相关论文: Learning Fricke signs from Maass form Coefficients

We describe a procedure for alleviating the fermion sign problem in which phase fluctuations are explicitly subtracted from the Boltzmann factor. Several ans\"atze for fluctuations are designed and compared. In the absence of a sufficiently…

We prove the murmuration phenomenon, which is a correlation between signs of functional equations and Fourier coefficients, in the case of modular forms in the weight aspect. We in particular improve the range of visibility of murmurations…

"Murmurations" are a recently-discovered type of fine structure in sums of Dirichlet coefficients averaged over families of $L$-functions. The root cause of this phenomenon remains mysterious. In the present paper, we demonstrate how…

Fractional learning algorithms are trending in signal processing and adaptive filtering recently. However, it is unclear whether the proclaimed superiority over conventional algorithms is well-grounded or is a myth as their performance has…

A density matrix describes the statistical state of a quantum system. It is a powerful formalism to represent both the quantum and classical uncertainty of quantum systems and to express different statistical operations such as measurement,…

We study statistical properties of Fourier coefficients of automorphic forms on GL(n). For most Hecke-Maass cusp forms, we give the asymptotic number of nonvanishing coefficients, show that there is a positive proportion of sign changes…

Pre-trained large language models (LLMs) exhibit impressive mathematical reasoning capabilities, yet how they compute basic arithmetic, such as addition, remains unclear. This paper shows that pre-trained LLMs add numbers using Fourier…

Recently, we showed that global root numbers of modular forms are biased toward +1. Together with Pharis, we also showed an initial bias of Fourier coefficients towards the sign of the root number. First, we prove analogous results with…

Embedding methods have achieved success in face recognition by comparing facial features in a latent semantic space. However, in a fully unconstrained face setting, the facial features learned by the embedding model could be ambiguous or…

Latent Dirichlet allocation (LDA) is useful in document analysis, image processing, and many information systems; however, its generalization performance has been left unknown because it is a singular learning machine to which regular…

Recently there has been a great deal of interest surrounding the calibration of quantum sensors using machine learning techniques. In this work, we explore the use of regression to infer a machine-learned point estimate of an unknown…

We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. When the underlying form is self-dual, we show that there are $\gg_\varepsilon X^{5/6-\varepsilon}$ sign changes among the…

Let $f$ be a half-integral weight cusp form of level $4N$ for odd and squarefree $N$ and let $a(n)$ denote its $n^{\rm th}$ normalized Fourier coefficient. Assuming that all the coefficients $a(n)$ are real, we study the sign of $a(n)$ when…

Machine learning of partial differential equations from data is a potential breakthrough to solve the lack of physical equations in complex dynamic systems, but because numerical differentiation is ill-posed to noise data, noise has become…

Data-driven discovery of "hidden physics" -- i.e., machine learning of differential equation models underlying observed data -- has recently been approached by embedding the discovery problem into a Gaussian Process regression of spatial…

Machine learning is rapidly making its path into natural sciences, including high-energy physics. We present the first study that infers, directly from experimental data, a functional form of fragmentation functions. The latter represent a…

We study the property of the Fused Lasso Signal Approximator (FLSA) for estimating a blocky signal sequence with additive noise. We transform the FLSA to an ordinary Lasso problem. By studying the property of the design matrix in the…

Modeling data uncertainty is important for noisy images, but seldom explored for face recognition. The pioneer work, PFE, considers uncertainty by modeling each face image embedding as a Gaussian distribution. It is quite effective.…

The fused lasso is an important method for signal processing when the hidden signals are sparse and blocky. It is often used in combination with the squared loss function. However, the squared loss is not suitable for heavy tail error…

Large language models (LLMs) have demonstrated strong performance on formal language tasks, yet whether this reflects genuine symbolic reasoning or pattern matching on familiar constructions remains unclear. We introduce a benchmark for…