arXiv subjects

David P. Woodruff

Publications and source records attributed to David P. Woodruff.

At least 55 records · Page 3Linked to original sources

arXiv2025

Guessing Efficiently for Constrained Subspace Approximation

In this paper we study constrained subspace approximation problem. Given a set of $n$ points $\{a_1,\ldots,a_n\}$ in $\mathbb{R}^d$, the goal of the {\em subspace approximation} problem is to find a $k$ dimensional subspace that best approximates the input points. More precisely, for a given $p\geq 1$, we aim to minimize the $p$th power of the $\ell_p$ norm of the error vector $(\|a_1-\bm{P}a_1\|,\ldots,\|a_n-\bm{P}a_n\|)$, where $\bm{P}$ denotes the projection matrix onto the subspace and the norms are Euclidean. In \emph{constrained} subspace approximation (CSA), we additionally have constraints on the projection matrix $\bm{P}$. In its most general form, we require $\bm{P}$ to belong to a given subset $\mathcal{S}$ that is described explicitly or implicitly. We introduce a general framework for constrained subspace approximation. Our approach, that we term coreset-guess-solve, yields either $(1+\varepsilon)$-multiplicative or $\varepsilon$-additive approximations for a variety of constraints. We show that it provides new algorithms for partition-constrained subspace approximation with applications to {\it fair} subspace approximation, $k$-means clustering, and projected non-negative matrix factorization, among others. Specifically, while we reconstruct the best known bounds for $k$-means clustering in Euclidean spaces, we improve the known results for the remainder of the problems.

cs.DS↗

arXiv2025

Tight Bounds for Heavy-Hitters and Moment Estimation in the Sliding Window Model

We consider the heavy-hitters and $F_p$ moment estimation problems in the sliding window model. For $F_p$ moment estimation with $1<p\leq 2$, we show that it is possible to give a $(1\pm ε)$ multiplicative approximation to the $F_p$ moment with $2/3$ probability on any given window of size $n$ using $\tilde{O}(\frac{1}{ε^p}\log^2 n + \frac{1}{ε^2}\log n)$ bits of space. We complement this result with a lower bound showing that our algorithm gives tight bounds up to factors of $\log\log n$ and $\log\frac{1}ε.$ As a consequence of our $F_2$ moment estimation algorithm, we show that the heavy-hitters problem can be solved on an arbitrary window using $O(\frac{1}{ε^2}\log^2 n)$ space which is tight.

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arXiv2025

Fast, Space-Optimal Streaming Algorithms for Clustering and Subspace Embeddings

We show that both clustering and subspace embeddings can be performed in the streaming model with the same asymptotic efficiency as in the central/offline setting. For $(k, z)$-clustering in the streaming model, we achieve a number of words of memory which is independent of the number $n$ of input points and the aspect ratio $Δ$, yielding an optimal bound of $\tilde{\mathcal{O}}\left(\frac{dk}{\min(\varepsilon^4,\varepsilon^{z+2})}\right)$ words for accuracy parameter $\varepsilon$ on $d$-dimensional points. Additionally, we obtain amortized update time of $d\,\log(k)\cdot\text{polylog}(\log(nΔ))$, which is an exponential improvement over the previous $d\,\text{poly}(k,\log(nΔ))$. Our method also gives the fastest runtime for $(k,z)$-clustering even in the offline setting. For subspace embeddings in the streaming model, we achieve $\mathcal{O}(d)$ update time and space-optimal constructions, using $\tilde{\mathcal{O}}\left(\frac{d^2}{\varepsilon^2}\right)$ words for $p\le 2$ and $\tilde{\mathcal{O}}\left(\frac{d^{p/2+1}}{\varepsilon^2}\right)$ words for $p>2$, showing that streaming algorithms can match offline algorithms in both space and time complexity.

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arXiv2025

Range Counting Oracles for Geometric Problems

In this paper, we study estimators for geometric optimization problems in the sublinear geometric model. In this model, we have oracle access to a point set with size $n$ in a discrete space $[Δ]^d$, where queries can be made to an oracle that responds to orthogonal range counting requests. The query complexity of an optimization problem is measured by the number of oracle queries required to compute an estimator for the problem. We investigate two problems in this framework, the Euclidean Minimum Spanning Tree (MST) and Earth Mover Distance (EMD). For EMD, we show the existence of an estimator that approximates the cost of EMD with $O(\log Δ)$-relative error and $O(\frac{nΔ}{s^{1+1/d}})$-additive error using $O(s\polylog Δ)$ range counting queries for any parameter $s$ with $1\leq s \leq n$. Moreover, we prove that this bound is tight. For MST, we demonstrate that the weight of MST can be estimated within a factor of $(1 \pm \eps)$ using $\tilde{O}(\sqrt{n})$ range counting queries.

cs.CG↗

arXiv2025

Perfect Sampling in Turnstile Streams Beyond Small Moments

Given a vector $x \in \mathbb{R}^n$ induced by a turnstile stream $S$, a non-negative function $G: \mathbb{R} \to \mathbb{R}$, a perfect $G$-sampler outputs an index $i$ with probability $\frac{G(x_i)}{\sum_{j\in[n]} G(x_j)}+\frac{1}{\text{poly}(n)}$. Jayaram and Woodruff (FOCS 2018) introduced a perfect $L_p$-sampler, where $G(z)=|z|^p$, for $p\in(0,2]$. In this paper, we solve this problem for $p>2$ by a sampling-and-rejection method. Our algorithm runs in $n^{1-2/p} \cdot \text{polylog}(n)$ bits of space, which is tight up to polylogarithmic factors in $n$. Our algorithm also provides a $(1+\varepsilon)$-approximation to the sampled item $x_i$ with high probability using an additional $\varepsilon^{-2} n^{1-2/p} \cdot \text{polylog}(n)$ bits of space. Interestingly, we show our techniques can be generalized to perfect polynomial samplers on turnstile streams, which is a class of functions that is not scale-invariant, in contrast to the existing perfect $L_p$ samplers. We also achieve perfect samplers for the logarithmic function $G(z)=\log(1+|z|)$ and the cap function $G(z)=\min(T,|z|^p)$. Finally, we give an application of our results to the problem of norm/moment estimation for a subset $\mathcal{Q}$ of coordinates of a vector, revealed only after the data stream is processed, e.g., when the set $\mathcal{Q}$ represents a range query, or the set $n\setminus\mathcal{Q}$ represents a collection of entities who wish for their information to be expunged from the dataset.

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arXiv2025

Ridge Leverage Score Sampling for $\ell_p$ Subspace Approximation

The $\ell_p$ subspace approximation problem is an NP-hard low rank approximation problem that generalizes the median hyperplane ($p = 1$), principal component analysis ($p = 2$), and center hyperplane problems ($p = \infty$). A popular approach to cope with the NP-hardness is to compute a strong coreset, which is a weighted subset of input points that simultaneously approximates the cost of every $k$-dimensional subspace, typically to $(1+ε)$ relative error for a small constant $ε$. We obtain an algorithm for constructing a strong coreset for $\ell_p$ subspace approximation of size $\tilde O(kε^{-4/p})$ for $p<2$ and $\tilde O(k^{p/2}ε^{-p})$ for $p>2$. This offers the following improvements over prior work: - We construct the first strong coresets with nearly optimal dependence on $k$ for all $p\neq 2$. In prior work, [SW18] constructed coresets of modified points with a similar dependence on $k$, while [HV20] constructed true coresets with polynomially worse dependence on $k$. - We recover or improve the best known $ε$ dependence for all $p$. In particular, for $p > 2$, the [SW18] coreset of modified points had a dependence of $ε^{-p^2/2}$ and the [HV20] coreset had a dependence of $ε^{-3p}$. Our algorithm is based on sampling by root ridge leverage scores, which admits fast algorithms, especially for sparse or structured matrices. Our analysis avoids the use of the representative subspace theorem [SW18], which is a critical component of all prior dimension-independent coresets for $\ell_p$ subspace approximation. Our techniques also lead to the first nearly optimal online strong coresets for $\ell_p$ subspace approximation with similar bounds as the offline setting, resolving a problem of [WY23]. All prior approaches lose $\mathrm{poly}(k)$ factors in this setting, even when allowed to modify the original points.

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arXiv2025

Lifting Linear Sketches: Optimal Bounds and Adversarial Robustness

We introduce a novel technique for ``lifting'' dimension lower bounds for linear sketches in the real-valued setting to dimension lower bounds for linear sketches with polynomially-bounded integer entries when the input is a polynomially-bounded integer vector. Using this technique, we obtain the first optimal sketching lower bounds for discrete inputs in a data stream, for classical problems such as approximating the frequency moments, estimating the operator norm, and compressed sensing. Additionally, we lift the adaptive attack of Hardt and Woodruff (STOC, 2013) for breaking any real-valued linear sketch via a sequence of real-valued queries, and show how to obtain an attack on any integer-valued linear sketch using integer-valued queries. This shows that there is no linear sketch in a data stream with insertions and deletions that is adversarially robust for approximating any $L_p$ norm of the input, resolving a central open question for adversarially robust streaming algorithms. To do so, we introduce a new pre-processing technique of independent interest which, given an integer-valued linear sketch, increases the dimension of the sketch by only a constant factor in order to make the orthogonal lattice to its row span smooth. This pre-processing then enables us to leverage results in lattice theory on discrete Gaussian distributions and reason that efficient discrete sketches imply efficient continuous sketches. Our work resolves open questions from the Banff '14 and '17 workshops on Communication Complexity and Applications, as well as the STOC '21 and FOCS '23 workshops on adaptivity and robustness.

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arXiv2025

Beyond Worst-Case Dimensionality Reduction for Sparse Vectors

We study beyond worst-case dimensionality reduction for $s$-sparse vectors. Our work is divided into two parts, each focusing on a different facet of beyond worst-case analysis: We first consider average-case guarantees. A folklore upper bound based on the birthday-paradox states: For any collection $X$ of $s$-sparse vectors in $\mathbb{R}^d$, there exists a linear map to $\mathbb{R}^{O(s^2)}$ which \emph{exactly} preserves the norm of $99\%$ of the vectors in $X$ in any $\ell_p$ norm (as opposed to the usual setting where guarantees hold for all vectors). We give lower bounds showing that this is indeed optimal in many settings: any oblivious linear map satisfying similar average-case guarantees must map to $Ω(s^2)$ dimensions. The same lower bound also holds for a wide class of smooth maps, including `encoder-decoder schemes', where we compare the norm of the original vector to that of a smooth function of the embedding. These lower bounds reveal a separation result, as an upper bound of $O(s \log(d))$ is possible if we instead use arbitrary (possibly non-smooth) functions, e.g., via compressed sensing algorithms. Given these lower bounds, we specialize to sparse \emph{non-negative} vectors. For a dataset $X$ of non-negative $s$-sparse vectors and any $p \ge 1$, we can non-linearly embed $X$ to $O(s\log(|X|s)/ε^2)$ dimensions while preserving all pairwise distances in $\ell_p$ norm up to $1\pm ε$, with no dependence on $p$. Surprisingly, the non-negativity assumption enables much smaller embeddings than arbitrary sparse vectors, where the best known bounds suffer exponential dependence. Our map also guarantees \emph{exact} dimensionality reduction for $\ell_{\infty}$ by embedding into $O(s\log |X|)$ dimensions, which is tight. We show that both the non-linearity of $f$ and the non-negativity of $X$ are necessary, and provide downstream algorithmic improvements.

cs.DS↗

arXiv2025

Understanding the Kronecker Matrix-Vector Complexity of Linear Algebra

We study the computational model where we can access a matrix $\mathbf{A}$ only by computing matrix-vector products $\mathbf{A}\mathrm{x}$ for vectors of the form $\mathrm{x} = \mathrm{x}_1 \otimes \cdots \otimes \mathrm{x}_q$. We prove exponential lower bounds on the number of queries needed to estimate various properties, including the trace and the top eigenvalue of $\mathbf{A}$. Our proofs hold for all adaptive algorithms, modulo a mild conditioning assumption on the algorithm's queries. We further prove that algorithms whose queries come from a small alphabet (e.g., $\mathrm{x}_i \in \{\pm1\}^n$) cannot test if $\mathbf{A}$ is identically zero with polynomial complexity, despite the fact that a single query using Gaussian vectors solves the problem with probability 1. In steep contrast to the non-Kronecker case, this shows that sketching $\mathbf{A}$ with different distributions of the same subguassian norm can yield exponentially different query complexities. Our proofs follow from the observation that random vectors with Kronecker structure have exponentially smaller inner products than their non-Kronecker counterparts.

cs.DS↗

arXiv2025

John Ellipsoids via Lazy Updates

We give a faster algorithm for computing an approximate John ellipsoid around $n$ points in $d$ dimensions. The best known prior algorithms are based on repeatedly computing the leverage scores of the points and reweighting them by these scores [CCLY19]. We show that this algorithm can be substantially sped up by delaying the computation of high accuracy leverage scores by using sampling, and then later computing multiple batches of high accuracy leverage scores via fast rectangular matrix multiplication. We also give low-space streaming algorithms for John ellipsoids using similar ideas.

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arXiv2024

Approximating the Top Eigenvector in Random Order Streams

When rows of an $n \times d$ matrix $A$ are given in a stream, we study algorithms for approximating the top eigenvector of the matrix ${A}^TA$ (equivalently, the top right singular vector of $A$). We consider worst case inputs $A$ but assume that the rows are presented to the streaming algorithm in a uniformly random order. We show that when the gap parameter $R = σ_1(A)^2/σ_2(A)^2 = Ω(1)$, then there is a randomized algorithm that uses $O(h \cdot d \cdot \operatorname{polylog}(d))$ bits of space and outputs a unit vector $v$ that has a correlation $1 - O(1/\sqrt{R})$ with the top eigenvector $v_1$. Here $h$ denotes the number of \emph{heavy rows} in the matrix, defined as the rows with Euclidean norm at least $\|{A}\|_F/\sqrt{d \cdot \operatorname{polylog}(d)}$. We also provide a lower bound showing that any algorithm using $O(hd/R)$ bits of space can obtain at most $1 - Ω(1/R^2)$ correlation with the top eigenvector. Thus, parameterizing the space complexity in terms of the number of heavy rows is necessary for high accuracy solutions. Our results improve upon the $R = Ω(\log n \cdot \log d)$ requirement in a recent work of Price and Xun (FOCS 2024). We note that the algorithm of Price and Xun works for arbitrary order streams whereas our algorithm requires a stronger assumption that the rows are presented in a uniformly random order. We additionally show that the gap requirements in their analysis can be brought down to $R = Ω(\log^2 d)$ for arbitrary order streams and $R = Ω(\log d)$ for random order streams. The requirement of $R = Ω(\log d)$ for random order streams is nearly tight for their analysis as we obtain a simple instance with $R = Ω(\log d/\log\log d)$ for which their algorithm, with any fixed learning rate, cannot output a vector approximating the top eigenvector $v_1$.

cs.DS↗

arXiv2024

Adversarially Robust Dense-Sparse Tradeoffs via Heavy-Hitters

In the adversarial streaming model, the input is a sequence of adaptive updates that defines an underlying dataset and the goal is to approximate, collect, or compute some statistic while using space sublinear in the size of the dataset. In 2022, Ben-Eliezer, Eden, and Onak showed a dense-sparse trade-off technique that elegantly combined sparse recovery with known techniques using differential privacy and sketch switching to achieve adversarially robust algorithms for $L_p$ estimation and other algorithms on turnstile streams. In this work, we first give an improved algorithm for adversarially robust $L_p$-heavy hitters, utilizing deterministic turnstile heavy-hitter algorithms with better tradeoffs. We then utilize our heavy-hitter algorithm to reduce the problem to estimating the frequency moment of the tail vector. We give a new algorithm for this problem in the classical streaming setting, which achieves additive error and uses space independent in the size of the tail. We then leverage these ingredients to give an improved algorithm for adversarially robust $L_p$ estimation on turnstile streams.

cs.DS↗

arXiv2024

On Socially Fair Low-Rank Approximation and Column Subset Selection

Low-rank approximation and column subset selection are two fundamental and related problems that are applied across a wealth of machine learning applications. In this paper, we study the question of socially fair low-rank approximation and socially fair column subset selection, where the goal is to minimize the loss over all sub-populations of the data. We show that surprisingly, even constant-factor approximation to fair low-rank approximation requires exponential time under certain standard complexity hypotheses. On the positive side, we give an algorithm for fair low-rank approximation that, for a constant number of groups and constant-factor accuracy, runs in $2^{\text{poly}(k)}$ time rather than the naïve $n^{\text{poly}(k)}$, which is a substantial improvement when the dataset has a large number $n$ of observations. We then show that there exist bicriteria approximation algorithms for fair low-rank approximation and fair column subset selection that run in polynomial time.

cs.LG↗

arXiv2024

Space Complexity of Minimum Cut Problems in Single-Pass Streams

We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. While graph sparsification in streams has been intensively studied, the specific application of finding minimum cuts in streams is less well-studied. To this end, we show upper and lower bounds on minimum cut problems in insertion-only streams for a variety of settings, including for both randomized and deterministic algorithms, for both arbitrary and random order streams, and for both approximate and exact algorithms. One of our main results is an $\widetilde{O}(n/\varepsilon)$ space algorithm with fast update time for approximating a spectral cut query with high probability on a stream given in an arbitrary order. Our result breaks the $Ω(n/\varepsilon^2)$ space lower bound required of a sparsifier that approximates all cuts simultaneously. Using this result, we provide streaming algorithms with near optimal space of $\widetilde{O}(n/\varepsilon)$ for minimum cut and approximate all-pairs effective resistances, with matching space lower-bounds. The amortized update time of our algorithms is $\widetilde{O}(1)$, provided that the number of edges in the input graph is at least $(n/\varepsilon^2)^{1+o(1)}$. We also give a generic way of incorporating sketching into a recursive contraction algorithm to improve the post-processing time of our algorithms. In addition to these results, we give a random-order streaming algorithm that computes the {\it exact} minimum cut on a simple, unweighted graph using $\widetilde{O}(n)$ space. Finally, we give an $Ω(n/\varepsilon^2)$ space lower bound for deterministic minimum cut algorithms which matches the best-known upper bound up to polylogarithmic factors.

cs.DS↗

arXiv2024

Tight Sampling Bounds for Eigenvalue Approximation

We consider the problem of estimating the spectrum of a symmetric bounded entry (not necessarily PSD) matrix via entrywise sampling. This problem was introduced by [Bhattacharjee, Dexter, Drineas, Musco, Ray '22], where it was shown that one can obtain an $εn$ additive approximation to all eigenvalues of $A$ by sampling a principal submatrix of dimension $\frac{\text{poly}(\log n)}{ε^3}$. We improve their analysis by showing that it suffices to sample a principal submatrix of dimension $\tilde{O}(\frac{1}{ε^2})$ (with no dependence on $n$). This matches known lower bounds and therefore resolves the sample complexity of this problem up to $\log\frac{1}ε$ factors. Using similar techniques, we give a tight $\tilde{O}(\frac{1}{ε^2})$ bound for obtaining an additive $ε\|A\|_F$ approximation to the spectrum of $A$ via squared row-norm sampling, improving on the previous best $\tilde{O}(\frac{1}{ε^{8}})$ bound. We also address the problem of approximating the top eigenvector for a bounded entry, PSD matrix $A.$ In particular, we show that sampling $O(\frac{1}ε)$ columns of $A$ suffices to produce a unit vector $u$ with $u^T A u \geq λ_1(A) - εn$. This matches what one could achieve via the sampling bound of [Musco, Musco'17] for the special case of approximating the top eigenvector, but does not require adaptivity. As additional applications, we observe that our sampling results can be used to design a faster eigenvalue estimation sketch for dense matrices resolving a question of [Swartworth, Woodruff'23], and can also be combined with [Musco, Musco'17] to achieve $O(1/ε^3)$ (adaptive) sample complexity for approximating the spectrum of a bounded entry PSD matrix to $εn$ additive error.

cs.DS↗

arXiv2024

LevAttention: Time, Space, and Streaming Efficient Algorithm for Heavy Attentions

A central problem related to transformers can be stated as follows: given two $n \times d$ matrices $Q$ and $K$, and a non-negative function $f$, define the matrix $A$ as follows: (1) apply the function $f$ to each entry of the $n \times n$ matrix $Q K^T$, and then (2) normalize each of the row sums of $A$ to be equal to $1$. The matrix $A$ can be computed in $O(n^2 d)$ time assuming $f$ can be applied to a number in constant time, but the quadratic dependence on $n$ is prohibitive in applications where it corresponds to long context lengths. For a large class of functions $f$, we show how to find all the ``large attention scores", i.e., entries of $A$ which are at least a positive value $\varepsilon$, in time with linear dependence on $n$ (i.e., $n \cdot \textrm{poly}(d/\varepsilon)$) for a positive parameter $\varepsilon > 0$. Our class of functions include all functions $f$ of the form $f(x) = |x|^p$, as explored recently in transformer models. Using recently developed tools from randomized numerical linear algebra, we prove that for any $K$, there is a ``universal set" $U \subset [n]$ of size independent of $n$, such that for any $Q$ and any row $i$, the large attention scores $A_{i,j}$ in row $i$ of $A$ all have $j \in U$. We also find $U$ in $n \cdot \textrm{poly}(d/\varepsilon)$ time. Notably, we (1) make no assumptions on the data, (2) our workspace does not grow with $n$, and (3) our algorithms can be computed in streaming and parallel settings. We call the attention mechanism that uses only the subset of keys in the universal set as LevAttention since our algorithm to identify the universal set $U$ is based on leverage scores. We empirically show the benefits of our scheme for vision transformers, showing how to train new models that use our universal set while training as well, showing that our model is able to consistently select ``important keys'' during training.

cs.LG↗

arXiv2024

A Strong Separation for Adversarially Robust $\ell_0$ Estimation for Linear Sketches

The majority of streaming problems are defined and analyzed in a static setting, where the data stream is any worst-case sequence of insertions and deletions that is fixed in advance. However, many real-world applications require a more flexible model, where an adaptive adversary may select future stream elements after observing the previous outputs of the algorithm. Over the last few years, there has been increased interest in proving lower bounds for natural problems in the adaptive streaming model. In this work, we give the first known adaptive attack against linear sketches for the well-studied $\ell_0$-estimation problem over turnstile, integer streams. For any linear streaming algorithm $\mathcal{A}$ that uses sketching matrix $\mathbf{A}\in \mathbb{Z}^{r \times n}$ where $n$ is the size of the universe, this attack makes $\tilde{\mathcal{O}}(r^8)$ queries and succeeds with high constant probability in breaking the sketch. We also give an adaptive attack against linear sketches for the $\ell_0$-estimation problem over finite fields $\mathbb{F}_p$, which requires a smaller number of $\tilde{\mathcal{O}}(r^3)$ queries. Finally, we provide an adaptive attack over $\mathbb{R}^n$ against linear sketches $\mathbf{A} \in \mathbb{R}^{r \times n}$ for $\ell_0$-estimation, in the setting where $\mathbf{A}$ has all nonzero subdeterminants at least $\frac{1}{\textrm{poly}(r)}$. Our results provide an exponential improvement over the previous number of queries known to break an $\ell_0$-estimation sketch.

cs.DS↗

arXiv2024

Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms

We study the problem of residual error estimation for matrix and vector norms using a linear sketch. Such estimates can be used, for example, to quickly assess how useful a more expensive low-rank approximation computation will be. The matrix case concerns the Frobenius norm and the task is to approximate the $k$-residual $\|A - A_k\|_F$ of the input matrix $A$ within a $(1+ε)$-factor, where $A_k$ is the optimal rank-$k$ approximation. We provide a tight bound of $Θ(k^2/ε^4)$ on the size of bilinear sketches, which have the form of a matrix product $SAT$. This improves the previous $O(k^2/ε^6)$ upper bound in (Andoni et al. SODA 2013) and gives the first non-trivial lower bound, to the best of our knowledge. In our algorithm, our sketching matrices $S$ and $T$ can both be sparse matrices, allowing for a very fast update time. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. For the vector case, we consider the $\ell_p$-norm for $p>2$, where the task is to approximate the $k$-residual $\|x - x_k\|_p$ up to a constant factor, where $x_k$ is the optimal $k$-sparse approximation to $x$. Such vector norms are frequently studied in the data stream literature and are useful for finding frequent items or so-called heavy hitters. We establish an upper bound of $O(k^{2/p}n^{1-2/p}\operatorname{poly}(\log n))$ for constant $ε$ on the dimension of a linear sketch for this problem. Our algorithm can be extended to the $\ell_p$ sparse recovery problem with the same sketching dimension, which seems to be the first such bound for $p > 2$. We also show an $Ω(k^{2/p}n^{1-2/p})$ lower bound for the sparse recovery problem, which is tight up to a $\mathrm{poly}(\log n)$ factor.

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