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William Watkins

Publications and source records attributed to William Watkins.

6 recordsLinked to original sources
arXiv2026

A unimodular bijection between harmonic vectors of 2-isomorphic graphs

Let $G$ and $H$ be connected graphs that are 2-isomorphic. It is known that their Laplacian matrices are congruent by a unimodular matrix $U$. In this paper we show (Thm. \ref{thm:main2}) that $U$ is a bijection between certain spaces of harmonic vectors on the vertices of $G$ and $H$. In particular (Cor. \ref{cor:main1}) if $u$ is a harmonic vector with respect to vertices $c, d$ in $H$ and the 2-isomorphism maps edge $(a,b)$ in $G$ to edge $(c,d)$ in $H$, then $uU$ is a harmonic vector with respect to vertices $a, b$ in $G$.

math.CO↗
arXiv2024

Quantum Privacy Aggregation of Teacher Ensembles (QPATE) for Privacy-preserving Quantum Machine Learning

The utility of machine learning has rapidly expanded in the last two decades and presents an ethical challenge. Papernot et. al. developed a technique, known as Private Aggregation of Teacher Ensembles (PATE) to enable federated learning in which multiple teacher models are trained on disjoint datasets. This study is the first to apply PATE to an ensemble of quantum neural networks (QNN) to pave a new way of ensuring privacy in quantum machine learning (QML) models.

quant-ph↗
arXiv2021

Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling

This paper introduces a framework for combining scientific knowledge of physics-based models with neural networks to advance scientific discovery. This framework, termed physics-guided neural networks (PGNN), leverages the output of physics-based model simulations along with observational features in a hybrid modeling setup to generate predictions using a neural network architecture. Further, this framework uses physics-based loss functions in the learning objective of neural networks to ensure that the model predictions not only show lower errors on the training set but are also scientifically consistent with the known physics on the unlabeled set. We illustrate the effectiveness of PGNN for the problem of lake temperature modeling, where physical relationships between the temperature, density, and depth of water are used to design a physics-based loss function. By using scientific knowledge to guide the construction and learning of neural networks, we are able to show that the proposed framework ensures better generalizability as well as scientific consistency of results. All the code and datasets used in this study have been made available on this link \url{https://github.com/arkadaw9/PGNN}.

cs.LG↗
arXiv2018

A note on Dehn colorings and invariant factors

If $A$ is an abelian group and $ϕ$ is an integer, let $A(ϕ)$ be the subgroup of $A$ consisting of elements $a \in A$ such that $ϕ\cdot a=0$. We prove that if $D$ is a diagram of a classical link $L$ and $0=ϕ_0,ϕ_1,\dots,ϕ_{n-1}$ are the invariant factors of an adjusted Goeritz matrix of $D$, then the group $\mathcal{D}_{A}(D)$ of Dehn colorings of $D$ with values in $A$ is isomorphic to the direct product of $A$ and $A=A(ϕ_{0}),A(ϕ_1),\dots,A(ϕ_{n-1})$. It follows that the Dehn coloring groups of $L$ are isomorphic to those of a connected sum of torus links $T_{(2,ϕ_1)} \text{ }\# \text{ } \cdots \text{ } \# \text{ } T_{(2,ϕ_{n-1})}$.

math.GT↗
arXiv2009

Sum of squares of degrees in a graph

Let $\G(v,e)$ be the set of all simple graphs with $v$ vertices and $e$ edges and let $P_2(G)=\sum d_i^2$ denote the sum of the squares of the degrees, $d_1, >..., d_v$, of the vertices of $G$. It is known that the maximum value of $P_2(G)$ for $G \in \G(v,e)$ occurs at one or both of two special graphs in $\G(v,e)$--the \qs graph or the \qc graph. For each pair $(v,e)$, we determine which of these two graphs has the larger value of $P_2(G)$. We also determine all pairs $(v,e)$ for which the values of $P_2(G)$ are the same for the \qs and the \qc graph. In addition to the \qs and \qc graphs, we find all other graphs in $\G(v,e)$ for which the maximum value of $P_2(G)$ is attained. Density questions posed by previous authors are examined.

math.CO↗