Coseparable Nonnegative Tensor Factorization With T-CUR Decomposition
Juefei Chen Longxiu Huang Yimin Wei
Nonnegative Matrix Factorization (NMF) is an important unsupervised learning
method to extract meaningful features from data. To address the NMF problem
within a polynomial time framework, researchers have introduced a separability
assumption, which has recently evolved into the concept of coseparability. This
advancement offers a more efficient core representation for the original data.
However, in the real world, the data is more natural to be represented as a
multi-dimensional array, such as images or videos. The NMF's application to
high-dimensional data involves vectorization, which risks losing essential
multi-dimensional correlations. To retain these inherent correlations in the
data, we turn to tensors (multidimensional arrays) and leverage the tensor
t-product. This approach extends the coseparable NMF to the tensor setting,
creating what we term coseparable Nonnegative Tensor Factorization (NTF). In
this work, we provide an alternating index selection method to select the
coseparable core. Furthermore, we validate the t-CUR sampling theory and
integrate it with the tensor Discrete Empirical Interpolation Method (t-DEIM)
to introduce an alternative, randomized index selection process. These methods
have been tested on both synthetic and facial analysis datasets. The results
demonstrate the efficiency of coseparable NTF when compared to coseparable NMF.