Stability of Sequential Lateration and of Stress Minimization in the Presence of Noise

TL;DR

This work develops a perturbation framework for stability of sequential lateration in noisy multidimensional scaling on lateration graphs. It proves that, under a noisy realizable model with $\sum_{(i,j)\in\mathcal{E}} \varepsilon_{ij}^2$ bounded, the sequentially embedded configuration $\{y_i\}$ stays close to the latent $\{x_i\}$ up to rigid motions, with $\min_{g} \sum_i \|y_i - g(x_i)\|^2$ controlled by a constant times the total squared noise. As a corollary, a similar bound holds for stress minimization, showing that any stress minimizer remains close to the latent configuration when the noise is small, leveraging rigidity theory and perturbation results. The paper further proves that large random geometric graphs are lateration graphs with high probability under mild sampling conditions, broadening applicability, and complements theory with numerical experiments validating the bounds and highlighting computational efficiency of sequential lateration over SMACOF/gradient methods. Collectively, the results enhance understanding of noise stability in MDS/localization and offer practical guarantees for anchor-free embedding in noisy settings.

Abstract

Sequential lateration is a class of methods for multidimensional scaling where a suitable subset of nodes is first embedded by some method, e.g., a clique embedded by classical scaling, and then the remaining nodes are recursively embedded by lateration. A graph is a lateration graph when it can be embedded by such a procedure. We provide a stability result for a particular variant of sequential lateration. We do so in a setting where the dissimilarities represent noisy Euclidean distances between nodes in a geometric lateration graph. We then deduce, as a corollary, a perturbation bound for stress minimization. To argue that our setting applies broadly, we show that a (large) random geometric graph is a lateration graph with high probability under mild conditions, extending a previous result of Aspnes et al (2006).

Figures (3)

• Figure 4.1: Examples of latent configurations $x_1, x_2, \dots, x_n$ and the embedding $y_1, y_2, \dots, y_n$ obtained from sequential lateration when (left) $h=0.5$ and $\kappa=1$, and when (right) $h=0.5$ and $\kappa=2$. The model is \ref{['setting']}, with $\varepsilon_{ij} \sim N(0, \varsigma^2)$ for $\varsigma^2=0.1$.
• Figure 4.2: Results of the numerical experiments. The vertical axis in all plots is the embedding error and the horizontal axis is the variance of the noise, $\varsigma^2$. The results are shown on a log-log scale. The dashed line in (a), (b) and (c) is the $45^\circ$ line corresponding to the mean perturbation $s(\varepsilon)^2$ defined in \ref{['s']}.
• Figure 4.3: (a) Comparison of the embedding error for SMACOF and Gradient Descent shown on a log-log scale. The black dashed line corresponds the mean perturbation, $s(\varepsilon)^2$, and the blue dashed line is a plot of $s(\varepsilon)^2/10^3$ which provides evidence of a lower bound for the embedding error. (b) Computational time for sequential lateration, SMACOF and Gradient Descent for varying sample sizes $n$.

Theorems & Definitions (10)

• Theorem 2.1
• Lemma 2.2: Corollary 2 in arias2020perturbation
• Lemma 2.3: Corollary 3 in arias2020perturbation
• proof : Proof of Theorem \ref{['thm:sequential lateration']}
• Theorem 3.1
• proof
• Proposition 3.2
• proof
• Theorem 4.1
• proof