arXiv subjects

Hoai-An Nguyen

Publications and source records attributed to Hoai-An Nguyen.

10 recordsLinked to original sources

arXiv2026

On Sketching Trimmed Statistics

We study sketching trimmed statistics of a frequency vector, including the $F_p$ moment of the top-$k$ coordinates and of the trimmed-$k$ vector. Despite their natural role in robust analytics, this is the first time these problems have been studied in any sublinear space setting. For $p \in [0,2]$, we obtain $poly(\log n/\varepsilon)$-space algorithms for both tasks when $k$ is moderately large, and for general $k$ we identify a sharp structural threshold that characterizes exactly when sublinear space is possible: in particular, it is actually determined by the ratio between $a_k^2$ and $\|x_{-k}\|_2^2/k$. We extend these results to $p > 2$ and present several applications including algorithms for thresholded $F_p$ estimation and generalized impact indices. Notably, we improve the space bounds of Govindan, Monemizadeh, and Muthukrishnan (PODS 2017) for computing the $h$-index.

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arXiv2026

Streaming Complexity Separations for Dense and Sparse Graphs

We identify a sharp separation in the streaming space complexity of Maximum Cut when the algorithm must output an approximate cut (rather than only the approximate value). For dense graphs, we show that $O(n/\varepsilon^2)$ space is sufficient and that $Ω(n)$ space is necessary. In contrast, for graphs with $Θ(n/\varepsilon^2)$ edges, the situation is markedly different: we show that the problem requires $Ω(n \log(\varepsilon^2 n)/\varepsilon^2)$ space for any $\varepsilon=ω(1/\sqrt{n})$, which is tight for the full range of $\varepsilon$. We also give an $Ω(n \log n/\varepsilon^2)$-space lower bound against deterministic algorithms for outputting a $(1-\varepsilon)$ approximation to the value of the maximum cut. Using similar techniques we prove an analogous sharp separation in the streaming space complexity of Densest Subgraph and show that for every constant-arity CSP over a constant-size alphabet and the Similarity problem the space complexity in dense streams can be improved by shaving a logarithmic factor.

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arXiv2026

Unbiased Insights: Optimal Streaming Algorithms for $\ell_p$ Sampling, the Forget Model, and Beyond

We study $\ell_p$ sampling and frequency moment estimation in a single-pass insertion-only data stream. For $p \in (0,2)$, we present a nearly space-optimal approximate $\ell_p$ sampler that uses $\widetilde{O}(\log n \log(1/δ))$ bits of space and for $p = 2$, we present a sampler with space complexity $\widetilde{O}(\log^2 n \log(1/δ))$. This space complexity is optimal for $p \in (0, 2)$ and improves upon prior work by a $\log n$ factor. We further extend our construction to a continuous $\ell_p$ sampler, which outputs a valid sample index at every point during the stream. Leveraging these samplers, we design nearly unbiased estimators for $F_p$ in data streams that include forget operations, which reset individual element frequencies and introduce significant non-linear challenges. As a result, we obtain near-optimal algorithms for estimating $F_p$ for all $p$ in this model, originally proposed by Pavan, Chakraborty, Vinodchandran, and Meel [PODS'24], resolving all three open problems they posed. Furthermore, we generalize this model to what we call the suffix-prefix deletion model, and extend our techniques to estimate entropy as a corollary of our moment estimation algorithms. Finally, we show how to handle arbitrary coordinate-wise functions during the stream, for any $g \in \mathbb{G}$, where $\mathbb{G}$ includes all (linear or non-linear) contraction functions.

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arXiv2025

Relative Error Fair Clustering in the Weak-Strong Oracle Model

We study fair clustering problems in a setting where distance information is obtained from two sources: a strong oracle providing exact distances, but at a high cost, and a weak oracle providing potentially inaccurate distance estimates at a low cost. The goal is to produce a near-optimal fair clustering on $n$ input points with a minimum number of strong oracle queries. This models the increasingly common trade-off between accurate but expensive similarity measures (e.g., large-scale embeddings) and cheaper but inaccurate alternatives. The study of fair clustering in the model is motivated by the important quest of achieving fairness with the presence of inaccurate information. We achieve the first $(1+\varepsilon)$-coresets for fair $k$-median clustering using $\text{poly}\left(\frac{k}{\varepsilon}\cdot\log n\right)$ queries to the strong oracle. Furthermore, our results imply coresets for the standard setting (without fairness constraints), and we could in fact obtain $(1+\varepsilon)$-coresets for $(k,z)$-clustering for general $z=O(1)$ with a similar number of strong oracle queries. In contrast, previous results achieved a constant-factor $(>10)$ approximation for the standard $k$-clustering problems, and no previous work considered the fair $k$-median clustering problem.

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arXiv2025

Entrywise Approximation for Matrix Inversion and Linear Systems

We study the bit complexity of inverting diagonally dominant matrices, which are associated with random walk quantities such as hitting times and escape probabilities. Such quantities can be exponentially small, even on undirected unit-weighted graphs. However, their nonnegativity suggests that they can be approximated entrywise, leading to a stronger notion of approximation than vector norm-based error. Under this notion of error, existing Laplacian solvers and fast matrix multiplication approaches have bit complexities of $mn^2$ and $n^{ω+1}$, respectively, where $m$ is the number of nonzero entries in the matrix, $n$ is its size, and $ω$ is the matrix multiplication exponent. We present algorithms that compute entrywise $\exp(ε)$-approximate inverses of row diagonally dominant $L$-matrices (RDDL) in two settings: (1) when the matrix entries are given in floating-point representation; (2) when they are given in fixed-point representation. For floating-point inputs, we present a cubic-time algorithm and show that it has an optimal running time under the all-pairs shortest paths (APSP) conjecture. For fixed-point inputs, we present several algorithms for solving linear systems and inverting RDDL and SDDM matrices, all with high probability. Omitting logarithmic factors: (1) For SDDM matrices, we provide an algorithm for solving a linear system with entrywise approximation guarantees using $\tilde{O}(m\sqrt{n})$ bit operations, and another for computing an entrywise approximate inverse using $\tilde{O}(mn)$ bit operations. (2) For RDDL matrices, we present an algorithm for solving a linear system using $\tilde{O}(mn^{1+o(1)})$ bit operations, and two algorithms for computing an entrywise approximate inverse: one using $\tilde{O}(n^{ω+0.5})$ bit operations, and the other using $\tilde{O}(mn^{1.5+o(1)})$ bit operations.

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arXiv2025

Numerical Linear Algebra in Linear Space

We present a randomized linear-space solver for general linear systems $\mathbf{A} \mathbf{x} = \mathbf{b}$ with $\mathbf{A} \in \mathbb{Z}^{n \times n}$ and $\mathbf{b} \in \mathbb{Z}^n$, without any assumption on the condition number of $\mathbf{A}$. For matrices whose entries are bounded by $\mathrm{poly}(n)$, the solver returns a $(1+ε)$-multiplicative entry-wise approximation to vector $\mathbf{x} \in \mathbb{Q}^{n}$ using $\widetilde{O}(n^2 \cdot \mathrm{nnz}(\mathbf{A}))$ bit operations and $O(n \log n)$ bits of working space (i.e., linear in the size of a vector), where $\mathrm{nnz}$ denotes the number of nonzero entries. Our solver works for right-hand vector $\mathbf{b}$ with entries up to $n^{O(n)}$. To our knowledge, this is the first linear-space linear system solver over the rationals that runs in $\widetilde{O}(n^2 \cdot \mathrm{nnz}(\mathbf{A}))$ time. We also present several applications of our solver to numerical linear algebra problems, for which we provide algorithms with efficient polynomial running time and near-linear space. In particular, we present results for linear regression, linear programming, eigenvalues and eigenvectors, and singular value decomposition.

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arXiv2025

Maximum Coverage in Turnstile Streams with Applications to Fingerprinting Measures

In the maximum coverage problem we are given $d$ subsets from a universe $[n]$, and the goal is to output $k$ subsets such that their union covers the largest possible number of distinct items. We present the first algorithm for maximum coverage in the turnstile streaming model, where updates which insert or delete an item from a subset come one-by-one. Notably our algorithm only uses $poly\log n$ update time. We also present turnstile streaming algorithms for targeted and general fingerprinting for risk management where the goal is to determine which features pose the greatest re-identification risk in a dataset. As part of our work, we give a result of independent interest: an algorithm to estimate the complement of the $p^{\text{th}}$ frequency moment of a vector for $p \geq 2$. Empirical evaluation confirms the practicality of our fingerprinting algorithms demonstrating a speedup of up to $210$x over prior work.

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arXiv2024

Entrywise Approximate Laplacian Solving

We study the escape probability problem in random walks over graphs. Given vertices, $s,t,$ and $p$, the problem asks for the probability that a random walk starting at $s$ will hit $t$ before hitting $p$. Such probabilities can be exponentially small even for unweighted undirected graphs with polynomial mixing time. Therefore current approaches, which are mostly based on fixed-point arithmetic, require $n$ bits of precision in the worst case. We present algorithms and analyses for weighted directed graphs under floating-point arithmetic and improve the previous best running times in terms of the number of bit operations. We believe our techniques and analysis could have a broader impact on the computation of random walks on graphs both in theory and in practice.

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arXiv2023

Provable Reset-free Reinforcement Learning by No-Regret Reduction

Reinforcement learning (RL) so far has limited real-world applications. One key challenge is that typical RL algorithms heavily rely on a reset mechanism to sample proper initial states; these reset mechanisms, in practice, are expensive to implement due to the need for human intervention or heavily engineered environments. To make learning more practical, we propose a generic no-regret reduction to systematically design reset-free RL algorithms. Our reduction turns the reset-free RL problem into a two-player game. We show that achieving sublinear regret in this two-player game would imply learning a policy that has both sublinear performance regret and sublinear total number of resets in the original RL problem. This means that the agent eventually learns to perform optimally and avoid resets. To demonstrate the effectiveness of this reduction, we design an instantiation for linear Markov decision processes, which is the first provably correct reset-free RL algorithm.

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arXiv2022

Asymptotically Optimal Bounds for Estimating H-Index in Sublinear Time with Applications to Subgraph Counting

The $h$-index is a metric used to measure the impact of a user in a publication setting, such as a member of a social network with many highly liked posts or a researcher in an academic domain with many highly cited publications. Specifically, the $h$-index of a user is the largest integer $h$ such that at least $h$ publications of the user have at least $h$ units of positive feedback. We design an algorithm that, given query access to the $n$ publications of a user and each publication's corresponding positive feedback number, outputs a $(1\pm \varepsilon)$-approximation of the $h$-index of this user with probability at least $1-δ$ in time \[ O(\frac{n \cdot \ln{(1/δ)}}{\varepsilon^2 \cdot h}), \] where $h$ is the actual $h$-index which is unknown to the algorithm a-priori. We then design a novel lower bound technique that allows us to prove that this bound is in fact asymptotically optimal for this problem in all parameters $n,h,\varepsilon,$ and $δ$. Our work is one of the first in sublinear time algorithms that addresses obtaining asymptotically optimal bounds, especially in terms of the error and confidence parameters. As such, we focus on designing novel techniques for this task. In particular, our lower bound technique seems quite general -- to showcase this, we also use our approach to prove an asymptotically optimal lower bound for the problem of estimating the number of triangles in a graph in sublinear time, which now is also optimal in the error and confidence parameters. This result improves upon prior lower bounds of Eden, Levi, Ron, and Seshadhri (FOCS'15) for this problem, as well as multiple follow-ups that extended this lower bound to other subgraph counting problems.

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