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Mingquan Ye

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5 papers

2 author rows

ICLR Conference 2025 Conference Paper

Efficient Alternating Minimization with Applications to Weighted Low Rank Approximation

Zhao Song 0002
Mingquan Ye
Junze Yin
Lichen Zhang 0003

Weighted low rank approximation is a fundamental problem in numerical linear algebra, and it has many applications in machine learning. Given a matrix $M \in \mathbb{R}^{n \times n}$, a non-negative weight matrix $W \in \mathbb{R}_{\geq 0}^{n \times n}$, a parameter $k$, the goal is to output two matrices $X,Y\in \mathbb{R}^{n \times k}$ such that $\\| W \circ (M - X Y^\top) \\|_F$ is minimized, where $\circ$ denotes the Hadamard product. It naturally generalizes the well-studied low rank matrix completion problem. Such a problem is known to be NP-hard and even hard to approximate assuming the Exponential Time Hypothesis. Meanwhile, alternating minimization is a good heuristic solution for weighted low rank approximation. In particular, [Li, Liang and Risteski, ICML'16] shows that, under mild assumptions, alternating minimization does provide provable guarantees. In this work, we develop an efficient and robust framework for alternating minimization that allows the alternating updates to be computed approximately. For weighted low rank approximation, this improves the runtime of [Li, Liang and Risteski, ICML'16] from $\\|W\\|_0k^2$ to $\\|W\\|_0 k$ where $\\|W\\|_0$ denotes the number of nonzero entries of the weight matrix. At the heart of our framework is a high-accuracy multiple response regression solver together with a robust analysis of alternating minimization.

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ICML Conference 2023 Conference Paper

A Nearly-Optimal Bound for Fast Regression with ℓ ∞ Guarantee

Zhao Song 0002
Mingquan Ye
Junze Yin
Lichen Zhang 0003

Given a matrix $A\in \mathbb{R}^{n\times d}$ and a vector $b\in \mathbb{R}^n$, we consider the regression problem with $\ell_\infty$ guarantees: finding a vector $x’\in \mathbb{R}^d$ such that $||x’-x^* ||_\infty \leq \frac{\epsilon}{\sqrt{d}}\cdot ||Ax^*-b||_2\cdot ||A^\dagger||$ with $x^*$ being the optimal solution to the regression $||Ax-b||_2$. One popular approach for solving $\ell_2$ regression problem is via sketching: picking a structured random matrix $S\in \mathbb{R}^{m\times n}$ with $m\ll n$ and $SA$ can be quickly computed, solve the “sketched” regression problem $x’=\mathrm{argmin} ||SAx-Sb||_2$. In this paper, we show that in order to obtain such $\ell_\infty$ guarantee for $\ell_2$ regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that, there exists a distribution of dense sketching matrices with $m=\epsilon^{-2}d\log^3(n/\delta)$ such that solving the sketched regression problem gives the $\ell_\infty$ guarantee, with probability at least $1-\delta$. Moreover, the matrix $SA$ can be computed in time $O(nd\log n)$. Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in [Price, Song and Woodruff, ICALP’17], in which $m=\Omega(\epsilon^{-2}d^{1+\gamma})$ for $\gamma\in (0, 1)$ is required. Moreover, we develop a novel analytical framework for $\ell_\infty$ guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in [Song and Yu, ICML’21]. Our analysis is much simpler and more general than that of [Price, Song and Woodruff, ICALP’17]. Leveraging this framework, we extend the $\ell_\infty$ guarantee regression result to dense sketching matrices for computing fast tensor product of vectors.

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SODA Conference 2022 Conference Paper

Universally-Optimal Distributed Shortest Paths and Transshipment via Graph-Based ℓ 1 -Oblivious Routing

Goran Zuzic
Gramoz Goranci
Mingquan Ye
Bernhard Haeupler
Xiaorui Sun

We provide universally-optimal distributed graph algorithms for (1+ ∊ )-approximate shortest path problems including shortest-path-tree and transshipment. The universal optimality of our algorithms guarantees that, on any n -node network G, our algorithm completes in T · n o (1) rounds whenever a T -round algorithm exists for G. This includes D · n o (1) -round algorithms for any planar or excluded-minor network. Our algorithms never require more than rounds, resulting in the first sub-linear-round distributed algorithm for transshipment. The key technical contribution leading to these results is the first efficient n o (1) -competitive linear ℓ 1 -oblivious routing operator that does not require the use of ℓ 1 -embeddings. Our construction is simple, solely based on low-diameter decompositions, and—in contrast to all known constructions—directly produces an oblivious flow instead of just an approximation of the optimal flow cost. This also has the benefit of simplifying the interaction with Sherman's multiplicative weight framework [SODA'17] in the distributed setting and its subsequent rounding procedures.

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FOCS Conference 2021 Conference Paper

Minor Sparsifiers and the Distributed Laplacian Paradigm

Sebastian Forster
Gramoz Goranci
Yang P. Liu
Richard Peng
Xiaorui Sun
Mingquan Ye

We study distributed algorithms built around minor-based vertex sparsifiers, and give the first algorithm in the CONGEST model for solving linear systems in graph Laplacian matrices to high accuracy. Our Laplacian solver has a round complexity of $O(n^{o(1)}(\sqrt{n}+D))$, and thus almost matches the lower bound of $\widetilde{\Omega}(\sqrt{n}+D)$, where $n$ is the number of nodes in the network and $D$ is its diameter. We show that our distributed solver yields new sublinear round algorithms for several cornerstone problems in combinatorial optimization. This is achieved by leveraging the powerful algorithmic framework of Interior Point Methods (IPMs) and the Laplacian paradigm in the context of distributed graph algorithms, which entails numerically solving optimization problems on graphs via a series of Laplacian systems. Problems that benefit from our distributed algorithmic paradigm include exact mincost flow, negative weight shortest paths, maxflow, and bipartite matching on sparse directed graphs. For the maxflow problem, this is the first exact distributed algorithm that applies to directed graphs, while the previous work by [Ghaffari et al. SICOMP'18] considered the approximate setting and works only for undirected graphs. For the mincost flow and the negative weight shortest path problems, our results constitute the first exact distributed algorithms running in a sublinear number of rounds. Given that the hybrid between IPMs and the Laplacian paradigm has proven useful for tackling numerous optimization problems in the centralized setting, we believe that our distributed solver will find future applications. At the heart of our distributed Laplacian solver is the notion of spectral subspace sparsifiers of [Li, Schild FOCS'18]. We present a nontrivial distributed implementation of their construction by (i) giving a parallel variant of their algorithm that avoids the sampling of random spanning trees and uses approximate leverage scores instead, and (ii) showing that the algorithm still produces a high-quality subspace spectral sparsifier by carefully setting up and analyzing matrix martingales. Combining this vertex reduction recursively with both tree and elimination-based preconditioners leads to our algorithm for solving Laplacian systems. The construction of the elimination-based preconditioners is based on computing short random walks, and we introduce a new technique for reducing the congestion incurred by the simulation of these walks on weighted graphs.

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AAAI Conference 2019 Conference Paper

On Geometric Alignment in Low Doubling Dimension

Hu Ding
Mingquan Ye

In real-world, many problems can be formulated as the alignment between two geometric patterns. Previously, a great amount of research focus on the alignment of 2D or 3D patterns, especially in the field of computer vision. Recently, the alignment of geometric patterns in high dimension finds several novel applications, and has attracted more and more attentions. However, the research is still rather limited in terms of algorithms. To the best of our knowledge, most existing approaches for high dimensional alignment are just simple extensions of their counterparts for 2D and 3D cases, and often suffer from the issues such as high complexities. In this paper, we propose an effective framework to compress the high dimensional geometric patterns and approximately preserve the alignment quality. As a consequence, existing alignment approach can be applied to the compressed geometric patterns and thus the time complexity is significantly reduced. Our idea is inspired by the observation that high dimensional data often has a low intrinsic dimension. We adopt the widely used notion “doubling dimension” to measure the extents of our compression and the resulting approximation. Finally, we test our method on both random and real datasets; the experimental results reveal that running the alignment algorithm on compressed patterns can achieve similar qualities, comparing with the results on the original patterns, but the running times (including the times cost for compression) are substantially lower.

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