arXiv subjects
Publications and source records attributed to Zhiqiang Xu.
We construct a finite connected simple cubic graph $F$ such that every signing of its edges yields a signed adjacency matrix with an eigenvalue outside $[-2\sqrt2,2\sqrt2]$. This disproves the Bilu--Linial signing conjecture for general regular graphs. The graph $F$ is not Ramanujan, and the conjecture restricted to Ramanujan base graphs remains open.
We prove that, for every integer $t\geq 1$, the unit two-sphere admits a spherical $t$-design consisting of exactly $54t^2$ points. More generally, such a design exists with exactly $6q^2$ points for every integer $q\geq 3t$. The proof builds on the topological degree method of Bondarenko, Radchenko and Viazovska, using an explicit equal-area partition of the sphere based on the map of Roşca and Plonka. By choosing the cell centers to minimize the average squared geodesic distance and deriving sharper sampling estimates, we obtain the stated quadratic bound on the size of spherical $t$-designs on $\Sph$.
Let $d\geq 2$ and let $\bf{f}_1,\ldots,\bf{f}_m\in\mathbb C^d$. We prove that if $m\leq 2d-1$, then the intensity measurement map \[ \bf{x}\longmapsto \bigl( |\langle \bf{x},\bf{f}_1\rangle|^2, \ldots, |\langle \bf{x},\bf{f}_m\rangle|^2 \bigr) \] fails to recover almost every signal in $\mathbb C^d$ uniquely up to a global phase factor. Combined with the known generic sufficiency of $2d$ measurements, our result establishes that the minimum number of measurements required for almost-everywhere phase retrieval in $\mathbb C^d$ is exactly $2d$. This resolves an open problem in phase retrieval by determining the exact measurement threshold for almost-everywhere phase retrieval in ${\mathbb C}^d$.
In this paper we focus on the stability of positive semidefinite matrix sensing maps $Φ_{\mathcal{A}}(X)=(\langle A_i,X\rangle)_{i=1}^m$ where $A_i\succeq 0$, $X\succeq0$ and $\operatorname{rank}(X)\le r$. We introduce the bi-Lipschitz constants of $Φ_{\mathcal{A}}(X)$ and define the global condition numbers as the ratio of upper and lower Lipschitz constants. We give deterministic universal lower bounds for these condition numbers, that depend only on the rank $r$ and on the underlying field. We then investigate the random rank-one Gaussian measurements and show that our lower bounds on condition numbers are asymptotically sharp, and therefore the random rank-one Gaussian measurements are asymptotically optimal. As an application, we derive the stability guarantees for an $\ell_1$-residual PhaseLift-type estimator at the optimal sampling scale.
Large language models (LLMs) are increasingly used to synthesize heuristic programs, yet most existing pipelines optimize solvers against fixed benchmark distributions. This static setup can obscure solver weaknesses and limit understanding of how LLM-designed algorithms adapt under distribution shift. We present Adversarial Co-Evolution (ACEvo), a closed-loop framework in which LLMs iteratively co-evolve two types of executable programs: heuristic solvers and problem generators. The generator proposes increasingly challenging instances, while the solver is refined to improve performance on the evolving distribution, forming an automated adversarial curriculum for program design and evaluation. We instantiate ACEvo on routing problems, including TSP, OP, and CVRP. Across these domains, the framework produces instance distributions that consistently induce larger optimality gaps than standard benchmarks and yields solver programs that outperform those obtained from static-training baselines under distribution shift. Beyond final performance, ACEvo provides a testbed for studying LLM-based algorithm design under evolving distributions, including how reflective mutation, adversarial feedback, and co-adaptation shape the evolution of both generators and solvers. These results suggest that closed-loop co-evolution is a promising paradigm for using language models not only to generate algorithms, but also to construct adaptive evaluation environments.
The Bilu-Linial conjecture asserts that every $d$-regular graph admits a signing $σ$ such that the spectral radius of the signed adjacency matrix $A_σ$ satisfies $ρ(A_σ)\le 2\sqrt{d-1}$. Bilu and Linial also proved the weaker bound $O(\sqrt{d\log^3 d})$ for graphs of maximum degree $d$. Marcus, Spielman, and Srivastava confirmed the conjecture in the case of $d$-regular bipartite graphs. In this paper, we prove that every graph of maximum degree $d$ has a signing $σ$ such that $$ρ(A_σ)\le 2\sqrt{3(d-1)}.$$ This removes the polylogarithmic factor from the estimate of Bilu and Linial and gives an explicit $2\sqrt{3(d-1)}$ two-sided spectral bound. The proof builds on the method of interlacing polynomials introduced by Marcus, Spielman, and Srivastava, together with results on mixed characteristic polynomials established by Marcus, Spielman, and Srivastava and by Bownik.
This paper investigates the one-dimensional Fourier phase retrieval problem for indicator functions of finite unions of intervals. Specifically, we study the recovery of a set $Ω= \bigcup_{j=1}^m I_j \subset\mathbb{R}$ from the magnitude of its Fourier transform $|\widehat{\mathbf{1}_Ω}|$, where each $I_j \subset \mathbb{R}$ is a bounded interval. For $m\le 2$, we prove that $Ω$ is uniquely determined by $ |\widehat{\mathbf{1}_Ω}|$ up to the natural ambiguities of translation and reflection, and we further establish a stability result for this reconstruction. In contrast, for $m\ge 3$, uniqueness fails in general. More precisely, for every $m\ge 3$, we explicitly construct functions $f_m,g_m\in\mathcal{I}_m$ such that $|\widehat{f_m}|=|\widehat{g_m}|,$ while $f_m$ cannot be obtained from $g_m$ by any translation or reflection, where $\mathcal{I}_m$ denotes the class of indicator functions of unions of exactly $m$ intervals. Furthermore, building on the theory of the turnpike problem, in which a finite integer set is uniquely determined by its multiset of pairwise differences under a collision-free condition, we establish an analogous result for finite subsets of $\mathbb{R}$. This, in turn, yields a sufficient condition for recovering indicator functions of finite unions of intervals. These results provide a complete characterization of the Fourier phase retrieval problem for indicator functions of finite unions of intervals and offer new insights into Fourier phase retrieval for indicator functions of more general domains in higher dimensions.
Unsupervised video domain adaptation (UVDA) is a practical but under-explored problem. In this paper, we propose a frustratingly easy UVDA method, called MetaTrans. Specifically, MetaTrans adopts a concise learning objective that contains only two fundamental loss terms. Despite the simplicity of the learning objective, MetaTrans embodies an advanced UVDA idea, that is, handling the spatial and temporal divergence of cross-domain videos separately, through a subtle model architecture design. By implementing a temporal-static subtraction module, MetaTrans effectively removes spatial and temporal divergence. Extensive empirical evaluations, particularly on various cross-domain action recognition tasks, show substantial absolute adaptation performance enhancement and significantly superior relative performance gain compared with state-of-the-art UVDA baselines.
This paper investigates the stability of the least squares approximation $P_m^n$ within the univariate polynomial space of degree $m$, denoted by ${\mathbb P}_m$. The approximation $P_m^n$ entails identifying a polynomial in ${\mathbb P}_m$ that approximates a function $f$ over a domain $X$ based on samples of $f$ taken at $n$ randomly selected points, according to a specified probability measure $ρ_X$. The primary goal is to determine the sampling rate necessary to ensure the stability of $P_m^n$. Assuming the sampling points are i.i.d. with respect to a Jacobi weight function, we present the sampling rate that guarantee the stability of $P_m^n$. Specifically, for uniform random sampling, we demonstrate that a sampling rate of $n \asymp m^2$ is required to maintain stability. By combining these findings with those of Cohen-Davenport-Leviatan, we conclude that, for uniform random sampling, the optimal sampling rate for guaranteeing the stability of $P_m^n$ is $n \asymp m^2$, up to a $\log n$ factor. Motivated by this result, we extend the impossibility theorem, previously applicable to equally spaced samples, to the case of random samples, illustrating the balance between accuracy and stability in recovering analytic functions.
This paper aims to characterize the optimal frame for phase retrieval, defined as the frame whose condition number for phase retrieval attains its minimal value. In the context of the two-dimensional real case, we reveal the connection between optimal frames for phase retrieval and the perimeter-maximizing isodiametric problem, originally proposed by Reinhardt in 1922. Our work establishes that every optimal solution to the perimeter-maximizing isodiametric problem inherently leads to an optimal frame in ${\mathbb R}^2$. By recasting the optimal polygons problem as one concerning the discrepancy of roots of unity, we characterize all optimal polygons. Building upon this connection, we then characterize all optimal frames with $m$ vectors in ${\mathbb R}^2$ for phase retrieval when $m \geq 3$ has an odd factor. As a key corollary, we show that the harmonic frame $E_m \subset {\mathbb R}^2$ is {\em not} optimal for any even integer $m \geq 4$. This finding disproves a conjecture proposed by Xia, Xu, and Xu [{\em Math. Comp.}, 94 (2025), pp.~2931--2960]. Previous work has established that $E_m$ is indeed optimal when $m$ is an odd integer.
Diffusion models have emerged as the leading paradigm in generative modeling, excelling in various applications. Despite their success, these models often misalign with human intentions and generate results with undesired properties or even harmful content. Inspired by the success and popularity of alignment in tuning large language models, recent studies have investigated aligning diffusion models with human expectations and preferences. This work mainly reviews alignment of diffusion models, covering advancements in fundamentals of alignment, alignment techniques of diffusion models, preference benchmarks, and evaluation for diffusion models. Moreover, we discuss key perspectives on current challenges and promising future directions on solving the remaining challenges in alignment of diffusion models. To the best of our knowledge, our work is the first comprehensive review paper for researchers and engineers to comprehend, practice, and research alignment of diffusion models.
Dataset distillation compresses large datasets into compact synthetic sets with comparable performance in training models. Despite recent progress on diffusion-based distillation, this type of method typically depends on heuristic guidance or prototype assignment, which comes with time-consuming sampling and trajectory instability and thus hurts downstream generalization especially under strong control or low IPC. We propose \emph{Path-Guided Flow Matching (PGFM)}, the first flow matching-based framework for generative distillation, which enables fast deterministic synthesis by solving an ODE in a few steps. PGFM conducts flow matching in the latent space of a frozen VAE to learn class-conditional transport from Gaussian noise to data distribution. Particularly, we develop a continuous path-to-prototype guidance algorithm for ODE-consistent path control, which allows trajectories to reliably land on assigned prototypes while preserving diversity and efficiency. Extensive experiments across high-resolution benchmarks demonstrate that PGFM matches or surpasses prior diffusion-based distillation approaches with fewer steps of sampling while delivering competitive performance with remarkably improved efficiency, e.g., 7.6$\times$ more efficient than the diffusion-based counterparts with 78\% mode coverage.
Real-world deployment of Vision-and-Language Navigation (VLN) agents is constrained by the scarcity of reliable supervision after offline training. While recent adaptation methods attempt to mitigate distribution shifts via environment-driven self-supervision (e.g., entropy minimization), these signals are often noisy and can cause the agent to amplify its own mistakes during long-horizon sequential decision-making. In this paper, we propose a paradigm shift that positions user feedback, specifically episode-level success confirmations and goal-level corrections, as a primary and general-purpose supervision signal for VLN. Unlike internal confidence scores, user feedback is intent-aligned and in-situ consistent, directly correcting the agent's decoupling from user instructions. To effectively leverage this supervision, we introduce a user-feedback-driven learning framework featuring a topology-aware trajectory construction pipeline. This mechanism lifts sparse, goal-level corrections into dense path-level supervision by generating feasible paths on the agent's incrementally built topological graph, enabling sample-efficient imitation learning without requiring step-by-step human demonstrations. Furthermore, we develop a persistent memory bank mechanism for warm-start initialization, supporting the reuse of previously acquired topology and cached representations across navigation sessions. Extensive experiments on the GSA-R2R benchmark demonstrate that our approach transforms sparse interaction into robust supervision, consistently outperforming environment-driven baselines while exhibiting strong adaptability across diverse instruction styles.
Predicting enzyme-substrate interactions has long been a fundamental problem in biochemistry and metabolic engineering. While existing methods could leverage databases of expert-curated enzyme-substrate pairs for models to learn from known pair interactions, the databases are often sparse, i.e., there are only limited and incomplete examples of such pairs, and also labor-intensive to maintain. This lack of sufficient training data significantly hinders the ability of traditional enzyme prediction models to generalize to unseen interactions. In this work, we try to exploit chemical reaction equations from domain-specific databases, given their easier accessibility and denser, more abundant data. However, interactions of multiple compounds, e.g., educts and products, with the same enzymes create complex relational data patterns that traditional models cannot easily capture. To tackle that, we represent chemical reaction equations as triples of (educt, enzyme, product) within a knowledge graph, such that we can take advantage of knowledge graph embedding (KGE) to infer missing enzyme-substrate pairs for graph completion. Particularly, in order to capture intricate relationships among compounds, we propose our knowledge-enhanced hypergraph model for enzyme prediction, i.e., Hyper-Enz, which integrates a hypergraph transformer with a KGE model to learn representations of the hyper-edges that involve multiple educts and products. Also, a multi-expert paradigm is introduced to guide the learning of enzyme-substrate interactions with both the proposed model and chemical reaction equations. Experimental results show a significant improvement, with up to a 88% relative improvement in average enzyme retrieval accuracy and 30% improvement in pair-level prediction compared to traditional models, demonstrating the effectiveness of our approach.
One of many impediments to applying graph neural networks (GNNs) to large-scale real-world graph data is the challenge of centralized training, which requires aggregating data from different organizations, raising privacy concerns. Federated graph learning (FGL) addresses this by enabling collaborative GNN model training without sharing private data. However, a core challenge in FGL systems is the variation in local training data distributions among clients, known as the data heterogeneity problem. Most existing solutions suffer from two problems: (1) The typical optimizer based on empirical risk minimization tends to cause local models to fall into sharp valleys and weakens their generalization to out-of-distribution graph data. (2) The prevalent dimensional collapse in the learned representations of local graph data has an adverse impact on the classification capacity of the GNN model. To this end, we formulate a novel optimization objective that is aware of the sharpness (i.e., the curvature of the loss surface) of local GNN models. By minimizing the loss function and its sharpness simultaneously, we seek out model parameters in a flat region with uniformly low loss values, thus improving the generalization over heterogeneous data. By introducing a regularizer based on the correlation matrix of local representations, we relax the correlations of representations generated by individual local graph samples, so as to alleviate the dimensional collapse of the learned model. The proposed \textbf{S}harpness-aware f\textbf{E}derated gr\textbf{A}ph \textbf{L}earning (SEAL) algorithm can enhance the classification accuracy and generalization ability of local GNN models in federated graph learning. Experimental studies on several graph classification benchmarks show that SEAL consistently outperforms SOTA FGL baselines and provides gains for more participants.
This paper introduces the concept of generalized interlacing families of polynomials, which extends the classical interlacing polynomial method to handle polynomials of varying degrees. We establish a fundamental property for these families, proving the existence of a polynomial with a desired degree whose smallest root is greater than or equal to the smallest root of the expected polynomial. Applying this framework to the generalized CUR matrix approximation problem, we derive a theoretical upper bound on the spectral norm of a residual matrix, expressed in terms of the largest root of the expected polynomial. We then explore two important special cases: the classical CUR matrix decompositions and the row subset selection problem. For classical CUR matrix decompositions, we derive an explicit upper bound for the largest root of the expected polynomial. This yields a tighter spectral norm error bound for the residual matrix compared to many existing results. Furthermore, we present a deterministic polynomial-time algorithm for solving the classical CUR problem under certain matrix conditions. For the row subset selection problem, we establish the first known spectral norm error bound. This paper extends the applicability of interlacing families and deepens the theoretical foundations of CUR matrix decompositions and related approximation problems.
Dataset distillation (DD) aims to construct compact synthetic datasets that allow models to achieve comparable performance to full-data training while substantially reducing storage and computation. Despite rapid empirical progress, its theoretical foundations remain limited: existing methods (gradient, distribution, trajectory matching) are built on heterogeneous surrogate objectives and optimization assumptions, which makes it difficult to analyze their common principles or provide general guarantees. Moreover, it is still unclear under what conditions distilled data can retain the effectiveness of full datasets when the training configuration, such as optimizer, architecture, or augmentation, changes. To answer these questions, we propose a unified theoretical framework, termed configuration--dynamics--error analysis, which reformulates major DD approaches under a common generalization-error perspective and provides two main results: (i) a scaling law that provides a single-configuration upper bound, characterizing how the error decreases as the distilled sample size increases and explaining the commonly observed performance saturation effect; and (ii) a coverage law showing that the required distilled sample size scales linearly with configuration diversity, with provably matching upper and lower bounds. In addition, our unified analysis reveals that various matching methods are interchangeable surrogates, reducing the same generalization error, clarifying why they can all achieve dataset distillation and providing guidance on how surrogate choices affect sample efficiency and robustness. Experiments across diverse methods and configurations empirically confirm the derived laws, advancing a theoretical foundation for DD and enabling theory-driven design of compact, configuration-robust dataset distillation.
Knowledge graph embedding (KGE) relies on the geometry of the embedding space to encode semantic and structural relations. Existing methods place all entities on one homogeneous manifold, Euclidean, spherical, hyperbolic, or their product/multi-curvature variants, to model linear, symmetric, or hierarchical patterns. Yet a predefined, homogeneous manifold cannot accommodate the sharply varying curvature that real-world graphs exhibit across local regions. Since this geometry is imposed a priori, any mismatch with the knowledge graph's local curvatures will distort distances between entities and hurt the expressiveness of the resulting KGE. To rectify this, we propose RicciKGE to have the KGE loss gradient coupled with local curvatures in an extended Ricci flow such that entity embeddings co-evolve dynamically with the underlying manifold geometry towards mutual adaptation. Theoretically, when the coupling coefficient is bounded and properly selected, we rigorously prove that i) all the edge-wise curvatures decay exponentially, meaning that the manifold is driven toward the Euclidean flatness; and ii) the KGE distances strictly converge to a global optimum, which indicates that geometric flattening and embedding optimization are promoting each other. Experimental improvements on link prediction and node classification benchmarks demonstrate RicciKGE's effectiveness in adapting to heterogeneous knowledge graph structures.