arXiv subjects

Mingchen Ma

Publications and source records attributed to Mingchen Ma.

21 records · Page 2Linked to original sources

arXiv2022

K-median: exact recovery in the extended stochastic ball model

We study exact recovery conditions for the linear programming relaxation of the k-median problem in the stochastic ball model (SBM). In Awasthi et al. (2015), the authors give a tight result for the k-median LP in the SBM, saying that exact recovery can be achieved as long as the balls are pairwise disjoint. We give a counterexample to their result, thereby showing that the k-median LP is not tight in low dimension. Instead, we give a near optimal result showing that the k-median LP in the SBM is tight in high dimension. We also show that, if the probability measure satisfies some concentration assumptions, then the k-median LP in the SBM is tight in every dimension. Furthermore, we propose a new model of data called extended stochastic ball model (ESBM), which significantly generalizes the well-known SBM. We then show that exact recovery can still be achieved in the ESBM.

math.OC↗

arXiv2022

Clustering with Queries under Semi-Random Noise

The seminal paper by Mazumdar and Saha \cite{MS17a} introduced an extensive line of work on clustering with noisy queries. Yet, despite significant progress on the problem, the proposed methods depend crucially on knowing the exact probabilities of errors of the underlying fully-random oracle. In this work, we develop robust learning methods that tolerate general semi-random noise obtaining qualitatively the same guarantees as the best possible methods in the fully-random model. More specifically, given a set of $n$ points with an unknown underlying partition, we are allowed to query pairs of points $u,v$ to check if they are in the same cluster, but with probability $p$, the answer may be adversarially chosen. We show that information theoretically $O\left(\frac{nk \log n} {(1-2p)^2}\right)$ queries suffice to learn any cluster of sufficiently large size. Our main result is a computationally efficient algorithm that can identify large clusters with $O\left(\frac{nk \log n} {(1-2p)^2}\right) + \text{poly}\left(\log n, k, \frac{1}{1-2p} \right)$ queries, matching the guarantees of the best known algorithms in the fully-random model. As a corollary of our approach, we develop the first parameter-free algorithm for the fully-random model, answering an open question by \cite{MS17a}.

cs.LG↗

arXiv2021

Proximity in Concave Integer Quadratic Programming

A classic result by Cook, Gerards, Schrijver, and Tardos provides an upper bound of $n Δ$ on the proximity of optimal solutions of an Integer Linear Programming problem and its standard linear relaxation. In this bound, $n$ is the number of variables and $Δ$ denotes the maximum of the absolute values of the subdeterminants of the constraint matrix. Hochbaum and Shanthikumar, and Werman and Magagnosc showed that the same upper bound is valid if a more general convex function is minimized, instead of a linear function. No proximity result of this type is known when the objective function is nonconvex. In fact, if we minimize a concave quadratic, no upper bound can be given as a function of $n$ and $Δ$. Our key observation is that, in this setting, proximity phenomena still occur, but only if we consider also approximate solutions instead of optimal solutions only. In our main result we provide upper bounds on the distance between approximate (resp., optimal) solutions to a Concave Integer Quadratic Programming problem and optimal (resp., approximate) solutions of its continuous relaxation. Our bounds are functions of $n, Δ$, and a parameter $ε$ that controls the quality of the approximation. Furthermore, we discuss how far from optimal are our proximity bounds.

math.OC↗