Many important applications, including signal reconstruction, parameter estimation, and signal processing in a compressed domain, rely on a low-dimensional representation of the dataset that preserves all pairwise distances between the data points and leverages the inherent geometric structure that is typically present. Hedge, Sankaranarayanan, Yin and Baraniuk proposed the first data-aware near-isometric linear embedding which achieves the best of both worlds. However, their method NuMax does not scale to large-scale datasets.
Our main contribution is a simple, data-aware, near-isometric linear dimensionality reduction method which significantly outperforms NuMax with respect to scalability while achieving high quality near-isometries. Furthermore, our method comes with strong worst-case theoretical guarantees that allow us to guarantee the quality of the obtained near-isometry. We verify experimentally the efficiency of our method on numerous real-world datasets, where we find that our method (<10 secs) is more than 3,000 faster than NuMax ($>$9 hours) on medium scale datasets with 60,000 datapoints in 784 dimensions. Finally, we use our method as a preprocessing step to increase the computational efficiency of a classification application and for speeding up approximate nearest neighbor queries.
Python code is available at Github.