These techniques calibrate an initial non-metric distance matrix estimated on incomplete data to a distance metric.

We also reiterated our contribution in the last paragraph of Introduction section.

We define the heat-geodesic dissimilarity based on V aradhan's formula and the two-sided heat

Such distance matrices are commonly computed in software packages and have applications to learning image manifolds, handwriting recognition, and multi-dimensional unfolding, among other things. In an attempt to reduce their description size, we study low rank approximation of such matrices. Our main result is to show that for any underlying distance metric $d$, it is possible to achieve an additive error low rank approximation in sublinear time. We note that it is provably impossible to achieve such a guarantee in sublinear time for arbitrary matrices $\AA$, and our proof exploits special properties of distance matrices. We develop a recursive algorithm based on additive projection-cost preserving sampling.

The distance matrix of a dataset $X$ of $n$ points with respect to a distance function $f$ represents all pairwise distances between points in $X$ induced by $f$. Due to their wide applicability, distance matrices and related families of matrices have been the focus of many recent algorithmic works. We continue this line of research and take a broad view of algorithm design for distance matrices with the goal of designing fast algorithms, which are specifically tailored for distance matrices, for fundamental linear algebraic primitives. Our results include efficient algorithms for computing matrix-vector products for a wide class of distance matrices, such as the $\ell_1$ metric for which we get a linear runtime, as well as an $\Omega(n^2)$ lower bound for any algorithm which computes a matrix-vector product for the $\ell_{\infty}$ case, showing a separation between the $\ell_1$ and the $\ell_{\infty}$ metrics. Our upper bound results in conjunction with recent works on the matrix-vector query model have many further downstream applications, including the fastest algorithm for computing a relative error low-rank approximation for the distance matrix induced by $\ell_1$ and $\ell_2^2$ functions and the fastest algorithm for computing an additive error low-rank approximation for the $\ell_2$ metric, in addition to applications for fast matrix multiplication among others. We also give algorithms for constructing distance matrices and show that one can construct an approximate $\ell_2$ distance matrix in time faster than the bound implied by the Johnson-Lindenstrauss lemma.

Identifying and comparing topological features, particularly cycles, across different topological objects remains a fundamental challenge in persistent homology and topological data analysis. This work introduces a novel framework for constructing cycle communities through two complementary approaches. First, a dendrogram-based methodology leverages merge-tree algorithms to construct hierarchical representations of homology classes from persistence intervals. The Wasserstein distance on merge trees is introduced as a metric for comparing dendrograms, establishing connections to hierarchical clustering frameworks. Through simulation studies, the discriminative power of dendrogram representations for identifying cycle communities is demonstrated. Second, an extension of Stratified Gradient Sampling simultaneously learns multiple filter functions that yield cycle barycenter functions capable of faithfully reconstructing distinct sets of cycles. The set of cycles each filter function can reconstruct constitutes cycle communities that are non-overlapping and partition the space of all cycles. Together, these approaches transform the problem of cycle matching into both a hierarchical clustering and topological optimization framework, providing principled methods to identify similar topological structures both within and across groups of topological objects.

Generative models augmented with external tools and update mechanisms (or \textit{agents}) have demonstrated capabilities beyond intelligent prompting of base models. As agent use proliferates, dynamic multi-agent systems have naturally emerged. Recent work has investigated the theoretical and empirical properties of low-dimensional representations of agents based on query responses at a single time point. This paper introduces the Temporal Data Kernel Perspective Space (TDKPS), which jointly embeds agents across time, and proposes several novel hypothesis tests for detecting behavioral change at the agent- and group-level in black-box multi-agent systems. We characterize the empirical properties of our proposed tests, including their sensitivity to key hyperparameters, in simulations motivated by a multi-agent system of evolving digital personas. Finally, we demonstrate via natural experiment that our proposed tests detect changes that correlate sensitively, specifically, and significantly with a real exogenous event. As far as we are aware, TDKPS is the first principled framework for monitoring behavioral dynamics in black-box multi-agent systems -- a critical capability as generative agent deployment continues to scale.

The Multi-Traveling Salesman Problem (MTSP) is a commonly used mathematical model for multi-agent task allocation. However, as the number of agents and task targets increases, existing optimization-based methods often incur prohibitive computational costs, posing significant challenges to large-scale coordination in unmanned systems. To address this issue, this paper proposes a K-means-inspired task allocation framework that reformulates the MTSP as a spatially constrained classification process. By leveraging spatial coherence, the proposed method enables fast estimation of path costs and efficient task grouping, thereby fundamentally reducing overall computational complexity. Extensive simulation results demonstrate that the framework can maintain high solution quality even in extremely large-scale scenarios-for instance, in tasks involving 1000 agents and 5000 targets. The findings indicate that this "cluster-then-route" decomposition strategy offers an efficient and reliable solution for large-scale multi-agent task allocation.

Transition states (TSs) are crucial for understanding reaction mechanisms, yet their exploration is limited by the complexity of experimental and computational approaches. Here we propose TS-DFM, a flow matching framework that predicts TSs from reactants and products. By operating in molecular distance geometry space, TS-DFM explicitly captures the dynamic changes of interatomic distances in chemical reactions. A network structure named TSDVNet is designed to learn the velocity field for generating TS geometries accurately. On the benchmark dataset Transition1X, TS-DFM outperforms the previous state-of-the-art method React-OT by 30\% in structural accuracy. These predicted TSs provide high-quality initial structures, accelerating the convergence of CI-NEB optimization. Additionally, TS-DFM can identify alternative reaction paths. In our experiments, even a more favorable TS with lower energy barrier is discovered. Further tests on RGD1 dataset confirm its strong generalization ability on unseen molecules and reaction types, highlighting its potential for facilitating reaction exploration.