Paper:
Structured Transforms for Small-Footprint Deep Learning
We consider the task of building compact deep learning pipelines suitable for deployment on storage and power constrained mobile devices. We propose a unified framework to learn a broad family of structured parameter matrices that are characterized by the notion of low displacement rank. Our structured transforms admit fast function and gradient evaluation, and span a rich range of parameter sharing configurations whose statistical modeling capacity can be explicitly tuned along a continuum from structured to unstructured. Experimental results show that these transforms can significantly accelerate inference and forward/backward passes during training, and offer superior accuracy-compactness-speed tradeoffs in comparison to a number of existing techniques. In keyword spotting applications in mobile speech recognition, our methods are much more effective than standard linear low-rank bottleneck layers and nearly retain the performance of state of the art models, while providing more than 3.5-fold compression.

Review:
In this paper, the author proposes a method for fast matrix-vector multiplication and gradient computation for structured matrices. The notion of structured matrices is a generalized version of the Toeplitz matrix. Nevertheless, the technique of this paper can also be generalized to capture Vandermonde and Cauchy matrices.The main idea behind the methodology is to represent the matrix as a product of Z_1(G) and Z_{-1}(H), where both G and H are tall matrices. This is a generalization of the low-rank factorization. The author shows that this new representation not only captures the low-rank matrices, but also the product, inverse and linear combination of Toeplitz matrices. It was also shown that the matrix-vector product can be computed efficiently through a small number of FFTs, so that it is order-of-magnitude faster than dense matrix-vector multiplication.The idea is clever and is promising in accelerating the training of neural networks. It is also useful for compressing the representation of a neural network on mobile devices.Here are some comment:1. Toeplitz is closely related to one-dimensional convolution, but in practice use the convolution is usually two-dimensional. It means that using the Toeplitz representation may lose some information captured by the ordinary CNN. To see the impact of this loss, it is important to include ordinary CNN as a baseline algorithm, and compare algorithms on datasets other than MNIST. Although one dimensional convolution may be sufficient for MNIST classification. It may not be the case for natural images.2. It is somewhat surprising that when n = 512, the implementation proposed in this paper is almost surely slower than the dense matrix multiplication. The author may want to discuss a little why it happens. It seems that the BLAS implementation of dense matrix-vector multiplication is 100x faster for executing task of the same computation complexity?3. In mobile search recognition, what is the specific algorithm for training these models? Does the author train a fully connected network at first, then shrink it to be a smaller one, or does the author train the smaller model directly? If it was the first case, then what is the algorithm for shrinking the network?