arXiv subjects

Zhiqiang Xu

Publications and source records attributed to Zhiqiang Xu.

At least 19 recordsLinked to original sources

arXiv2026

CaliSplat: Sparse Gaussian Fitting and Calibrated Rate Allocation for Image Compression

2D Gaussian representations provide an explicit and lightweight alternative for image representation, enabling fast deterministic decoding without a heavy neural synthesis network. However, existing per-image Gaussian codecs still require costly image-specific optimization to determine how limited Gaussian capacity should be distributed across heterogeneous regions. We introduce CaliSplat, an optimization-efficient Gaussian image codec that selectively explores candidate representations and globally allocates a constrained Gaussian budget. We exploit the closed-form reconstruction error of the single-Gaussian case as an inexpensive measure of local difficulty, fitting higher-capacity candidates only in regions where additional Gaussians are likely to be useful. To reduce selection errors caused by mismatched convergence rates across candidate sizes, a small set of longer-horizon fits is used to calibrate short-horizon distortion estimates before global budget-aware selection. The selected representation is then compressed into a bitstream. Across standard and large image datasets, CaliSplat achieves competitive or superior rate--distortion performance while reducing per-image fitting time by $3.0\times$--$15.7\times$ compared with representative Gaussian codecs. It further maintains high-throughput deterministic decoding at $175.9$--$478.2$~MP/s. These results demonstrate that content-adaptive Gaussian compression can substantially reduce per-image optimization cost while preserving strong reconstruction quality and efficient explicit decoding.

cs.AI↗

arXiv2026

Optimal Condition Numbers in Low-Rank Positive Semidefinite Matrix Sensing

In this paper we focus on the stability of positive semidefinite matrix sensing maps $Φ_{\mathcal{A}}=(\langle A_i,X\rangle)_{i=1}^m$ where $A_i\succeq 0$, $X\succeq0$ and $\operatorname{rank}(X)\le r$. We introduce the bi-Lipschitz constants of $Φ_{\mathcal{A}}$ 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 show that rank-one Gaussian measurements asymptotically attain these bounds, except for the operator-norm condition number over $\mathbb R$ when $r=1$. For this exceptional case, we establish a sharper universal lower bound and prove its asymptotic sharpness. As an application, we derive stability guarantees for an $\ell_1$-residual PhaseLift-type estimator at the optimal sampling scale. These results identify the intrinsic limits of conditioning in positive semidefinite matrix sensing and connect these limits to stable low-rank recovery.

cs.IT↗

arXiv2026

WAR-CRAFT: Refining Web API Recommendations through Requirement Auditing

Web API recommenders return a set of APIs for a mashup request but do not check it against provider documentation, the main public evidence of an API's capabilities. The set can therefore contain unsupported or redundant APIs and miss requested capabilities. Filling every apparent gap does not fix this, because a gap may already be served, may not need its own API, or may have no documented provider. We study post-recommendation refinement, which revises a frozen recommender's output using only the request, the catalog, and provider documentation. We propose WAR-CRAFT, which treats refinement as three decisions. An LLM removes unsupported or subsumed APIs by judging candidates jointly, then audits the remaining set against a requirement checklist derived from the request. For each uncovered requirement, a fine-tuned catalog-constrained Resolver proposes a catalog API or abstains, and an independent documentation check admits only supported proposals. On ProgrammableWeb with 7 heterogeneous upstream recommenders, WAR-CRAFT raises mean SetF1 from 0.558 to 0.644, above the strongest of 5 refinement baselines (62.5%), and roughly halves functional redundancy (0.127 to 0.065) without changing mean functional coverage. Gains range from +2.2 points on generative recommenders to +18.2 on fixed-cutoff rankers. At matched set sizes, WAR-CRAFT still outperforms truncating the upstream output, so the gain is not explained by set-size calibration alone. Our code is available at https://github.com/ZsXu-enrico/WAR-CRAFT.

cs.AI↗

arXiv2026

A deterministic polynomial-time algorithm for the Bilu--Linial signing problem on bipartite graphs

We give a deterministic polynomial-time algorithm for the Bilu--Linial signing problem on bipartite graphs. For every finite simple bipartite graph of maximum degree at most an integer $Δ\ge3$, the algorithm assigns signs to its edges so that the signed adjacency matrix has operator norm strictly less than $2\sqrt{Δ-1}$. Our algorithm builds on the randomized recursive repair framework of Jadbabaie, Saberi, and Sra~\cite{JSS26}, with deterministic rules for sign selection and vertex deletion.

cs.DS↗

arXiv2026

KGOT: Unified Knowledge Graph and Optimal Transport Pseudo-Labeling for Molecule-Protein Interaction Prediction

Predicting molecule-protein interactions (MPIs) is a fundamental task in computational biology, with crucial applications in drug discovery and molecular function annotation. However, existing MPI models face two major challenges. First, the scarcity of labeled molecule-protein pairs significantly limits model performance, as available datasets capture only a small fraction of biological relevant interactions. Second, most methods rely solely on molecular and protein features, ignoring broader biological context such as genes, metabolic pathways, and functional annotations that could provide essential complementary information. To address these limitations, our framework first aggregates diverse biological datasets, including molecular, protein, genes and pathway-level interactions, and then develop an optimal transport-based approach to generate high-quality pseudo-labels for unlabeled molecule-protein pairs, leveraging the underlying distribution of known interactions to guide label assignment. By treating pseudo-labeling as a mechanism for bridging disparate biological modalities, our approach enables the effective use of heterogeneous data to enhance MPI prediction. We evaluate our framework on multiple MPI datasets including virtual screening tasks and protein retrieval tasks, demonstrating substantial improvements over state-of-the-art methods in prediction accuracies and zero shot ability across unseen interactions. Beyond MPI prediction, our approach provides a new paradigm for leveraging diverse biological data sources to tackle problems traditionally constrained by single- or bi-modal learning, paving the way for future advances in computational biology and drug discovery.

cs.LG↗

arXiv2026

Hadamard Transforms for Dimension Reduction

Using the Hadamard matrix for dimension reduction can accelerate the computation due to the fast Hadamard transform. However, the previous target dimension bounds in terms of the Gaussian width depend on the ambient dimension. In this paper, we obtain the following ambient dimension free bound using three Hadamard-diagonal blocks. Let $H\in\mathbb R^{d\times d}$ be the normalized Walsh-Hadamard matrix, and let $D_1,D_2,D_3\in\mathbb R^{d\times d}$ be independent Rademacher diagonal matrices. We prove that for any symmetric set $T\subset \mathbb{S}^{d-1}$, \[ \mathbb P\left\{\sup_{\boldsymbol t\in T}\big|\frac dm\|P_ΩHD_3HD_2HD_1\boldsymbol t\|_2^2-1\big|\le\varepsilon\right\} \ge1-δ\] whenever \[ m\gtrsim\varepsilon^{-2}w(T)^2\log^2\frac eδ\,\log^4\!\left(e+\frac{w(T)}\varepsilon\log\frac eδ\right). \] Here $w(T)$ is the Gaussian width of $T$, and $Ω$ is a set of $m$ distinct coordinates chosen uniformly. This is the first ambient dimension free bound on the target dimension using Hadamard transforms, for an arbitrary symmetric set in terms of its Gaussian width. We also prove that it is impossible to obtain an ambient dimension free bound for an arbitrary set in terms of its Gaussian width using two Hadamard-diagonal blocks.

cs.DS↗

arXiv2026

A 3-regular counterexample to the Bilu--Linial signing conjecture

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.

math.CO↗

arXiv2026

Spherical $t$-Designs on $\mathbb S^2$ with $54t^2$ Points

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 $\Sph$.

math.CO↗

arXiv2026

The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval

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 ${\mathbb C}^d$.

cs.IT↗

arXiv2026

ACEvo: Adversarial Co-Evolution of Problem Distributions and Solvers for Combinatorial Optimization

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.

cs.AI↗

arXiv2026

An Improved Upper Bound for the Bilu-Linial Conjecture via Interlacing Families

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.

math.CO↗

arXiv2026

Fourier Phase Retrieval for Finite Unions of Intervals

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.

math.FA↗

arXiv2026

Return of Frustratingly Easy Unsupervised Video Domain Adaptation

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.

cs.CV↗

arXiv2026

Stability of Least Squares Approximation under Random Sampling

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.

math.NA↗

arXiv2026

Optimal Frames for Phase Retrieval from Edge Vectors of Optimal Polygons

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.

cs.IT↗

arXiv2026

Alignment of Diffusion Models: Fundamentals, Challenges, and Future

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.

cs.LG↗

arXiv2026

Path-Guided Flow Matching for Dataset Distillation

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.

cs.LG↗

arXiv2026

User-Feedback-Driven Adaptation for Vision-and-Language Navigation

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.

cs.AI↗