arXiv subjects

Erik Waingarten

Publications and source records attributed to Erik Waingarten.

At least 37 records · Page 2Linked to original sources

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.

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arXiv2023

A Near-Linear Time Algorithm for the Chamfer Distance

For any two point sets $A,B \subset \mathbb{R}^d$ of size up to $n$, the Chamfer distance from $A$ to $B$ is defined as $\text{CH}(A,B)=\sum_{a \in A} \min_{b \in B} d_X(a,b)$, where $d_X$ is the underlying distance measure (e.g., the Euclidean or Manhattan distance). The Chamfer distance is a popular measure of dissimilarity between point clouds, used in many machine learning, computer vision, and graphics applications, and admits a straightforward $O(d n^2)$-time brute force algorithm. Further, the Chamfer distance is often used as a proxy for the more computationally demanding Earth-Mover (Optimal Transport) Distance. However, the \emph{quadratic} dependence on $n$ in the running time makes the naive approach intractable for large datasets. We overcome this bottleneck and present the first $(1+ε)$-approximate algorithm for estimating the Chamfer distance with a near-linear running time. Specifically, our algorithm runs in time $O(nd \log (n)/\varepsilon^2)$ and is implementable. Our experiments demonstrate that it is both accurate and fast on large high-dimensional datasets. We believe that our algorithm will open new avenues for analyzing large high-dimensional point clouds. We also give evidence that if the goal is to \emph{report} a $(1+\varepsilon)$-approximate mapping from $A$ to $B$ (as opposed to just its value), then any sub-quadratic time algorithm is unlikely to exist.

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arXiv2022

Streaming Euclidean MST to a Constant Factor

We study streaming algorithms for the fundamental geometric problem of computing the cost of the Euclidean Minimum Spanning Tree (MST) on an $n$-point set $X \subset \mathbb{R}^d$. In the streaming model, the points in $X$ can be added and removed arbitrarily, and the goal is to maintain an approximation in small space. In low dimensions, $(1+ε)$ approximations are possible in sublinear space [Frahling, Indyk, Sohler, SoCG '05]. However, for high dimensional spaces the best known approximation for this problem was $\tilde{O}(\log n)$, due to [Chen, Jayaram, Levi, Waingarten, STOC '22], improving on the prior $O(\log^2 n)$ bound due to [Indyk, STOC '04] and [Andoni, Indyk, Krauthgamer, SODA '08]. In this paper, we break the logarithmic barrier, and give the first constant factor sublinear space approximation to Euclidean MST. For any $ε\geq 1$, our algorithm achieves an $\tilde{O}(ε^{-2})$ approximation in $n^{O(ε)}$ space. We complement this by proving that any single pass algorithm which obtains a better than $1.10$-approximation must use $Ω(\sqrt{n})$ space, demonstrating that $(1+ε)$ approximations are not possible in high-dimensions, and that our algorithm is tight up to a constant. Nevertheless, we demonstrate that $(1+ε)$ approximations are possible in sublinear space with $O(1/ε)$ passes over the stream. More generally, for any $α\geq 2$, we give a $α$-pass streaming algorithm which achieves a $(1+O(\frac{\log α+ 1}{ αε}))$ approximation in $n^{O(ε)} d^{O(1)}$ space. Our streaming algorithms are linear sketches, and therefore extend to the massively-parallel computation model (MPC). Thus, our results imply the first $(1+ε)$-approximation to Euclidean MST in a constant number of rounds in the MPC model.

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arXiv2022

Estimation of Entropy in Constant Space with Improved Sample Complexity

Recent work of Acharya et al. (NeurIPS 2019) showed how to estimate the entropy of a distribution $\mathcal D$ over an alphabet of size $k$ up to $\pmε$ additive error by streaming over $(k/ε^3) \cdot \text{polylog}(1/ε)$ i.i.d. samples and using only $O(1)$ words of memory. In this work, we give a new constant memory scheme that reduces the sample complexity to $(k/ε^2)\cdot \text{polylog}(1/ε)$. We conjecture that this is optimal up to $\text{polylog}(1/ε)$ factors.

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arXiv2022

Polylogarithmic Sketches for Clustering

Given $n$ points in $\ell_p^d$, we consider the problem of partitioning points into $k$ clusters with associated centers. The cost of a clustering is the sum of $p^{\text{th}}$ powers of distances of points to their cluster centers. For $p \in [1,2]$, we design sketches of size poly$(\log(nd),k,1/ε)$ such that the cost of the optimal clustering can be estimated to within factor $1+ε$, despite the fact that the compressed representation does not contain enough information to recover the cluster centers or the partition into clusters. This leads to a streaming algorithm for estimating the clustering cost with space poly$(\log(nd),k,1/ε)$. We also obtain a distributed memory algorithm, where the $n$ points are arbitrarily partitioned amongst $m$ machines, each of which sends information to a central party who then computes an approximation of the clustering cost. Prior to this work, no such streaming or distributed-memory algorithm was known with sublinear dependence on $d$ for $p \in [1,2)$.

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arXiv2021

New Streaming Algorithms for High Dimensional EMD and MST

We study streaming algorithms for two fundamental geometric problems: computing the cost of a Minimum Spanning Tree (MST) of an $n$-point set $X \subset \{1,2,\dots,Δ\}^d$, and computing the Earth Mover Distance (EMD) between two multi-sets $A,B \subset \{1,2,\dots,Δ\}^d$ of size $n$. We consider the turnstile model, where points can be added and removed. We give a one-pass streaming algorithm for MST and a two-pass streaming algorithm for EMD, both achieving an approximation factor of $\tilde{O}(\log n)$ and using polylog$(n,d,Δ)$-space only. Furthermore, our algorithm for EMD can be compressed to a single pass with a small additive error. Previously, the best known sublinear-space streaming algorithms for either problem achieved an approximation of $O(\min\{ \log n , \log (Δd)\} \log n)$ [Andoni-Indyk-Krauthgamer '08, Backurs-Dong-Indyk-Razenshteyn-Wagner '20]. For MST, we also prove that any constant space streaming algorithm can only achieve an approximation of $Ω(\log n)$, analogous to the $Ω(\log n)$ lower bound for EMD of [Andoni-Indyk-Krauthgamer '08]. Our algorithms are based on an improved analysis of a recursive space partitioning method known generically as the Quadtree. Specifically, we show that the Quadtree achieves an $\tilde{O}(\log n)$ approximation for both EMD and MST, improving on the $O(\min\{ \log n , \log (Δd)\} \log n)$ approximation of [Andoni-Indyk-Krauthgamer '08, Backurs-Dong-Indyk-Razenshteyn-Wagner '20].

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arXiv2021

Approximating the Distance to Monotonicity of Boolean Functions

We design a nonadaptive algorithm that, given oracle access to a function $f: \{0,1\}^n \to \{0,1\}$ which is $α$-far from monotone, makes poly$(n, 1/α)$ queries and returns an estimate that, with high probability, is an $\widetilde{O}(\sqrt{n})$-approximation to the distance of $f$ to monotonicity. The analysis of our algorithm relies on an improvement to the directed isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018). Furthermore, we rule out a poly$(n, 1/α)$-query nonadaptive algorithm that approximates the distance to monotonicity significantly better by showing that, for all constant $κ> 0,$ every nonadaptive $n^{1/2 - κ}$-approximation algorithm for this problem requires $2^{n^κ}$ queries. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. We obtain our lower bound by proving an analogous bound for erasure-resilient (and tolerant) testers. Our method also yields the same lower bounds for unateness and being a $k$-junta.

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arXiv2021

Random Restrictions of High-Dimensional Distributions and Uniformity Testing with Subcube Conditioning

We give a nearly-optimal algorithm for testing uniformity of distributions supported on $\{-1,1\}^n$, which makes $\tilde O (\sqrt{n}/\varepsilon^2)$ queries to a subcube conditional sampling oracle (Bhattacharyya and Chakraborty (2018)). The key technical component is a natural notion of random restriction for distributions on $\{-1,1\}^n$, and a quantitative analysis of how such a restriction affects the mean vector of the distribution. Along the way, we consider the problem of mean testing with independent samples and provide a nearly-optimal algorithm.

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arXiv2020

Learning and Testing Junta Distributions with Subcube Conditioning

We study the problems of learning and testing junta distributions on $\{-1,1\}^n$ with respect to the uniform distribution, where a distribution $p$ is a $k$-junta if its probability mass function $p(x)$ depends on a subset of at most $k$ variables. The main contribution is an algorithm for finding relevant coordinates in a $k$-junta distribution with subcube conditioning [BC18, CCKLW20]. We give two applications: 1. An algorithm for learning $k$-junta distributions with $\tilde{O}(k/ε^2) \log n + O(2^k/ε^2)$ subcube conditioning queries, and 2. An algorithm for testing $k$-junta distributions with $\tilde{O}((k + \sqrt{n})/ε^2)$ subcube conditioning queries. All our algorithms are optimal up to poly-logarithmic factors. Our results show that subcube conditioning, as a natural model for accessing high-dimensional distributions, enables significant savings in learning and testing junta distributions compared to the standard sampling model. This addresses an open question posed by Aliakbarpour, Blais, and Rubinfeld [ABR17].

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arXiv2019

Optimal Adaptive Detection of Monotone Patterns

We investigate adaptive sublinear algorithms for detecting monotone patterns in an array. Given fixed $2 \leq k \in \mathbb{N}$ and $\varepsilon > 0$, consider the problem of finding a length-$k$ increasing subsequence in an array $f \colon [n] \to \mathbb{R}$, provided that $f$ is $\varepsilon$-far from free of such subsequences. Recently, it was shown that the non-adaptive query complexity of the above task is $Θ((\log n)^{\lfloor \log_2 k \rfloor})$. In this work, we break the non-adaptive lower bound, presenting an adaptive algorithm for this problem which makes $O(\log n)$ queries. This is optimal, matching the classical $Ω(\log n)$ adaptive lower bound by Fischer [2004] for monotonicity testing (which corresponds to the case $k=2$), and implying in particular that the query complexity of testing whether the longest increasing subsequence (LIS) has constant length is $Θ(\log n)$.

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arXiv2019

Finding monotone patterns in sublinear time

We study the problem of finding monotone subsequences in an array from the viewpoint of sublinear algorithms. For fixed $k \in \mathbb{N}$ and $\varepsilon > 0$, we show that the non-adaptive query complexity of finding a length-$k$ monotone subsequence of $f \colon [n] \to \mathbb{R}$, assuming that $f$ is $\varepsilon$-far from free of such subsequences, is $Θ((\log n)^{\lfloor \log_2 k \rfloor})$. Prior to our work, the best algorithm for this problem, due to Newman, Rabinovich, Rajendraprasad, and Sohler (2017), made $(\log n)^{O(k^2)}$ non-adaptive queries; and the only lower bound known, of $Ω(\log n)$ queries for the case $k = 2$, followed from that on testing monotonicity due to Ergün, Kannan, Kumar, Rubinfeld, and Viswanathan (2000) and Fischer (2004).

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arXiv2019

Nearly optimal edge estimation with independent set queries

We study the problem of estimating the number of edges of an unknown, undirected graph $G=([n],E)$ with access to an independent set oracle. When queried about a subset $S\subseteq [n]$ of vertices the independent set oracle answers whether $S$ is an independent set in $G$ or not. Our first main result is an algorithm that computes a $(1+ε)$-approximation of the number of edges $m$ of the graph using $\min(\sqrt{m},n / \sqrt{m})\cdot\textrm{poly}(\log n,1/ε)$ independent set queries. This improves the upper bound of $\min(\sqrt{m},n^2/m)\cdot\textrm{poly}(\log n,1/ε)$ by Beame et al. \cite{BHRRS18}. Our second main result shows that ${\min(\sqrt{m},n/\sqrt{m}))/\textrm{polylog}(n)}$ independent set queries are necessary, thus establishing that our algorithm is optimal up to a factor of $\textrm{poly}(\log n, 1/ε)$.

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arXiv2019

Testing Unateness Nearly Optimally

We present an $\tilde{O}(n^{2/3}/ε^2)$-query algorithm that tests whether an unknown Boolean function $f\colon\{0,1\}^n\rightarrow \{0,1\}$ is unate (i.e., every variable is either non-decreasing or non-increasing) or $ε$-far from unate. The upper bound is nearly optimal given the $\tildeΩ(n^{2/3})$ lower~bound of [CWX17a]. The algorithm builds on a novel use of the binary search procedure and its analysis over long random paths.

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arXiv2019

On Mean Estimation for General Norms with Statistical Queries

We study the problem of mean estimation for high-dimensional distributions, assuming access to a statistical query oracle for the distribution. For a normed space $X = (\mathbb{R}^d, \|\cdot\|_X)$ and a distribution supported on vectors $x \in \mathbb{R}^d$ with $\|x\|_{X} \leq 1$, the task is to output an estimate $\hatμ \in \mathbb{R}^d$ which is $ε$-close in the distance induced by $\|\cdot\|_X$ to the true mean of the distribution. We obtain sharp upper and lower bounds for the statistical query complexity of this problem when the the underlying norm is symmetric as well as for Schatten-$p$ norms, answering two questions raised by Feldman, Guzmán, and Vempala (SODA 2017).

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arXiv2018

Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs

We introduce a new model for testing graph properties which we call the \emph{rejection sampling model}. We show that testing bipartiteness of $n$-nodes graphs using rejection sampling queries requires complexity $\widetildeΩ(n^2)$. Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions of the form $f\colon\{0,1\}^n\to \{0,1\}$: $\bullet$Tolerant $k$-junta testing with \emph{non-adaptive} queries requires $\widetildeΩ(k^2)$ queries. $\bullet$Tolerant unateness testing requires $\widetildeΩ(n)$ queries. $\bullet$Tolerant unateness testing with \emph{non-adaptive} queries requires $\widetildeΩ(n^{3/2})$ queries. Given the $\widetilde{O}(k^{3/2})$-query non-adaptive junta tester of Blais \cite{B08}, we conclude that non-adaptive tolerant junta testing requires more queries than non-tolerant junta testing. In addition, given the $\widetilde{O}(n^{3/4})$-query unateness tester of Chen, Waingarten, and Xie \cite{CWX17b} and the $\widetilde{O}(n)$-query non-adaptive unateness tester of Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri \cite{BCPRS17}, we conclude that tolerant unateness testing requires more queries than non-tolerant unateness testing, in both adaptive and non-adaptive settings. These lower bounds provide the first separation between tolerant and non-tolerant testing for a natural property of Boolean functions.

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arXiv2017

Beyond Talagrand Functions: New Lower Bounds for Testing Monotonicity and Unateness

We prove a lower bound of $\tildeΩ(n^{1/3})$ for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is monotone or far from monotone. This improves the recent bound of $\tildeΩ(n^{1/4})$ for the same problem by Belovs and Blais [BB15]. Our result builds on a new family of random Boolean functions that can be viewed as a two-level extension of Talagrand's random DNFs. Beyond monotonicity, we also prove a lower bound of $\tildeΩ(n^{2/3})$ for any two-sided and adaptive algorithm, and a lower bound of $\tildeΩ(n)$ for any one-sided and non-adaptive algorithm for testing unateness, a natural generalization of monotonicity. The latter matches the recent linear upper bounds by Khot and Shinkar [KS15] and by Chakrabarty and Seshadhri [CS16].

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arXiv2017

Boolean Unateness Testing with $\widetilde{O}(n^{3/4})$ Adaptive Queries

We give an adaptive algorithm which tests whether an unknown Boolean function $f\colon \{0, 1\}^n \to\{0, 1\}$ is unate, i.e. every variable of $f$ is either non-decreasing or non-increasing, or $ε$-far from unate with one-sided error using $\widetilde{O}(n^{3/4}/ε^2)$ queries. This improves on the best adaptive $O(n/ε)$-query algorithm from Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova and Seshadhri when $1/ε\ll n^{1/4}$. Combined with the $\widetildeΩ(n)$-query lower bound for non-adaptive algorithms with one-sided error of [CWX17, BCPRS17], we conclude that adaptivity helps for the testing of unateness with one-sided error. A crucial component of our algorithm is a new subroutine for finding bi-chromatic edges in the Boolean hypercube called adaptive edge search.

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arXiv2017

Approximate Near Neighbors for General Symmetric Norms

We show that every symmetric normed space admits an efficient nearest neighbor search data structure with doubly-logarithmic approximation. Specifically, for every $n$, $d = n^{o(1)}$, and every $d$-dimensional symmetric norm $\|\cdot\|$, there exists a data structure for $\mathrm{poly}(\log \log n)$-approximate nearest neighbor search over $\|\cdot\|$ for $n$-point datasets achieving $n^{o(1)}$ query time and $n^{1+o(1)}$ space. The main technical ingredient of the algorithm is a low-distortion embedding of a symmetric norm into a low-dimensional iterated product of top-$k$ norms. We also show that our techniques cannot be extended to general norms.

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