  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 {\em all} pairwise
distances between the data points and leverages the inherent geometric
structure that is typically present. Recently Hedge, Sankaranarayanan, Yin and
Baraniuk \cite{hedge2015} 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 a
state-of-the-art method \cite{hedge2015} 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 ($<$10
secs) is more than 3\,000$\times$ faster than the state-of-the-art method
\cite{hedge2015} ($>$9 hours) on medium scale datasets with 60\,000 data points
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.

Machine Learning;Machine Learning