Continuous Multidimensional Scaling
<|reference_start|>Continuous Multidimensional Scaling: Multidimensional scaling (MDS) is the act of embedding proximity information about a set of $n$ objects in $d$-dimensional Euclidean space. As originally conceived by the psychometric community, MDS was concerned with embedding a fixed set of proximities associated with a fixed set of objects. Modern concerns, e.g., that arise in developing asymptotic theories for statistical inference on random graphs, more typically involve studying the limiting behavior of a sequence of proximities associated with an increasing set of objects. Standard results from the theory of point-to-set maps imply that, if $n$ is fixed and a sequence of proximities converges, then the limit of the embedded structures is the embedded structure of the limiting proximities. But what if $n$ increases? It then becomes necessary to reformulate MDS so that the entire sequence of embedding problems can be viewed as a sequence of optimization problems in a fixed space. We present such a reformulation and derive some consequences.<|reference_end|>
@article{trosset2024continuous,
    title={Continuous Multidimensional Scaling},
    author={Michael W. Trosset, Carey E. Priebe},
    journal={arXiv preprint arXiv:2402.04436},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.04436},
    primaryClass={stat.ML cs.LG}
}