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We propose a fast, distance-preserving, binary embedding algorithm totransform a high-dimensional dataset $\mathcal{T}\subseteq\mathbb{R}^n$ intobinary sequences in the cube $\{\pm 1\}^m$. When $\mathcal{T}$ consists ofwell-spread (i.e., non-sparse) vectors, our embedding method applies a stablenoise-shaping quantization scheme to $A x$ where $A\in\mathbb{R}^{m\times n}$is a sparse Gaussian random matrix. This contrasts with most binary embeddingmethods, which usually use $x\mapsto \mathrm{sign}(Ax)$ for the embedding.Moreover, we show that Euclidean distances among the elements of $\mathcal{T}$are approximated by the $\ell_1$ norm on the images of $\{\pm 1\}^m$ under afast linear transformation. This again contrasts with standard methods, wherethe Hamming distance is used instead. Our method is both fast and memoryefficient, with time complexity $O(m)$ and space complexity $O(m)$. Further, weprove that the method is accurate and its associated error is comparable tothat of a continuous valued Johnson-Lindenstrauss embedding plus a quantizationerror that admits a polynomial decay as the embedding dimension $m$ increases.Thus the length of the binary codes required to achieve a desired accuracy isquite small, and we show it can even be compressed further without compromisingthe accuracy. To illustrate our results, we test the proposed method on naturalimages and show that it achieves strong performance.

Faster Binary Embeddings for Preserving Euclidean Distances