Dimension Thresholds for Relative-Error Representations of Finite Metrics in Pseudo-Euclidean Spaces
A finite metric space with $N$ points admits an exact representation in a pseudo-Euclidean space of dimension at most $N-1$. We study the minimum dimension required for relative-error approximation of squared distances. Our main result gives an upper bound for arbitrary metrics in terms of local distance ratios: for $4/5<\varepsilon<1$, dimension $O((N/k)\log N+kρ_k^4\log N/(\varepsilon-4/5)^2)$ suffices, where $ρ_k$ is the largest ratio, over all points, between the distance to the $k$th closest point (counting the point itself) and the nearest-neighbour distance. The proof combines a low-rank landmark approximation with a sparse correction controlled by its factorization norm. For metrics with distances in $[1,Φ]$, where $Φ\ge5$ is fixed, the worst-case dimension is $Θ(N)$ for $0<\varepsilon<4/5$, lies between $Ω(\sqrt N)$ and $O(\sqrt N\log N)$ for $4/5<\varepsilon<12/13$, and is $Θ(\log N)$ for $\varepsilon>(Φ^2-1)/(Φ^2+1)$. For $Φ>5$, an intermediate range remains without matching bounds. Sign-pattern counting yields the $Ω(\sqrt N)$ lower bound for $4/5<\varepsilon<12/13$, while Hadamard matrices give explicit linear lower bounds for $\varepsilon<4/5$ via a known approximate-rank estimate. We also construct configurations with null squared intervals whose minimum representation dimension remains $N-1$ for every $\varepsilon<1$. Finally, for every fixed $0<\varepsilon<3/5$, we show that the quadratic dependence on the parameter $C$ in the dimension bound derived from Deng et al.'s pseudo-Euclidean Johnson-Lindenstrauss lemma is necessary up to a logarithmic factor.