ADAGIO: Fast Data-aware Near-Isometric Linear Embeddings
This provides a scalable solution for applications like signal reconstruction and classification, though it is incremental as it builds on prior work to improve speed.
The paper tackles the problem of creating fast, data-aware near-isometric linear embeddings for dimensionality reduction, achieving over 3,000x speedup compared to the state-of-the-art method on medium-scale datasets, with strong theoretical guarantees.
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.