arXiv subjects

David P. Woodruff

Publications and source records attributed to David P. Woodruff.

At least 73 records · Page 4Linked to original sources

arXiv2024

Faster Algorithms for Schatten-p Low Rank Approximation

We study algorithms for the Schatten-$p$ Low Rank Approximation (LRA) problem. First, we show that by using fast rectangular matrix multiplication algorithms and different block sizes, we can improve the running time of the algorithms in the recent work of Bakshi, Clarkson and Woodruff (STOC 2022). We then show that by carefully combining our new algorithm with the algorithm of Li and Woodruff (ICML 2020), we can obtain even faster algorithms for Schatten-$p$ LRA. While the block-based algorithms are fast in the real number model, we do not have a stability analysis which shows that the algorithms work when implemented on a machine with polylogarithmic bits of precision. We show that the LazySVD algorithm of Allen-Zhu and Li (NeurIPS 2016) can be implemented on a floating point machine with only logarithmic, in the input parameters, bits of precision. As far as we are aware, this is the first stability analysis of any algorithm using $O((k/\sqrt{\varepsilon})\text{poly}(\log n))$ matrix-vector products with the matrix $A$ to output a $1+\varepsilon$ approximate solution for the rank-$k$ Schatten-$p$ LRA problem.

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arXiv2024

Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-Cut

In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is to approximate cuts in balanced directed graphs. In this problem, the goal is to build a data structure that $(1 \pm ε)$-approximates cut values in graphs with $n$ vertices. For arbitrary directed graphs, such a data structure requires $Ω(n^2)$ bits even for constant $ε$. To circumvent this, recent works study $β$-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most $β$ times that in the other direction. We consider two models: the {\em for-each} model, where the goal is to approximate each cut with constant probability, and the {\em for-all} model, where all cuts must be preserved simultaneously. We improve the previous $Ω(n \sqrt{β/ε})$ lower bound to $\tildeΩ(n \sqrtβ/ε)$ in the for-each model, and we improve the previous $Ω(n β/ε)$ lower bound to $Ω(n β/ε^2)$ in the for-all model. This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is to approximate the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We improve the previous $Ω\bigl(\frac{m}{k}\bigr)$ query complexity lower bound to $Ω\bigl(\min\{m, \frac{m}{ε^2 k}\}\bigr)$ for this problem, where $m$ is the number of edges, $k$ is the size of the minimum cut, and we seek a $(1+ε)$-approximation. In addition, we show that existing upper bounds with slight modifications match our lower bound up to logarithmic factors.

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arXiv2024

Fast Sampling Based Sketches for Tensors

We introduce a new approach for applying sampling-based sketches to two and three mode tensors. We illustrate our technique to construct sketches for the classical problems of $\ell_0$ sampling and producing $\ell_1$ embeddings. In both settings we achieve sketches that can be applied to a rank one tensor in $(\mathbb{R}^d)^{\otimes q}$ (for $q=2,3$) in time scaling with $d$ rather than $d^2$ or $d^3$. Our main idea is a particular sampling construction based on fast convolution which allows us to quickly compute sums over sufficiently random subsets of tensor entries.

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arXiv2024

Fast White-Box Adversarial Streaming Without a Random Oracle

Recently, the question of adversarially robust streaming, where the stream is allowed to depend on the randomness of the streaming algorithm, has gained a lot of attention. In this work, we consider a strong white-box adversarial model (Ajtai et al. PODS 2022), in which the adversary has access to all past random coins and the parameters used by the streaming algorithm. We focus on the sparse recovery problem and extend our result to other tasks such as distinct element estimation and low-rank approximation of matrices and tensors. The main drawback of previous work is that it requires a random oracle, which is especially problematic in the streaming model since the amount of randomness is counted in the space complexity of a streaming algorithm. Also, the previous work suffers from large update time. We construct a near-optimal solution for the sparse recovery problem in white-box adversarial streams, based on the subexponentially secure Learning with Errors assumption. Importantly, our solution does not require a random oracle and has a polylogarithmic per item processing time. We also give results in a related white-box adversarially robust distributed model. Our constructions are based on homomorphic encryption schemes satisfying very mild structural properties that are currently satisfied by most known schemes.

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arXiv2024

Streaming Algorithms with Few State Changes

In this paper, we study streaming algorithms that minimize the number of changes made to their internal state (i.e., memory contents). While the design of streaming algorithms typically focuses on minimizing space and update time, these metrics fail to capture the asymmetric costs, inherent in modern hardware and database systems, of reading versus writing to memory. In fact, most streaming algorithms write to their memory on every update, which is undesirable when writing is significantly more expensive than reading. This raises the question of whether streaming algorithms with small space and number of memory writes are possible. We first demonstrate that, for the fundamental $F_p$ moment estimation problem with $p\ge 1$, any streaming algorithm that achieves a constant factor approximation must make $Ω(n^{1-1/p})$ internal state changes, regardless of how much space it uses. Perhaps surprisingly, we show that this lower bound can be matched by an algorithm that also has near-optimal space complexity. Specifically, we give a $(1+\varepsilon)$-approximation algorithm for $F_p$ moment estimation that uses a near-optimal $\widetilde{\mathcal{O}}_\varepsilon(n^{1-1/p})$ number of state changes, while simultaneously achieving near-optimal space, i.e., for $p\in[1,2]$, our algorithm uses $\text{poly}\left(\log n,\frac{1}{\varepsilon}\right)$ bits of space, while for $p>2$, the algorithm uses $\widetilde{\mathcal{O}}_\varepsilon(n^{1-2/p})$ space. We similarly design streaming algorithms that are simultaneously near-optimal in both space complexity and the number of state changes for the heavy-hitters problem, sparse support recovery, and entropy estimation. Our results demonstrate that an optimal number of state changes can be achieved without sacrificing space complexity.

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arXiv2024

High-Dimensional Geometric Streaming for Nearly Low Rank Data

We study streaming algorithms for the $\ell_p$ subspace approximation problem. Given points $a_1, \ldots, a_n$ as an insertion-only stream and a rank parameter $k$, the $\ell_p$ subspace approximation problem is to find a $k$-dimensional subspace $V$ such that $(\sum_{i=1}^n d(a_i, V)^p)^{1/p}$ is minimized, where $d(a, V)$ denotes the Euclidean distance between $a$ and $V$ defined as $\min_{v \in V}\|{a - v}\|_{\infty}$. When $p = \infty$, we need to find a subspace $V$ that minimizes $\max_i d(a_i, V)$. For $\ell_{\infty}$ subspace approximation, we give a deterministic strong coreset construction algorithm and show that it can be used to compute a $\text{poly}(k, \log n)$ approximate solution. We show that the distortion obtained by our coreset is nearly tight for any sublinear space algorithm. For $\ell_p$ subspace approximation, we show that suitably scaling the points and then using our $\ell_{\infty}$ coreset construction, we can compute a $\text{poly}(k, \log n)$ approximation. Our algorithms are easy to implement and run very fast on large datasets. We also use our strong coreset construction to improve the results in a recent work of Woodruff and Yasuda (FOCS 2022) which gives streaming algorithms for high-dimensional geometric problems such as width estimation, convex hull estimation, and volume estimation.

cs.DS↗

arXiv2024

Reweighted Solutions for Weighted Low Rank Approximation

Weighted low rank approximation (WLRA) is an important yet computationally challenging primitive with applications ranging from statistical analysis, model compression, and signal processing. To cope with the NP-hardness of this problem, prior work considers heuristics, bicriteria, or fixed parameter tractable algorithms to solve this problem. In this work, we introduce a new relaxed solution to WLRA which outputs a matrix that is not necessarily low rank, but can be stored using very few parameters and gives provable approximation guarantees when the weight matrix has low rank. Our central idea is to use the weight matrix itself to reweight a low rank solution, which gives an extremely simple algorithm with remarkable empirical performance in applications to model compression and on synthetic datasets. Our algorithm also gives nearly optimal communication complexity bounds for a natural distributed problem associated with this problem, for which we show matching communication lower bounds. Together, our communication complexity bounds show that the rank of the weight matrix provably parameterizes the communication complexity of WLRA. We also obtain the first relative error guarantees for feature selection with a weighted objective.

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arXiv2024

Coresets for Multiple $\ell_p$ Regression

A coreset of a dataset with $n$ examples and $d$ features is a weighted subset of examples that is sufficient for solving downstream data analytic tasks. Nearly optimal constructions of coresets for least squares and $\ell_p$ linear regression with a single response are known in prior work. However, for multiple $\ell_p$ regression where there can be $m$ responses, there are no known constructions with size sublinear in $m$. In this work, we construct coresets of size $\tilde O(\varepsilon^{-2}d)$ for $p<2$ and $\tilde O(\varepsilon^{-p}d^{p/2})$ for $p>2$ independently of $m$ (i.e., dimension-free) that approximate the multiple $\ell_p$ regression objective at every point in the domain up to $(1\pm\varepsilon)$ relative error. If we only need to preserve the minimizer subject to a subspace constraint, we improve these bounds by an $\varepsilon$ factor for all $p>1$. All of our bounds are nearly tight. We give two application of our results. First, we settle the number of uniform samples needed to approximate $\ell_p$ Euclidean power means up to a $(1+\varepsilon)$ factor, showing that $\tildeΘ(\varepsilon^{-2})$ samples for $p = 1$, $\tildeΘ(\varepsilon^{-1})$ samples for $1 < p < 2$, and $\tildeΘ(\varepsilon^{1-p})$ samples for $p>2$ is tight, answering a question of Cohen-Addad, Saulpic, and Schwiegelshohn. Second, we show that for $1<p<2$, every matrix has a subset of $\tilde O(\varepsilon^{-1}k)$ rows which spans a $(1+\varepsilon)$-approximately optimal $k$-dimensional subspace for $\ell_p$ subspace approximation, which is also nearly optimal.

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arXiv2024

Learning the Positions in CountSketch

We consider sketching algorithms which first compress data by multiplication with a random sketch matrix, and then apply the sketch to quickly solve an optimization problem, e.g., low-rank approximation and regression. In the learning-based sketching paradigm proposed by~\cite{indyk2019learning}, the sketch matrix is found by choosing a random sparse matrix, e.g., CountSketch, and then the values of its non-zero entries are updated by running gradient descent on a training data set. Despite the growing body of work on this paradigm, a noticeable omission is that the locations of the non-zero entries of previous algorithms were fixed, and only their values were learned. In this work, we propose the first learning-based algorithms that also optimize the locations of the non-zero entries. Our first proposed algorithm is based on a greedy algorithm. However, one drawback of the greedy algorithm is its slower training time. We fix this issue and propose approaches for learning a sketching matrix for both low-rank approximation and Hessian approximation for second order optimization. The latter is helpful for a range of constrained optimization problems, such as LASSO and matrix estimation with a nuclear norm constraint. Both approaches achieve good accuracy with a fast running time. Moreover, our experiments suggest that our algorithm can still reduce the error significantly even if we only have a very limited number of training matrices.

cs.LG↗

arXiv2024

A New Information Complexity Measure for Multi-pass Streaming with Applications

We introduce a new notion of information complexity for multi-pass streaming problems and use it to resolve several important questions in data streams. In the coin problem, one sees a stream of $n$ i.i.d. uniform bits and one would like to compute the majority with constant advantage. We show that any constant pass algorithm must use $Ω(\log n)$ bits of memory, significantly extending an earlier $Ω(\log n)$ bit lower bound for single-pass algorithms of Braverman-Garg-Woodruff (FOCS, 2020). This also gives the first $Ω(\log n)$ bit lower bound for the problem of approximating a counter up to a constant factor in worst-case turnstile streams for more than one pass. In the needle problem, one either sees a stream of $n$ i.i.d. uniform samples from a domain $[t]$, or there is a randomly chosen needle $α\in[t]$ for which each item independently is chosen to equal $α$ with probability $p$, and is otherwise uniformly random in $[t]$. The problem of distinguishing these two cases is central to understanding the space complexity of the frequency moment estimation problem in random order streams. We show tight multi-pass space bounds for this problem for every $p < 1/\sqrt{n \log^3 n}$, resolving an open question of Lovett and Zhang (FOCS, 2023); even for $1$-pass our bounds are new. To show optimality, we improve both lower and upper bounds from existing results. Our information complexity framework significantly extends the toolkit for proving multi-pass streaming lower bounds, and we give a wide number of additional streaming applications of our lower bound techniques, including multi-pass lower bounds for $\ell_p$-norm estimation, $\ell_p$-point query and heavy hitters, and compressed sensing problems.

cs.CC↗

arXiv2024

Optimal Communication for Classic Functions in the Coordinator Model and Beyond

In the coordinator model of communication with $s$ servers, given an arbitrary non-negative function $f$, we study the problem of approximating the sum $\sum_{i \in [n]}f(x_i)$ up to a $1 \pm \varepsilon$ factor. Here the vector $x \in R^n$ is defined to be $x = x(1) + \cdots + x(s)$, where $x(j) \ge 0$ denotes the non-negative vector held by the $j$-th server. A special case of the problem is when $f(x) = x^k$ which corresponds to the well-studied problem of $F_k$ moment estimation in the distributed communication model. We introduce a new parameter $c_f[s]$ which captures the communication complexity of approximating $\sum_{i\in [n]} f(x_i)$ and for a broad class of functions $f$ which includes $f(x) = x^k$ for $k \ge 2$ and other robust functions such as the Huber loss function, we give a two round protocol that uses total communication $c_f[s]/\varepsilon^2$ bits, up to polylogarithmic factors. For this broad class of functions, our result improves upon the communication bounds achieved by Kannan, Vempala, and Woodruff (COLT 2014) and Woodruff and Zhang (STOC 2012), obtaining the optimal communication up to polylogarithmic factors in the minimum number of rounds. We show that our protocol can also be used for approximating higher-order correlations. Apart from the coordinator model, algorithms for other graph topologies in which each node is a server have been extensively studied. We argue that directly lifting protocols leads to inefficient algorithms. Hence, a natural question is the problems that can be efficiently solved in general graph topologies. We give communication efficient protocols in the so-called personalized CONGEST model for solving linear regression and low rank approximation by designing composable sketches. Our sketch construction may be of independent interest and can implement any importance sampling procedure that has a monotonicity property.

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arXiv2024

The $\ell_p$-Subspace Sketch Problem in Small Dimensions with Applications to Support Vector Machines

In the $\ell_p$-subspace sketch problem, we are given an $n\times d$ matrix $A$ with $n>d$, and asked to build a small memory data structure $Q(A,ε)$ so that, for any query vector $x\in\mathbb{R}^d$, we can output a number in $(1\pmε)\|Ax\|_p^p$ given only $Q(A,ε)$. This problem is known to require $\tildeΩ(dε^{-2})$ bits of memory for $d=Ω(\log(1/ε))$. However, for $d=o(\log(1/ε))$, no data structure lower bounds were known. We resolve the memory required to solve the $\ell_p$-subspace sketch problem for any constant $d$ and integer $p$, showing that it is $Ω(ε^{-2(d-1)/(d+2p)})$ bits and $\tilde{O} (ε^{-2(d-1)/(d+2p)})$ words. This shows that one can beat the $Ω(ε^{-2})$ lower bound, which holds for $d = Ω(\log(1/ε))$, for any constant $d$. We also show how to implement the upper bound in a single pass stream, with an additional multiplicative $\operatorname{poly}(\log \log n)$ factor and an additive $\operatorname{poly}(\log n)$ cost in the memory. Our bounds can be applied to point queries for SVMs with additive error, yielding an optimal bound of $\tildeΘ(ε^{-2d/(d+3)})$ for every constant $d$. This is a near-quadratic improvement over the $Ω(ε^{-(d+1)/(d+3)})$ lower bound of (Andoni et al. 2020). Our techniques rely on a novel connection to low dimensional techniques from geometric functional analysis.

cs.DS↗

arXiv2024

Pseudorandom Hashing for Space-bounded Computation with Applications in Streaming

We revisit Nisan's classical pseudorandom generator (PRG) for space-bounded computation (STOC 1990) and its applications in streaming algorithms. We describe a new generator, HashPRG, that can be thought of as a symmetric version of Nisan's generator over larger alphabets. Our generator allows a trade-off between seed length and the time needed to compute a given block of the generator's output. HashPRG can be used to obtain derandomizations with much better update time and \emph{without sacrificing space} for a large number of data stream algorithms, such as $F_p$ estimation in the parameter regimes $p > 2$ and $0 < p < 2$ and CountSketch with tight estimation guarantees as analyzed by Minton and Price (SODA 2014) which assumed access to a random oracle. We also show a recent analysis of Private CountSketch can be derandomized using our techniques. For a $d$-dimensional vector $x$ being updated in a turnstile stream, we show that $\|x\|_{\infty}$ can be estimated up to an additive error of $\varepsilon\|x\|_{2}$ using $O(\varepsilon^{-2}\log(1/\varepsilon)\log d)$ bits of space. Additionally, the update time of this algorithm is $O(\log 1/\varepsilon)$ in the Word RAM model. We show that the space complexity of this algorithm is optimal up to constant factors. However, for vectors $x$ with $\|x\|_{\infty} = Θ(\|x\|_{2})$, we show that the lower bound can be broken by giving an algorithm that uses $O(\varepsilon^{-2}\log d)$ bits of space which approximates $\|x\|_{\infty}$ up to an additive error of $\varepsilon\|x\|_{2}$. We use our aforementioned derandomization of the CountSketch data structure to obtain this algorithm, and using the time-space trade off of HashPRG, we show that the update time of this algorithm is also $O(\log 1/\varepsilon)$ in the Word RAM model.

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arXiv2024

Sharper Bounds for $\ell_p$ Sensitivity Sampling

In large scale machine learning, random sampling is a popular way to approximate datasets by a small representative subset of examples. In particular, sensitivity sampling is an intensely studied technique which provides provable guarantees on the quality of approximation, while reducing the number of examples to the product of the VC dimension $d$ and the total sensitivity $\mathfrak S$ in remarkably general settings. However, guarantees going beyond this general bound of $\mathfrak S d$ are known in perhaps only one setting, for $\ell_2$ subspace embeddings, despite intense study of sensitivity sampling in prior work. In this work, we show the first bounds for sensitivity sampling for $\ell_p$ subspace embeddings for $p > 2$ that improve over the general $\mathfrak S d$ bound, achieving a bound of roughly $\mathfrak S^{2-2/p}$ for $2<p<\infty$. Furthermore, our techniques yield further new results in the study of sampling algorithms, showing that the root leverage score sampling algorithm achieves a bound of roughly $d$ for $1\leq p<2$, and that a combination of leverage score and sensitivity sampling achieves an improved bound of roughly $d^{2/p}\mathfrak S^{2-4/p}$ for $2<p<\infty$. Our sensitivity sampling results yield the best known sample complexity for a wide class of structured matrices that have small $\ell_p$ sensitivity.

cs.DS↗

arXiv2023

HyperAttention: Long-context Attention in Near-Linear Time

We present an approximate attention mechanism named HyperAttention to address the computational challenges posed by the growing complexity of long contexts used in Large Language Models (LLMs). Recent work suggests that in the worst-case scenario, quadratic time is necessary unless the entries of the attention matrix are bounded or the matrix has low stable rank. We introduce two parameters which measure: (1) the max column norm in the normalized attention matrix, and (2) the ratio of row norms in the unnormalized attention matrix after detecting and removing large entries. We use these fine-grained parameters to capture the hardness of the problem. Despite previous lower bounds, we are able to achieve a linear time sampling algorithm even when the matrix has unbounded entries or a large stable rank, provided the above parameters are small. HyperAttention features a modular design that easily accommodates integration of other fast low-level implementations, particularly FlashAttention. Empirically, employing Locality Sensitive Hashing (LSH) to identify large entries, HyperAttention outperforms existing methods, giving significant speed improvements compared to state-of-the-art solutions like FlashAttention. We validate the empirical performance of HyperAttention on a variety of different long-context length datasets. For example, HyperAttention makes the inference time of ChatGLM2 50\% faster on 32k context length while perplexity increases from 5.6 to 6.3. On larger context length, e.g., 131k, with causal masking, HyperAttention offers 5-fold speedup on a single attention layer.

cs.LG↗

arXiv2023

Space-Optimal Profile Estimation in Data Streams with Applications to Symmetric Functions

We revisit the problem of estimating the profile (also known as the rarity) in the data stream model. Given a sequence of $m$ elements from a universe of size $n$, its profile is a vector $ϕ$ whose $i$-th entry $ϕ_i$ represents the number of distinct elements that appear in the stream exactly $i$ times. A classic paper by Datar and Muthukrishan from 2002 gave an algorithm which estimates any entry $ϕ_i$ up to an additive error of $\pm εD$ using $O(1/ε^2 (\log n + \log m))$ bits of space, where $D$ is the number of distinct elements in the stream. In this paper, we considerably improve on this result by designing an algorithm which simultaneously estimates many coordinates of the profile vector $ϕ$ up to small overall error. We give an algorithm which, with constant probability, produces an estimated profile $\hatϕ$ with the following guarantees in terms of space and estimation error: - For any constant $τ$, with $O(1 / ε^2 + \log n)$ bits of space, $\sum_{i=1}^τ|ϕ_i - \hatϕ_i| \leq εD$. - With $O(1/ ε^2\log (1/ε) + \log n + \log \log m)$ bits of space, $\sum_{i=1}^m |ϕ_i - \hatϕ_i| \leq εm$. In addition to bounding the error across multiple coordinates, our space bounds separate the terms that depend on $1/ε$ and those that depend on $n$ and $m$. We prove matching lower bounds on space in both regimes. Application of our profile estimation algorithm gives estimates within error $\pm εD$ of several symmetric functions of frequencies in $O(1/ε^2 + \log n)$ bits. This generalizes space-optimal algorithms for the distinct elements problems to other problems including estimating the Huber and Tukey losses as well as frequency cap statistics.

cs.DS↗

arXiv2023

Near-Linear Time and Fixed-Parameter Tractable Algorithms for Tensor Decompositions

We study low rank approximation of tensors, focusing on the tensor train and Tucker decompositions, as well as approximations with tree tensor networks and more general tensor networks. For tensor train decomposition, we give a bicriteria $(1 + \eps)$-approximation algorithm with a small bicriteria rank and $O(q \cdot \nnz(A))$ running time, up to lower order terms, which improves over the additive error algorithm of \cite{huber2017randomized}. We also show how to convert the algorithm of \cite{huber2017randomized} into a relative error algorithm, but their algorithm necessarily has a running time of $O(qr^2 \cdot \nnz(A)) + n \cdot \poly(qk/\eps)$ when converted to a $(1 + \eps)$-approximation algorithm with bicriteria rank $r$. To the best of our knowledge, our work is the first to achieve polynomial time relative error approximation for tensor train decomposition. Our key technique is a method for obtaining subspace embeddings with a number of rows polynomial in $q$ for a matrix which is the flattening of a tensor train of $q$ tensors. We extend our algorithm to tree tensor networks. In addition, we extend our algorithm to tensor networks with arbitrary graphs (which we refer to as general tensor networks), by using a result of \cite{ms08_simulating_quantum_tensor_contraction} and showing that a general tensor network of rank $k$ can be contracted to a binary tree network of rank $k^{O(°(G)\tw(G))}$, allowing us to reduce to the case of tree tensor networks. Finally, we give new fixed-parameter tractable algorithms for the tensor train, Tucker, and CP decompositions, which are simpler than those of \cite{swz19_tensor_low_rank} since they do not make use of polynomial system solvers. Our technique of Gaussian subspace embeddings with exactly $k$ rows (and thus exponentially small success probability) may be of independent interest.

cs.DS↗

arXiv2023

Computing Approximate $\ell_p$ Sensitivities

Recent works in dimensionality reduction for regression tasks have introduced the notion of sensitivity, an estimate of the importance of a specific datapoint in a dataset, offering provable guarantees on the quality of the approximation after removing low-sensitivity datapoints via subsampling. However, fast algorithms for approximating $\ell_p$ sensitivities, which we show is equivalent to approximate $\ell_p$ regression, are known for only the $\ell_2$ setting, in which they are termed leverage scores. In this work, we provide efficient algorithms for approximating $\ell_p$ sensitivities and related summary statistics of a given matrix. In particular, for a given $n \times d$ matrix, we compute $α$-approximation to its $\ell_1$ sensitivities at the cost of $O(n/α)$ sensitivity computations. For estimating the total $\ell_p$ sensitivity (i.e. the sum of $\ell_p$ sensitivities), we provide an algorithm based on importance sampling of $\ell_p$ Lewis weights, which computes a constant factor approximation to the total sensitivity at the cost of roughly $O(\sqrt{d})$ sensitivity computations. Furthermore, we estimate the maximum $\ell_1$ sensitivity, up to a $\sqrt{d}$ factor, using $O(d)$ sensitivity computations. We generalize all these results to $\ell_p$ norms for $p > 1$. Lastly, we experimentally show that for a wide class of matrices in real-world datasets, the total sensitivity can be quickly approximated and is significantly smaller than the theoretical prediction, demonstrating that real-world datasets have low intrinsic effective dimensionality.

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