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Sjoerd de Vries

Publications and source records attributed to Sjoerd de Vries.

8 recordsLinked to original sources

Traces of Hecke operators on Drinfeld modular forms for $\mathrm{GL}_2(\mathbb{F}_q[T])$

In this paper, we study traces of Hecke operators on Drinfeld modular forms of level 1 in the case $A = \mathbb{F}_q[T]$. We deduce closed-form expressions for traces of Hecke operators corresponding to primes of degree at most 2 and provide algorithms for primes of higher degree. We improve the Ramanujan bound and deduce the decomposition of cusp forms of level $Γ_0(\mathfrak{p})$ into oldforms and newforms, as conjectured by Bandini-Valentino, under the hypothesis that each Hecke eigenvalue has multiplicity less than $p$.

math.NT

Drinfeld Quasi-Modular Forms of Higher Level

We study the structure of the vector space of Drinfeld quasi-modular forms for congruence subgroups. We provide representations as polynomials in the false Eisenstein series with coefficients in the space of Drinfeld modular forms (the $E$-expansion), and, whenever possible, as sums of hyperderivatives of Drinfeld modular forms. \\ Moreover, we introduce and study the double-slash operator, and use it to provide a well-posed definition for Hecke operators on Drinfeld quasi-modular forms. We characterize eigenforms and, for the special case of Hecke congruence subgroups $Γ_0(\mathfrak n)$, we give explicit formulas for the Hecke action on $E$-expansions.

math.NT

A Ramanujan bound for Drinfeld modular forms

We prove a Lefschetz trace formula for Böckle-Pink crystals on tame Deligne-Mumford stacks of finite type over $\mathbb{F}_q$ and apply it to the crystal associated to the universal Drinfeld module. Combined with the Eichler-Shimura theory developed by Böckle, this leads to a trace formula for Hecke operators on Drinfeld modular forms. As an application, we deduce a Ramanujan bound on the traces of Hecke operators.

math.NT

Computing Largest Subsets of Points Whose Convex Hulls have Bounded Area and Diameter

We study the problem of computing a convex region with bounded area and diameter that contains the maximum number of points from a given point set $P$. We show that this problem can be solved in $O(n^6k)$ time and $O(n^3k)$ space, where $n$ is the size of $P$ and $k$ is the maximum number of points in the found region. We experimentally compare this new algorithm with an existing algorithm that does the same but without the diameter constraint, which runs in $O(n^3k)$ time. For the new algorithm, we use different diameters. We use both synthetic data and data from an application in cancer detection, which motivated our research.

cs.CG

On Newton's identities in positive characteristic

Newton's identities provide a way to express elementary symmetric polynomials in terms of power polynomials over fields of characteristic zero. In this article, we study the failure of this relation in positive characteristic and what can be recovered. In particular, we show how one can write the elementary symmetric polynomials as rational functions in the power polynomials over any commutative unital ring.

math.CO

Learning with Confidence: Training Better Classifiers from Soft Labels

In supervised machine learning, models are typically trained using data with hard labels, i.e., definite assignments of class membership. This traditional approach, however, does not take the inherent uncertainty in these labels into account. We investigate whether incorporating label uncertainty, represented as discrete probability distributions over the class labels -- known as soft labels -- improves the predictive performance of classification models. We first demonstrate the potential value of soft label learning (SLL) for estimating model parameters in a simulation experiment, particularly for limited sample sizes and imbalanced data. Subsequently, we compare the performance of various wrapper methods for learning from both hard and soft labels using identical base classifiers. On real-world-inspired synthetic data with clean labels, the SLL methods consistently outperform hard label methods. Since real-world data is often noisy and precise soft labels are challenging to obtain, we study the effect that noisy probability estimates have on model performance. Alongside conventional noise models, our study examines four types of miscalibration that are known to affect human annotators. The results show that SLL methods outperform the hard label methods in the majority of settings. Finally, we evaluate the methods on a real-world dataset with confidence scores, where the SLL methods are shown to match the traditional methods for predicting the (noisy) hard labels while providing more accurate confidence estimates.

cs.LG

Generating the Ground Truth: Synthetic Data for Soft Label and Label Noise Research

In many real-world classification tasks, label noise is an unavoidable issue that adversely affects the generalization error of machine learning models. Additionally, evaluating how methods handle such noise is complicated, as the effect label noise has on their performance cannot be accurately quantified without clean labels. Existing research on label noise typically relies on either noisy or oversimplified simulated data as a baseline, into which additional noise with known properties is injected. In this paper, we introduce SYNLABEL, a framework designed to address these limitations by creating noiseless datasets informed by real-world data. SYNLABEL supports defining a pre-specified or learned function as the ground truth function, which can then be used for generating new clean labels. Furthermore, by repeatedly resampling values for selected features within the domain of the function, evaluating the function and aggregating the resulting labels, each data point can be assigned a soft label or label distribution. These distributions capture the inherent uncertainty present in many real-world datasets and enable the direct injection and quantification of label noise. The generated datasets serve as a clean baseline of adjustable complexity, into which various types of noise can be introduced. Additionally, they facilitate research into soft label learning and related applications. We demonstrate the application of SYNLABEL, showcasing its ability to precisely quantify label noise and its improvement over existing methodologies.

cs.LG