arXiv subjects

Erik Waingarten

Publications and source records attributed to Erik Waingarten.

At least 19 recordsLinked to original sources

arXiv2026

Learning Partition Trees for Nearest Neighbor Search

We study nearest neighbor search from the perspective of data-driven algorithm design: given a dataset $P \subset \mathbb{R}^d$ of size $n$ and sample access to a query distribution over $\mathbb{R}^d$, the goal is to learn a data structure optimized for queries drawn from that specific distribution. We focus on the class of balanced halfspace trees, which naturally abstracts space-partitioning frameworks like locality-sensitive hashing. Assuming Gaussian-like marginal conditions on the dataset and query distribution, we give an efficient algorithm that learns a tree achieving $o(nd)$ query time, provided that a perfect tree exists. At the core of our algorithmic approach is the balanced halfspace cut problem, where we are given a distribution over $\mathbb{R}^d \times \mathbb{R}^d$ and must find a balanced halfspace that minimizes the fraction of cut pairs. We prove that without distributional assumptions, finding the optimal balanced halfspace is NP-hard. To circumvent this computational barrier, we design an efficient improper learning algorithm: if the optimal halfspace cuts an $\alpha$ fraction of pairs, our algorithm outputs a balanced polynomial threshold function of degree $\tilde{O}(1/\varepsilon^2)$ that cuts at most an $O(\sqrt{\alpha+\varepsilon})$ fraction.

cs.DS↗

arXiv2026

Locally Approximating the Top Eigenvector of Bounded Entry Matrices

We provide a local computation algorithm to approximate the top eigenvector $x \in \mathbb{R}^n$ of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries between $-1$ and $1$, building on the work of Swartworth and Woodruff [SODA 25] who show how to approximate the eigenvalues up to additive-$\varepsilon n$ error using $\tilde{O}(1/\varepsilon^4)$ queries. Our local computation algorithm has a preprocessing complexity of $\tilde{O}(1/\varepsilon^4)$ and per-coordinate query complexity of $\tilde{O}(1/\varepsilon^2)$ for an additive-$\varepsilon n$ approximation whenever {$|\lambda_{\min}(A)| = O(\lambda_{\max}(A))$. When $\lambda_{\min}(A)$ greatly exceeds $\lambda_{\max}(A)$, our complexity degrades to at most $\tilde{O}(1/\varepsilon^{6.\overline{6}})$ in preprocessing and $\tilde{O}(1/\varepsilon^{3.\overline{3}})$ per query. Furthermore, we show a lower bound of $\Omega(n/\varepsilon^2)$ on the total number of queries needed to output an approximately top eigenvector (implying that the per-coordinate query complexity of $\Omega(1/\varepsilon^2)$ is necessary). As an application, we use our algorithm to provide local computation algorithms for the sparsest-cut and max-cut problems in the dense graph model of Goldreich, Goldwasser, Ron [JACM 98]. By accessing the top eigenvectors (of an approximate normalized adjacency), we implement local versions of Cheeger's inequality and Trevisan's algorithm [SICOMP 12] to obtain "square-root-opt" approximations in polynomial time (as opposed to exponential-in-$\text{poly}(1/\varepsilon)$ time which is incurred in Goldreich, Goldwasser, Ron.

cs.DS↗

arXiv2026

Prune, Don't Rebuild: Efficiently Tuning $\alpha$-Reachable Graphs for Nearest Neighbor Search

Vector similarity search is an essential primitive in modern AI and ML applications. Most vector databases adopt graph-based approximate nearest neighbor (ANN) search algorithms, such as DiskANN (Subramanya et al., 2019), which have demonstrated state-of-the-art empirical performance. DiskANN's graph construction is governed by a reachability parameter $\alpha$, which gives a trade-off between construction time, query time, and accuracy. However, adaptively tuning this trade-off typically requires rebuilding the index for different $\alpha$ values, which is prohibitive at scale. In this work, we propose RP-Tuning, an efficient post-hoc routine, based on DiskANN's pruning step, to adjust the $\alpha$ parameter without reconstructing the full index. Within the $\alpha$-reachability framework of prior theoretical works (Indyk and Xu, 2023; Gollapudi et al., 2025), we prove that pruning an initially $\alpha$-reachable graph with RP-Tuning preserves worst-case reachability guarantees in general metrics and improved guarantees in Euclidean metrics. Empirically, we show that RP-Tuning accelerates DiskANN tuning on four public datasets by up to $43\times$ with negligible overhead.

cs.DS↗

arXiv2025

A Polynomial Space Lower Bound for Diameter Estimation in Dynamic Streams

We study the space complexity of estimating the diameter of a subset of points in an arbitrary metric space in the dynamic (turnstile) streaming model. The input is given as a stream of updates to a frequency vector $x \in \mathbb{Z}_{\geq 0}^n$, where the support of $x$ defines a multiset of points in a fixed metric space $M = ([n], \mathsf{d})$. The goal is to estimate the diameter of this multiset, defined as $\max\{\mathsf{d}(i,j) : x_i, x_j > 0\}$, to a specified approximation factor while using as little space as possible. In insertion-only streams, a simple $O(\log n)$-space algorithm achieves a 2-approximation. In sharp contrast to this, we show that in the dynamic streaming model, any algorithm achieving a constant-factor approximation to diameter requires polynomial space. Specifically, we prove that a $c$-approximation to the diameter requires $n^{Ω(1/c)}$ space. Our lower bound relies on two conceptual contributions: (1) a new connection between dynamic streaming algorithms and linear sketches for {\em scale-invariant} functions, a class that includes diameter estimation, and (2) a connection between linear sketches for diameter and the {\em minrank} of graphs, a notion previously studied in index coding. We complement our lower bound with a nearly matching upper bound, which gives a $c$-approximation to the diameter in general metrics using $n^{O(1/c)}$ space.

cs.DS↗

arXiv2025

Approximating High-Dimensional Earth Mover's Distance as Fast as Closest Pair

We give a reduction from $(1+\varepsilon)$-approximate Earth Mover's Distance (EMD) to $(1+\varepsilon)$-approximate Closest Pair (CP). As a consequence, we improve the fastest known approximation algorithm for high-dimensional EMD. Here, given $p\in [1, 2]$ and two sets of $n$ points $X,Y \subseteq (\mathbb R^d,\ell_p)$, their EMD is the minimum cost of a perfect matching between $X$ and $Y$, where the cost of matching two vectors is their $\ell_p$ distance. Further, CP is the basic problem of finding a pair of points realizing $\min_{x \in X, y\in Y} ||x-y||_p$. Our contribution is twofold: we show that if a $(1+\varepsilon)$-approximate CP can be computed in time $n^{2-ϕ}$, then a $1+O(\varepsilon)$ approximation to EMD can be computed in time $n^{2-Ω(ϕ)}$; plugging in the fastest known algorithm for CP [Alman, Chan, Williams FOCS'16], we obtain a $(1+\varepsilon)$-approximation algorithm for EMD running in time $n^{2-\tildeΩ(\varepsilon^{1/3})}$ for high-dimensional point sets, which improves over the prior fastest running time of $n^{2-Ω(\varepsilon^2)}$ [Andoni, Zhang FOCS'23]. Our main technical contribution is a sublinear implementation of the Multiplicative Weights Update framework for EMD. Specifically, we demonstrate that the updates can be executed without ever explicitly computing or storing the weights; instead, we exploit the underlying geometric structure to perform the updates implicitly.

cs.DS↗

arXiv2025

Instance-Optimal Uniformity Testing and Tracking

In the uniformity testing task, an algorithm is provided with samples from an unknown probability distribution over a (known) finite domain, and must decide whether it is the uniform distribution, or, alternatively, if its total variation distance from uniform exceeds some input distance parameter. This question has received a significant amount of interest and its complexity is, by now, fully settled. Yet, we argue that it fails to capture many scenarios of interest, and that its very definition as a gap problem in terms of a prespecified distance may lead to suboptimal performance. To address these shortcomings, we introduce the problem of uniformity tracking, whereby an algorithm is required to detect deviations from uniformity (however they may manifest themselves) using as few samples as possible, and be competitive against an optimal algorithm knowing the distribution profile in hindsight. Our main contribution is a $\operatorname{polylog}(\operatorname{opt})$-competitive uniformity tracking algorithm. We obtain this result by leveraging new structural results on Poisson mixtures, which we believe to be of independent interest.

cs.DS↗

arXiv2025

Sparse Navigable Graphs for Nearest Neighbor Search: Algorithms and Hardness

We initiate the study of approximation algorithms and computational barriers for constructing sparse $α$-navigable graphs [IX23, DGM+24], a core primitive underlying recent advances in graph-based nearest neighbor search. Given an $n$-point dataset $P$ with an associated metric $\mathsf{d}$ and a parameter $α\geq 1$, the goal is to efficiently build the sparsest graph $G=(P, E)$ that is $α$-navigable: for every distinct $s, t \in P$, there exists an edge $(s, u) \in E$ with $\mathsf{d}(u, t) < \mathsf{d}(s, t)/α$. We consider two natural sparsity objectives: minimizing the maximum out-degree and minimizing the total size. We first show a strong negative result: the slow-preprocessing version of DiskANN (analyzed in [IX23] for low-doubling metrics) can yield solutions whose sparsity is $\widetildeΩ(n)$ times larger than optimal, even on Euclidean instances. We then show a tight approximation-preserving equivalence between the Sparsest Navigable Graph problem and the classic Set Cover problem, obtaining an $O(n^3)$-time $(\ln n + 1)$-approximation algorithm, as well as establishing NP-hardness of achieving an $o(\ln n)$-approximation. Building on this equivalence, we develop faster $O(\ln n)$-approximation algorithms. The first runs in $\widetilde{O}(n \cdot \mathrm{OPT})$ time and is thus much faster when the optimal solution is sparse. The second, based on fast matrix multiplication, is a bicriteria algorithm that computes an $O(\ln n)$-approximation to the sparsest $2α$-navigable graph, running in $\widetilde{O}(n^ω)$ time. Finally, we complement our upper bounds with a query complexity lower bound, showing that any $o(n)$-approximation requires examining $Ω(n^2)$ distances. This result shows that in the regime where $\mathrm{OPT} = \widetilde{O}(n)$, our $\widetilde{O}(n \cdot \mathrm{OPT})$-time algorithm is essentially best possible.

cs.DS↗

arXiv2025

Randomized Dimensionality Reduction for Euclidean Maximization and Diversity Measures

Randomized dimensionality reduction is a widely-used algorithmic technique for speeding up large-scale Euclidean optimization problems. In this paper, we study dimension reduction for a variety of maximization problems, including max-matching, max-spanning tree, max TSP, as well as various measures for dataset diversity. For these problems, we show that the effect of dimension reduction is intimately tied to the \emph{doubling dimension} $λ_X$ of the underlying dataset $X$ -- a quantity measuring intrinsic dimensionality of point sets. Specifically, we prove that a target dimension of $O(λ_X)$ suffices to approximately preserve the value of any near-optimal solution,which we also show is necessary for some of these problems. This is in contrast to classical dimension reduction results, whose dependence increases with the dataset size $|X|$. We also provide empirical results validating the quality of solutions found in the projected space, as well as speedups due to dimensionality reduction.

cs.DS↗

arXiv2025

Average-Distortion Sketching

We introduce average-distortion sketching for metric spaces. As in (worst-case) sketching, these algorithms compress points in a metric space while approximately recovering pairwise distances. The novelty is studying average-distortion: for any fixed (yet, arbitrary) distribution $μ$ over the metric, the sketch should not over-estimate distances, and it should (approximately) preserve the average distance with respect to draws from $μ$. The notion generalizes average-distortion embeddings into $\ell_1$ [Rabinovich '03, Kush-Nikolov-Tang '21] as well as data-dependent locality-sensitive hashing [Andoni-Razenshteyn '15, Andoni-Naor-Nikolov-et-al. '18], which have been recently studied in the context of nearest neighbor search. $\bullet$ For all $p \in (2, \infty)$ and any $c$ larger than a fixed constant, we give an average-distortion sketch for $([Δ]^d, \ell_p)$ with approximation $c$ and bit-complexity $\text{poly}(2^{p/c} \cdot \log(dΔ))$, which is provably impossible in (worst-case) sketching. $\bullet$ As an application, we improve on the approximation of sublinear-time data structures for nearest neighbor search over $\ell_p$ (for large $p > 2$). The prior best approximation was $O(p)$ [Andoni-Naor-Nikolov-et-al. '18, Kush-Nikolov-Tang '21], and we show it can be any $c$ larger than a fixed constant (irrespective of $p$) by using $n^{O(p/c)}$ space. We give some evidence that $2^{Ω(p/c)}$ space may be necessary by giving a lower bound on average-distortion sketches which produce a certain probabilistic certificate of farness (which our sketches crucially rely on).

cs.DS↗

arXiv2025

Streaming and Massively Parallel Algorithms for Euclidean Max-Cut

Given a set of vectors $X = \{ x_1,\dots, x_n \} \subset \mathbb{R}^d$, the Euclidean max-cut problem asks to partition the vectors into two parts so as to maximize the sum of Euclidean distances which cross the partition. We design new algorithms for Euclidean max-cut in models for massive datasets: $\bullet$ We give a fully-scalable constant-round MPC algorithm using $O(nd) + n \cdot \text{poly}( \log(n) / ε)$ total space which gives a $(1+ε)$-approximate Euclidean max-cut. $\bullet$ We give a dynamic streaming algorithm using $\text{poly}(d \log Δ/ ε)$ space when $X \subseteq [Δ]^d$, which provides oracle access to a $(1+ε)$-approximate Euclidean max-cut. Recently, Chen, Jiang, and Krauthgamer $[\text{STOC}~'23]$ gave a dynamic streaming algorithm with space $\text{poly}(d\logΔ/ε)$ to approximate the value of the Euclidean max-cut, but could not provide oracle access to an approximately optimal cut. This was left open in that work, and we resolve it here. Both algorithms follow from the same framework, which analyzes a ``parallel'' and ``subsampled'' (Euclidean) version of a greedy algorithm of Mathieu and Schudy $[\text{SODA}~'08]$ for dense max-cut.

cs.DS↗

arXiv2025

Monotonicity Testing of High-Dimensional Distributions with Subcube Conditioning

We study monotonicity testing of high-dimensional distributions on $\{-1,1\}^n$ in the model of subcube conditioning, suggested and studied by Canonne, Ron, and Servedio~\cite{CRS15} and Bhattacharyya and Chakraborty~\cite{BC18}. Previous work shows that the \emph{sample complexity} of monotonicity testing must be exponential in $n$ (Rubinfeld, Vasilian~\cite{RV20}, and Aliakbarpour, Gouleakis, Peebles, Rubinfeld, Yodpinyanee~\cite{AGPRY19}). We show that the subcube \emph{query complexity} is $\tildeΘ(n/\varepsilon^2)$, by proving nearly matching upper and lower bounds. Our work is the first to use directed isoperimetric inequalities (developed for function monotonicity testing) for analyzing a distribution testing algorithm. Along the way, we generalize an inequality of Khot, Minzer, and Safra~\cite{KMS18} to real-valued functions on $\{-1,1\}^n$. We also study uniformity testing of distributions that are promised to be monotone, a problem introduced by Rubinfeld, Servedio~\cite{RS09} , using subcube conditioning. We show that the query complexity is $\tildeΘ(\sqrt{n}/\varepsilon^2)$. Our work proves the lower bound, which matches (up to poly-logarithmic factors) the uniformity testing upper bound for general distributions (Canonne, Chen, Kamath, Levi, Waingarten~\cite{CCKLW21}). Hence, we show that monotonicity does not help, beyond logarithmic factors, in testing uniformity of distributions with subcube conditional queries.

math.ST↗

arXiv2024

Nearly Tight Bounds on Testing of Metric Properties

Given a non-negative $n \times n$ matrix viewed as a set of distances between $n$ points, we consider the property testing problem of deciding if it is a metric. We also consider the same problem for two special classes of metrics, tree metrics and ultrametrics. For general metrics, our paper is the first to consider these questions. We prove an upper bound of $O(n^{2/3}/ε^{4/3})$ on the query complexity for this problem. Our algorithm is simple, but the analysis requires great care in bounding the variance on the number of violating triangles in a sample. When $ε$ is a slowly decreasing function of $n$ (rather than a constant, as is standard), we prove a lower bound of matching dependence on $n$ of $Ω(n^{2/3})$, ruling out any property testers with $o(n^{2/3})$ query complexity unless their dependence on $1/ε$ is super-polynomial. Next, we turn to tree metrics and ultrametrics. While there were known upper and lower bounds, we considerably improve these bounds showing essentially tight bounds of $\tilde{O}(1/ε)$ on the sample complexity. We also show a lower bound of $Ω( 1/ε^{4/3} )$ on the query complexity. Our upper bounds are derived by doing a more careful analysis of a natural, simple algorithm. For the lower bounds, we construct distributions on NO instances, where it is hard to find a witness showing that these are not ultrametrics.

cs.DM↗

arXiv2024

Lower Bounds for Convexity Testing

We consider the problem of testing whether an unknown and arbitrary set $S \subseteq \mathbb{R}^n$ (given as a black-box membership oracle) is convex, versus $\varepsilon$-far from every convex set, under the standard Gaussian distribution. The current state-of-the-art testing algorithms for this problem make $2^{\tilde{O}(\sqrt{n})\cdot \mathrm{poly}(1/\varepsilon)}$ non-adaptive queries, both for the standard testing problem and for tolerant testing. We give the first lower bounds for convexity testing in the black-box query model: - We show that any one-sided tester (which may be adaptive) must use at least $n^{Ω(1)}$ queries in order to test to some constant accuracy $\varepsilon>0$. - We show that any non-adaptive tolerant tester (which may make two-sided errors) must use at least $2^{Ω(n^{1/4})}$ queries to distinguish sets that are $\varepsilon_1$-close to convex versus $\varepsilon_2$-far from convex, for some absolute constants $0<\varepsilon_1<\varepsilon_2$. Finally, we also show that for any constant $c>0$, any non-adaptive tester (which may make two-sided errors) must use at least $n^{1/4 - c}$ queries in order to test to some constant accuracy $\varepsilon>0$.

cs.CC↗

arXiv2024

Data-Dependent LSH for the Earth Mover's Distance

We give new data-dependent locality sensitive hashing schemes (LSH) for the Earth Mover's Distance ($\mathsf{EMD}$), and as a result, improve the best approximation for nearest neighbor search under $\mathsf{EMD}$ by a quadratic factor. Here, the metric $\mathsf{EMD}_s(\mathbb{R}^d,\ell_p)$ consists of sets of $s$ vectors in $\mathbb{R}^d$, and for any two sets $x,y$ of $s$ vectors the distance $\mathsf{EMD}(x,y)$ is the minimum cost of a perfect matching between $x,y$, where the cost of matching two vectors is their $\ell_p$ distance. Previously, Andoni, Indyk, and Krauthgamer gave a (data-independent) locality-sensitive hashing scheme for $\mathsf{EMD}_s(\mathbb{R}^d,\ell_p)$ when $p \in [1,2]$ with approximation $O(\log^2 s)$. By being data-dependent, we improve the approximation to $\tilde{O}(\log s)$. Our main technical contribution is to show that for any distribution $μ$ supported on the metric $\mathsf{EMD}_s(\mathbb{R}^d, \ell_p)$, there exists a data-dependent LSH for dense regions of $μ$ which achieves approximation $\tilde{O}(\log s)$, and that the data-independent LSH actually achieves a $\tilde{O}(\log s)$-approximation outside of those dense regions. Finally, we show how to "glue" together these two hashing schemes without any additional loss in the approximation. Beyond nearest neighbor search, our data-dependent LSH also gives optimal (distributional) sketches for the Earth Mover's Distance. By known sketching lower bounds, this implies that our LSH is optimal (up to $\mathrm{poly}(\log \log s)$ factors) among those that collide close points with constant probability.

cs.DS↗

arXiv2024

A Quasi-Monte Carlo Data Structure for Smooth Kernel Evaluations

In the kernel density estimation (KDE) problem one is given a kernel $K(x, y)$ and a dataset $P$ of points in a Euclidean space, and must prepare a data structure that can quickly answer density queries: given a point $q$, output a $(1+ε)$-approximation to $μ:=\frac1{|P|}\sum_{p\in P} K(p, q)$. The classical approach to KDE is the celebrated fast multipole method of [Greengard and Rokhlin]. The fast multipole method combines a basic space partitioning approach with a multidimensional Taylor expansion, which yields a $\approx \log^d (n/ε)$ query time (exponential in the dimension $d$). A recent line of work initiated by [Charikar and Siminelakis] achieved polynomial dependence on $d$ via a combination of random sampling and randomized space partitioning, with [Backurs et al.] giving an efficient data structure with query time $\approx \mathrm{poly}{\log(1/μ)}/ε^2$ for smooth kernels. Quadratic dependence on $ε$, inherent to the sampling methods, is prohibitively expensive for small $ε$. This issue is addressed by quasi-Monte Carlo methods in numerical analysis. The high level idea in quasi-Monte Carlo methods is to replace random sampling with a discrepancy based approach -- an idea recently applied to coresets for KDE by [Phillips and Tai]. The work of Phillips and Tai gives a space efficient data structure with query complexity $\approx 1/(εμ)$. This is polynomially better in $1/ε$, but exponentially worse in $1/μ$. We achieve the best of both: a data structure with $\approx \mathrm{poly}{\log(1/μ)}/ε$ query time for smooth kernel KDE. Our main insight is a new way to combine discrepancy theory with randomized space partitioning inspired by, but significantly more efficient than, that of the fast multipole methods. We hope that our techniques will find further applications to linear algebra for kernel matrices.

cs.DS↗

arXiv2023

Simple, Scalable and Effective Clustering via One-Dimensional Projections

Clustering is a fundamental problem in unsupervised machine learning with many applications in data analysis. Popular clustering algorithms such as Lloyd's algorithm and $k$-means++ can take $Ω(ndk)$ time when clustering $n$ points in a $d$-dimensional space (represented by an $n\times d$ matrix $X$) into $k$ clusters. In applications with moderate to large $k$, the multiplicative $k$ factor can become very expensive. We introduce a simple randomized clustering algorithm that provably runs in expected time $O(\mathrm{nnz}(X) + n\log n)$ for arbitrary $k$. Here $\mathrm{nnz}(X)$ is the total number of non-zero entries in the input dataset $X$, which is upper bounded by $nd$ and can be significantly smaller for sparse datasets. We prove that our algorithm achieves approximation ratio $\smash{\widetilde{O}(k^4)}$ on any input dataset for the $k$-means objective. We also believe that our theoretical analysis is of independent interest, as we show that the approximation ratio of a $k$-means algorithm is approximately preserved under a class of projections and that $k$-means++ seeding can be implemented in expected $O(n \log n)$ time in one dimension. Finally, we show experimentally that our clustering algorithm gives a new tradeoff between running time and cluster quality compared to previous state-of-the-art methods for these tasks.

cs.LG↗

arXiv2023

Fast Algorithms for a New Relaxation of Optimal Transport

We introduce a new class of objectives for optimal transport computations of datasets in high-dimensional Euclidean spaces. The new objectives are parametrized by $ρ\geq 1$, and provide a metric space $\mathcal{R}_ρ(\cdot, \cdot)$ for discrete probability distributions in $\mathbb{R}^d$. As $ρ$ approaches $1$, the metric approaches the Earth Mover's distance, but for $ρ$ larger than (but close to) $1$, admits significantly faster algorithms. Namely, for distributions $μ$ and $ν$ supported on $n$ and $m$ vectors in $\mathbb{R}^d$ of norm at most $r$ and any $ε> 0$, we give an algorithm which outputs an additive $εr$-approximation to $\mathcal{R}_ρ(μ, ν)$ in time $(n+m) \cdot \mathrm{poly}((nm)^{(ρ-1)/ρ} \cdot 2^{ρ/ (ρ-1)} / ε)$.

cs.DS↗

arXiv2023

The Johnson-Lindenstrauss Lemma for Clustering and Subspace Approximation: From Coresets to Dimension Reduction

We study the effect of Johnson-Lindenstrauss transforms in various projective clustering problems, generalizing recent results which only applied to center-based clustering [MMR19]. We ask the general question: for a Euclidean optimization problem and an accuracy parameter $ε\in (0, 1)$, what is the smallest target dimension $t \in \mathbb{N}$ such that a Johnson-Lindenstrauss transform $Π\colon \mathbb{R}^d \to \mathbb{R}^t$ preserves the cost of the optimal solution up to a $(1+ε)$-factor. We give a new technique which uses coreset constructions to analyze the effect of the Johnson-Lindenstrauss transform. Our technique, in addition applying to center-based clustering, improves on (or is the first to address) other Euclidean optimization problems, including: $\bullet$ For $(k,z)$-subspace approximation: we show that $t = \tilde{O}(zk^2 / ε^3)$ suffices, whereas the prior best bound, of $O(k/ε^2)$, only applied to the case $z = 2$ [CEMMP15]. $\bullet$ For $(k,z)$-flat approximation: we show $t = \tilde{O}(zk^2/ε^3)$ suffices, completely removing the dependence on $n$ from the prior bound $\tilde{O}(zk^2 \log n/ε^3)$ of [KR15]. $\bullet$ For $(k,z)$-line approximation: we show $t = O((k \log \log n + z + \log(1/ε)) / ε^3)$ suffices, and ours is the first to give any dimension reduction result.

cs.DS↗