Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.19629

Lifting Linear Sketches: Optimal Bounds and Adversarial Robustness

Elena Gribelyuk·Honghao Lin·David P. Woodruff·Huacheng Yu·Samson Zhou
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Elena Gribelyuk, Honghao Lin, David P. Woodruff, Huacheng Yu, Samson Zhou. 2025-03-25. Lifting Linear Sketches: Optimal Bounds and Adversarial Robustness. https://arxiv.org/abs/2503.19629

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗