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版本更新 · v4arXiv:2306.04169

Efficient Alternating Minimization with Applications to Weighted Low Rank Approximation

Zhao Song, Mingquan Ye, Junze Yin, Lichen Zhang

原文摘要

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 [GG11, RSW16]. Meanwhile, alternating minimization is a good heuristic solution for weighted low rank approximation. In particular, [LLR16] 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 cost of each iteration of [LLR16] 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. We also replace the random initialization of [LLR16], whose claimed guarantee fails, by subspace iteration from a Gaussian start, so our total running time of $\widetilde O((\|W\|_0 k+nk^3)\log(1/ε))$ includes the initialization. In exchange, our guarantee needs the weight matrix to be within spectral distance $γn$ of the all-ones matrix for a $γ$ that is polynomially small in $n$.

cs.LG · math.OC

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arXiv ↗PDF ↗
版本更新 · v4arXiv:2302.11068

Low Rank Matrix Completion via Robust Alternating Minimization in Nearly Linear Time

Yuzhou Gu, Zhao Song, Junze Yin, Lichen Zhang

原文摘要

Given a matrix $M\in \mathbb{R}^{m\times n}$, the low rank matrix completion problem asks us to find a rank-$k$ approximation of $M$ as $UV^\top$ for $U\in \mathbb{R}^{m\times k}$ and $V\in \mathbb{R}^{n\times k}$ by only observing a few entries specified by a set of entries $Ω\subseteq [m]\times [n]$. In particular, we examine an approach that is widely used in practice -- the alternating minimization framework. Jain, Netrapalli, and Sanghavi [JNS13] showed that if $M$ has incoherent rows and columns, then alternating minimization provably recovers the matrix $M$ by observing a nearly linear in $n$ number of entries. While the sample complexity has been subsequently improved [GLZ17], alternating minimization steps are required to be computed exactly. This hinders the development of more efficient algorithms and fails to depict the practical implementation of alternating minimization, where the updates are usually performed approximately in favor of efficiency. In this paper, we take a major step towards a more efficient and error-robust alternating minimization framework. To this end, we develop an analytical framework for alternating minimization that can tolerate a moderate amount of errors caused by approximate updates. Moreover, our algorithm runs in time $\widetilde O(|Ω| k)$, which is nearly linear in the time to verify the solution while preserving the sample complexity. This improves upon all prior known alternating minimization approaches which require $\widetilde O(|Ω| k^2)$ time.

cs.LG · cs.DS · math.OC · stat.ML

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2304.03426

Convex Minimization with Integer Minima in $\widetilde O(n^4)$ Time

Haotian Jiang, Yin Tat Lee, Zhao Song, Lichen Zhang

原文摘要

Given a convex function $f$ on $\mathbb{R}^n$ whose set of minimizers is nonempty and closed, lies in a box of radius $R$ with $\log R\le\mathrm{poly}(n)$, and has integral extreme points, we give a randomized algorithm that finds an integral minimizer of $f$ with high probability using $O(n^2\log n+n\log R)$ calls to a separation oracle, $O(n+\log R/\log n)$ calls to an approximate shortest vector routine such as the LLL algorithm, and $O(n^4\log(nR)\log\log n)$ other arithmetic operations. The previous best polynomial time algorithm [Jiang, SODA 2021, JACM 2022] achieves $O(n(n\log\log n/\log n+\log R))$ oracle complexity, but its overall runtime is at least $\widetildeΩ(n^8)$. We obtain our speedup by combining a variant of the volumetric center cutting plane method of [Vaidya, FOCS 1989], run on the hybrid barrier of [Lee, Sidford, Wong, FOCS 2015] with the fast implementation of [Jiang, Lee, Song, Wong, STOC 2020], and a variant of the LLL algorithm that reduces segments of consecutive basis vectors recursively. The method runs in blocks, so that shortest vectors are computed only $O(n+\log R/\log n)$ times, and each computation takes $O(n^3\log^3 n)$ arithmetic operations in exact arithmetic. For submodular function minimization, our result implies an algorithm using $O(n^3\log n)$ calls to an evaluation oracle and $O(n^4\log^3 n)$ arithmetic operations in total. Its oracle complexity is better than that of the algorithms of [Lee, Sidford, Wong, FOCS 2015] and [Dadush, Végh, Zambelli, SODA 2018, MOR 2021], which had the best running time before this work.

cs.DS · cs.DM · math.OC

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2509.16915

Differential Privacy for Euclidean Jordan Algebra with Applications to Private Symmetric Cone Programming

Zhao Song, Jianfei Xue, Lichen Zhang

原文摘要

In this paper, we study differentially private mechanisms for functions whose outputs lie in a Euclidean Jordan algebra. Euclidean Jordan algebras capture many important mathematical structures and form the foundation of linear programming, second-order cone programming, and semidefinite programming. Our main contribution is a generic Gaussian mechanism for such functions, with sensitivity measured in $\ell_2$, $\ell_1$, and $\ell_\infty$ norms. Notably, this framework includes the important case where the function outputs are symmetric matrices, and sensitivity is measured in the Frobenius, nuclear, or spectral norm. We further derive private algorithms for solving symmetric cone programs under various settings, using a combination of the multiplicative weights update method and our generic Gaussian mechanism. As an application, we present differentially private algorithms for semidefinite programming, resolving a major open question posed by [Hsu, Roth, Roughgarden, and Ullman, ICALP 2014].

math.OC · cs.CR · cs.DS · cs.LG

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2608.26552

Hadamard Flattening and Gaussian Pooling Sketch for Least Squares with Coordinate-wise Guarantee

Zhao Song, Lichen Zhang

原文摘要

Randomized sketch-and-solve algorithms accelerate overconstrained $\ell_2$ regression by replacing the input with a much smaller, randomly projected problem. While standard subspace embeddings nearly preserve the regression cost, coordinate-wise accuracy is more delicate: rather than the objective value, we want the solution vector itself to be close to the optimum in $\ell_\infty$ norm. Specifically, we seek $x'\in\mathbb{R}^d$ with $\|x'-x^\star\|_\infty\leq\fracε{\sqrt d}\|Ax^\star-b\|_2\|A^\dagger\|_{\mathrm{op}}$, where $A\in\mathbb{R}^{n\times d}$ is the design matrix, $b\in\mathbb{R}^n$ the label vector, $x^\star$ the optimal solution and $A^\dagger$ the pseudo-inverse of $A$. Price, Song and Woodruff introduced this problem and claimed that the subsampled randomized Hadamard transform (SRHT) with $O(ε^{-2}d^{1+Θ(\sqrt{\log\log n/\log d})})$ rows achieves this guarantee; we point out a gap in the proof. Song, Ye, Yin and Zhang later claimed $O(ε^{-2}d\log^3 n)$ rows, but their proof relies on an independence assumption that does not hold in general, and we exhibit an explicit instance on which it fails. To achieve a truly nearly-linear-in-$d$ row count, we introduce a new fast, dense randomized transform combining a randomized Hadamard flattening, a random permutation, and balanced, disjoint Gaussian pooling. Conditioned on the Hadamard-and-permutation stage, the sketched problem becomes an exact Gaussian regression whose noise is independent of the entire sketched design, exactly what Song, Ye, Yin and Zhang's argument was missing. Our sketch yields the $\ell_\infty$ guarantee with $m=O(ε^{-2}d\log d)$ rows, using one Hadamard pass with padded internal dimension $N=\widetilde{O}(n+ε^{-2}d^3)$. It is also efficient to apply: $(SA,Sb)$ can be computed in $O(Nd\log N)=\widetilde{O}(nd+ε^{-2}d^4)$ time.

cs.DS · cs.LG · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2509.16801

Sublinear Time Quantum Sensitivity Sampling

Zhao Song, David P. Woodruff, Lichen Zhang

原文摘要

We present a unified quantum sensitivity sampling framework that streamlines coreset construction for many classical approximation problems, significantly speeding up clustering, regression, and low-rank approximation. We assume $n,d\geq\mathrm{poly}(k/ε)$ and, for regression, $n\geq\mathrm{poly}(d)$: - $k$-median/$k$-means: For $n$ points in $\mathbb{R}^d$, we build an $ε$-coreset of input points of size $\tilde O(k^{4/3}ε^{-2})$ (resp. $\tilde O(k^{3/2}ε^{-2})$) in time $\tilde O(ε^{-1}n^{0.5}dk^{1.67})$ (resp. $\tilde O(ε^{-1}n^{0.5}dk^{1.75})$). Our better $d$- and linear $1/ε$-dependence improve on [Xue, Chen, Li and Jiang, ICML'23] for $d\geqΩ(k^{7/3})$ (resp. $Ω(k^{2.5})$). - $\ell_p$ regression: For $\min_x\|Ax-b\|_p$, $A\in\mathbb{R}^{n\times d}$, we build an $ε$-coreset of size $\tilde O_p(d^{\max\{1,p/2\}}ε^{-2})$ in time $\tilde O_p(n^{0.5}d^{1.5}ε^{-1})$ for $p\leq 2$ and $\tilde O_p(n^{0.5}d^{p/4+1}(ε^{-1}+d^{0.5}))$ for $p>2$, matching [Liu and Ji, ICML'26] for $p\leq 2$ and improving on the prior best quantum sampling approach [Apers and Gribling, SICOMP'26] for all $p>2$. - Low-rank approximation: We give the first quantum sublinear-time algorithm for Frobenius-norm rank-$k$ approximation of $A\in\mathbb{R}^{d\times n}$ free of data-dependent parameters, running in $\tilde O(n^{0.5}dk^{0.5}ε^{-0.5})$ time with factored output (plus $\tilde O(nk^2ε^{-2})$ to materialize them). We also give sublinear algorithms for kernel and tensor variants. - Lower bound: For generic sensitivity sampling with total sensitivity $S$, any quantum algorithm outputting an $ε$-coreset of size $o(\sqrt{nS})$ needs $Ω(\sqrt{nS})$ queries, so our query complexity is optimal up to $\mathrm{poly}(ε^{-1},\log n)$ and VC-dimension factors.

cs.DS · cs.LG · quant-ph

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.07980

Solving Linear Programs in the Matrix Multiplication Time

Zhao Song

原文摘要

We give a randomized algorithm for feasible bounded-box linear programs with $n$ variables and $d\le n=O(d)$ linearly independent equality constraints. For additive accuracy $ε$, its expected running time is $O((n^ω+n^2\log n)\log(1/ε))$, where $ω$ is the matrix multiplication exponent. This answers the matrix-multiplication-time open question posted in Song's Ph.D. thesis [Son19] in this setting.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35228

Improved Quasi-Polynomial-Time Approximation for Maximum Independent Set

Zhao Song

原文摘要

We give a deterministic quasi-polynomial-time algorithm for the maximum independent set problem. If an $N$-vertex graph contains an independent set of size $2^{-t}N$, where $t$ is an integer with $1\le t\le\sqrt{\log N}/12$, then our algorithm finds an independent set of size $2^t$ in time $N^{O(t)}$. This gives an approximation ratio of $N/2^{Ω(\sqrt{\log N})}$ in time $N^{O(\sqrt{\log N})}$, improving on the approximation ratio $N/2^{Ω((\log N)^{1/3})}$ obtained by Lovett [Lov26]. A limitation of this form of analysis follows from the integrality gaps of Tulsiani [Tul09]. These gaps imply that a universal guarantee of the form $\operatorname{Las}_{R(t)}(G)\ge2^{-t}N\Longrightarrowα(G)\ge2^t$, at $2^{o(\sqrt{\log N\log\log N})}$ levels, cannot hold over a range extending beyond $t=O(\sqrt{\log N\log\log N})$. Thus, the exponent $1/2$ is the limit of this form of analysis, up to a factor of $\sqrt{\log\log N}$.

cs.DS · cs.CC

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arXiv ↗PDF ↗