Embedding-based Representation of Signal Geometry

Low-dimensional embeddings have emerged as a key component in modern signal processing theory and practice. In particular, embeddings transform signals in a way that preserves their geometric relationship but makes processing more convenient. The literature has, for the most part, focused on lowering the dimensionality of the signal space while preserving distances between signals. However, there has also been work exploring the effects of quantization, as well as on transforming geometric quantities, such as distances and inner products, to metrics easier to compute on modern computers, such as the Hamming distance. Embeddings are particularly suited for modern signal processing applications, in which the fidelity of information represented by the signals is of interest, instead of the fidelity of the signal itself. Most typically, this information is encoded in the relationship of the signal to other signals and templates, as encapsulated in the geometry of the signal space. Thus, embeddings are very good tools to capture the geometry, while reducing the processing burden. In this chapter, we provide a concise overview of the area, including foundational results and recent developments. Our goal is to expose the field to a wider community, to provide, as much as possible, a unifying view of the literature, and to demonstrate the usefulness and applicability of the results.