Text-to-video (T2V) models have shown remarkable performance in generating visually reasonable scenes, while their capability to leverage world knowledge for ensuring semantic consistency and factual accuracy remains largely understudied. In response to this challenge, we propose T2VWorldBench, the first systematic evaluation framework for evaluating the world knowledge generation abilities of text-to-video models, covering 6 major categories, 60 subcategories, and 1,200 prompts across a wide range of domains, including physics, nature, activity, culture, causality, and object. To address both human preference and scalable evaluation, our benchmark incorporates both human evaluation and automated evaluation using vision-language models (VLMs). We evaluated the 10 most advanced text-to-video models currently available, ranging from open source to commercial models, and found that most models are unable to understand world knowledge and generate truly correct videos. These findings point out a critical gap in the capability of current text-to-video models to leverage world knowledge, providing valuable research opportunities and entry points for constructing models with robust capabilities for commonsense reasoning and factual generation.

Marton's covering conjecture studies finite sets in high-dimensional binary spaces whose pairwise sums create relatively few new elements. It predicts that every such set can be described efficiently by shifted copies of one linear subspace of comparable size. Gowers, Green, Manners, and Tao [GGMT25] proved the conjecture with exponent 12. Liao [L24] improved the exponent to 9. We improve it further to 8.873.

Randomized sketch-and-solve algorithms accelerate overconstrained ℓ_2 regression by replacing the input with a smaller problem. Standard subspace embeddings guarantee that the cost of the regression is nearly preserved, but coordinate-wise accuracy of the solution is more delicate: we want the solution vector itself to be close to the optimal solution in ℓ_∞ norm. In particular, we want to find a vector x'∈ℝ^d such that x'-x^*_∞≤√(d)·Ax^⋆-b_2·A^†_ op. Price, Song and Woodruff initiated the study of this problem and showed that the subsampled randomized Hadamard transform (SRHT) with O(ε^-2 d^1+Θ(√(loglog n/log d))) rows achieves this guarantee. A subsequent work of Song, Ye, Yin and Zhang claimed to improve the row count to O(ε^-2dlog^3 n). Unfortunately, their proof relies on an independence assumption that does not hold in general, and we exhibit an explicit instance on which it fails. To achieve a truly nearly-linear-in-d row count, we introduce a new fast, dense randomized transform, which combines a randomized Hadamard flattening, a random permutation, and balanced, disjoint Gaussian pooling. Conditioned on the Hadamard-and-permutation stage, the sketched problem becomes an exact Gaussian regression in which the noise is independent of the entire sketched design; this conditional independence is exactly what the earlier argument was missing. Our sketch yields the ℓ_∞ guarantee with m=O(ε^-2dlog d) rows, uses one Hadamard pass with a padded internal dimension N=O(n+ε^-2d^3), and is efficient to apply: the sketched pair (SA, Sb) can be computed in O(Ndlog N)=O(nd+ε^-2d^4) time.

Brand, Nanongkai, and Saranurak introduced a conjecture known as the Hinted Mv Conjecture. Although it was originally formulated for the matrix case, we generalize it here to the tensor setting.

Matrix concentration for Markov chains was initiated in the expander-walk setting by Garg, Lee, Song, and Srivastava'18 [GLSS18]. However, the constant obtained in [GLSS18] is quite loose, and it is natural to ask whether a tighter proof can yield the same constant as in the independent matrix concentration setting. In this paper, we provide a positive answer to this question. Our Hoeffding exponent is sharp, as shown by a scalar obstruction. Our Chernoff and Bernstein constants improve upon those in [GLSS18] and Neeman, Shi, and Ward'24 [NSW24], respectively.

We give a randomized algorithmic version of the Balog--Szemer\'edi--Gowers theorem for sets of integers. Let $A\subseteq[N]$ have size $n:=|A|\geq2$, let $1\leq K\leq n$, and suppose that its additive energy satisfies $E(A)\geq n^3/K$. Reiher and Schoen proved existentially that, for every fixed $\epsilon \in (0,1/2)$, there is a subset $A'\subseteq A$ satisfying $|A'|\geq(1-\epsilon)n/\sqrt K$ and $|A'-A'|\leq O_{\epsilon}(K^4)|A'|$; the size scale $n/\sqrt K$ is essentially optimal. We give an algorithmic counterpart: with probability at least $1-n^{-10}$, our algorithm runs in time $nKN^{o(1)}$ and returns a subset $A'\subseteq A$ satisfying $|A'|\geq cn/\sqrt K$ and $|A'-A'|\leq CK^4|A'|$, where $c,C>0$ are absolute constants. Thus the output attains the essentially optimal subset-size scale and the best-known $K^4$ dependence for the normalized difference set.

For n unit vectors x_1,…,x_n ∈ℝ^d, we study the continuous ReLU derivative Gram matrix H, whose entries are obtained by averaging pairwise gated inner products over a standard Gaussian direction. Writing Δ_± := min_i ≠ jmin{x_i-x_j_2, x_i+x_j_2 } for their projective separation, we prove the universal dimension-free lower bound λ_min(H) = Ω( Δ_±/√(log n) ). Conversely, we construct worst-case families satisfying the matching upper bound λ_min(H) = O( Δ_±/√(log n) ), showing that this rate is tight up to universal constants.

Space complexity is a critical factor in various computational models, including streaming, parallel/distributed computing, and communication complexity. We study the space complexity of the minimum-cost flow problem, a generalization of the st-max flow problem, focusing on computing flows in subquadratic space. In the general case with arbitrary capacities, minimum cost and st-maximum flows can use up to Ω(n^2) edges, so computing the flow on each edge (rather than just the size/cost) seems impossible in subquadratic space. Indeed, there are lower bounds proving quadratic space is needed to store the flow on every edge, which has been used to prove lower bounds on streaming algorithms. However, we show that these lower bounds can be circumvented, opening up improvements for streaming and communication complexity. For a directed graph with integer capacities and costs bounded by W, we provide a Õ(n^1.5log (W/ε))-space Õ(√(n)log(W/ε))-pass streaming algorithm, which during the last pass returns the flow on each edge up to an additive error of ε. Crucially, the algorithm does not return the flow at the end of the last pass but returns the flow on an edge, as the edge is read in the stream. This allows us to circumvent existing Ω(n^2) space lower bounds. In the 2-party communication model, our algorithm implies Õ(n^1.5log^2 W) bits of communication.

This paper studies the computational challenges of large-scale attention-based models in artificial intelligence by introducing innovative sampling methods in the streaming setting. Inspired by the classical definition of the $\ell_2$ sampler and the recent progress of the attention scheme in Large Language Models (LLMs), we propose the definition of the attention sampler. Our approach significantly reduces the computational burden of traditional attention mechanisms. We demonstrate the effectiveness of the attention sampler from a theoretical perspective, including space and update time. Additionally, our framework exhibits scalability and broad applicability across various model architectures and domains.

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 ∈ℝ^n, outputs an index i with probability f_i^p/f_p^p+1/poly(n). In this paper, we resolve the problem of perfect L_p sampling for all p≥ 1 in the distributed monitoring model. Specifically, our algorithm runs in k^p-1·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+ε)-approximation to f_1^p+…+f_n^p. Our algorithm uses k^p-1/ε^2·polylog(n) bits of communication for all p≥ 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.

Given the query, key and value matrices $Q, K, V\in \mathbb{R}^{n\times d}$, the attention matrix is defined as $\mathrm{Att}(Q, K, V)=D^{-1}AV$ where $A=\exp(QK^\top/\sqrt{d})$ with $\exp(\cdot)$ applied entrywise, $D=\mathrm{diag}(A{\bf 1}_n)$. The attention matrix is the backbone of modern transformers and large language models, but explicitly forming the softmax matrix $D^{-1}A$ incurs $\Omega(n^2)$, motivating numerous approximation schemes that reduce runtime to $\widetilde O(nd)$ via sparsity or low-rank factorization. We propose a quantum data structure that approximates any row of $\mathrm{Att}(Q, K, V)$ using only row queries to $Q, K, V$. Our algorithm preprocesses these matrices in $\widetilde{O}\left( \epsilon^{-1} n^{0.5} \left( s_\lambda^{2.5} + s_\lambda^{1.5} d + \alpha^{0.5} d \right) \right)$ time, where $\epsilon$ is the target accuracy, $s_\lambda$ is the $\lambda$-statistical dimension of the exponential kernel defined by $Q$ and $K$, and $\alpha$ measures the row distortion of $V$ that is at most $d/{\rm srank}(V)$, the stable rank of $V$. Each row query can be answered in $\widetilde{O}(s_\lambda^2 + s_\lambda d)$ time. To our knowledge, this is the first quantum data structure that approximates rows of the attention matrix in sublinear time with respect to $n$. Our approach relies on a quantum Nystr{\"o}m approximation of the exponential kernel, quantum multivariate mean estimation for computing $D$, and quantum leverage score sampling for the multiplication with $V$.

In this paper, we show how to generalize the lazy update regime from dynamic matrix product [Cohen, Lee, Song STOC 2019, JACM 2021] to dynamic kronecker product. We provide an algorithm that uses n^ω( ⌈ k/2 ⌉, ⌊ k/2 ⌋, a )-a amortized update time and n^ω( ⌈(k-s)/2 ⌉, ⌊ (k-s)/2 ⌋,a ) worst case query time for dynamic kronecker product problem. Unless tensor MV conjecture is false, there is no algorithm that can use both n^ω( ⌈ k/2 ⌉, ⌊ k/2 ⌋, a )-a-Ω(1) amortized update time, and n^ω( ⌈(k-s)/2 ⌉, ⌊ (k-s)/2 ⌋,a )-Ω(1) worst case query time.

We analyze exact-metric, Metropolis-adjusted Dikin walks by keeping the proposal determinant and reverse quadratic form together. Their leading uncentered terms cancel in the complete logarithmic acceptance ratio, leaving centered fluctuations that can be controlled with second-order tools. For a polytope given by n inequalities and a convex L-Lipschitz potential, this yields warm-start mixing in O((d^2+dL^2R^2)log(w/δ)) steps for the regularized Lee–Sidford walk. For a spectrahedron with n× n blocks, the log-det walk mixes in O((ψ^⋆ nd+dL^2R^2)log(w/δ)) steps, where ψ^⋆ measures matrix leverage. The two analyses share an acceptance-to-mixing reduction. A proposal-comparison argument transfers the polytope bound to an appropriately padded O(1/d)-accurate metric computed from high-precision Lewis weights. For spectrahedra, given ^⋆, a direct-or-two-seed TensorSRHT construction gives an exact-arithmetic implementation with ψ^⋆ replaced by in the mixing bound.

Diffusion models have become the de facto standard for modern visual generation, including well-established frameworks such as latent diffusion and flow matching. Recently, modeling high-order dynamics has emerged as a promising frontier in generative modeling. Rather than only learning the first-order velocity field that transports random noise to a target data distribution, these approaches simultaneously learn higher-order derivatives, such as acceleration and jerk, yielding a diverse family of higher-order diffusion variants. To represent higher-order derivatives, naive approaches instantiate separate neural networks for each order, which scales the parameter space linearly with the derivative order. To overcome this computational bottleneck, we introduce cascading low-rank fitting, an ordinary differential equation inspired method that approximates successive derivatives by applying a shared base function augmented with sequentially accumulated low-rank components. Theoretically, we analyze the rank dynamics of these successive matrix differences. We prove that if the initial difference is linearly decomposable, the generic ranks of high-order derivatives are guaranteed to be monotonically non-increasing. Conversely, we demonstrate that without this structural assumption, the General Leibniz Rule allows ranks to strictly increase. Furthermore, we establish that under specific conditions, the sequence of derivative ranks can be designed to form any arbitrary permutation. Finally, we present a straightforward algorithm to efficiently compute the proposed cascading low-rank fitting.

We present a unified framework for quantum sensitivity sampling, extending the advantages of quantum computing to a broad class of classical approximation problems. Our unified framework provides a streamlined approach for constructing coresets and offers significant runtime improvements in applications such as clustering, regression, and low-rank approximation. Our contributions include: * k-median and k-means clustering: For n points in d-dimensional Euclidean space, we give an algorithm that constructs an ε-coreset in time O(n^0.5dk^2.5 poly(ε^-1)) for k-median and k-means clustering. Our approach achieves a better dependence on d and constructs smaller coresets that only consist of points in the dataset, compared to recent results of [Xue, Chen, Li and Jiang, ICML'23]. * ℓ_p regression: For ℓ_p regression problems, we construct an ε-coreset of size O_p(d^max{1, p/2}ε^-2) in time O_p(n^0.5d^max{0.5, p/4}+1(ε^-3+d^0.5)), improving upon the prior best quantum sampling approach of [Apers and Gribling, QIP'24] for all p∈ (0, 2)∪ (2, 22], including the widely studied least absolute deviation regression (ℓ_1 regression). * Low-rank approximation with Frobenius norm error: We introduce the first quantum sublinear-time algorithm for low-rank approximation that does not rely on data-dependent parameters, and runs in O(nd^0.5k^0.5ε^-1) time. Additionally, we present quantum sublinear algorithms for kernel low-rank approximation and tensor low-rank approximation, broadening the range of achievable sublinear time algorithms in randomized numerical linear algebra.

The application of transformer-based models on time series forecasting (TSF) tasks has long been popular to study. However, many of these works fail to beat the simple linear residual model, and the theoretical understanding of this issue is still limited. In this work, we propose the first theoretical explanation of the inefficiency of transformers on TSF tasks. We attribute the mechanism behind it to Asymmetric Learning in training attention networks. When the sign of the previous step is inconsistent with the sign of the current step in the next-step-prediction time series, attention fails to learn the residual features. This makes it difficult to generalize on out-of-distribution (OOD) data, especially on the sign-inconsistent next-step-prediction data, with the same representation pattern, whereas a linear residual network could easily accomplish it. We hope our theoretical insights provide important necessary conditions for designing the expressive and efficient transformer-based architecture for practitioners.

Understanding the expressive ability of a specific model is essential for grasping its capacity limitations. Recently, several studies have established circuit complexity bounds for Transformer architecture. Besides, the Visual AutoRegressive (VAR) model has risen to be a prominent method in the field of image generation, outperforming previous techniques, such as Diffusion Transformers, in generating high-quality images. We investigate the circuit complexity of the VAR model and establish a bound in this study. Our primary result demonstrates that the VAR model is equivalent to a simulation by a uniform 𝖳𝖢^0 threshold circuit with hidden dimension d ≤ O(n) and poly(n) precision. This is the first study to rigorously highlight the limitations in the expressive power of VAR models despite their impressive performance. We believe our findings will offer valuable insights into the inherent constraints of these models and guide the development of more efficient and expressive architectures in the future.

Recently, 1-bit Large Language Models (LLMs) have emerged, showcasing an impressive combination of efficiency and performance that rivals traditional LLMs. Research by Wang et al. [1], Ma et al. [2] indicates that the performance of these 1-bit LLMs progressively improves as the number of parameters increases, hinting at the potential existence of a Scaling Law in 1-bit Neural Networks. This paper presents the first theoretical result that rigorously establishes this scaling law for 1-bit models. Our analysis starts with initializing a 1-bit two-layer linear network. We prove that, despite the constraint of weights restricted to {-1, +1}, its training dynamics inevitably align with kernel behavior as the network width grows. This theoretical breakthrough guarantees convergence of the 1-bit model to an arbitrarily small loss as width increases. Furthermore, we introduce the concept of the generalization difference, defined as the gap between the outputs of 1-bit networks and their full-precision counterparts, and demonstrate that this difference maintains a negligible level under the over-parameterization setting. Building on the work of Kaplan et al. [3], we examine how the training loss scales as a power-law function of the model size, dataset size, and computational resources utilized for training. Our findings underscore the promising potential of scaling 1-bit neural networks, suggesting that int1 could become the standard in future neural network precision.

For choosing initial cluster centers, $k$-means++ is an important algorithm. Given $n$ data points in $\mathbb{R}^{d}$, the current state-of-the-art algorithm runs in $\widetilde{O}(k)$ iterations, and each iteration takes $\widetilde{O}(n d k)$ time. The overall running time is thus $\widetilde{O}\left(n d k^{2}\right)$. In this work, we present a new algorithm that can solve the $k$ means++ problem with nearly optimal running time. We propose a new algorithm that only takes $\widetilde{O}\left(n d+n k^{2}\right)$ time in total.

Thanks to recent advancements in scalable deep architectures and large-scale pretraining, text-to-video generation has achieved unprecedented capabilities in producing high-fidelity, instruction-following content across a wide range of styles, enabling applications in advertising, entertainment, and education. However, these models' ability to render precise on-screen text, such as captions or mathematical formulas, remains largely untested, posing significant challenges for applications requiring exact textual accuracy. In this work, we introduce T2VTextBench, the first human-evaluation benchmark dedicated to evaluating on-screen text fidelity and temporal consistency in text-to-video models. Our suite of prompts integrates complex text strings with dynamic scene changes, testing each model's ability to maintain detailed instructions across frames. We evaluate ten state-of-the-art systems, ranging from open-source solutions to commercial offerings, and find that most struggle to generate legible, consistent text. These results highlight a critical gap in current video generators and provide a clear direction for future research aimed at enhancing textual manipulation in video synthesis.