  Feature extraction for tensor data serves as an important step in many tasks
such as anomaly detection, process monitoring, image classification, and
quality control. Although many methods have been proposed for tensor feature
extraction, there are still two challenges that need to be addressed: 1) how to
reduce the computation cost for high dimensional and large volume tensor data;
2) how to interpret the output features and evaluate their significance. {The
most recent methods in deep learning, such as Convolutional Neural Network
(CNN), have shown outstanding performance in analyzing tensor data, but their
wide adoption is still hindered by model complexity and lack of
interpretability. To fill this research gap, we propose to use CP-decomposition
to approximately compress the convolutional layer (CPAC-Conv layer) in deep
learning. The contributions of our work could be summarized into three aspects:
(1) we adapt CP-decomposition to compress convolutional kernels and derive the
expressions of both forward and backward propagations for our proposed
CPAC-Conv layer; (2) compared with the original convolutional layer, the
proposed CPAC-Conv layer can reduce the number of parameters without decaying
prediction performance. It can combine with other layers to build novel deep
Neural Networks; (3) the value of decomposed kernels indicates the significance
of the corresponding feature map, which provides us with insights to guide
feature selection.
