Euclidean distance matrix completion via semidefinite facial reduction

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Revision as of 20:20, 2 December 2009

Nathan Krislock

Euclidean distance matrix completion

Let LaTeX: \mathcal E^n be the set of LaTeX: n \times n Euclidean distance matrices, and let LaTeX: \mathcal S^n_+ be the set of (symmetric) positive semidefinite matrices. Defining LaTeX: \mathcal K : \mathcal S^n \rightarrow \mathcal S^n by

LaTeX: \mathcal K(Y)_{ij} := Y_{ii} + Y_{jj} - 2Y_{ij} ~~\mbox{for all}~ i,j = 1, \ldots, n

we have that LaTeX: \mathcal K(\mathcal S^n_+) = \mathcal E^n. A matrix LaTeX: D \in \mathcal S^n is a Euclidean distance matrix with embedding dimension LaTeX: r\, if and only if there exists LaTeX: P \in \mathbb R^{n \times r} such that LaTeX: D = \mathcal K(PP^T).

Suppose LaTeX: D\, is a partial Euclidean distance matrix with embedding dimension LaTeX: r\,. The low-dimensional Euclidean distance matrix completion problem is

LaTeX: \begin{array}{rl}\mbox{find}&P \in \mathbb R^{n \times r} \\ \mbox{s.t.}&H \circ \mathcal K(PP^T) = H \circ D \\ & P^Te = 0, \end{array}

where LaTeX: e \in \mathbb R^n is the vector of all ones, and LaTeX: H\, is the adjacency matrix of the graph LaTeX: G = (V,E)\, associated with the partial Euclidean distance matrix LaTeX: D\,.


Semidefinite programming relaxation of the low-dimensional Euclidean distance matrix completion problem

Using the substitution LaTeX: Y = PP^T\,, and relaxing the condition that LaTeX: \mathrm{rank}(Y) = r\,, we obtain the semidefinite programming relaxation

LaTeX: \begin{array}{rl}\mbox{find}& Y \in \mathcal S^n_+ \\ \mbox{s.t.}& H \circ \mathcal K(Y) = H \circ D \\ & Ye = 0. \end{array}


Single Clique Facial Reduction Theorem [1]

Let LaTeX: C \subseteq V be a clique in the graph LaTeX: G\, such that the embedding dimension of LaTeX: D[C]\, is LaTeX: r\,. Then there exists LaTeX: U \in \mathbb R^{n \times (n-|C|+r)} such that

LaTeX: \mathbf{face} \left\{ Y \in \mathcal S^n_+ : \mathcal{K}(Y[C]) = D[C] \right\} = U \mathcal S^{n-|C|+r}_+ U^T.
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