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Carol Wu

Publications and source records attributed to Carol Wu.

6 recordsLinked to original sources
arXiv2026

Expressivity of Shallow Neural Networks Over Finite Fields

We study the expressivity of shallow polynomial neural networks (PNNs) with monomial activation functions over finite fields. For a given architecture, we define a neuromanifold as the image of the map from all possible network weights into the product of polynomial rings. We quantify the expressivity by the cardinality of the neuromanifold, and derive a natural lower and upper bound. This leads to counting rational points over finite fields, a problem closely linked to the Weil conjectures. Finally, we present an architecture that exhibits a striking difference in the neuromanifolds when considered over a characteristic zero versus a finite-characteristic field, illustrating the critical role of field characteristic in the notion of expressivity.

math.AG↗
arXiv2026

Generalization of Pearl's Front-Door Criterion

Pearl's front-door criterion provides a set of sufficient conditions for estimating the total causal effect from observational data in the presence of latent confounding, using the functional P(y | do(x := x*)) = \sum_z P(z | x*) \sum_x P(y | x, z) P(x). An open question is whether these conditions can be generalized to be both necessary and sufficient for the validity of this functional, similar to the generalization achieved for the back-door adjustment criterion by Shpitser. In this paper, we present a new, weakened set of graph-based conditions sufficient for the front-door formula to estimate the total causal effect, expanding the scope of problems amenable to front-door identification.

math.ST↗
arXiv2025

MAARTA:Multi-Agentic Adaptive Radiology Teaching Assistant

Radiology students often struggle to develop perceptual expertise due to limited expert mentorship time, leading to errors in visual search and diagnostic interpretation. These perceptual errors, such as missed fixations, short dwell times, or misinterpretations, are not adequately addressed by current AI systems, which focus on diagnostic accuracy but fail to explain how and why errors occur. To address this gap, we introduce MAARTA (Multi-Agentic Adaptive Radiology Teaching Assistant), a multi-agent framework that analyzes gaze patterns and radiology reports to provide personalized feedback. Unlike single-agent models, MAARTA dynamically selects agents based on error complexity, enabling adaptive and efficient reasoning. By comparing expert and student gaze behavior through structured graphs, the system identifies missed findings and assigns Perceptual Error Teacher agents to analyze discrepancies. MAARTA then uses step-by-step prompting to help students understand their errors and improve diagnostic reasoning, advancing AI-driven radiology education.

cs.CY↗
arXiv2024

On smooth gaps between primes using the Maynard-Tao sieve

In 1999, Balog, Brüdern, and Wooley (1999) showed there are infinitely many prime gaps $p-q$ that are $(\log p)^{\frac{3}{4}}$-smooth, and infinitely many consecutive prime gaps that are $(\log p)^\frac{7}{8}$-smooth. Advancements made since then by Zhang (2014), Maynard (2014), and Polymath8b (2014) towards resolving the twin prime conjecture have given us the tools to lower the bounds made by Balog, Brüdern, and Wooley to 47. Moreover, we can show there are infinitely many $m$-tuples of primes whose gaps are all $y_m$-smooth for a calculable prime $y_m$.

math.NT↗
arXiv2024

Higher a-numbers in $\mathbf{Z}_p$-towers via Counting Lattice Points

Booher, Cais, Kramer-Miller and Upton study a class of $\mathbf{Z}_p$-tower of curves in characteristic $p$ with ramification controlled by an integer $d$. In the special case that $d$ divides $p-1$, they prove a formula for the higher $a$-numbers of these curves involving the number of lattice points in a complicated region of the plane. Booher and Cais had previously conjectured that for $n$ sufficiently large the higher $a$-numbers of the $n$th curve are given by formulae of the form $\alpha(n) p^{2n} + \beta(n) p^n + \lambda_r(n) n + \nu(n) $, where $\alpha,\beta,\nu,\lambda_r$ are periodic functions of $n$. This is an example of a new kind of Iwasawa theory. We establish this conjecture by carefully studying these lattice points.

math.NT↗