Summary
The paper looks at how many dimensions are needed to represent distances between points approximately in a type of space that allows both positive and negative distances (pseudo-Euclidean space). It shows formulas connecting the required number of dimensions to how distances relate locally among points and the allowed approximation error. The authors also identify ranges where the number of dimensions needed is very large or can be much smaller. They prove these limits using mathematical tools like matrix theory and pattern counting. This helps understand when low-dimensional approximate representations are possible for complex distance relationships.
What this means in practice
- •For machine learning engineers: Determine the minimum number of dimensions needed for low-error metric embeddings to improve efficiency in training and storing complex data representations.
- •For graphics software developers: Inform design of algorithms that approximate distances in virtual environments where maintaining relative distance accuracy is critical.
A theory result. No direct application yet.
Abstract
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.