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

Enhanced Expressive Power and Fast Training of Neural Networks by Random Projections