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A non-Euclidean three-point dissimilarity produces a negative MDS eigenvalue

Example

Take the symmetric dissimilarity matrix

$D = (\begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & 3 & 0 \end{array}) .$

It is nonnegative with zero diagonal and is not Euclidean because $D_{13} + D_{12} = 2 < D_{23} = 3$ . Its classical-MDS matrix $B = - \frac{1}{2} H_{3} D^{\circ 2} H_{3}$ has a negative eigenvalue, so positive-eigenvalue truncation is not exact distance recovery.

Example references

Hastie, Tibshirani, and Friedman, ESL II, §14.8, printed pp. 570–572

Cameron, Linear and Nonlinear Dimensionality Reduction, §3, PDF pp. 4–6

Facts & Assumptions

Given: The stated dissimilarity matrix $D$ and the classical-MDS construction.

[F1]

Classical MDS squares the entries of $D$ and forms $B = - \frac{1}{2} H_{n} D^{\circ 2} H_{n}$ (Classical multidimensional scaling from a finite dissimilarity matrix).

[F2]

A Euclidean distance matrix yields a positive-semidefinite classical-MDS matrix; the general symmetric matrix has a real orthonormal eigendecomposition (Double-centering Euclidean squared distances recovers their centered Gram matrix).

Verification

1.1givenF1algebra

Put $u = (1, 1, - 2)^{T}$ and $v = (1, - 1, 0)^{T}$ . Both are zero-sum, so $H_{3} u = u$ and $H_{3} v = v$ . Here $D^{\circ 2} = (\begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 9 \\ 1 & 9 & 0 \end{array})$ , so direct multiplication gives $u^{T} B u = - \frac{1}{2} u^{T} D^{\circ 2} u = 19$ , $v^{T} B v = 1$ , and $u^{T} B v = - \frac{1}{2} u^{T} D^{\circ 2} v = - 8$ . Thus $w = u + 8 v$ satisfies $w^{T} B w = 19 + 16 (- 8) + 64 = - 45 < 0$ .

2.1step 1.1F2algebra

By [F2], $B$ is symmetric and has an orthonormal eigenbasis. If every eigenvalue were nonnegative, expressing $w$ in that eigenbasis would give $w^{T} B w \geq 0$ , contradicting step 1.1. Therefore $B$ has a negative eigenvalue.

3.1givenstep 2.1F2∎

The stated triangle inequality fails, since $D_{13} + D_{12} = 2 < D_{23} = 3$ , so no Euclidean point configuration realizes $D$ . This matches the negative eigenvalue and the Euclidean-only exactness hypothesis of [F2].

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Sources

The Elements of Statistical Learning, second edition, corrected 12th printing (standard reference, not scraped)
Maria Cameron, Linear and Nonlinear Dimensionality Reduction, lecture notes (standard reference, not scraped)