Multidimensional scaling (MDS) is a fundamental tool for both data visualization and dimensionality reduction. Given a finite collection of points and distances between them, MDS finds a map of these points into Euclidean space which optimally preserves distances. Once in Euclidean space, it is often easier to both visualize and apply standard data analytics and machine learning algorithms. Crucially, MDS automatically provides a measure of how well distances are preserved as a function of the dimension of the target space. In this paper we show that when MDS is applied to a set of points on which a group $G$ acts and the metric defining distance is invariant with respect to the action of the group, then MDS can be understood (and its output calculated) in terms of the representation theory of $G$. In particular when our set of points is $G$ itself, this means that the MDS embedding can be calculated using the Fourier transform on groups. We propose this as an alternative implementation of MDS. We investigate an example in which we apply MDS to permutations from the symmetric group and where distances between permutations are calculated by either the Hamming distance, Cayley distance, or Coxeter distance.

This paper applies fast sparse multidimensional scaling (MDS) to a large graph of music similarity, with 267K vertices that represent artists, albums, and tracks; and 3.22M edges that represent similarity between those entities. Once vertices are assigned locations in a Euclidean space, the locations can be used to browse music and to generate playlists. MDS on very large sparse graphs can be effectively performed by a family of algorithms called Rectangular Dijsktra (RD) MDS algorithms. These RD algorithms operate on a dense rectangular slice of the distance matrix, created by calling Dijsktra a constant number of times. Two RD algorithms are compared: Landmark MDS, which uses the Nyström approximation to perform MDS; and a new algorithm called Fast Sparse Embedding, which uses FastMap. These algorithms compare favorably to Laplacian Eigenmaps, both in terms of speed and embedding quality.

This paper applies fast sparse multidimensional scaling (MDS) to a large graph of music similarity, with 267K vertices that represent artists, albums, and tracks; and 3.22M edges that represent similarity between those entities. Once vertices are assigned locations in a Euclidean space, the locations can be used to browse music and to generate playlists. MDS on very large sparse graphs can be effectively performed by a family of algorithms called Rectangular Dijsktra (RD) MDS algorithms. These RD algorithms operate on a dense rectangular slice of the distance matrix, created by calling Dijsktra a constant number of times. Two RD algorithms are compared: Landmark MDS, which uses the Nyström approximation to perform MDS; and a new algorithm called Fast Sparse Embedding, which uses FastMap. These algorithms compare favorably to Laplacian Eigenmaps, both in terms of speed and embedding quality.

This paper applies fast sparse multidimensional scaling (MDS) to a large graph of music similarity, with 267K vertices that represent artists, albums, andtracks; and 3.22M edges that represent similarity between those entities. Once vertices are assigned locations in a Euclidean space, the locations can be used to browse music and to generate playlists. MDS on very large sparse graphs can be effectively performed by a family of algorithms called Rectangular Dijsktra (RD) MDS algorithms. These RD algorithms operate on a dense rectangular slice of the distance matrix, created by calling Dijsktra a constant number of times. Two RD algorithms are compared: Landmark MDS, which uses the Nyström approximation toperform MDS; and a new algorithm called Fast Sparse Embedding, which uses FastMap. These algorithms compare favorably to Laplacian Eigenmaps, both in terms of speed and embedding quality.