= Metric embedding
{title2=$f:(X,d_X)\hookrightarrow(Y,d_Y)$}
{wiki}
A <metric embedding> represents the points of one <metric space> $(X,d_X)$ inside another $(Y,d_Y)$ by an <injective function> $f:X\to Y$. The geometric question is how faithfully the representation preserves pairwise distances. For example, <binary strings> can be treated as <vectors> so that tools from <linear algebra> become available, but the chosen <norm> determines what their separation means. A drawing of the points alone does not specify a <metric embedding>: both <metrics> matter.
An <isometric embedding> preserves every distance exactly. A <metric embedding> that is a <bi-Lipschitz map> allows multiplicative error while keeping its <bi-Lipschitz distortion> bounded over all pairs. A <coarse embedding> instead controls separation through possibly nonlinear functions of distance. These are different requirements: the <Hamming cube as an L1 and L2 metric> below preserves <Hamming distance> exactly in one representation and takes its <square root> in another.
The usefulness is that a complicated distance comparison becomes a calculation with familiar <vectors> and <norms>. This also helps assess a visualization: nearby points should not acquire arbitrarily large separations, and far-apart points should not collapse together. Matoušek's 2013 lecture notes describe how motivations from <computer science>, especially <approximation algorithms>, accelerated the study of <metric embeddings> from the 1990s onward. <Lecture notes on metric embeddings by Jiří Matoušek>[Jiří Matoušek, Lecture notes on metric embeddings, preface and Section 1.1]{ref}
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