arXiv subjects

Nicolas Menand

Publications and source records attributed to Nicolas Menand.

3 recordsLinked to original sources

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.

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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 $O(nd) + n \cdot \text{poly}( \log(n) / \epsilon)$ total space which gives a $(1+\epsilon)$-approximate Euclidean max-cut. $\bullet$ We give a dynamic streaming algorithm using $\text{poly}(d \log \Delta / \epsilon)$ space when $X \subseteq [\Delta]^d$, which provides oracle access to a $(1+\epsilon)$-approximate Euclidean max-cut. Recently, Chen, Jiang, and Krauthgamer $[\text{STOC}~'23]$ gave a dynamic streaming algorithm with space $\text{poly}(d\log\Delta/\epsilon)$ 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.

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arXiv2024

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 $\mu$ over the metric, the sketch should not over-estimate distances, and it should (approximately) preserve the average distance with respect to draws from $\mu$. 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 $([\Delta]^d, \ell_p)$ with approximation $c$ and bit-complexity $\text{poly}(2^{p/c} \cdot \log(d\Delta))$, 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 $2^{\Omega(p/c)}$ 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).

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