arXiv subjects

Honghao Lin

Publications and source records attributed to Honghao Lin.

At least 19 recordsLinked to original sources

arXiv2026

Near-Optimal Space and Time Cut Sparsification in Turnstile Streams

A seminal result of Ahn, Guha, and McGregor (PODS'12) showed that a cut sparsifier of an unweighted graph can be computed from a near-linear number of linear measurements. Later work extended this to spectral sparsifiers (Kapralov et al., FOCS'14) and to near-linear recovery time (Kapralov et al., SODA'20). For weighted graphs, these sketches rely on grouping edges by weight, which general turnstile streams do not allow. Chen, Khanna, and Li (FOCS'22) removed this limitation, using $\tilde{O}(n/\varepsilon^3)$ bits of space. We improve the space to $\tilde{O}(n/\varepsilon^2)$ bits. Our algorithm makes one pass over a general turnstile stream of a weighted graph, processes each update in $\tilde{O}(1)$ time, and recovers, in $\tilde{O}(n/\varepsilon^2)$ time, a reweighted subgraph with $O(n\varepsilon^{-2}\log^2 n)$ edges that preserves every cut within a factor of $1\pm\varepsilon$. Its space, update time, and recovery time are all optimal up to polylogarithmic factors.

cs.DS↗

arXiv2026

On the Accuracy of Approximate Leave-One-Out with Blockwise Nonsmooth Regularization in High-dimensions

Approximate leave-one-out (ALO) refers to methods that provide computationally efficient and statistically accurate approximations to leave-one-out risk estimation. These methods have recently attracted increased attention because of their connections to other important problems, including machine unlearning and data valuation. For empirical risk minimization, ALO methods often rely on a few Newton steps and therefore require smooth loss functions and regularizers. In high-dimensional problems, however, regularizers are frequently non-smooth. This has motivated the development of heuristic ALO methods for non-smooth regularization, whose empirical accuracy is often strong but whose theoretical justification remains limited. In this paper, we study the accuracy of a heuristic ALO method for a broad class of non-differentiable penalties that promote block sparsity in the proportional high-dimensional regime. Assuming uniformly bounded block sizes, we prove that the expected average absolute discrepancy between the exact leave-one-out loss and its ALO approximation is $O(\log^{3/2}(en)/\sqrt{n})$. Our proof technique differs substantially from existing approaches and relies on obtaining sharp bounds on the number of active groups whose status changes between the full-data and leave-one-out estimators. In addition to yielding stronger guarantees under weaker assumptions than prior work, our analysis provides new insight into the stability of sparse estimators under perturbations of the training data.

math.ST↗

arXiv2026

Optimal and Efficient Online Inverse Optimization

In online inverse linear optimization, a learner recommends an action and then observes the choice of an expert who maximizes a fixed, unknown linear objective on $\mathbb{R}^{d}$; the goal is to learn to optimize this objective without observing it. Sakaue recently obtained the optimal regret $O(\sqrt d)$ with a randomized algorithm making $(dT)^{O(d)}$ linear optimizations per round, and asked whether it can be attained in polynomial time. We answer positively: our deterministic algorithm has regret $O(\sqrt d)$ for every horizon $T$ and runs in time polynomial in $d$ and $T$. It is a variant of the variable-metric algorithms of Sakaue et al.\ and Cai et al., in which a metric update is revoked once the query point moves far enough from where the update was made.

cs.LG↗

arXiv2026

A Proof of the Most Informative Boolean Function Conjecture

Let $X$ be uniform on $\{-1,1\}^n$, let $Y$ be obtained by passing its coordinates independently through a binary symmetric channel with crossover probability $p$, and let $g:\{-1,1\}^n\to\{0,1\}$ be a Boolean function. We give a computer-assisted proof of the Courtade--Kumar conjecture $I(g(X);Y)\le1-H_2(p)$, where $H_2$ is binary entropy, with equality attained by dictator functions. The present work builds on the differential-equation method, itself a limiting form of the auxiliary-receiver approach in network information theory using a continuum of degraded receivers. The proof proceeds from a local inequality to a dimension-independent bound on entropy production. Differentiation along the Boolean noise semigroup expresses entropy production as an average of edge costs. The key estimate is therefore an unrestricted Bellman inequality with two mean constraints and two entropy constraints, allowing arbitrary couplings of the edge variables. This paper and its supplement provide the proofs and computational verification records. The document is lengthy because it is designed to be entirely self-contained, deriving all proofs from first principles and reproducing the proofs of cited results. We also give a self-contained expository note explaining the reduction to a low-dimensional inequality and the ideas behind the key lower bounds. The entire proof, including all numerical certificates, has been formally verified in Lean end-to-end, and is available online.

cs.DS↗

arXiv2026

Stellar Colosseum: A Many-Agent Harness for Long-Horizon Research in Mathematics and Theoretical Computer Science

Language models can produce plausible short proofs, but may still be unreliable on long-horizon research problems, where progress depends on a sequence of uncertain and interdependent decisions. We introduce Stellar Colosseum, a model-agnostic harness for allocating inference across research in mathematics and theoretical computer science. Colosseum explores alternative strategies before proof construction, uses a readiness gate to decide when a route is mature enough to decompose, represents the proof plan as interdependent section-level subproblems, and routes verifier findings back to the affected part of the argument. Across these stages, it generates candidates in parallel, attacks them with targeted falsification, and combines candidates and their critiques into a single research artifact through overlapping random-sample tree aggregation. The Colosseum workflow has been integrated into Google Antigravity's Teamwork framework as the Long Proof pattern. We demonstrate the capabilities of Colosseum through open-ended research and evaluations on theorem-proving and competitive programming benchmarks. Using Colosseum with Gemini 3.1 Pro, we obtain several new results that address open problems arising from papers published at top venues such as FOCS and JMLR. On TCS-Bench, a benchmark of research-level theorem-proving tasks drawn from papers published at FOCS, STOC, and SODA, Colosseum achieves 71.0% accuracy using Gemini 3.1 Pro and Gemini 3.7 Flash. In a separate Codeforces evaluation using Gemini 3.1 Pro, the proof-oriented pipeline with execution feedback solves 218 of 222 problems.

cs.AI↗

arXiv2026

TCS-BENCH: Benchmarking State-of-the-Art Generative AI Theoretical Computer Science Research Ability

We introduce TCS-Bench, a benchmark for evaluating Large Language Models (LLMs) on research-level Theoretical Computer Science (TCS) proof generation. TCS-Bench consists of theorem-proving tasks from papers published at top theoretical computer science venues (STOC, FOCS, and SODA). Each task provides the necessary context to derive a self-contained proof for a target result. We evaluate state-of-the-art models on this benchmark. We verify the correctness of generated proofs via a verification agent, and further benchmark the verifier against human-expert proof judgements on a set of target statements and generated proofs pairs. Our reference verifier achieves over 90% accuracy on the expert labeled set.

cs.CL↗

arXiv2026

Nearly Optimal Strong Coresets for $\ell_p$ Subspace Approximation

We study strong coresets for $\ell_p$ subspace approximation. Given a matrix $A\in\mathbb{R}^{n\times d}$, the goal is to sample and rescale a small number of its rows to obtain $SA$ such that $\left\|SA(I-P_F)\right\|_{p,2}^p=(1\pm\varepsilon)\left\|A(I-P_F)\right\|_{p,2}^p$ simultaneously for every subspace $F\subseteq\mathbb{R}^d$ of dimension at most $k$, where $P_F$ is the orthogonal projector onto $F$. Woodruff and Yasuda [WY25] (FOCS 2025) obtained coreset sizes $\widetilde{O}_p(k\varepsilon^{-4/p})$ for $1\leq p<2$ and $\widetilde{O}_p(k^{p/2}\varepsilon^{-p})$ for $p>2$. We improve these bounds to $\widetilde{O}_p(k\varepsilon^{-2})$ and $\widetilde{O}_p(k^{p/2}\varepsilon^{-2})$, respectively. For $1\leq p<2$, our algorithm runs in $\widetilde{O}_p(\mathrm{nnz}(A)+d^ω+k\varepsilon^{-2})$ time. The resulting coreset size matches the sampling lower bound [LWW21] up to logarithmic factors when $k+1\geq C\log(1/\varepsilon)$ for an absolute constant $C$. For $p>2$, our algorithm runs in $\widetilde{O}_p(\mathrm{nnz}(A)+d^ω)$ time, matching the running time of the framework of Woodruff and Yasuda. We use different techniques in the two regimes. For $1\leq p<2$, we combine a bicriteria low-rank split with Lewis-weight sampling and empirical-process bounds independent of the output dimension. For $p>2$, we give a sharper analysis of the Woodruff-Yasuda construction. By retaining the truncation in its sampling probabilities throughout the row-count recurrence, we show that it achieves the improved $\varepsilon^{-2}$ dependence.

cs.DS↗

arXiv2026

A Near-Optimal Lower Bound for Prefix-Matrix Factorizations

For the $n\times n$ lower-triangular all-ones matrix $Q$, we prove a near-optimal lower bound \[ γ_{2,1}(Q) := \inf_{Q=AB} \|A\|_{2\to\infty}\|B\|_{1\to1} = Ω\!\left( \frac{\log^{3/2}n}{(\log\log n)^{3/2}} \right), \] where the infimum ranges over real factorizations of arbitrary finite inner dimension. This cost is a central parameter in space bounds for factorization-based rank and quantile estimation in turnstile streams and in error bounds for matrix mechanisms for continual counting under pure differential privacy. The proof combines right-sided Haar projections with a scale-dependent numerical-sparsity decomposition of the rows of $B$. At each scale, a rank--Frobenius argument shows that the numerically sparse rows cannot account for all of the required Schatten $2/3$ mass, while a Haar projection estimate bounds the contribution of the remaining rows. Summing these bounds over the dyadic scales yields the result. The proof was obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors verified the proof and made minor revisions.

cs.DS↗

arXiv2026

Pairwise-Independent Dithering for Single-Stage Hadamard Quantization

Quantizing high-dimensional vectors is fundamental to similarity search, distributed learning, and model compression. Feng, Indyk, Kapralov, Krachun, and Prokhorov established sharp guarantees for an unbiased dithered quantizer based on a randomized Hadamard transform [FIK+26]. Their $1/d$-scale inner-product estimator, however, uses a second randomized transform and residual quantization, increasing both communication and the leading constant in the proved bound. We show that this extra stage is unnecessary: pairwise-independent dithers across Hadamard coordinates suffice. The resulting unbiased single-stage estimator uses $b$ bits per coordinate and achieves \[ \mathbb{E}\!\left[ \left|\left\langle y,\widehat{x}-x\right\rangle\right|^2 \right] \leq \left(\frac{3π\sqrt{3}}{2}+o(1)\right) \frac{\lVert y\rVert_2^2}{d\,4^b}, \] as $b\to\infty$, with a dimension-free $o(1)$ term uniform over unit inputs and fixed queries. Compared with the two-stage construction of Feng et al., it eliminates the residual-stage $O(d)$-bit payload and reduces the leading upper-bound constant by a factor of approximately $5.93$. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

cs.DS↗

arXiv2026

The Condition-Number Barrier in Sparse Least Squares

In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis in the weighted regular-graph formulation of Raghavendra, Steurer, and Tulsiani [RST12]. Concretely, for every fixed $γ\in(0,1]$, there is no randomized polynomial-time algorithm that, with probability at least $2/3$, returns a vector $x$ such that, writing $s=\lVert x\rVert_0$, \[ \lVert Ax-b\rVert_2^2 \leq \min_{\lVert z\rVert_0\leq k}\lVert Az-b\rVert_2^2+\varepsilon \quad\text{and}\quad s=O\!\left(k\,κ_{s+k}^{\,1-γ}\right), \] where $κ_r$ is the restricted condition number at sparsity level $r$. The result holds even on rational instances with $A$ of full column rank. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

cs.DS↗

arXiv2026

Near-Optimal Dimension Lower Bounds for Single-Vector Embeddings of Maximum Inner Product Similarity

Multi-vector embeddings represent items by point clouds and compare query and document point clouds using Chamfer similarity, whereas single-vector embeddings use ordinary inner products. For singleton queries, Chamfer becomes maximum inner product similarity (MAX-IP). In our setting, MUVERA gives dimension $m^{O(1/ε^2)}$ [DHJ+24], whereas the previous lower bound $(ε^2m)^{Ω(1/ε)}$ [Jay26] left a gap between $1/ε$ and $1/ε^2$ in the exponent of $m$. We nearly close this gap. For every fixed $δ\in(0,1)$, there are constants $A_δ,c_δ>0$ such that, for all sufficiently small $ε>0$ and every $m\ge(1/ε)^{A_δ}$, there exist unit query vectors and document point clouds of at most $m$ unit vectors for which every single-vector approximation of all pairwise MAX-IP values to additive error $ε$ has dimension \[ D \ge m^{c_δ/ε^{2-2δ}}. \] This holds even for fully data-dependent representations chosen after seeing the dataset. It also applies to Chamfer because all queries are singletons. Since $δ$ can be arbitrarily small, the exponent approaches the $O(1/ε^2)$ dependence of the upper bound. The proof combines Sherstov's pattern matrix method with polynomial-size, constant-width DNF formulas computing functions of approximate degree $Ω(k^{1-δ})$. Uniform-width padding and a block encoding create an $Ω(ε)$ gap. A dummy coordinate then equalizes all false inputs, yielding a unit-sphere MAX-IP matrix that is an exact two-valued affine image of the DNF pattern matrix with gap at least $8ε$. This allows the approximate-rank bound to apply. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

cs.DS↗

arXiv2026

Adversarial Robustness for Small Frequency Moments and a Weak Equivalence Theorem for Turnstile Streams

We study adversarially robust algorithms for insertion-deletion (turnstile) streams, where future updates may depend on past algorithm outputs. While recent work achieved a robust $(1+ε)$-approximation for the second moment $F_2$ in polylogarithmic space, achieving high accuracy for other frequency moments remained a major open question; for $p\in[0,2)$, including the fundamental distinct elements problem ($F_0$), only constant-factor approximations were known in sublinear space. We close this gap, showing that $(1+ε)$-approximate robustness can be achieved in polylogarithmic space for all $p\in[0,2]$. Our approach generalizes the estimator-corrector-learner framework to non-Hilbert spaces by dynamically maintaining implicit isometric embeddings into $L_2$ and performing regularized kernel ridge regression over adaptively discovered hard queries, yielding the first insertion-deletion algorithms that approximate: (1) the $p$-th frequency moment $F_p$ up to a $(1+ε)$-factor in poly$(1/ε, \log n)$ space for all $p\in[0,2]$, including the support size $F_0$, (2) metric and information-theoretic quantities, including the Earth Mover Distance (EMD) and $k$-median clustering cost over $[Δ]^d$ up to an $O(d \log Δ)$-factor, and the Shannon entropy up to an $ε$-additive error, and (3) non-normed symmetric losses defined by Bernstein functions up to a $(1+ε)$-factor. For the $F_p$ moments, our algorithm is optimal up to poly$(1/ε, \log n)$ factors. Furthermore, we establish a weak equivalence between classical oblivious sketching and adversarial robustness. We prove that for any sub-multiplicative norm, the existence of an efficient classical linear sketch is equivalent to the existence of an efficient robust turnstile algorithm, up to polynomial factors, formalizing $L_1$ embeddability as the fundamental mechanism governing both models.

cs.DS↗

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.

cs.DS↗

arXiv2026

Adversarial Robustness on Insertion-Deletion Streams

We study adversarially robust algorithms for insertion-deletion (turnstile) streams, where future updates may depend on past algorithm outputs. While robust algorithms exist for insertion-only streams with only a polylogarithmic overhead in memory over non-robust algorithms, it was widely conjectured that turnstile streams of length polynomial in the universe size $n$ require space linear in $n$. We refute this conjecture, showing that robustness can be achieved using space which is significantly sublinear in $n$. Our framework combines multiple linear sketches in a novel estimator-corrector-learner framework, yielding the first insertion-deletion algorithms that approximate: (1) the second moment $F_2$ up to a $(1+\varepsilon)$-factor in polylogarithmic space, (2) any symmetric function $\cal{F}$ with an $\mathcal{O}(1)$-approximate triangle inequality up to a $2^{\mathcal{O}(C)}$ factor in $\tilde{\mathcal{O}}(n^{1/C}) \cdot S(n)$ bits of space, where $S$ is the space required to approximate $\cal{F}$ non-robustly; this includes a broad class of functions such as the $L_1$-norm, the support size $F_0$, and non-normed losses such as the $M$-estimators, and (3) $L_2$ heavy hitters. For the $F_2$ moment, our algorithm is optimal up to $\textrm{poly}((\log n)/\varepsilon)$ factors. Given the recent results of Gribelyuk et al. (STOC, 2025), this shows an exponential separation between linear sketches and non-linear sketches for achieving adversarial robustness in turnstile streams.

cs.DS↗

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.

cs.DS↗

arXiv2025

$L_p$ Sampling in Distributed Data Streams with Applications to Adversarial Robustness

In the distributed monitoring model, a data stream over a universe of size $n$ is distributed over $k$ servers, who must continuously provide certain statistics of the overall dataset, while minimizing communication with a central coordinator. In such settings, the ability to efficiently collect a random sample from the global stream is a powerful primitive, enabling a wide array of downstream tasks such as estimating frequency moments, detecting heavy hitters, or performing sparse recovery. Of particular interest is the task of producing a perfect $L_p$ sample, which given a frequency vector $f \in \mathbb{R}^n$, outputs an index $i$ with probability $\frac{f_i^p}{\|f\|_p^p}+\frac{1}{\mathrm{poly}(n)}$. In this paper, we resolve the problem of perfect $L_p$ sampling for all $p\ge 1$ in the distributed monitoring model. Specifically, our algorithm runs in $k^{p-1} \cdot \mathrm{polylog}(n)$ bits of communication, which is optimal up to polylogarithmic factors. Utilizing our perfect $L_p$ sampler, we achieve adversarially-robust distributed monitoring protocols for the $F_p$ moment estimation problem, where the goal is to provide a $(1+\varepsilon)$-approximation to $f_1^p+\ldots+f_n^p$. Our algorithm uses $\frac{k^{p-1}}{\varepsilon^2}\cdot\mathrm{polylog}(n)$ bits of communication for all $p\ge 2$ and achieves optimal bounds up to polylogarithmic factors, matching lower bounds by Woodruff and Zhang (STOC 2012) in the non-robust setting. Finally, we apply our framework to achieve near-optimal adversarially robust distributed protocols for central problems such as counting, frequency estimation, heavy-hitters, and distinct element estimation.

cs.DS↗

arXiv2025

Lifting Linear Sketches: Optimal Bounds and Adversarial Robustness

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.

cs.DS↗

arXiv2024

Space Complexity of Minimum Cut Problems in Single-Pass Streams

We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. While graph sparsification in streams has been intensively studied, the specific application of finding minimum cuts in streams is less well-studied. To this end, we show upper and lower bounds on minimum cut problems in insertion-only streams for a variety of settings, including for both randomized and deterministic algorithms, for both arbitrary and random order streams, and for both approximate and exact algorithms. One of our main results is an $\widetilde{O}(n/\varepsilon)$ space algorithm with fast update time for approximating a spectral cut query with high probability on a stream given in an arbitrary order. Our result breaks the $Ω(n/\varepsilon^2)$ space lower bound required of a sparsifier that approximates all cuts simultaneously. Using this result, we provide streaming algorithms with near optimal space of $\widetilde{O}(n/\varepsilon)$ for minimum cut and approximate all-pairs effective resistances, with matching space lower-bounds. The amortized update time of our algorithms is $\widetilde{O}(1)$, provided that the number of edges in the input graph is at least $(n/\varepsilon^2)^{1+o(1)}$. We also give a generic way of incorporating sketching into a recursive contraction algorithm to improve the post-processing time of our algorithms. In addition to these results, we give a random-order streaming algorithm that computes the {\it exact} minimum cut on a simple, unweighted graph using $\widetilde{O}(n)$ space. Finally, we give an $Ω(n/\varepsilon^2)$ space lower bound for deterministic minimum cut algorithms which matches the best-known upper bound up to polylogarithmic factors.

cs.DS↗