Each hidden layer in neural networks can be formulated as one HOPE model [1]

• The HOPE model combines a linear orthogonal projection and a mixture model under a unified generative modelling framework;
• The HOPE model can be used as a novel tool to probe why and how NNs work;
• The HOPE model provides several new learning algorithms to learn NNs in either supervised or unsupervised ways.

Reference:
[1] Shiliang Zhang and Hui Jiang, “Hybrid Orthogonal Projection and Estimation (HOPE): A New Framework to Probe and Learn Neural Networks,” arXiv:1502.00702.

[2] Shiliang Zhang, Hui Jiang, Lirong Dai, “The New HOPE Way to Learn Neural Networks”, Proc. of Deep Learning Workshop at ICML 2015, July 2015. (paper)

[3] S. Zhang, H. Jiang, L. Dai, “ Hybrid Orthogonal Projection and Estimation (HOPE): A New Framework to Learn Neural Networks,” Journal of Machine Learning Research (JMLR), 17(37):1−33, 2016.

Software:
The matlab codes to reproduce the MNIST results in [1,2] can be downloaded here.