The nearest point map of a real algebraic variety with respect to Euclidean distance is
an algebraic function. The Euclidean distance degree is the number of critical points of
this optimization problem. We focus on varieties seen in engineering applications, and we discuss exact computational methods. Our running example is the Eckart-Young Theorem which states that the nearest point map for low rank matrices is given by the singular value decomposition. This is joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani, and Rekha Thomas.
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