Dimensionality reduction with subgaussian matrices: a unified theory
This work provides a general framework for dimensionality reduction, impacting fields like machine learning and signal processing, though it is incremental as it builds on existing restricted isometry and Johnson-Lindenstrauss results.
The paper presents a unified theory for Euclidean dimensionality reduction using subgaussian matrices, recovering and improving results for sparse vectors, low-rank matrices, tensors, and smooth manifolds, and establishes a new Johnson-Lindenstrauss embedding for infinite unions of subspaces.
We present a theory for Euclidean dimensionality reduction with subgaussian matrices which unifies several restricted isometry property and Johnson-Lindenstrauss type results obtained earlier for specific data sets. In particular, we recover and, in several cases, improve results for sets of sparse and structured sparse vectors, low-rank matrices and tensors, and smooth manifolds. In addition, we establish a new Johnson-Lindenstrauss embedding for data sets taking the form of an infinite union of subspaces of a Hilbert space.