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Mahalanobis pseudometric induced by a positive-semidefinite matrix

Definition

Fix an integer $d \geq 1$ . A real matrix $M \in R^{d \times d}$ is positive semidefinite, written $M ⪰ 0$ , when it is symmetric and $v^{T} M v \geq 0$ for every $v \in R^{d}$ . For $x, x^{'} \in R^{d}$ , define the squared Mahalanobis distance and its nonnegative square root by

$q_{M} (x, x^{'}) : = (x - x^{'})^{T} M (x - x^{'}), d_{M} (x, x^{'}) : = \sqrt{q_{M} (x, x^{'})} .$

The square root is defined because $q_{M} (x, x^{'}) \geq 0$ . A pseudometric is a nonnegative symmetric function that vanishes on the diagonal and satisfies the triangle inequality, but may assign distance zero to distinct points. The function $d_{M}$ is called the Mahalanobis pseudometric associated with $M$ ; $q_{M} = d_{M}^{2}$ is its squared-distance form. The theorem on this page proves the pseudometric properties and the equivalent linear-feature representation.

Definition references

Existence and uniqueness of nonnegative square roots

Weinberger and Saul, §2.1, printed pp. 209–210

Bellet, Habrard, and Sebban, Mahalanobis distance and interpretation, printed pp. 9–10

Depends on

Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$; the positives are $\{x^2 : x \neq 0\}$

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kilian Q. Weinberger and Lawrence K. Saul, Distance Metric Learning for Large Margin Nearest Neighbor Classification, JMLR 10 (2009), 207–244 (standard reference, not scraped)
Aurélien Bellet, Amaury Habrard, and Marc Sebban, A Survey on Metric Learning for Feature Vectors and Structured Data, arXiv:1306.6709v4 (2014) (standard reference, not scraped)