Continuous Multidimensional Scaling
arxiv(2024)
摘要
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, 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.
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