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Xiaolong Han

Publications and source records attributed to Xiaolong Han.

At least 19 recordsLinked to original sources
arXiv2026

Schrodinger group along the horocycles and Egorov's theorem on compact hyperbolic surfaces

In this paper, we study Egorov's theorem for Laplacian eigenfunctions on compact hyperbolic surfaces in Zelditch's quantization. In this quantization, the remainder in Egorov's theorem for eigenfunctions is given explicitly by the second derivatives of the symbols along the weak stable foliation of the geodesic flow. Forward in time, we show that Egorov's theorem for eigenfunctions is valid to all logarithmic times, far beyond the Ehrenfest time T_E=|log h|/2, where h going to 0 is the semiclassical parameter. Backward in time, we introduce the Schrodinger group along the stable horocycles, which absorbs the exponentially growing part of the remainder. We show that Egorov's theorem for eigenfunctions is valid up to -T_E+T_h for any Th going to infinity with 0<=T_h<=T_E. Moreover, the limit of the remainder in Egorov's theorem at the Ehrenfest time -T_E is exactly the difference between the semiclassical measure and the Liouville measure, tested against the symbols propagated by the Schrodinger group. Consequently, Egorov's theorem for a sequence of eigenfunctions holds at -T_E (and in fact, at -T_E-T for any fixed T in R if and only if the sequence tends to be equidistributed in the phase space. Thus, the quantum unique ergodicity (QUE) conjecture is equivalent to this Egorov property holding for every eigenbasis. Furthermore, QUE holds with a polynomial rate if and only if Egorov's theorem holds at -T_E with a polynomial remainder. In this case, Egorov's theorem holds beyond -T_E with a polynomial remainder, and the stronger the rate is in QUE, the longer the time is in Egorov's theorem.

math.SP↗
arXiv2026

SEAR: Spoofing Evidence-Grounded Audio Reasoning Benchmark for Audio Language Models

Audio language models (ALMs) are increasingly used for audio deepfake detection (ADD), yet existing benchmarks assess their verdicts or rationale plausibility without verifying the underlying acoustic evidence. To address this issue, we first introduce spoofing evidence-grounded audio reasoning (SEAR), a four-task AQA benchmark to evaluate ALM-based ADD through acoustic evidence identification and quantification, deepfake detection, and forensic rationale generation. We further propose a bona-fide-based acoustic evidence agent (BAEA), which equips a frozen ALM with controlled acoustic tools under \textsc{fixed} or \textsc{adaptive} evidence-acquisition policies. Experiments with six ALMs reveal a clear gap between plausible rationales and verifiable acoustic evidence reasoning, while BAEA-\textsc{Fixed} improves final verdicts and forensic rationales on both evaluation partitions. Controlled interventions further show that misleading evidence degrades both detection and grounding performance.

cs.SD↗
arXiv2026

Spherical harmonics with maximal Lp norm growth

In this paper, we show that for every p>2, there is an orthonormal basis of spherical harmonics on Sn with n>=2 which achieves the maximal Lp norm growth. More precisely, one basis achieves this growth for all p<=2(n+1)/(n-1), and another for all p>=2(n+1)/(n-1). This extends the results in Han (arXiv:1404.5016) to all dimensions and all p>2, and improves the positive density subsequence there to a whole basis.

math.CA↗
arXiv2026

Spherical harmonics and point configurations on the sphere

We construct spherical harmonics on the two-dimensional unit sphere as superpositions of Gaussian beams whose poles form well-separated point configurations. The distributional and analytic properties of the resulting spherical harmonics are determined by the geometry of these poles: when the configuration is equidistributed, the sequence of harmonics exhibits quantum ergodicity, while their supremum norms are quantitatively controlled by the maximal clustering of poles within small neighborhoods of great circles.

math.CA↗
arXiv2026

Fractal uncertainty principle for random Cantor sets

We continue our investigation of the fractal uncertainty principle (FUP) for random fractal sets. In the prequel (arXiv:2107.08276), we considered the Cantor sets in the discrete setting with alphabets randomly chosen from a base of digits so the dimension d is in (0,2/3). We proved that, with overwhelming probability, the FUP with an exponent >=1/2-3d/4- holds for these discrete Cantor sets with random alphabets. In this sequel, we construct random Cantor sets with dimension d in (0,2/3) in R via a different random procedure from the one in the prequel. We prove that, with overwhelming probability, the FUP with an exponent >=1/2-3d/4- holds. The proof follows from establishing a Fourier decay estimate of the corresponding random Cantor measures, which is in turn based on a concentration of measure phenomenon in an appropriate probability space for the random Cantor sets.

math.CA↗
arXiv2026

The semiclassical measure of certain spherical harmonics on S3

Bourgain used the Rudin-Shapiro sequences to construct a basis of uniformly bounded holomorphic functions on the unit sphere in C2. They are also spherical harmonics (i.e., Laplacian eigenfunctions) on S3 in R4. In this paper, we identify the semiclassical measure associated to these spherical harmonics by a measure supported on the family of Clifford tori in S3. In particular, these functions equidistribute on S3 in the semiclassical limit.

math.CA↗
arXiv2026

W2T: LoRA Weights Already Know What They Can Do

Each LoRA checkpoint compactly stores task-specific updates in low-rank weight matrices, offering an efficient way to adapt large language models to new tasks and domains. In principle, these weights already encode what the adapter does and how well it performs. In this paper, we ask whether this information can be read directly from the weights, without running the base model or accessing training data. A key obstacle is that a single LoRA update can be factorized in infinitely many ways. Without resolving this ambiguity, models trained on the factors may fit the particular factorization rather than the underlying update. To this end, we propose \methodfull, which maps each LoRA update to a provably canonical form via QR decomposition followed by SVD, so that all equivalent factorizations share the same representation. The resulting components are then tokenized and processed by a Transformer to produce a weight-space embedding. Across language and vision LoRA collections, W2T achieves strong results on attribute classification, performance prediction, and adapter retrieval, demonstrating that LoRA weights reliably indicate model behavior once factorization ambiguity is removed. Code is available at https://github.com/xiaolonghan2000/Weight2Token.

cs.LG↗
arXiv2026

A Survey of Weight Space Learning: Understanding, Representation, and Generation

Neural network weights are typically viewed as the end product of training, while most deep learning research focuses on data, features, and architectures. However, recent advances show that the set of all possible weight values (weight space) itself contains rich structure: pretrained models form organized distributions, exhibit symmetries, and can be embedded, compared, or even generated. Understanding such structures has tremendous impact on how neural networks are analyzed and compared, and on how knowledge is transferred across models, beyond individual training instances. This emerging research direction, which we refer to as Weight Space Learning (WSL), treats neural weights as a meaningful domain for analysis and modeling. This survey provides the first unified taxonomy of WSL. We categorize existing methods into three core dimensions: Weight Space Understanding (WSU), which studies the geometry and symmetries of weights; Weight Space Representation (WSR), which learns embeddings over model weights; and Weight Space Generation (WSG), which synthesizes new weights through hypernetworks or generative models. We further show how these developments enable practical applications, including model retrieval, continual and federated learning, neural architecture search, and data-free reconstruction. By consolidating fragmented progress under a coherent framework, this survey highlights weight space as a learnable, structured domain with growing impact across model analysis, transferring, and weight generation. We release an accompanying resource at https://github.com/Zehong-Wang/Awesome-Weight-Space-Learning.

cs.LG↗
arXiv2026

Molecular Representations in Implicit Functional Space via Hyper-Networks

Molecular representations fundamentally shape how machine learning systems reason about molecular structure and physical properties. Most existing approaches adopt a discrete pipeline: molecules are encoded as sequences, graphs, or point clouds, mapped to fixed-dimensional embeddings, and then used for task-specific prediction. This paradigm treats molecules as discrete objects, despite their intrinsically continuous and field-like physical nature. We argue that molecular learning can instead be formulated as learning in function space. Specifically, we model each molecule as a continuous function over three-dimensional (3D) space and treat this molecular field as the primary object of representation. From this perspective, conventional molecular representations arise as particular sampling schemes of an underlying continuous object. We instantiate this formulation with MolField, a hyper-network-based framework that learns distributions over molecular fields. To ensure physical consistency, these functions are defined over canonicalized coordinates, yielding invariance to global SE(3) transformations. To enable learning directly over functions, we introduce a structured weight tokenization and train a sequence-based hyper-network to model a shared prior over molecular fields. We evaluate MolField on molecular dynamics and property prediction. Our results show that treating molecules as continuous functions fundamentally changes how molecular representations generalize across tasks and yields downstream behavior that is stable to how molecules are discretized or queried.

cs.LG↗
arXiv2023

Heterogeneous Graph Contrastive Multi-view Learning

Inspired by the success of contrastive learning (CL) in computer vision and natural language processing, graph contrastive learning (GCL) has been developed to learn discriminative node representations on graph datasets. However, the development of GCL on Heterogeneous Information Networks (HINs) is still in the infant stage. For example, it is unclear how to augment the HINs without substantially altering the underlying semantics, and how to design the contrastive objective to fully capture the rich semantics. Moreover, early investigations demonstrate that CL suffers from sampling bias, whereas conventional debiasing techniques are empirically shown to be inadequate for GCL. How to mitigate the sampling bias for heterogeneous GCL is another important problem. To address the aforementioned challenges, we propose a novel Heterogeneous Graph Contrastive Multi-view Learning (HGCML) model. In particular, we use metapaths as the augmentation to generate multiple subgraphs as multi-views, and propose a contrastive objective to maximize the mutual information between any pairs of metapath-induced views. To alleviate the sampling bias, we further propose a positive sampling strategy to explicitly select positives for each node via jointly considering semantic and structural information preserved on each metapath view. Extensive experiments demonstrate HGCML consistently outperforms state-of-the-art baselines on five real-world benchmark datasets.

cs.LG↗
arXiv2021

Fractal uncertainty principle for discrete Cantor sets with random alphabets

In this paper, we investigate the fractal uncertainty principle (FUP) for discrete Cantor sets, which are determined by an alphabet from a base of digits. Consider the base of M digits and the alphabets of cardinality A such that all the corresponding Cantor sets have a fixed dimension 0<log A/log M<2/3. We prove that the FUP with an improved exponent over Dyatlov-Jin (arXiv:1608.02238) holds for almost all alphabets, asymptotically as M tends to infinity. Our result provides the best possible exponent when the Cantor sets enjoy either the strongest Fourier decay assumption or strongest additive energy assumption. The proof is based on a concentration of measure phenomenon in the space of alphabets.

math.CA↗
arXiv2021

Riemann moduli spaces are quantum ergodic

In this note we show that the Riemann moduli spaces $M_{g, n}$ equipped with the Weil--Petersson metric are quantum ergodic for $3g+n \geq 4$. We also provide other examples of singular spaces with ergodic geodesic flow for which quantum ergodicity holds.

math.AP↗
arXiv2019

Equidistribution of random waves on small balls

In this paper, we investigate the small scale equidistribution properties of randomised sums of Laplacian eigenfunctions (i.e. random waves) on a compact manifold. We prove small scale expectation and variance results for random waves on all compact manifolds. Here, "small scale" refers to balls of radius $r(λ)\to 0$ such that $r/r_{\text{Planck}}\to\infty$, where $r_{\text{Planck}}$ is the Planck scale. For balls at a larger scale (although still $r(λ)\to 0$) we also obtain estimates showing that the probability that a random wave fails to equidistribute decays exponentially with the eigenvalue.

math.SP↗
arXiv2018

Small scale quantum ergodicity in cat maps. I

In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps"). In Part I of the series, we prove quantum ergodicity at various scales. Let $N=1/h$, in which $h$ is the Planck constant. First, for all integers $N\in\mathbb{N}$, we show quantum ergodicity at logarithmical scales $|\log h|^{-α}$ for some $α>0$. Second, we show quantum ergodicity at polynomial scales $h^α$ for some $α>0$, in two special cases: $N\in S(\mathbb{N})$ of a full density subset $S(\mathbb{N})$ of integers and Hecke eigenbasis for all integers.

math-ph↗
arXiv2018

Tomas-Stein restriction estimates on convex cocompact hyperbolic manifolds. I

In this paper, we investigate the Tomas-Stein restriction estimates on convex cocompact hyperbolic manifolds $Γ\backslash\mathbb{H}^{n+1}$. Via the spectral measure of the Laplacian, we prove that the Tomas-Stein restriction estimate holds when the limit set has Hausdorff dimension $δ_Γ<n/2$. This provides an example for which restriction estimate holds in the presence of hyperbolic geodesic trapping.

math.CA↗
arXiv2018

Lp bilinear quasimode estimates

In this paper, we investigate the $L^p$ bilinear quasimode estimates on compact Riemannian manifolds. We obtain results in the full range $p\ge2$ on all $n$-dimensional manifolds with $n\ge2$. This in particular implies the $L^p$ bilinear eigenfunction estimates. We further show that all of these estimates are sharp by constructing various quasimodes and eigenfunctions that saturate our estimates.

math.AP↗