Given a graph G = (V,E), a subgraph H is an additive +β spanner if dist_H(u,v) ≤ dist_G(u,v) + β for all u, v ∈ V. A pairwise spanner is a spanner for which the above inequality is only required to hold for specific pairs P ⊆ V × V given on input; when the pairs have the structure P = S × S for some S ⊆ V, it is called a subsetwise spanner. Additive spanners in unweighted graphs have been studied extensively in the literature, but have only recently been generalized to weighted graphs. In this paper, we consider a multi-level version of the subsetwise additive spanner in weighted graphs motivated by multi-level network design and visualization, where the vertices in S possess varying level, priority, or quality of service (QoS) requirements. The goal is to compute a nested sequence of spanners with the minimum total number of edges. We first generalize the +2 subsetwise spanner of [Pettie 2008, Cygan et al., 2013] to the weighted setting. We experimentally measure the performance of this and several existing algorithms by [Ahmed et al., 2020] for weighted additive spanners, both in terms of runtime and sparsity of the output spanner, when applied as a subroutine to multi-level problem. We provide an experimental evaluation on graphs using several different random graph generators and show that these spanner algorithms typically achieve much better guarantees in terms of sparsity and additive error compared with the theoretical maximum. By analyzing our experimental results, we additionally developed a new technique of changing a certain initialization parameter which provides better spanners in practice at the expense of a small increase in running time.

Low rank tensor approximation is a fundamental tool in modern machine learning and data science. In this paper, we study the characterization, perturbation analysis, and an efficient sampling strategy for two primary tensor CUR approximations, namely Chidori and Fiber CUR. We characterize exact tensor CUR decompositions for low multilinear rank tensors. We also present theoretical error bounds of the tensor CUR approximations when (adversarial or Gaussian) noise appears. Moreover, we show that low cost uniform sampling is sufficient for tensor CUR approximations if the tensor has an incoherent structure. Empirical performance evaluations, with both synthetic and real-world datasets, establish the speed advantage of the tensor CUR approximations over other state-of-the-art low multilinear rank tensor approximations.

This paper considers the use of robust principal component analysis (RPCA) in a CUR decomposition framework and applications thereof. Our main algorithms produce a robust version of column-row factorizations of matrices $D=L+S$, where $L$ is low-rank and $S$ contains sparse outliers. These methods yield interpretable factorizations at low computational cost and provide new CUR decompositions that are robust to sparse outliers, in contrast to previous methods. We consider two key imaging applications of RPCA: video foreground-background separation and face modeling. This paper examines the qualitative behavior of our robust CUR decompositions on the benchmark videos and face datasets and finds that our method works as well as standard RPCA while being significantly faster. Additionally, we consider hybrid randomized and deterministic sampling methods which produce a compact CUR decomposition of a given matrix and apply this to video sequences to produce canonical frames thereof.

Robust principal component analysis (RPCA) is a widely used tool for dimension reduction. In this work, we propose a novel non-convex algorithm, coined Iterated Robust CUR (IRCUR), for solving RPCA problems, which dramatically improves the computational efficiency in comparison with the existing algorithms. IRCUR achieves this acceleration by employing CUR decomposition when updating the low rank component, which allows us to obtain an accurate low rank approximation via only three small submatrices. Consequently, IRCUR is able to process only the small submatrices and avoid the expensive computing on full matrix through the entire algorithm. Numerical experiments establish the computational advantage of IRCUR over the state-of-art algorithms on both synthetic and real-world datasets.

We study the multi-level Steiner tree problem: a generalization of the Steiner tree problem in graphs where terminals T require varying priority, level, or quality of service. In this problem, we seek to find a minimum cost tree containing edges of varying rates such that any two terminals u, v with priorities P(u), P(v) are connected using edges of rate min{P(u),P(v)} or better. The case where edge costs are proportional to their rate is approximable to within a constant factor of the optimal solution. For the more general case of non-proportional costs, this problem is hard to approximate with ratio c log log n, where n is the number of vertices in the graph. A simple greedy algorithm by Charikar et al., however, provides a min{2(ln |T|+1), 𝓁 ρ}-approximation in this setting, where ρ is an approximation ratio for a heuristic solver for the Steiner tree problem and 𝓁 is the number of priorities or levels (Byrka et al. give a Steiner tree algorithm with ρ≈1.39, for example). In this paper, we describe a natural generalization to the multi-level case of the classical (single-level) Steiner tree approximation algorithm based on Kruskal’s minimum spanning tree algorithm. We prove that this algorithm achieves an approximation ratio at least as good as Charikar et al., and experimentally performs better with respect to the optimum solution. We develop an integer linear programming formulation to compute an exact solution for the multi-level Steiner tree problem with non-proportional edge costs and use it to evaluate the performance of our algorithm on both random graphs and multi-level instances derived from SteinLib.

This paper introduces a clustering algorithm, called principal coordinate clustering. It takes in a similarity matrix SWof a data matrix W and computes the singular value decomposition of SWto determine the principal coordinates to convert the clustering problem to a simpler domain. It is a relative of spectral clustering, however, principal coordinate clustering is easier to interpret, and gives a clear understanding of why it performs well. In a fashion, this gives intuition behind why spectral clustering works from a more simple, linear algebra perspective, beyond the typical explanations via graph cuts, or other techniques. Moreover, it was demonstrated through experimentation on real and synthetic data that the proposed method performs equally well on average as spectral clustering, and that the method has the ability to scale quite easily to truly large data.