We establish a generic reduction from nonlinear spectral gaps of metric spaces to data-dependent Locality-Sensitive Hashing, yielding a new approach to the high-dimensional Approximate Near Neighbor Search problem (ANN) under various distance functions. Using this reduction, we obtain the following results: For general d-dimensional normed spaces and n-point datasets, we obtain a cell-probe ANN data structure with approximation O(log ε2 d), space dO(1)n1+ε, and dO(1)nε cell probes per query, for any > 0. No non-trivial approximation was known before in this generality other than the O(d) bound which follows from embedding a general norm into ℓ2. For ℓp and Schatten-p norms, we improve the data structure further, to obtain approximation O(p) and sublinear query time. For ℓp, this improves upon the previous best approximation 2O(p) (which required polynomial as opposed to near-linear in n space). For the Schatten-p norm, no non-trivial ANN data structure was known before this work. Previous approaches to the ANN problem either exploit the low dimensionality of a metric, requiring space exponential in the dimension, or circumvent the curse of dimensionality by embedding a metric into a “tractable” space, such as ℓ1. Our new generic reduction proceeds differently from both of these approaches using a novel partitioning method.