Abstract. We consider the problem of embedding a subset of [Formula: see text] into a low-dimensional Hamming cube in an almost isometric way. We construct a simple, data-oblivious, and computationally efficient map that achieves this task with high probability; we first apply a specific structured random matrix, which we call the double circulant matrix; using that a matrix requires linear storage and matrix-vector multiplication that can be performed in near-linear time. We then binarize each vector by comparing each of its entries to a random threshold, selected uniformly at random from a well-chosen interval. We estimate the number of bits required for this encoding scheme in terms of two natural geometric complexity parameters of the set: its Euclidean covering numbers and its localized Gaussian complexity. The estimate we derive turns out to be the best that one can hope for, up to logarithmic terms. The key to the proof is a phenomenon of independent interest: we show that the double circulant matrix mimics the behavior of the Gaussian matrix in two important ways. First, it maps an arbitrary set in [Formula: see text] into a set of well-spread vectors. Second, it yields a fast near-isometric embedding of any finite subset of [Formula: see text] into [Formula: see text]. This embedding achieves the same dimension reduction as the Gaussian matrix in near-linear time, under an optimal condition—up to logarithmic factors—on the number of points to be embedded. This improves a well-known construction due to Ailon and Chazelle.

Neural networks with random weights appear in a variety of machine learning applications, most prominently as the initialization of many deep learning algorithms and as a computationally cheap alternative to fully learned neural networks. In the present article, we enhance the theoretical understanding of random neural networks by addressing the following data separation problem: under what conditions can a random neural network make two classes $\mathcal{X}^-, \mathcal{X}^+ \subset \mathbb{R}^d$ (with positive distance) linearly separable? We show that a sufficiently large two-layer ReLU-network with standard Gaussian weights and uniformly distributed biases can solve this problem with high probability. Crucially, the number of required neurons is explicitly linked to geometric properties of the underlying sets $\mathcal{X}^-, \mathcal{X}^+$ and their mutual arrangement. This instance-specific viewpoint allows us to overcome the usual curse of dimensionality (exponential width of the layers) in non-pathological situations where the data carries low-complexity structure. We quantify the relevant structure of the data in terms of a novel notion of mutual complexity (based on a localized version of Gaussian mean width), which leads to sound and informative separation guarantees. We connect our result with related lines of work on approximation, memorization, and generalization.