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The Mahalanobis distance is a pseudometric and comes from a linear feature map

Statement

Let $d \geq 1$ and let $M \in R^{d \times d}$ be real symmetric and positive semidefinite, meaning $v^{T} M v \geq 0$ for every $v \in R^{d}$ . Define $q_{M} (x, x^{'}) = (x - x^{'})^{T} M (x - x^{'})$ and $d_{M} (x, x^{'}) = \sqrt{q_{M} (x, x^{'})}$ . Then there is a real matrix $L \in R^{d \times d}$ with $M = L^{T} L$ and, for all $x, x^{'} \in R^{d}$ , $d_{M} (x, x^{'}) = ∥ L x - L x^{'} ∥_{2}$ . Consequently $d_{M}$ is nonnegative, symmetric, zero on the diagonal, and satisfies the triangle inequality, without asserting separation of distinct points.

Statement references

Definition of q_M and d_M

Positive semidefinite forms and nonnegative operators over the reals

Factorization of nonnegative operators

Euclidean inner product and norm

Triangle inequality for the Euclidean norm

Nonnegative square-root uniqueness

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

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

Facts & Assumptions

Given: An integer $d \geq 1$ , a real symmetric positive-semidefinite matrix $M \in R^{d \times d}$ , and $x, x^{'} \in R^{d}$ .

[F1]

For a finite-dimensional real inner-product space, a symmetric bilinear form is positive semidefinite exactly when its associated self-adjoint operator is nonnegative (Over the reals, non-negative and positive operators correspond exactly to positive semidefinite and positive definite symmetric forms).

[F2]

Every nonnegative endomorphism of a finite-dimensional real inner-product space has the form $S^{*} S$ for a linear endomorphism $S$ (A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S).

[F3]

The Euclidean inner product is $⟨ u, v ⟩ = \sum_{j = 1}^{d} u_{j} v_{j}$ and its induced norm satisfies $∥ u ∥_{2}^{2} = ⟨ u, u ⟩$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F4]

The Euclidean norm is nonnegative, symmetric under sign change, vanishes only at zero, and satisfies the triangle inequality (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F5]

Every nonnegative real has a unique nonnegative square root (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

Proof

1.1givenF1

Regard $M$ as the matrix of an endomorphism $T_{M} : R^{d} \to R^{d}$ , $T_{M} v = M v$ . The associated symmetric bilinear form is $(u, v) \mapsto u^{T} M v$ , so its positive-semidefinite hypothesis and [F1] imply that $T_{M}$ is nonnegative.

2.1step 1.1F2algebra

By [F2], there is an endomorphism $S$ with $T_{M} = S^{*} S$ . Let $L$ be its real matrix in the standard basis; the Euclidean adjoint has matrix $L^{T}$ , hence $M = L^{T} L$ .

3.1step 2.1F3F5algebra

For $h = x - x^{'}$ , steps 2.1 and [F3] give $q_{M} (x, x^{'}) = h^{T} M h = h^{T} L^{T} L h = ⟨ L h, L h ⟩ = ∥ L h ∥_{2}^{2}$ . Both $d_{M} (x, x^{'})$ and $∥ L h ∥_{2}$ are nonnegative and their squares agree, so [F5] gives $d_{M} (x, x^{'}) = ∥ L x - L x^{'} ∥_{2}$ .

4.1step 3.1F4∎

Nonnegativity, symmetry, and zero diagonal follow from the corresponding properties of the Euclidean norm in [F4]. For any $x, y, z \in R^{d}$ , $d_{M} (x, z) = ∥ L x - L z ∥_{2} \leq ∥ L x - L y ∥_{2} + ∥ L y - L z ∥_{2} = d_{M} (x, y) + d_{M} (y, z)$ by the triangle inequality in [F4]. The pseudometric axioms follow.

Depends on

Used by

A rank-deficient Mahalanobis matrix identifies distinct points Example

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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)