We firstly propose a new independence criterion that has linear-time complexity. We establish that our independence criterion is an upper bound of the Hirschfeld-Gebelein-Renyi maximum correlation coefficient between tested variables. A finite set of basis functions is employed to approximate the mapping functions that can achieve the maximal correlation. We show that the Finite Set Independence Criterion is closely related to our criterion. Benchmark experiments based on independent component analysis are performed with a comprehensive set of source distributions. The experimental results demonstrate that our independence criterion performs on-par with the state-of-the-art Hilbert-Schmidt Independence Criterion while being more computationally efficient. The experiments also validate the potential application of our criterion in the objective functions of deep neural networks. Secondly, we will propose a multi-channel speech separation algorithm applicable in the real world setting based on deep neural networks. The algorithm should be easily scalable with the number of microphones and capable of handling interfering reverberant speech, noisy background and intermittence during speech conversations. Finally we will propose a new disentanglement metric free of ground truth factors and develop a new disentanglement model that can exceed the state-of the-art methods according to the new metric.