Ke Wang

Jian-Feng Cai, Dong Li, Jiaze Sun et al.

Random projections are able to perform dimension reduction efficiently for datasets with nonlinear low-dimensional structures. One well-known example is that random matrices embed sparse vectors into a low-dimensional subspace nearly isometrically, known as the restricted isometric property in compressed sensing. In this paper, we explore some applications of random projections in deep neural networks. We provide the expressive power of fully connected neural networks when the input data are sparse vectors or form a low-dimensional smooth manifold. We prove that the number of neurons required for approximating a Lipschitz function with a prescribed precision depends on the sparsity or the dimension of the manifold and weakly on the dimension of the input vector. The key in our proof is that random projections embed stably the set of sparse vectors or a low-dimensional smooth manifold into a low-dimensional subspace. Based on this fact, we also propose some new neural network models, where at each layer the input is first projected onto a low-dimensional subspace by a random projection and then the standard linear connection and non-linear activation are applied. In this way, the number of parameters in neural networks is significantly reduced, and therefore the training of neural networks can be accelerated without too much performance loss.

Jiaxi Tang, Ke Wang

We propose a novel way to train ranking models, such as recommender systems, that are both effective and efficient. Knowledge distillation (KD) was shown to be successful in image recognition to achieve both effectiveness and efficiency. We propose a KD technique for learning to rank problems, called \emph{ranking distillation (RD)}. Specifically, we train a smaller student model to learn to rank documents/items from both the training data and the supervision of a larger teacher model. The student model achieves a similar ranking performance to that of the large teacher model, but its smaller model size makes the online inference more efficient. RD is flexible because it is orthogonal to the choices of ranking models for the teacher and student. We address the challenges of RD for ranking problems. The experiments on public data sets and state-of-the-art recommendation models showed that RD achieves its design purposes: the student model learnt with RD has a model size less than half of the teacher model while achieving a ranking performance similar to the teacher model and much better than the student model learnt without RD.

Sean O'Rourke, Van Vu, Ke Wang

The Davis-Kahan-Wedin $\sin Θ$ theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic version of the Davis-Kahan-Wedin $\sin Θ$ theorem when the perturbation is a Gaussian random matrix. Under certain structural assumptions, we obtain an optimal bound that significantly improves upon the classic Davis-Kahan-Wedin $\sin Θ$ theorem. One of our key tools is a new perturbation bound for the singular values, which may be of independent interest.