Manuel E. Lladser

Rafael Frongillo, Manuel Lladser, Anish Thilagar et al.

Beginning with Witkowski et al. [2022], recent work on forecasting competitions has addressed incentive problems with the common winner-take-all mechanism. Frongillo et al. [2021] propose a competition mechanism based on follow-the-regularized-leader (FTRL), an online learning framework. They show that their mechanism selects an $ε$-optimal forecaster with high probability using only $O(\log(n)/ε^2)$ events. These works, together with all prior work on this problem thus far, assume that events are independent. We initiate the study of forecasting competitions for correlated events. To quantify correlation, we introduce a notion of block correlation, which allows each event to be strongly correlated with up to $b$ others. We show that under distributions with this correlation, the FTRL mechanism retains its $ε$-optimal guarantee using $O(b^2 \log(n)/ε^2)$ events. Our proof involves a novel concentration bound for correlated random variables which may be of broader interest.

Manuel E. Lladser, Alexander J. Paradise

A subset of points in a metric space is said to resolve it if each point in the space is uniquely characterized by its distance to each point in the subset. In particular, resolving sets can be used to represent points in abstract metric spaces as Euclidean vectors. Importantly, due to the triangle inequality, points close by in the space are represented as vectors with similar coordinates, which may find applications in classification problems of symbolic objects under suitably chosen metrics. In this manuscript, we address the resolvability of Jaccard spaces, i.e., metric spaces of the form $(2^X,\text{Jac})$, where $2^X$ is the power set of a finite set $X$, and $\text{Jac}$ is the Jaccard distance between subsets of $X$. Specifically, for different $a,b\in 2^X$, $\text{Jac}(a,b)=|aΔb|/|a\cup b|$, where $|\cdot|$ denotes size (i.e., cardinality) and $Δ$ denotes the symmetric difference of sets. We combine probabilistic and linear algebra arguments to construct highly likely but nearly optimal (i.e., of minimal size) resolving sets of $(2^X,\text{Jac})$. In particular, we show that the metric dimension of $(2^X,\text{Jac})$, i.e., the minimum size of a resolving set of this space, is $Θ(|X|/\ln|X|)$. In addition, we show that a much smaller subset of $2^X$ suffices to resolve, with high probability, all different pairs of subsets of $X$ of cardinality at most $\sqrt{|X|}/\ln|X|$, up to a factor.

Richard D. Tillquist, Manuel E. Lladser

The minimal number of nodes required to multilaterate a network endowed with geodesic distance (i.e., to uniquely identify all nodes based on shortest path distances to the selected nodes) is called its metric dimension. This quantity is related to a useful technique for embedding graphs in low-dimensional Euclidean spaces and representing the nodes of a graph numerically for downstream analyses such as vertex classification via machine learning. While metric dimension has been studied for many kinds of graphs, its behavior on the Stochastic Block Model (SBM) ensemble has not. The simple community structure of graphs in this ensemble make them interesting in a variety of contexts. Here we derive probabilistic bounds for the metric dimension of random graphs generated according to the SBM, and describe algorithms of varying complexity to find---with high probability---subsets of nodes for multilateration. Our methods are tested on SBM ensembles with parameters extracted from real-world networks. We show that our methods scale well with increasing network size as compared to the state-of-the-art Information Content Heuristic algorithm for metric dimension approximation.