ADAGIO: Fast Data-aware Near-Isometric Linear Embeddings

TL;DR

ADAGIO is a scalable, data-aware linear embedding technique that preserves pairwise distances efficiently, outperforming previous methods in speed and maintaining high-quality near-isometries for large datasets.

Contribution

We introduce a simple, scalable, data-aware near-isometric linear embedding method with strong theoretical guarantees, significantly outperforming prior approaches like NuMax in speed and efficiency.

Findings

Our method is over 3,000 times faster than NuMax on medium datasets.

It achieves high-quality near-isometries with strong theoretical guarantees.

Experimental results confirm efficiency on real-world datasets.

Abstract

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.