions the decomposition is essentially unique.
- Once at least one factor matrix has been correctly estimated, all the other factors can often be uniquely computed [PDF].
- By imposing proper constraints on all/partial factors, the property of uniqueness may be improved.

# Nonnegative Tensor Factorizations (Tucker model and CP model)

G. Zhou, A. Cichocki, Q. Zhao, S. Xie, Efficient Nonnegative Tucker Decompositions: Algorithms and Uniqueness,

*[ArXiv e-prints]: http://arxiv.org/abs/1404.4412*G. Zhou, A. Cichocki, Q. Zhao, S. Xie, Nonnegative Matrix and Tensor Factorizations: An algorithmic perspective,

*IEEE Signal Processing Magazine*, vol.31, no.3, pp.54--65, May 2014. [Pseudo-code]

- Nonnegative Matrix Factorization (NMF) based on
*Low-rank Approximation(LRA)*, including the Multiplicative Update (MU), the Accelerated Proximal Gradient (APG), and the Hierarchical ALS (HALS) method. - Nonnegative Tensor Factorization (NTF, i.e. CP Decompositions with Nonnegativity Constraints), include: [1] CP_HALS performs CPD with or without nonnegativity constraints; [2] FastNTFAPG performs NTF based on the LRA and accelerated proximal gradient methods; [3] lraNTF allows to select the update rule from MU/HALS/AGP.
- Nonnegative Tucker Decomposition (NTD) based on
*Low-rank Approximation*by Zhou [New!].