arXiv subjects
Publications and source records attributed to Zhao Song.
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 $\ell_2$ 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 $\ell_\infty$ norm. In particular, we want to find a vector $x'\in \mathbb{R}^d$ such that $\|x'-x^*\|_\infty\leq \frac{\epsilon}{\sqrt d}\cdot \|Ax^\star-b\|_2\cdot \|A^\dagger\|_{\rm op}$. Price, Song and Woodruff initiated the study of this problem and showed that the subsampled randomized Hadamard transform (SRHT) with $O(\epsilon^{-2} d^{1+\Theta(\sqrt{\log\log n/\log d})})$ rows achieves this guarantee. A subsequent work of Song, Ye, Yin and Zhang claimed to improve the row count to $O(\epsilon^{-2}d\log^3 n)$. 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 $\ell_\infty$ guarantee with $m=O(\epsilon^{-2}d\log d)$ rows, uses one Hadamard pass with a padded internal dimension $N=\widetilde{O}(n+\epsilon^{-2}d^3)$, and is efficient to apply: the sketched pair $(SA, Sb)$ can be computed in $O(Nd\log N)=\widetilde{O}(nd+\epsilon^{-2}d^4)$ 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 $\widetilde O((d^{2}+dL^{2}R^{2})\log(w/\delta))$ steps for the regularized Lee--Sidford walk. For a spectrahedron with $n\times n$ blocks, the log-det walk mixes in $\widetilde O((\psi^\star nd+dL^{2}R^{2})\log(w/\delta))$ steps, where $\psi^\star$ 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 $\widehat\psi\ge\psi^\star$, a direct-or-two-seed TensorSRHT construction gives an exact-arithmetic implementation with $\psi^\star$ replaced by $\widehat\psi$ in the mixing bound.
For $n$ unit vectors $x_1,\ldots,x_n \in \mathbb{R}^d$, 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 $ \Delta_\pm := \min_{i \neq j} \min\{ \|x_i-x_j\|_2, \|x_i+x_j\|_2 \} $ for their projective separation, we prove the universal dimension-free lower bound $ \lambda_{\min}(H) = \Omega( \Delta_\pm/\sqrt{\log n} ) $. Conversely, we construct worst-case families satisfying the matching upper bound $ \lambda_{\min}(H) = O( \Delta_\pm/\sqrt{\log n} ) $, showing that this rate is tight up to universal constants.
For a Boolean communication matrix $M$, let $D(M)$ denote its deterministic communication complexity and let $r(M):={\mathrm{rank}}_{\mathbb{R}}(M)$. The log-rank conjecture asks whether $D(M)$ is polynomial in $\log r(M)$. The best known general upper bound, due to Sudakov and Tomon'25, is $D(M)=O(\sqrt{r(M)})$. On the lower-bound side, G{\"o}{\"o}s, Pitassi, and Watson'18 constructed explicit matrices satisfying $D(M)=\Omega((\log r(M))^2/(\log\log r(M))^2)$. We improve the lower bound to $D(M)=\Omega((\log r(M))^2/\log\log r(M))$. Our construction revisits their pointer function over its original non-Boolean alphabet and lifts it with an alphabet-valued Index gadget, via the multicolor simulation theorem stated by Roughgarden and Weinstein'16. Compared with the quantitatively explicit GPW bound, the alphabet-preserving lift removes one factor of $\log\log r$. We also give a self-contained proof of the multicolor simulation theorem in the parameter regime required by the construction.
We investigate how peer and institutional incentives jointly shape the evolution of cooperation, social welfare, and enforcement efficiency in social dilemmas. In a Prisoners Dilemma with four strategies, unconditional cooperators (C), defectors (D), social punishers (SP), and social rewarders (SR), we allow decentralised peer incentives and centralised institutional incentives to act simultaneously, with the institution able to reward or punish any subset of strategies. In infinite well-mixed populations, we analyse the resulting four-strategy replicator dynamics, and in structured populations we use agent-based simulations on square lattices to study spatial effects and network reciprocity. Intervention schemes are evaluated by equilibrium states and evolutionary flow for infinite well-mixed populations, by cooperation levels and social welfare for structured populations, defined as aggregate population payoff net of institutional cost. We find that peer punishment most strongly promotes cooperation, whereas peer reward is more beneficial for social welfare. Institutionally rewarding peer incentive strategies substantially improves both cooperation and welfare, while subsidising unconditional cooperators has little impact. Under institutional punishment, directly penalising defectors is the only consistently effective policy; punishing peer incentive strategies dismantles decentralised incentives, reduces cooperation, and harms social welfare, showing that maximising cooperation does not necessarily optimise overall societal benefit. Our findings provide design principles for institutions seeking to balance cooperation promotion with welfare maximisation.
We prove an improved restricted isometry bound for Gaussian partial circulant matrices with arbitrary prescribed sampling sets. There is a universal constant $C>0$ such that the following holds. Let $1\leq K\leq m\leq N$ be positive integers, let $\Omega\subset\mathbb Z_N$ be any fixed set with $|\Omega|=m$, and let $g\sim\mathcal N(0,I_N)$. For every $\delta,\eta\in(0,1)$, the normalized partial circulant matrix generated by $g$ has the RIP of order $K$ with constant at most $\delta$, with probability at least $1-\eta$ over the draw of $g$, provided \[ m\geq C\delta^{-2}K \max\{\log^2(eK)\log(2N)\log(em),\log(2/\eta)\}. \] The proof refines the Maurey entropy step in the chaos-process argument by combining a noncommutative Khintchine inequality with a Schatten moment estimate controlled by $m$, replacing one factor $\log(2N)$ in the Krahmer--Mendelson--Rauhut bound by $\log(em)$.
AI-assisted cybersecurity systems are characterised by continuous adaptation between attackers and defenders, making evolutionary game theory a natural framework for studying their long-term behaviour. However, existing evolutionary cybersecurity models have primarily focused on homogeneous interactions, providing limited understanding of how population structure influences cyber attack-defence dynamics. In this paper, we develop a mixed-role evolutionary game in which adaptive cyber agents can exhibit both offensive and defensive behaviours, and investigate its dynamics in well-mixed and structured populations. The proposed framework combines stochastic evolutionary analysis with large-scale agent-based simulations to examine how interaction structure shapes long-run strategic behaviour. Our results reveal a fundamental difference between global and local interactions. While well-mixed populations exhibit broad coexistence between attacking and defensive strategies, structured populations generate well-defined evolutionary regimes in which secure defensive behaviour becomes dominant over a much larger region of the parameter space. Spatial analysis further shows that neighbouring defenders naturally form resilient clusters that suppress persistent attacks through network reciprocity. These findings demonstrate that interaction structure is a fundamental determinant of AI-assisted cybersecurity evolution and suggest that organising defensive agents through local networked interactions can substantially improve long-term cyber resilience without requiring additional defensive incentives.
Institutional incentives are widely used to promote cooperation among autonomous, self-regarding agents, from human societies to multi-agent and AI systems. Existing work typically treats incentive design as a bi-objective problem: minimise institutional cost while achieving a high long-run frequency of cooperation. Whether such schemes also maximise social welfare - total population payoff net of institutional expenditure - has remained largely unexplored. We develop a welfare-centric framework for institutional incentives in finite, well-mixed populations playing a social dilemma (Donation Game and Public Goods Game), considering both rewards for cooperators and punishments for defectors. For each mechanism, we derive explicit expressions for expected social welfare and characterise how it depends on incentive efficiency and selection intensity. Analytically, we identify parameter regimes where social welfare has a single optimal incentive level and regimes with qualitative phase transitions, in which welfare becomes non-monotonic with multiple local optima. We prove that any welfare-maximising incentive is either zero or concentrated around a simple closed-form target, and we provide an efficient algorithm to compute these optima. Comparing reward and punishment, we further derive close-formed conditions under which reward outperform punishment in terms of social welfare for any given budget. Overall, our results reveal a systematic gap between incentives optimised for cost or cooperation frequency and those that maximise welfare.
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 $\Omega(n^2)$ 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 $\tilde O(n^{1.5}\log (W/\epsilon))$-space $\tilde O(\sqrt{n} \log(W/\epsilon))$-pass streaming algorithm, which during the last pass returns the flow on each edge up to an additive error of $\epsilon$. 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 $\Omega(n^2)$ space lower bounds. In the 2-party communication model, our algorithm implies $\tilde O(n^{1.5}\log^2 W)$ bits of communication.
The growing integration of AI into cybersecurity is reshaping the balance between attackers and defenders. When access to advanced AI-enabled defence tools is uneven, resource-limited defenders may be unable to adopt effective protection, creating persistent system vulnerabilities. We study the impact of differential AI access using an evolutionary game-theoretic model in a finite population. We first show that when high-capability defence is costly, the population is driven toward low-cost, weak-defence behaviour, sustaining attacks and weakening long-run security. To address this problem, we introduce differential access to AI defence tools by allowing defenders to choose between low- and high-capability protection based on their resources. We then examine the role of a small group of committed defenders who always adopt strong defence and influence others through social learning. Although commitment increases the prevalence of strong defence, it alone cannot stabilise secure outcomes due to high defence costs. We therefore incorporate a targeted subsidy to remove the cost disadvantage from committed defenders. Our analysis shows that subsidised commitment significantly increases strong defence adoption, suppresses successful attacks, and improves overall system resilience. Simulations across a broad parameter space confirm that subsidies consistently outperform commitment alone. In addition, social-welfare analysis shows improved defender outcomes while keeping attacker gains low. These findings suggest that targeted support for key defenders can be an effective mechanism for stabilising cybersecurity in AI-driven environments and provide a theoretical bridge between cybersecurity policy, AI governance, and strategic allocation of defensive AI capabilities.
The rapid uptake of generative artificial intelligence (AI) in higher education is reshaping assessment practices and intensifying concerns around academic integrity, fairness, and learning quality. While institutional responses increasingly emphasise policy guidance and ethical principles, there remains limited formal understanding of how collective norms of responsible or opportunistic AI use emerge and stabilise within student cohorts. This paper reframes student AI use in assessment as a coordination problem shaped by peer expectations and assessment design rather than individual compliance alone. We develop a coordination-based evolutionary game-theoretic framework that captures learning value, effort, perceived fairness, and transparency, with institutional AI governance modelled implicitly through reflective assessment incentives. We use analytical results and finite-population simulations to reveal threshold-driven behavioural transitions in student AI use: small, well-calibrated changes in reflective assessment incentives can trigger rapid shifts towards responsible, learning-oriented AI-use norms, whereas weak or misaligned incentives allow opportunistic practices to persist. These non-linear dynamics explain why policy statements alone often fail to change behaviour, while modest assessment redesigns can have disproportionate effects. By providing a mechanism-level account of how assessment structures shape collective AI-use practices, this work offers higher education institutions an analytically grounded tool for Future Facing Learning, supporting proportionate, pedagogy-led AI governance without reliance on surveillance or punitive enforcement.
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.
Content creation for major video-sharing platforms demands significant manual labor, particularly for long-form documentary videos spanning one to two hours. In this work, we introduce Sima 1.0, a multi-agent system designed to optimize the weekly production pipeline for high-quality video generation. The framework partitions the production process into an 11-step pipeline distributed across a hybrid workforce. While foundational creative tasks and physical recording are executed by a human operator, time-intensive editing, caption refinement, and supplementary asset integration are delegated to specialized junior and senior-level AI agents. By systematizing tasks from script annotation to final asset exportation, Sima 1.0 significantly reduces the production workload, empowering a single creator to efficiently sustain a rigorous weekly publishing schedule.
As the capabilities and adoption of Artificial Intelligence (AI) systems grow, trust in these AI systems is an increasingly urgent concern. Much research has focused on models of AI governance and has primarily examined incentives for safe development and effective regulation. Hence they typically represented users trust as a one-shot adoption choice rather than as a dynamic, evolving process shaped by repeated interactions. We instead model trust as the dynamic choice of reduced monitoring in a repeated, asymmetric interaction between users and AI developers, where checking developers' behaviour is costly. Using evolutionary game theory, we study how users' strategies of trust and developers' strategies of providing safe (compliant) or unsafe (non-compliant) AI co-evolve under different levels of monitoring cost and institutional regimes. We conduct the analysis on both imitation-based and learning-based perspectives, with the stochastic finite-population dynamics, the infinite-population replicator analysis and the reinforcement learning analysis. We find three robust long-run regimes: no adoption by users while developers provide unsafe AI, unsafe but widely adopted systems, and safe systems that are widely adopted. Only the last is desirable, and it arises when penalties for unsafe behaviour exceed the extra cost of safety and users can still afford to monitor at least occasionally. Our results formally support governance proposals that emphasise transparency, low-cost monitoring, and meaningful sanctions, and they show that neither regulation alone nor blind user trust is sufficient to prevent the drift towards unsafe or low-adoption outcomes.
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 $n^{\omega( \lceil k/2 \rceil, \lfloor k/2 \rfloor, a )-a}$ amortized update time and $ n^{\omega( \lceil(k-s)/2 \rceil, \lfloor (k-s)/2 \rfloor,a )}$ worst case query time for dynamic kronecker product problem. Unless tensor MV conjecture is false, there is no algorithm that can use both $n^{\omega( \lceil k/2 \rceil, \lfloor k/2 \rfloor, a )-a-\Omega(1)}$ amortized update time, and $ n^{\omega( \lceil(k-s)/2 \rceil, \lfloor (k-s)/2 \rfloor,a )-\Omega(1)}$ worst case query time.
Designing protocols enhancing cooperation for multi-agent systems remains a grand challenge. Cheap talk, defined as costless, non-binding communication before formal action, serves as a pivotal solution. However, existing theoretical frameworks often exclude random exploration, or noise, for analytical tractability, leaving its functional impact on system performance largely unexplored. To bridge this gap, we propose a two-stage evolutionary game-theoretical model, integrating signalling with a donation game, with exploration explicitly incorporated into the decision-making. Our agent-based simulations across topologies reveal a universal optimal exploration rate that maximises system-wide cooperation. Mechanistically, moderate exploration undermines the stability of defection and catalyses the self-organised cooperative alliances, facilitating their cyclic success. Moreover, the cooperation peak is enabled by the delicate balance between oscillation period and amplification. Our findings suggest that rather than pursuing deterministic rigidity, embracing strategic exploration, as a form of engineered randomness, is essential to sustain cooperation and realise optimal performance in communication-based intelligent systems.
Fairness in hybrid societies hinges on a simple choice: should AI be a generous host or a strict gatekeeper? Moving beyond symmetric models, we show that asymmetric social structures--like those in hiring, regulation, and negotiation--AI that guards fairness outperforms AI that gifts it. We bridge this gap with a bipartite hybrid population model of the Ultimatum Game, separating humans and AI into distinct proposer and receiver groups. We first introduce Samaritan AI agents, which act as either unconditional fair proposers or strict receivers. Our results reveal a striking asymmetry: Samaritan AI receivers drive population-wide fairness far more effectively than Samaritan AI proposers. To overcome the limitations of the Samaritan AI proposer, we design the Discriminatory AI proposer, which predicts co-players' expectations and only offers fair portions to those with high acceptance thresholds. Our results demonstrate that this Discriminatory AI outperforms both types of Samaritan AI, especially in strong selection scenarios. It not only sustains fairness across both populations but also significantly lowers the critical mass of agents required to reach an equitable steady state. By transitioning from unconditional modelling to strategic enforcement, our work provides a pivotal framework for deploying asymmetric AIs in the increasingly hybrid society.