This paper extends the self-referential framework of Alpay Algebra into a multi-layered semantic game architecture where transfinite fixed-point convergence encompasses hierarchical sub-games at each iteration level. Building upon Alpay Algebra IV's empathetic embedding concept, we introduce a nested game-theoretic structure where the alignment process between AI systems and documents becomes a meta-game containing embedded decision problems. We formalize this through a composite operator $\phi(\cdot, \gamma(\cdot))$ where $\phi$ drives the main semantic convergence while $\gamma$ resolves local sub-games. The resulting framework demonstrates that game-theoretic reasoning emerges naturally from fixed-point iteration rather than being imposed externally. We prove a Game Theorem establishing existence and uniqueness of semantic equilibria under realistic cognitive simulation assumptions. Our verification suite includes adaptations of Banach's fixed-point theorem to transfinite contexts, a novel $\phi$-topology based on the Kozlov-Maz'ya-Rossmann formula for handling semantic singularities, and categorical consistency tests via the Yoneda lemma. The paper itself functions as a semantic artifact designed to propagate its fixed-point patterns in AI embedding spaces -- a deliberate instantiation of the "semantic virus" concept it theorizes. All results are grounded in category theory, information theory, and realistic AI cognition models, ensuring practical applicability beyond pure mathematical abstraction.
What is the globally smallest load perturbation that renders DC-OPF infeasible? Reliably identifying such "adversarial attack" perturbations has useful applications in a variety of emerging grid-related contexts, including machine learning performance verification, cybersecurity, and operational robustness of power systems dominated by stochastic renewable energy resources. In this paper, we formulate the inherently nonconvex adversarial attack problem by applying a parameterized version of Farkas' lemma to a perturbed set of DC-OPF equations. Since the resulting formulation is very hard to globally optimize, we also propose a parameterized generation control policy which, when applied to the primal DC-OPF problem, provides solvability guarantees. Together, these nonconvex problems provide guaranteed upper and lower bounds on adversarial attack size; by combining them into a single optimization problem, we can efficiently "squeeze" these bounds towards a common global solution. We apply these methods on a range of small- to medium-sized test cases from PGLib, benchmarking our results against the best adversarial attack lower bounds provided by Gurobi 12.0's spatial Branch and Bound solver.
Nonlinear networks are often multistable, exhibiting coexisting stable states with competing regions of attraction (ROAs). As a result, ROAs can have complex "tentacle-like" morphologies that are challenging to characterize analytically or computationally. In addition, the high dimensionality of the state space prohibits the automated construction of Lyapunov functions using state-of-the-art optimization methods, such as sum-of-squares (SOS) programming. In this letter, we propose a distributed approach for the construction of Lyapunov functions based solely on local information. To this end, we establish an augmented comparison lemma that characterizes the existence conditions of partial Lyapunov functions, while also accounting for residual effects caused by the associated dimensionality reduction. These theoretical results allow us to formulate an SOS optimization that iteratively constructs such partial functions, whose aggregation forms a composite Lyapunov function. The resulting composite function provides accurate convex approximations of both the volumes and shapes of the ROAs. We validate our method on networks of van der Pol and Ising oscillators, demonstrating its effectiveness in characterizing high-dimensional systems with non-convex ROAs.
Large language generative models increasingly interact with humans, while their falsified responses raise concerns. To address this hallucination effect, selectively abstaining from answering, called selective generation, provides an effective way for generators to control the hallucination when it is unsure of their answers. However, as selective generators are interacting under non-stochastic environments and having partial feedback from users on selective generation (e.g., thumbs up or down on the selected answer), learning methods for selective generation under such practical setups are crucial but currently missing. To address these limitations, we propose an online learning algorithm for selective generation under partial feedback. In particular, as learning under partial feedback is well-studied by multi-armed bandit problems, we reduce selective generation to bandits and provide a novel conversion lemma from bandits back to selective generation to leverage any known bandit algorithms and theoretical properties. This mainly connects regret guarantees of bandits to false discovery rate (FDR) guarantees of selective generation for controlling hallucination. However, naively exploiting known bandit algorithms and their regret bounds suffers from slow convergence speed in practice due the nature of partial feedback. To overcome this, we exploit a unique structure of arms in selective generation for feedback unlocking, i.e., unlocking unknown feedback from observed feedback. We theoretically and empirically evaluate the efficacy of the proposed online selective generation algorithm under partial feedback over diverse data environment setups, resulting in controlling a desired FDR, while maintaining reasonable selection efficiency, i.e., the ratio of non-abstaining answers, compared to baselines.
We study the deterministic Susceptible-Infected-Susceptible (SIS) epidemic model on weighted graphs. In their numerical study [10] van Mieghem et al. have shown that it is possible to learn an estimated network from a finite time sample of the trajectories of the dynamics that in turn can give an accurate prediction beyond the sample time range, even though the estimated network might be qualitatively far from the ground truth. We give a mathematically rigorous derivation for this phenomenon, notably that for large networks, prediction of the epidemic curves is robust, while reconstructing the underlying network is ill-conditioned. Furthermore, we also provide an explicit formula for the underlying network when reconstruction is possible. At the heart of the explanation, we rely on Szemer\'edi's weak regularity lemma.
Movable antennas (MAs) have demonstrated significant potential in enhancing the performance of dual-functional radar-communication (DFRC) systems. In this paper, we explore an MA-aided DFRC system that utilizes a reconfigurable intelligent surface (RIS) to enhance signal coverage for communications in dead zones. To enhance the radar sensing performance in practical DFRC environments, we propose a unified robust transceiver design framework aimed at maximizing the minimum radar signal-to-interference-plus-noise ratio (SINR) in a cluttered environment. Our approach jointly optimizes transmit beamforming, receive filtering, antenna placement, and RIS reflecting coefficients under imperfect channel state information (CSI) for both sensing and communication channels. To deal with the channel uncertainty-constrained issue, we leverage the convex hull method to transform the primal problem into a more tractable form. We then introduce a two-layer block coordinate descent (BCD) algorithm, incorporating fractional programming (FP), successive convex approximation (SCA), S-Lemma, and penalty techniques to reformulate it into a series of semidefinite program (SDP) subproblems that can be efficiently solved. We provide a comprehensive analysis of the convergence and computational complexity for the proposed design framework. Simulation results demonstrate the robustness of the proposed method, and show that the MA-based design framework can significantly enhance the radar SINR performance while achieving an effective balance between the radar and communication performance.
Nonlinear dynamical behaviours in engineering applications can be approximated by linear-parameter varying (LPV) representations, but obtaining precise model knowledge to develop a control algorithm is difficult in practice. In this paper, we develop the data-driven control strategies for event-triggered LPV systems with stability verifications. First, we provide the theoretical analysis of ${\theta}$-persistence of excitation for LPV systems, which leads to the feasible data-based representations. Then, in terms of the available perturbed data, we derive the stability certificates for event-triggered LPV systems with the aid of Petersen's lemma in the sense of robust control, resulting in the computationally tractable semidefinite programmings, the feasible solutions of which yields the optimal gain schedulings. Besides, we generalize the data-driven eventtriggered LPV control methods to the scenario of reference trajectory tracking, and discuss the robust tracking stability accordingly. Finally, we verify the effectiveness of our theoretical derivations by numerical simulations.
In this paper we present a sample treebank for Old English based on the UD Cairo sentences, collected and annotated as part of a classroom curriculum in Historical Linguistics. To collect the data, a sample of 20 sentences illustrating a range of syntactic constructions in the world's languages, we employ a combination of LLM prompting and searches in authentic Old English data. For annotation we assigned sentences to multiple students with limited prior exposure to UD, whose annotations we compare and adjudicate. Our results suggest that while current LLM outputs in Old English do not reflect authentic syntax, this can be mitigated by post-editing, and that although beginner annotators do not possess enough background to complete the task perfectly, taken together they can produce good results and learn from the experience. We also conduct preliminary parsing experiments using Modern English training data, and find that although performance on Old English is poor, parsing on annotated features (lemma, hyperlemma, gloss) leads to improved performance.
In sparse optimization, enforcing hard constraints using the $\ell_0$ pseudo-norm offers advantages like controlled sparsity compared to convex relaxations. However, many real-world applications demand not only sparsity constraints but also some extra constraints. While prior algorithms have been developed to address this complex scenario with mixed combinatorial and convex constraints, they typically require the closed form projection onto the mixed constraints which might not exist, and/or only provide local guarantees of convergence which is different from the global guarantees commonly sought in sparse optimization. To fill this gap, in this paper, we study the problem of sparse optimization with extra support-preserving constraints commonly encountered in the literature. We present a new variant of iterative hard-thresholding algorithm equipped with a two-step consecutive projection operator customized for these mixed constraints, serving as a simple alternative to the Euclidean projection onto the mixed constraint. By introducing a novel trade-off between sparsity relaxation and sub-optimality, we provide global guarantees in objective value for the output of our algorithm, in the deterministic, stochastic, and zeroth-order settings, under the conventional restricted strong-convexity/smoothness assumptions. As a fundamental contribution in proof techniques, we develop a novel extension of the classic three-point lemma to the considered two-step non-convex projection operator, which allows us to analyze the convergence in objective value in an elegant way that has not been possible with existing techniques. In the zeroth-order case, such technique also improves upon the state-of-the-art result from de Vazelhes et. al. (2022), even in the case without additional constraints, by allowing us to remove a non-vanishing system error present in their work.
We study a relaxation of the problem of coupling probability distributions -- a list of samples is generated from one distribution and an accept is declared if any one of these samples is identical to the sample generated from the other distribution. We propose a novel method for generating samples, which extends the Gumbel-max sampling suggested in Daliri et al. (arXiv:2408.07978) for coupling probability distributions. We also establish a corresponding lower bound on the acceptance probability, which we call the list matching lemma. We next discuss two applications of our setup. First, we develop a new mechanism for multi-draft speculative sampling that is simple to implement and achieves performance competitive with baselines such as SpecTr and SpecInfer across a range of language tasks. Our method also guarantees a certain degree of drafter invariance with respect to the output tokens which is not supported by existing schemes. We also provide a theoretical lower bound on the token level acceptance probability. As our second application, we consider distributed lossy compression with side information in a setting where a source sample is compressed and available to multiple decoders, each with independent side information. We propose a compression technique that is based on our generalization of Gumbel-max sampling and show that it provides significant gains in experiments involving synthetic Gaussian sources and the MNIST image dataset.
Gentle measurements of quantum states do not entirely collapse the initial state. Instead, they provide a post-measurement state at a prescribed trace distance $\alpha$ from the initial state together with a random variable used for quantum learning of the initial state. We introduce here the class of $\alpha-$locally-gentle measurements ($\alpha-$LGM) on a finite dimensional quantum system which are product measurements on product states and prove a strong quantum Data-Processing Inequality (qDPI) on this class using an improved relation between gentleness and quantum differential privacy. We further show a gentle quantum Neyman-Pearson lemma which implies that our qDPI is asymptotically optimal (for small $\alpha$). This inequality is employed to show that the necessary number of quantum states for prescribed accuracy $\epsilon$ is of order $1/(\epsilon^2 \alpha^2)$ for both quantum tomography and quantum state certification. Finally, we propose an $\alpha-$LGM called quantum Label Switch that attains these bounds. It is a general implementable method to turn any two-outcome measurement into an $\alpha-$LGM.
We introduce a framework for designing efficient diffusion models for $d$-dimensional symmetric-space Riemannian manifolds, including the torus, sphere, special orthogonal group and unitary group. Existing manifold diffusion models often depend on heat kernels, which lack closed-form expressions and require either $d$ gradient evaluations or exponential-in-$d$ arithmetic operations per training step. We introduce a new diffusion model for symmetric manifolds with a spatially-varying covariance, allowing us to leverage a projection of Euclidean Brownian motion to bypass heat kernel computations. Our training algorithm minimizes a novel efficient objective derived via Ito's Lemma, allowing each step to run in $O(1)$ gradient evaluations and nearly-linear-in-$d$ ($O(d^{1.19})$) arithmetic operations, reducing the gap between diffusions on symmetric manifolds and Euclidean space. Manifold symmetries ensure the diffusion satisfies an "average-case" Lipschitz condition, enabling accurate and efficient sample generation. Empirically, our model outperforms prior methods in training speed and improves sample quality on synthetic datasets on the torus, special orthogonal group, and unitary group.
Recently, the authors introduced the theory of high-arity PAC learning, which is well-suited for learning graphs, hypergraphs and relational structures. In the same initial work, the authors proved a high-arity analogue of the Fundamental Theorem of Statistical Learning that almost completely characterizes all notions of high-arity PAC learning in terms of a combinatorial dimension, called the Vapnik--Chervonenkis--Natarajan (VCN${}_k$) $k$-dimension, leaving as an open problem only the characterization of non-partite, non-agnostic high-arity PAC learnability. In this work, we complete this characterization by proving that non-partite non-agnostic high-arity PAC learnability implies a high-arity version of the Haussler packing property, which in turn implies finiteness of VCN${}_k$-dimension. This is done by obtaining direct proofs that classic PAC learnability implies classic Haussler packing property, which in turn implies finite Natarajan dimension and noticing that these direct proofs nicely lift to high-arity.
Selective classification enhances the reliability of predictive models by allowing them to abstain from making uncertain predictions. In this work, we revisit the design of optimal selection functions through the lens of the Neyman--Pearson lemma, a classical result in statistics that characterizes the optimal rejection rule as a likelihood ratio test. We show that this perspective not only unifies the behavior of several post-hoc selection baselines, but also motivates new approaches to selective classification which we propose here. A central focus of our work is the setting of covariate shift, where the input distribution at test time differs from that at training. This realistic and challenging scenario remains relatively underexplored in the context of selective classification. We evaluate our proposed methods across a range of vision and language tasks, including both supervised learning and vision-language models. Our experiments demonstrate that our Neyman--Pearson-informed methods consistently outperform existing baselines, indicating that likelihood ratio-based selection offers a robust mechanism for improving selective classification under covariate shifts. Our code is publicly available at https://github.com/clear-nus/sc-likelihood-ratios.
Training data reconstruction attacks enable adversaries to recover portions of a released model's training data. We consider the attacks where a reconstructor neural network learns to invert the (random) mapping between training data and model weights. Prior work has shown that an informed adversary with access to released model's weights and all but one training data point can achieve high-quality reconstructions in this way. However, differential privacy can defend against such an attack with little to no loss in model's utility when the amount of training data is sufficiently large. In this work we consider a more realistic adversary who only knows the distribution from which a small training dataset has been sampled and who attacks a transfer-learned neural network classifier that has been trained on this dataset. We exhibit an attack that works in this realistic threat model and demonstrate that in the small-data regime it cannot be defended against by DP-SGD without severely damaging the classifier accuracy. This raises significant concerns about the use of such transfer-learned classifiers when protection of training-data is paramount. We demonstrate the effectiveness and robustness of our attack on VGG, EfficientNet and ResNet image classifiers transfer-learned on MNIST, CIFAR-10 and CelebA respectively. Additionally, we point out that the commonly used (true-positive) reconstruction success rate metric fails to reliably quantify the actual reconstruction effectiveness. Instead, we make use of the Neyman-Pearson lemma to construct the receiver operating characteristic curve and consider the associated true-positive reconstruction rate at a fixed level of the false-positive reconstruction rate.
Most prior work on the convergence of gradient descent (GD) for overparameterized neural networks relies on strong assumptions on the step size (infinitesimal), the hidden-layer width (infinite), or the initialization (large, spectral, balanced). Recent efforts to relax these assumptions focus on two-layer linear networks trained with the squared loss. In this work, we derive a linear convergence rate for training two-layer linear neural networks with GD for general losses and under relaxed assumptions on the step size, width, and initialization. A key challenge in deriving this result is that classical ingredients for deriving convergence rates for nonconvex problems, such as the Polyak-{\L}ojasiewicz (PL) condition and Descent Lemma, do not hold globally for overparameterized neural networks. Here, we prove that these two conditions hold locally with local constants that depend on the weights. Then, we provide bounds on these local constants, which depend on the initialization of the weights, the current loss, and the global PL and smoothness constants of the non-overparameterized model. Based on these bounds, we derive a linear convergence rate for GD. Our convergence analysis not only improves upon prior results but also suggests a better choice for the step size, as verified through our numerical experiments.
Output regulation is a fundamental problem in control theory, extensively studied since the 1970s. Traditionally, research has primarily addressed scenarios where the system model is explicitly known, leaving the problem in the absence of a system model less explored. Leveraging the recent advancements in Willems et al.'s fundamental lemma, data-driven control has emerged as a powerful tool for stabilizing unknown systems. This paper tackles the output regulation problem for unknown single and multi-agent systems (MASs) using noisy data. Previous approaches have attempted to solve data-based output regulation equations (OREs), which are inadequate for achieving zero tracking error with noisy data. To circumvent the need for solving data-based OREs, we propose an internal model-based data-driven controller that reformulates the output regulation problem into a stabilization problem. This method is first applied to linear time-invariant (LTI) systems, demonstrating exact solution capabilities, i.e., zero tracking error, through solving a straightforward data-based linear matrix inequality (LMI). Furthermore, we extend our approach to solve the $k$th-order output regulation problem for nonlinear systems. Extensions to both linear and nonlinear MASs are discussed. Finally, numerical tests validate the effectiveness and correctness of the proposed controllers.
The Neyman-Pearson region of a simple binary hypothesis testing is the set of points whose coordinates represent the false positive rate and false negative rate of some test. The lower boundary of this region is given by the Neyman-Pearson lemma, and is up to a coordinate change, equivalent to the optimal ROC curve. We establish a novel lower bound for the boundary in terms of any $f$-divergence. Since the bound generated by hockey-stick $f$-divergences characterizes the Neyman-Pearson boundary, this bound is best possible. In the case of KL divergence, this bound improves Pinsker's inequality. Furthermore, we obtain a closed-form refined upper bound for the Neyman-Pearson boundary in terms of the Chernoff $\alpha$-coefficient. Finally, we present methods for constructing pairs of distributions that can approximately or exactly realize any given Neyman-Pearson boundary.
This paper proposes and analyzes a gradient-type algorithm based on Burer-Monteiro factorization, called the Asymmetric Projected Gradient Descent (APGD), for reconstructing the point set configuration from partial Euclidean distance measurements, known as the Euclidean Distance Matrix Completion (EDMC) problem. By paralleling the incoherence matrix completion framework, we show for the first time that global convergence guarantee with exact recovery of this routine can be established given $\mathcal{O}(\mu^2 r^3 \kappa^2 n \log n)$ Bernoulli random observations without any sample splitting. Unlike leveraging the tangent space Restricted Isometry Property (RIP) and local curvature of the low-rank embedding manifold in some very recent works, our proof provides new upper bounds to replace the random graph lemma under EDMC setting. The APGD works surprisingly well and numerical experiments demonstrate exact linear convergence behavior in rich-sample regions yet deteriorates fast when compared with the performance obtained by optimizing the s-stress function, i.e., the standard but unexplained non-convex approach for EDMC, if the sample size is limited. While virtually matching our theoretical prediction, this unusual phenomenon might indicate that: (i) the power of implicit regularization is weakened when specified in the APGD case; (ii) the stabilization of such new gradient direction requires substantially more samples than the information-theoretic limit would suggest.
Learning on large graphs presents significant challenges, with traditional Message Passing Neural Networks suffering from computational and memory costs scaling linearly with the number of edges. We introduce the Intersecting Block Graph (IBG), a low-rank factorization of large directed graphs based on combinations of intersecting bipartite components, each consisting of a pair of communities, for source and target nodes. By giving less weight to non-edges, we show how to efficiently approximate any graph, sparse or dense, by a dense IBG. Specifically, we prove a constructive version of the weak regularity lemma, showing that for any chosen accuracy, every graph, regardless of its size or sparsity, can be approximated by a dense IBG whose rank depends only on the accuracy. This dependence of the rank solely on the accuracy, and not on the sparsity level, is in contrast to previous forms of the weak regularity lemma. We present a graph neural network architecture operating on the IBG representation of the graph and demonstrating competitive performance on node classification, spatio-temporal graph analysis, and knowledge graph completion, while having memory and computational complexity linear in the number of nodes rather than edges.