A short list of popular frameworks:
I'm curious about one thing -- now that we discussed that ASIC can provide high performance with low power consumption, is it possible to use ASICs to assist neural network computation / training? I think it's viable since quantization is pretty common in neural networks, and the input data is sometimes 8 or 16-bit integers only. Since smartphones are already putting ASICs for photo processing, why not put another for, say, matrix multiplication for neural networks, such that phones can support much smarter pattern recognition?
There are ASICs for neural network computation! Here is an example: http://www.cognimem.com/products/chips-and-modules/CM1K-Chip/