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This paper provides a one-line proof of Frequent Directions (FD) for sketching streams of matrices. The simpler proof arises from sketching the covariance of the stream of matrices rather than the stream itself.
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We adapt a well known streaming algorithm for approximating item frequencies to the matrix sketching setting. The algorithm receives the rows of a large matrix $A \in \R^{n \times m}$ one after the other in a streaming fashion. It maintains a sketch matrix $B \in \R^ {1/\eps \times m}$ such that for any unit vector $x$ [\|Ax\|^2 \ge \|Bx\|^2 \ge \|Ax\|^2 - \eps \|A\|_{f}^2 \.] Sketch updates per row in $A$ require $O(m/\eps^2)$ operations in the worst case. A slight modification of the algorithm allows for an amortized update time of $O(m/\eps)$ operations per row. The presented algorithm stan…
We describe a new algorithm called Frequent Directions for deterministic matrix sketching in the row-updates model. The algorithm is presented an arbitrary input matrix $A \in R^{n \times d}$ one row at a time. It performed $O(d \times \ell)$ operations per row and maintains a sketch matrix $B \in R^{\ell \times d}$ such that for any $k < \ell$ $\|A^TA - B^TB \|_2 \leq \|A - A_k\|_F^2 / (\ell-k)$ and $\|A - π_{B_k}(A)\|_F^2 \leq \big(1 + \frac{k}{\ell-k}\big) \|A-A_k\|_F^2 $ . Here, $A_k$ stands for the minimizer of $\|A - A_k\|_F$ over all rank $k$ matrices (similarly $B_k$) and $π_{B_k}(A)$ …
Deterministic Lower Bounds for $k$-Edge Connectivity in the Distributed Sketching Model
本文提出了分布式草图模型中k边连通性问题的第一个确定性下界，证明了最坏情况下的消息长度为Ω(k)位。
We introduce a Generalized LU-Factorization (\textbf{GLU}) for low-rank matrix approximation. We relate this to past approaches and extensively analyze its approximation properties. The established deterministic guarantees are combined with sketching ensembles satisfying Johnson-Lindenstrauss properties to present complete bounds. Particularly good performance is shown for the sub-sampled randomized Hadamard transform (SRHT) ensemble. Moreover, the factorization is shown to unify and generalize many past algorithms. It also helps to explain the effect of sketching on the growth factor during G…
On Deterministic Sketching and Streaming for Sparse Recovery and Norm Estimation
我们研究经典的流和稀疏恢复问题，使用确定性的线性草图，包括l1/l1和linf/l1稀疏恢复问题（后者也被称为l1重打击者），范数估计，近似内积。我们专注于设计一个固定的矩阵A在R^{mxn}和一个确定性的恢复/估计过程，同时工作的所有可能的输入向量。我们的结果改进现有的工作，以下是我们的主要贡献：* 证明linf/l1稀疏恢复和内积估计是等价的，并且可以使用非相干矩阵来解决这两个问题。我们的测量次数的上限是m=O（eps^{-2}*min{log n，（log n / log（1/eps））^2}）。利用快速Johnson-Lindenstrauss变换还可以得到快速的草图绘制和恢复算法。我们的运行时间和测量次数都比以前的工作有所改进。我们还可以获得更好的误差保证比以前的工作在一个较小的尾部的输入向量。* 求解l1/l1稀疏恢复所需线性测量数的新下界。我们证明了需要Omega（k/eps^2 + klog（n/k）/eps）测量来恢复x'，|x - x'|（1+eps）|x_{tail(k)}|_1，其中x_{tail(k)}是x投影到除了其最大k坐标之外的所有坐标上。* m = Theta（eps^{-2}log（eps^2 n））对求解确定性范数估计所需的测量数的紧界，即，以回收|x| _2 +/- eps |x| _1。对于我们研究的所有问题，紧界已经知道从以前的工作随机复…
\emph{Cardinality Estimation} (aka \emph{Distinct Elements}) is a classic problem in sketching with many industrial applications. Although sketching \emph{algorithms} are fairly simple, analyzing the cardinality \emph{estimators} is notoriously difficult, and even today the state-of-the-art sketches such as HyperLogLog and (compressed) \PCSA{} are not covered in graduate level Big Data courses. In this paper we define a class of \emph{generalized remaining area} (\tGRA) estimators, and observe that HyperLogLog, LogLog, and some estimators for PCSA are merely instantiations of \tGRA{} for vario…
A longstanding observation, which was partially proven in \cite{LNW14,AHLW16}, is that any turnstile streaming algorithm can be implemented as a linear sketch (the reverse is trivially true). We study the relationship between turnstile streaming and linear sketching algorithms in more detail, giving both new separations and new equivalences between the two models. It was shown in \cite{LNW14} that, if a turnstile algorithm works for arbitrarily long streams with arbitrarily large coordinates at intermediate stages of the stream, then the turnstile algorithm is equivalent to a linear sketch. We…
Distributed Least Squares in Small Space via Sketching and Bias Reduction
针对矩阵草图在最小二乘估计中的根本限制，提出通过最小化估计量偏差的稀疏草图方法，以最优空间和两遍数据扫描获得近无偏估计，并改进分布式平均算法。
A new algorithm, Guidesort, for sorting in the uniprocessor variant of the parallel disk model (PDM) of Vitter and Shriver is presented. The algorithm is deterministic and executes a number of (parallel) I/O operations that comes within a constant factor $C$ of the optimum. The algorithm and its analysis are simpler than those proposed in previous work, and the achievable constant factor $C$ of essentially 3 appears to be smaller than for all other known deterministic algorithms, at least for plausible parameter values.
Polynomial Lower Bounds for Distributed Graph Sketching with Tiny Error: Connectivity and Spanning Tree Construction
针对分布式图草图模型，我们证明了在微小误差下解决连通性和生成树构造问题需要多项式长度的消息，显著缩小了现有下界与上界之间的差距，并扩展至k边连通性。
We study the streaming model for approximate matrix multiplication (AMM). We are interested in the scenario that the algorithm can only take one pass over the data with limited memory. The state-of-the-art deterministic sketching algorithm for streaming AMM is the co-occurring directions (COD), which has much smaller approximation errors than randomized algorithms and outperforms other deterministic sketching methods empirically. In this paper, we provide a tighter error bound for COD whose leading term considers the potential approximate low-rank structure and the correlation of input matrice…
Optimal Matrix Sketching over Sliding Windows
本文提出DS-FD算法，在滑动窗口上实现矩阵素描的最优O(d/ε)空间界，并给出多种窗口模型的匹配上下界，通过实验验证了其正确性与有效性。
We provide a deterministic space-efficient algorithm for estimating ridge regression. For $n$ data points with $d$ features and a large enough regularization parameter, we provide a solution within $\varepsilon$ L$_2$ error using only $O(d/\varepsilon)$ space. This is the first $o(d^2)$ space deterministic streaming algorithm with guaranteed solution error and risk bound for this classic problem. The algorithm sketches the covariance matrix by variants of Frequent Directions, which implies it can operate in insertion-only streams and a variety of distributed data settings. In comparisons to ra…
Revisiting Approximate Leverage Score Sketching for Matrix Least Squares
本文重新审视了用于矩阵最小二乘问题的近似杠杆分数草图法，通过理论分析改进了相关常数，并提出了混合确定性和随机采样方案以减少样本需求。
Stream monitoring is fundamental in many data stream applications, such as financial data trackers, security, anomaly detection, and load balancing. In that respect, quantiles are of particular interest, as they often capture the user's utility. For example, if a video connection has high tail latency, the perceived quality will suffer, even if the average and median latencies are low. In this work, we consider the problem of approximating the per-item quantiles. Elements in our stream are (ID, latency) tuples, and we wish to track the latency quantiles for each ID. Existing quantile sketches …
We present a simple deterministic distributed algorithm that computes a $(Δ+1)$-vertex coloring in $O(\log^2 Δ\cdot \log n)$ rounds. The algorithm can be implemented with $O(\log n)$-bit messages. The algorithm can also be extended to the more general $(degree+1)$-list coloring problem. Obtaining a polylogarithmic-time deterministic algorithm for $(Δ+1)$-vertex coloring had remained a central open question in the area of distributed graph algorithms since the 1980s, until a recent network decomposition algorithm of Rozhoň and Ghaffari [STOC'20]. The current state of the art is based on an impr…
Spectral Sparsification by Deterministic Discrepancy Walk
本文将确定性差异游走框架从向量差异推广到矩阵差异，统一处理多种谱稀疏化问题，并以确定性分析给出更优结果与更简证明。
Deterministic algorithms for the Lovasz Local Lemma: simpler, more general, and more parallel
本文提出更简单、更一般、更并行的确定性算法，适用于 Shearer 准则的 LLL，并给出 MT 分布的确定性版本，应用于非重复顶点着色等问题，匹配最佳随机算法。
In the decremental $(1+ε)$-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph $G=(V,E)$ with $n = |V|, m = |E|$, undergoing edge deletions, and a distinguished source $s \in V$, and we are asked to process edge deletions efficiently and answer queries for distance estimates $\widetilde{\mathbf{dist}}_G(s,v)$ for each $v \in V$, at any stage, such that $\mathbf{dist}_G(s,v) \leq \widetilde{\mathbf{dist}}_G(s,v) \leq (1+ ε)\mathbf{dist}_G(s,v)$. In the decremental $(1+ε)$-approximate All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for distance est…
Finding 4-Additive Spanners: Faster, Stronger, and Simpler
本文提出一种构造 $4$-加性伸缩器的确定性算法，在保持 $\tilde{O}(n^{7/5})$ 边数界的同时，将运行时间改进至 $\tilde{O}(\min\{mn^{3/5},n^{11/5}\})$，且更简单易实现。
This paper resolves one of the longest standing basic problems in the streaming computational model. Namely, optimal construction of quantile sketches. An $\varepsilon$ approximate quantile sketch receives a stream of items $x_1,\ldots,x_n$ and allows one to approximate the rank of any query up to additive error $\varepsilon n$ with probability at least $1-δ$. The rank of a query $x$ is the number of stream items such that $x_i \le x$. The minimal sketch size required for this task is trivially at least $1/\varepsilon$. Felber and Ostrovsky obtain a $O((1/\varepsilon)\log(1/\varepsilon))$ spac…
The algorithm and complexity of approximating the permanent of a matrix is an extensively studied topic. Recently, its connection with quantum supremacy and more specifically BosonSampling draws special attention to the average-case approximation problem of the permanent of random matrices with zero or small mean value for each entry. Eldar and Mehraban (FOCS 2018) gave a quasi-polynomial time algorithm for random matrices with mean at least $1/\mathbf{\mathrm{polyloglog}} (n)$. In this paper, we improve the result by designing a deterministic quasi-polynomial time algorithm and a PTAS for ran…
Simpler Proofs by Symbolic Perturbation
在算法分析中，大量的精力往往花在讨论特殊情况上。例如，当分析分别考虑X和Y的情况时<Y and X>，我们可能必须特别注意当X=Y时会发生什么。另一方面，经验告诉我们，当一个尚未被重视的这类特殊情况被发现时，人们几乎总是能找到一种方法来处理它，通常是通过对分析和/或算法进行非常轻微的修改来完成的。在这篇文章中，我们从理论上证实了这一观察。我们专注于加权组合优化问题的确定性算法。这类问题实例由其结构和权重向量定义。空情况的概念被引入作为问题实例的集合，其权重向量在所有可能的权重配置的空间中构成无处开集（或空集）。一个算法被称为鲁棒的，如果任何空的情况下，可以忽略它的解决方案的质量和资源需求的分析。我们表明，实现鲁棒性只是以正确的方式打破关系的问题。更具体地说，我们表明，从几何算法领域已知的符号扰动的概念，保证不会发生任何意外的情况下，空。我们认为，对于一个巨大的类的组合优化算法，很容易验证，他们隐含地使用符号扰动打破关系，从而可以分析的假设下，一些任意空的情况下从来没有发生。最后，我们证明了存在一个符号扰动平局打破任何算法的政策。
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