We consider the following oblivious sketching problem: given $ε\in (0,1/3)$ and $n \geq d/ε^2$, design a distribution $\mathcal{D}$ over $\mathbb{R}^{k \times nd}$ and a function $f: \mathbb{R}^k \times \mathbb{R}^{nd} \rightarrow \mathbb{R}$, so that for any $n \times d$ matrix $A$, $$\Pr_{S \sim \mathcal{D}} [(1-ε) \|A\|_{op} \leq f(S(A),S) \leq (1+ε)\|A\|_{op}] \geq 2/3,$$ where $\|A\|_{op}$ is the operator norm of $A$ and $S(A)$ denotes $S \cdot A$, interpreting $A$ as a vector in $\mathbb{R}^{nd}$. We show a tight lower bound of $k = Ω(d^2/ε^2)$ for this problem. Our result considerably strengthens the result of Nelson and Nguyen (ICALP, 2014), as it (1) applies only to estimating the operator norm, which can be estimated given any OSE, and (2) applies to distributions over general linear operators $S$ which treat $A$ as a vector and compute $S(A)$, rather than the restricted class of linear operators corresponding to matrix multiplication. Our technique also implies the first tight bounds for approximating the Schatten $p$-norm for even integers $p$ via general linear sketches, improving the previous lower bound from $k = Ω(n^{2-6/p})$ [Regev, 2014] to $k = Ω(n^{2-4/p})$. Importantly, for sketching the operator norm up to a factor of $α$, where $α- 1 = Ω(1)$, we obtain a tight $k = Ω(n^2/α^4)$ bound, matching the upper bound of Andoni and Nguyen (SODA, 2013), and improving the previous $k = Ω(n^2/α^6)$ lower bound. Finally, we also obtain the first lower bounds for approximating Ky Fan norms.
In this paper, we study the distributed sketching complexity of connectivity. In distributed graph sketching, an $n$-node graph $G$ is distributed to $n$ players such that each player sees the neighborhood of one vertex. The players then simultaneously send one message to the referee, who must compute some function of $G$ with high probability. For connectivity, the referee must output whether $G$ is connected. The goal is to minimize the message lengths. Such sketching schemes are equivalent to one-round protocols in the broadcast congested clique model. We prove that the expected average mes…
Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e.g., estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot…
Given an $n \times d$ matrix $A$, its Schatten-$p$ norm, $p \geq 1$, is defined as $\|A\|_p = \left (\sum_{i=1}^{\textrm{rank}(A)}σ_i(A)^p \right )^{1/p}$, where $σ_i(A)$ is the $i$-th largest singular value of $A$. These norms have been studied in functional analysis in the context of non-commutative $\ell_p$-spaces, and recently in data stream and linear sketching models of computation. Basic questions on the relations between these norms, such as their embeddability, are still open. Specifically, given a set of matrices $A^1, \ldots, A^{\operatorname{poly}(nd)} \in \mathbb{R}^{n \times d}$,…
An outstanding open question posed by Guha and Indyk in 2006 asks to characterize metric spaces in which distances can be estimated using efficient sketches. Specifically, we say that a sketching algorithm is efficient if it achieves constant approximation using constant sketch size. A well-known result of Indyk (J. ACM, 2006) implies that a metric that admits a constant-distortion embedding into $\ell_p$ for $p\in(0,2]$ also admits an efficient sketching scheme. But is the converse true, i.e., is embedding into $\ell_p$ the only way to achieve efficient sketching? We address these questions f…
We study the problem of constructing a linear sketch of minimum dimension that allows approximation of a given real-valued function $f \colon \mathbb{F}_2^n \rightarrow \mathbb R$ with small expected squared error. We develop a general theory of linear sketching for such functions through which we analyze their dimension for most commonly studied types of valuation functions: additive, budget-additive, coverage, $α$-Lipschitz submodular and matroid rank functions. This gives a characterization of how many bits of information have to be stored about the input $x$ so that one can compute $f$ und…
We initiate a systematic study of linear sketching over $\mathbb F_2$. For a given Boolean function $f \colon \{0,1\}^n \to \{0,1\}$ a randomized $\mathbb F_2$-sketch is a distribution $\mathcal M$ over $d \times n$ matrices with elements over $\mathbb F_2$ such that $\mathcal Mx$ suffices for computing $f(x)$ with high probability. We study a connection between $\mathbb F_2$-sketching and a two-player one-way communication game for the corresponding XOR-function. Our results show that this communication game characterizes $\mathbb F_2$-sketching under the uniform distribution (up to dependenc…
We initiate the study of data dimensionality reduction, or sketching, for the $q\to p$ norms. Given an $n \times d$ matrix $A$, the $q\to p$ norm, denoted $\|A\|_{q \to p} = \sup_{x \in \mathbb{R}^d \backslash \vec{0}} \frac{\|Ax\|_p}{\|x\|_q}$, is a natural generalization of several matrix and vector norms studied in the data stream and sketching models, with applications to datamining, hardness of approximation, and oblivious routing. We say a distribution $S$ on random matrices $L \in \mathbb{R}^{nd} \rightarrow \mathbb{R}^k$ is a $(k,α)$-sketching family if from $L(A)$, one can approximate…
We study classic streaming and sparse recovery problems using deterministic linear sketches, including l1/l1 and linf/l1 sparse recovery problems (the latter also being known as l1-heavy hitters), norm estimation, and approximate inner product. We focus on devising a fixed matrix A in R^{m x n} and a deterministic recovery/estimation procedure which work for all possible input vectors simultaneously. Our results improve upon existing work, the following being our main contributions: * A proof that linf/l1 sparse recovery and inner product estimation are equivalent, and that incoherent matrices…
We consider the problem of sketching the $p$-th frequency moment of a vector, $p>2$, with multiplicative error at most $1\pm ε$ and \emph{with high confidence} $1-δ$. Despite the long sequence of work on this problem, tight bounds on this quantity are only known for constant $δ$. While one can obtain an upper bound with error probability $δ$ by repeating a sketching algorithm with constant error probability $O(\log(1/δ))$ times in parallel, and taking the median of the outputs, we show this is a suboptimal algorithm! Namely, we show optimal upper and lower bounds of $Θ(n^{1-2/p} \log(1/δ) + n^…
UC San Diego（加州大学圣地亚哥分校）；Carnegie Mellon University（卡内基梅隆大学）；Tsinghua University（清华大学）
We study statistical problems, such as planted clique, its variants, and sparse principal component analysis in the context of average-case communication complexity. Our motivation is to understand the statistical-computational trade-offs in streaming, sketching, and query-based models. Communication complexity is the main tool for proving lower bounds in these models, yet many prior results do not hold in an average-case setting. We provide a general reduction method that preserves the input distribution for problems involving a random graph or matrix with planted structure. Then, we derive t…
Bounded Independence for $k$-Min-Wise Hashing: Tight Bounds and Limitations of Structured Hashing
University of Science and Technology of China（中国科学技术大学）；Center on Frontiers of Computing Studies, Peking University（北京大学前沿计算研究中心）；Department of Computer Science, Johns Hopkins University（约翰斯·霍普金斯大学计算机系）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
We consider the problem of minimizing the number of matrix-vector queries needed for accurate trace estimation in the dynamic setting where our underlying matrix is changing slowly, such as during an optimization process. Specifically, for any $m$ matrices $A_1,...,A_m$ with consecutive differences bounded in Schatten-$1$ norm by $α$, we provide a novel binary tree summation procedure that simultaneously estimates all $m$ traces up to $ε$ error with $δ$ failure probability with an optimal query complexity of $\widetilde{O}\left(m α\sqrt{\log(1/δ)}/ε+ m\log(1/δ)\right)$, improving the dependenc…
Almost Tight Error Bounds on Differentially Private Continual Counting
Faculty of Computer Science, University of Vienna（维也纳大学计算机学院）；Rutgers University（罗格斯大学）；Indian Statistical Institute, Delhi（印度统计研究所德里中心）；Fujitsu Research of America（美国富士通研究所）
We initiate a broad study of classical problems in the streaming model with insertions and deletions in the setting where we allow the approximation factor $α$ to be much larger than $1$. Such algorithms can use significantly less memory than the usual setting for which $α= 1+ε$ for an $ε\in (0,1)$. We study large approximations for a number of problems in sketching and streaming and the following are some of our results. For the $\ell_p$ norm/quasinorm $\|x\|_p$ of an $n$-dimensional vector $x$, $0 < p \le 2$, we show that obtaining a $\poly(n)$-approximation requires the same amount of memor…
