Mathematics (Faculty of)

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A 2-Approximation for the Height of Maximal Outerplanar Graph Drawings
(University of Waterloo, 2016-08-18) Demontigny, Philippe; Biedl, Therese
In this thesis, we study drawings of maximal outerplanar graphs that place vertices on integer coordinates. We introduce a new class of graphs, called umbrellas, and a new method of splitting maximal outerplanar graphs into systems of umbrellas. By doing so, we generate a new graph parameter, called the umbrella depth (ud), that can be used to approximate the optimal height of a drawing of a maximal outerplanar graph. We show that for any maximal outerplanar graph G, we can create a flat visibility representation of G with height at most 2ud(G) + 1. This drawing can be transformed into a straight-line drawing of the same height. We then prove that the height of any drawing of G is at least ud(G) + 1, which makes our result a 2-approximation for the optimal height. The best previously known approximation algorithm gave a 4-approximation. In addition, we provide an algorithm for finding the umbrella depth of G in linear time. Lastly, we compare the umbrella depth to other graph parameters such as the pathwidth and the rooted pathwidth, which have been used in the past for outerplanar graph drawing algorithms.
A 2-class maintenance model with dynamic server behavior
(Springer, 2019-04-29) Granville, Kevin; Drekic, Steve
We analyze a 2-class maintenance system within a single-server polling model framework. There are C+f machines in the system, where C is the cap on the number of machines that can be turned on simultaneously (and hence, be at risk of failure), and the excess f machines comprise a maintenance float which can be used to replace machines that are taken down for repair. The server’s behavior is dynamic, capable of switching queues upon a machine failure or service completion depending on both queue lengths. This generalized server behavior permits the analysis of several classic service policies, including preemptive resume priority, non-preemptive priority, and exhaustive. More complicated polices can also be considered, such as threshold-based ones and a version of the Bernoulli service rule. The system is modeled as a level-dependent quasi-birth-and-death process and matrix analytic methods are used to find the steady-state joint queue length distribution, as well as the distribution for the sojourn time of a broken machine. An upper bound on the expected number of working machines as a function of C is derived, and Little’s Law is used to find the relationship between the expected number of working machines and the expected sojourn time of a failed machine when f=0 or f≥1. Several numerical examples are presented, including how one might optimize an objective function depending on the mean number of working machines, with penalty costs attributed to increasing C or f.
2-crossing critical graphs with a V8 minor
(University of Waterloo, 2012-01-17T20:51:50Z) Austin, Beth Ann
The crossing number of a graph is the minimum number of pairwise crossings of edges among all planar drawings of the graph. A graph G is k-crossing critical if it has crossing number k and any proper subgraph of G has a crossing number less than k. The set of 1-crossing critical graphs is is determined by Kuratowski’s Theorem to be {K5, K3,3}. Work has been done to approach the problem of classifying all 2-crossing critical graphs. The graph V2n is a cycle on 2n vertices with n intersecting chords. The only remaining graphs to find in the classification of 2-crossing critical graphs are those that are 3-connected with a V8 minor but no V10 minor. This paper seeks to fill some of this gap by defining and completely describing a class of graphs called fully covered. In addition, we examine other ways in which graphs may be 2-crossing critical. This discussion classifies all known examples of 3-connected, 2-crossing critical graphs with a V8 minor but no V10 minor.
2-Semilattices: Residual Properties and Applications to the Constraint Satisfaction Problem
(University of Waterloo, 2017-08-22) Payne, Ian; Willard, Ross
Semilattices are algebras known to have an important connection to partially ordered sets. In particular, if a partially ordered set $(A,\leq)$ has greatest lower bounds, a semilattice $(A;\wedge)$ can be associated to the order where $a\wedge b$ is the greatest lower bound of $a$ and $b$. In this thesis, we study a class of algebras known as 2-semilattices, which is a generalization of the class of semilattices. Similar to the correspondence between partial orders and semilattices, there is a correspondence between certain digraphs and 2-semilattices. That is, to every 2-semilattice, there is an associated digraph which holds information about the 2-semilattice. Making frequent use of this correspondence, we explore the class of 2-semilattices from three perspectives: (i) Tame Congruence Theory, (ii) the ``residual character" of the class of 2-semilattices, and (iii), the constraint satisfaction problem. Tame Congruence Theory, developed in [29], is a structure theory on finite algebras driven by understanding their prime congruence quotients. The theory assigns to each such quotient a type from 1 to 5. We show that types 3, 4, and 5 can occur in the class of 2-semilattices, but type 4 can not occur in a finite simple 2-semilattice. Classes of algebras contain ``subdirectly irreducible" members which hold information about the class. Specifically, the size of these members has been of interest to many authors. We show for certain subclasses of the class of 2-semilattices that there is no cardinal bound on the size of the irreducible members in that subclass. The ``fixed template constraint satisfaction problem" can be identified with the decision problem hom$(\mathbb{A})$ where $\mathbb{A}$ is a fixed finite relational structure. The input to hom$(\mathbb{A})$ is a finite structure $\mathbb{B}$ similar to $\mathbb{A}$. The question asked is ``does there exist a homomorphism from $\mathbb{B}$ to $\mathbb{A}$?" Feder and Vardi [22] conjectured that for fixed $\mathbb{A}$, this decision problem is either NP-complete or solvable in polynomial time. Bulatov [15] confirmed this conjecture in the case that $\mathbb{A}$ is invariant under a 2-semilattice operation. We extend this result.
2D Surface-Only Liquid-Solid Coupling
(University of Waterloo, 2025-01-06) Kim, Clara; Batty, Christopher
Liquid simulations typically involve solving the fluid equations over the simulated liquid domain via some volumetric discretization of the domain. Consequently, the majority of schemes which aim to simulate the interaction between liquids and freely moving solids are built assuming a volumetrically discretized liquid model. However, storing and manipulating a surface mesh rather than a volumetric discretization, on top of allowing us to avoid storing interior volume data, has the potential to reduce the number of unknowns in the systems necessary to resolve fluid flow. Motivated by this potential, we present a method for simulating the two-way coupled interactions between solid rigid bodies and an inviscid liquid, where the liquid domain is represented and simulated entirely by its surface. Our work builds off of the surface-only liquids method first proposed by Da et al. [2016]. We are concerned with the 2D version of the liquid-solid coupling problem. As such, the liquid surface is represented as a series of point vertices connected by line segment edges, with the velocity data stored at the vertices. The surface-only liquid simulation method integrates outside forces, such as forces caused by scripted solids in contact with the liquid, by performing a boundary element method (BEM) solve for fluid pressures using surface tension and solid velocities to set boundary conditions. We perform liquid-solid coupling in a single unified solve by modifying this force integration step to account for solids with dynamic velocities by modeling the momentum exchange that occurs at the solid-liquid interface. We show several examples demonstrating our method’s ability to handle liquid-rigid body dynamics, as well as validate our method against analytical solutions derived using the fluid mechanics concept of added mass. We also demonstrate our method’s ability to support multiple solid rigid bodies interacting with each other through the liquid domain without need for direct contact between the solids. We hope that our work encourages further investigations into the surface-only liquids framework in the future, allowing for the simulation of an even wider range of interesting liquid phenomena using only a surface discretization of the simulated domain.
3-D Reconstruction from Single Projections, with Applications to Astronomical Images
(University of Waterloo, 2013-08-23T18:52:01Z) Cormier, Michael
A variety of techniques exist for three-dimensional reconstruction when multiple views are available, but less attention has been given to reconstruction when only a single view is available. Such a situation is normal in astronomy, when a galaxy (for example) is so distant that it is impossible to obtain views from significantly different angles. In this thesis I examine the problem of reconstructing the three-dimensional structure of a galaxy from this single viewpoint. I accomplish this by taking advantage of the image formation process, symmetry relationships, and other structural assumptions that may be made about galaxies. Most galaxies are approximately symmetric in some way. Frequently, this symmetry corresponds to symmetry about an axis of rotation, which allows strong statements to be made about the relationships between luminosity at each point in the galaxy. It is through these relationships that the number of unknown values needed to describe the structure of the galaxy can be reduced to the number of constraints provided by the image so the optimal reconstruction is well-defined. Other structural properties can also be described under this framework. I provide a mathematical framework and analyses that prove the uniqueness of solutions under certain conditions and to show how uncertainty may be precisely and explicitly expressed. Empirical results are shown using real and synthetic data. I also show a comparison to a state-of-the-art two-dimensional modelling technique to demonstrate the contrasts between the two frameworks and show the important advantages of the three-dimensional approach. In combination, the theoretical and experimental aspects of this thesis demonstrate that the proposed framework is versatile, practical, and novel---a contribution to both computer science and astronomy.
3D Online Multi-Object Tracking for Autonomous Driving
(University of Waterloo, 2019-08-29) Balasubramanian, Venkateshwaran; Czarnecki, Krzysztof
This research work focuses on exploring a novel 3D multi-object tracking architecture: 'FANTrack: 3D Multi-Object Tracking with Feature Association Network' for autonomous driving, based on tracking by detection and online tracking strategies using deep learning architectures for data association. The problem of multi-target tracking aims to assign noisy detections to a-priori unknown and time-varying number of tracked objects across a sequence of frames. A majority of the existing solutions focus on either tediously designing cost functions or formulating the task of data association as a complex optimization problem that can be solved effectively. Instead, we exploit the power of deep learning to formulate the data association problem as inference in a CNN. To this end, we propose to learn a similarity function that combines cues from both image and spatial features of objects. The proposed approach consists of a similarity network that predicts the similarity scores of the object pairs and builds a local similarity map. Another network formulates the data association problem as inference in a CNN by using the similarity scores and spatial information. The model learns to perform global assignments in 3D purely from data, handles noisy detections and a varying number of targets, and is easy to train. Experiments on the challenging Kitti dataset show competitive results with the state of the art. The model is finally implemented in ROS and deployed on our autonomous vehicle to show the robustness and online tracking capabilities. The proposed tracker runs alongside the object detector utilizing the resources efficiently.
3D Pointing with Everyday Devices: Speed, Occlusion, Fatigue
(University of Waterloo, 2015-07-24) Pietroszek, Krzysztof
In recent years, display technology has evolved to the point where displays can be both non-stereoscopic and stereoscopic, and 3D environments can be rendered realistically on many types of displays. From movie theatres and shopping malls to conference rooms and research labs, 3D information can be deployed seamlessly. Yet, while 3D environments are commonly displayed in desktop settings, there are virtually no examples of interactive 3D environments deployed within ubiquitous environments, with the exception of console gaming. At the same time, immersive 3D environments remain - in users' minds - associated with professional work settings and virtual reality laboratories. An excellent opportunity for 3D interactive engagements is being missed not because of economic factors, but due to the lack of interaction techniques that are easy to use in ubiquitous, everyday environments. In my dissertation, I address the lack of support for interaction with 3D environments in ubiquitous settings by designing, implementing, and evaluating 3D pointing techniques that leverage a smartphone or a smartwatch as an input device. I show that mobile and wearable devices may be especially beneficial as input devices for casual use scenarios, where specialized 3D interaction hardware may be impractical, too expensive or unavailable. Such scenarios include interactions with home theatres, intelligent homes, in workplaces and classrooms, with movie theatre screens, in shopping malls, at airports, during conference presentations and countless other places and situations. Another contribution of my research is to increase the potential of mobile and wearable devices for efficient interaction at a distance. I do so by showing that such interactions are feasible when realized with the support of a modern smartphone or smartwatch. I also show how multimodality, when realized with everyday devices, expands and supports 3D pointing. In particular, I show how multimodality helps to address the challenges of 3D interaction: performance issues related to the limitations of the human motor system, interaction with occluded objects and related problem of perception of depth on non-stereoscopic screens, and user subjective fatigue, measured with NASA TLX as perceived workload, that results from providing spatial input for a prolonged time. I deliver these contributions by designing three novel 3D pointing techniques that support casual, "walk-up-and-use" interaction at a distance and are fully realizable using off-the-shelf mobile and wearable devices available today. The contributions provide evidence that democratization of 3D interaction can be realized by leveraging the pervasiveness of a device that users already carry with them: a smartphone or a smartwatch.
5-Choosability of Planar-plus-two-edge Graphs
(University of Waterloo, 2018-01-02) Mahmoud, Amena; Richter, Bruce
We prove that graphs that can be made planar by deleting two edges are 5-choosable. To arrive at this, first we prove an extension of a theorem of Thomassen. Second, we prove an extension of a theorem Postle and Thomas. The difference between our extensions and the theorems of Thomassen and of Postle and Thomas is that we allow the graph to contain an inner 4-list vertex. We also use a colouring technique from two papers by Dvořák, Lidický and Škrekovski, and independently by Compos and Havet.
5G RAN/MEC Slicing and Admission Control using Deep Reinforcement Learning
(University of Waterloo, 2023-01-19) Moayyedi, Arash; Boutaba, Raouf
The 5G RAN functions can be virtualized and distributed across the radio unit (RU), distributed unit (DU), and centralized unit (CU) to facilitate flexible resource management. Complemented by multi-access edge computing (MEC), these components create network slices tailored for applications with diverse quality of service (QoS) requirements. However, as the requests for various slices arrive dynamically over time and the network resources are limited, it is non-trivial for an infrastructure provider (InP) to optimize its long-term revenue from real-time admission and embedding of slice requests. Prior works have leveraged Deep Reinforcement Learning (DRL) to address this problem, however, these solutions either do not scale to realistic topologies, require re-training of the DRL agents when facing topology changes, or do not consider the slice admission and embedding problems jointly. In this thesis, we use multi-agent DRL and Graph Attention Networks (GATs) to address these limitations. Specifically, we propose novel topology-independent admission and slicing agents that are scalable and generalizable to large and different metropolitan networks. Results show that the proposed approach converges faster and achieves up to 35.2% and 20% gain in revenue compared to heuristics and other DRL-based approaches, respectively. Additionally, we demonstrate that our approach is generalizable to scenarios and substrate networks previously unseen during training, as it maintains superior performance without re-training or re-tuning. Finally, we extract the attention maps of the GAT, and analyze them to detect potential bottlenecks and efficiently improve network performance and InP revenue through eliminating them.
A First-Principles Framework for Simulating Light and Snow Interactions
(University of Waterloo, 2025-02-25) Varsa, Petri Matthew; Baranoski, Gladimir
Interactions between light and matter give rise to an abundance of natural phenomena. Common examples include those between light and atmospheric ice crystals producing halos, and between light and liquid water droplets producing rainbows. However, interesting effects may also be observed when light impinges upon more dense materials such as snow. These may be noted as changes to the appearance of the material resulting from variations in the characteristics of the material itself. In some cases, these appearance changes may even manifest themselves as dramatic changes in colour. In this thesis, we study snow as a material and reproduce such phenomena by simulating light interactions with virtual snow samples. Accordingly, this work presents a first-principles framework for simulating light transport through snow. Data and information that describe the characteristics of snowpacks are obtained from the literature and used to devise a digital representation of them suitable for predicatively modelling light interactions with snow. The employed formulation allows for different virtual snow samples to be investigated. Simulated rays of light are cast into a virtual snow sample, and these rays are reflected and refracted until they exit from the surface of the sample, are transmitted through the sample or are absorbed. The modelling results are recorded as spectral response datasets for evaluation and analysis. These datasets are then compared with measured data and observations reported in the literature in order to assess the simulations’ fidelity. There are a number of study areas where such a framework can make a contribution. In this thesis, we discuss such contributions to two fields of research, namely, computer graphics and remote sensing. From a computer graphics perspective, the outputs of simulating light interactions with snow may be used in natural phenomena visualizations employed for educational and entertainment purposes. From a remote sensing perspective, the simulations may be used to conduct in silico experiments that help to shed light on topics that are actively being studied. Furthermore, the simulation outputs may also be used as data products in themselves, to make comparisons against remotely acquired data and support other modelling initiatives. The proposed framework presented in this thesis encapsulates a body of work that is expected to advance the state of the art of snow appearance modelling using a multi-faceted approach. The foundation of the framework is a novel radiative transfer model of light impingement on snow, whose predictive capabilities are extensively evaluated. Then, data products produced by this framework are used to address open questions in the two fields of interest, i.e., computer graphics and remote sensing. In particular, we describe a method to include the complex, visual phenomena that are predicted by the radiative transfer model introduced here into a traditional rendering pipeline. We also make use of the proposed framework to investigate open problems (e.g., the absorption of solar radiation by snow and the effect that this has on avalanche prediction) with potential interdisciplinary applications.
A Longitudinal Analysis Of Replicas in the Wild Wild Android
(University of Waterloo, 2024-09-24) Abbas Zaidi, Syeda Mashal; Aafer, Yousra
In this thesis, we report and study a phenomenon that contributes to Android API sprawls. We observe that OEM developers introduce private APIs that are composed by copy-paste-editing full or partial code from AOSP and other OEM APIs – we call such APIs, Replicas. To quantify the prevalence of Replicas in the wildly fragmented Android ecosystem, we perform the first large-scale (security) measurement study, aiming at detecting and evaluating Replicas across 342 ROMs, manufactured by 10 vendors and spanning 7 versions. Our study is motivated by the intuition that Replicas contribute to the production of bloated custom Android codebases, add to the complexity of the Android access control mechanism and updates process, and hence may lead to access control vulnerabilities. Our study is facilitated by RepFinder, a tool we develop. It infers the core functionality of an API and detects syntactically and semantically similar APIs using static program paths. RepFinder reveals that Replicas are commonly introduced by OEMs and more importantly, they unnecessarily introduce security enforcement anomalies. Specifically, RepFinder reports an average of 141 Replicas per the studied ROMs, accounting for 9% to 17% of custom APIs – where 37% (on average) are identified as under-protected. Our study thus points to the urgent need to debloat Replicas.
A Security Analysis of the Multi-User Ecosystem in Android Framework
(University of Waterloo, 2024-10-23) Khan, Muhammad Shahpar Nafees; Aafer, Yousra
The Android framework’s multi-user ecosystem introduces significant security challenges, particularly in the enforcement of user-specific access control checks. While previous research has highlighted flaws in Android’s access control mechanism, these efforts often overlook the complexities introduced by vendor customization and the unique demands of a multi-user environment. In this thesis, we conduct a systematic analysis of the Android Open Source Project (AOSP), identifying key patterns regulating multi-user access control implementations. We use these patterns to develop MVP, a static analysis tool that examines vendor ROMs for missing user-specific access control checks in custom ROMs. For example, our analysis reveals that Android’s multi-user environment is susceptible to cross-user attacks; sensitive data can be shared between profiles, and non-privileged users can manipulate privileged system settings. These findings underscore the need for rigorous enforcement of access control mechanisms to mitigate security risks in Android’s multi-user environment.
A Study of Statistical Methods for Modelling Longevity and Climate Risks
(University of Waterloo, 2025-03-27) Guo, Yiping
In recent years, two pivotal risks have emerged and taken a significant position in modern actuarial science: longevity risk and climate risk. Longevity risk, or the risk of individuals living longer than expected, poses a severe challenge to both private insurance companies and public pension systems, potentially destabilizing financial structures built on assumptions of life expectancy. On the other hand, climate risk, associated with fluctuations and extreme conditions in weather, has substantial implications for various sectors such as agriculture, energy, and insurance, particularly in the era of increasing climate change impacts. The Society of Actuaries (SOA) has recognized the growing importance of these risks, advocating for innovative research and solutions to manage them effectively. Furthermore, statistical modelling plays an indispensable role in understanding, quantifying, and managing these risks. The development of sophisticated and robust statistical methods enables practitioners and researchers to capture complex risk patterns and make reliable predictions, thereby informing risk management strategies. This thesis, composed of four distinct projects, explores statistical methods for modelling longevity and weather risk, contributing valuable insights to these fields. The first part in this thesis studies the statistical methods for modelling longevity risk, and in particular, modelling mortality rates. In the first chapter, we study parameter estimation of the Lee-Carter model and its multi-population extensions. Although the impact of outliers on stochastic mortality modelling has been examined, previous studies on this topic focus on how outliers in the estimated time-varying indexes may be detected and/or modelled, with little attention being paid to the adverse effects of outliers on estimation robustness, particularly that pertaining to age-specific parameters. In this chapter, we propose a robust estimation method for the Lee-Carter model, through a reformulation of the model into a probabilistic principal component analysis with multivariate t-distributions and an efficient expectation-maximization algorithm for implementation. The proposed method yields significantly more robust parameter estimates, while preserving the fundamental interpretation for the bilinear term in the model as the first principal component and the flexibility of pairing the estimated time-varying parameters with any appropriate time-series process. We also extend the proposed method for use with multi-population generalizations of the Lee-Carter model, allowing for a wider range of applications such as quantification of population basis risk in index-based longevity hedges. Using a combination of real and pseudo datasets, we demonstrate that the superiority of the proposed method relative to conventional estimation approaches such as singular value decomposition and maximum likelihood. Next, we move onto parameter estimation of the Renshaw-Haberman model, a cohort-based extension to the Lee-Carter model. In mortality modelling, cohort effects are often taken into consideration as they add insights about variations in mortality across different generations. Statistically speaking, models such as the Renshaw-Haberman model may provide a better fit to historical data compared to their counterparts that incorporate no cohort effects. However, when such models are estimated using an iterative maximum likelihood method in which parameters are updated one at a time, convergence is typically slow and may not even be reached within a reasonably established maximum number of iterations. Among others, the slow convergence problem hinders the study of parameter uncertainty through bootstrapping methods. In this chapter, we propose an intuitive estimation method that minimizes the sum of squared errors between actual and fitted log central death rates. The complications arising from the incorporation of cohort effects are overcome by formulating part of the optimization as a principal component analysis with missing values. Using mortality data from various populations, we demonstrate that our proposed method produces satisfactory estimation results and is significantly more efficient compared to the traditional likelihood-based approach. The third part of this thesis continues our exploration of the efficient computational algorithm of the Renshaw-Haberman model. Existing software packages and estimation algorithms often rely on maximum likelihood estimation with iterative Newton-Raphson methods, which can be computationally intensive and prone to convergence issues. In this chapter, we present the R package RHals, offering an efficient alternative with an alternating least squares method for fitting a generalized class of Renshaw-Haberman models, including configurations with multiple age-period terms. We extend this method to multi-population settings, allowing for shared or population-specific age effects under various configurations. The full modelling workflow and functionalities of RHals are demonstrated using mortality data from England and Wales. Lastly, we turn to modelling climate risk in the final chapter of the thesis. The use of weather index insurances is subject to spatial basis risk, which arises from the fact that the location of the user's risk exposure is not the same as the location of any of the weather stations where an index can be measured. To gauge the effectiveness of weather index insurances, spatial interpolation techniques such as kriging can be adopted to estimate the relevant weather index from observations taken at nearby locations. In this chapter, we study the performance of various statistical methods, ranging from simple nearest neighbor to more advanced trans-Gaussian kriging, in spatial interpolations of daily precipitations with data obtained from the US National Oceanic and Atmospheric Administration. We also investigate how spatial interpolations should be implemented in practice when the insurance is linked to popular weather indexes including annual consecutive dry days (CDD) and maximum five-day precipitation in one month (MFP). It is found that although spatially interpolating the raw weather variables on a daily basis is more sophisticated and computationally demanding, it does not necessarily yield superior results compared to direct interpolations of CDD/MFP on a yearly/monthly basis. This intriguing outcome can be explained by the statistical properties of the weather indexes and the underlying weather variables.
A Study of the Opportunities and Challenges of Using Edge Computing to Accelerate Cloud Applications
(University of Waterloo, 2025-02-18) Qadi, Hala; Al-Kiswany, Samer
I explore the viability of using edge clusters to host latency-sensitive applications and to run services that can improve end-to-end communication performance across both wide area networks (WANs) and 5G environments. The study examines the viability of using edge clusters in three scenarios: accelerating TCP communications through TCP splitting in 5G deployments, hosting an entire application-level service or the latency-sensitive part of an application on an edge cluster, and deploying a TCP splitting service on edge clusters to support WAN communication. I explore these scenarios while varying packet drop rates, communication stacks, congestion control protocols, and TCP buffer sizes. My findings bring new insights about these deployment scenarios. I show that edge computing, especially through TCP splitting, can significantly improve end-to-end communication performance over the classical communication stack. TCP splitting over the 5G communication stack does not bring any benefit and can reduce throughput. This is because of the unique characteristics of the 5G communication stack. Furthermore, over the classical communication stack, TCP splitting brings higher benefit for flows larger than 64 KB. These findings provide valuable insights into how edge clusters can accelerate TCP communication in different network environments and identify high-impact research ideas for future work.
A Survival-Driven Machine Learning Framework for Donor-Recipient Matching in Liver Transplantation: Predictive Ranking and Optimal Donor Profiling
(University of Waterloo, 2025-01-27) Wang, Yingke; He, Xi; Rambhatla, Sirisha
Liver transplantation is a life-saving treatment for patients with end-stage liver disease. However, donor organ scarcity and patient heterogeneity make finding the optimal donor-recipient matching a persistent challenge. Existing models and clinical scores are shown to be ineffective for large national datasets such as the United Network for Organ Sharing (UNOS). In this study, I present a comprehensive machine-learning-based approach to predict posttransplant survival probabilities at discrete clinical important time points and to derive a ranking score for donor-recipient compatibility. Furthermore, I developed a recipient-specific "optimal donor profile," enabling clinicians to quickly compare waiting-list patients to their ideal standard, streamlining allocation decisions. Empirical results demonstrate that my score’s discriminative performance outperforms traditional methods while maintaining clinical interpretability. I further validate that the top compatibility list generated by our proposed scoring method is non-trivial, demonstrating statistically significant differences from the list produced by the traditional approach. By integrating these advances into a cohesive framework, our approach supports more nuanced donor-recipient matching and facilitates practical decision-making in real-world clinical settings.
A+ Indexes: Highly Flexible Adjacency Lists in Graph Database Management Systems
(University of Waterloo, 2019-09-17) Khaliq, Shahid; Salihoglu, Semih
Adjacency lists are the most fundamental storage structure in existing graph database management systems (GDBMSs) to index input graphs. Adjacency lists are universally linked-list like per-vertex structures that allow access to a set of edges that are all adjacent to a vertex. In several systems, adjacency lists can also allow efficient access to subsets of a vertex’s adjacent edges that satisfy a fixed set of predicates, such as those that have the same label, and support a fixed set of ordering criteria, such as sorting by the ID of destination vertices of the edges. This thesis describes a highly-flexible indexing subsystem for GDBMSs, which consists of two components. The primary component called A+ indexes store adjacency lists, which compared to existing adjacency lists, provide flexibility to users in three aspects: (1) in addition to per-vertex adjacency lists, users can define per-edge adjacency lists; (2) users can define adjacency lists for sets of edges that satisfy a wide range of predicates; and (3) provide flexible sorting criteria. Indexes in existing GDBMS, such as adjacency list, B+ tree, or hash indexes, index as elements the vertices or edges in the input graph. The second component of our indexing sub-system is secondary B+ tree and bitmap indexes that index aggregate properties of adjacency lists in A+ indexes. Therefore, our secondary indexes effectively index adjacency lists as elements. We have implemented our indexing sub-system on top of the Graphflow GDBMS. We describe our indexes, the modifications we had to do to Graphflow’s optimizer, and our implementation. We provide extensive experiments demonstrating both the flexibility and efficiency of our indexes on a large suite of queries from several application domains.
Abelian, amenable operator algebras are similar to C∗ -algebras
(Duke University Press, 2016-12) Marcoux, Laurent W.; Popov, Alexey I.
Suppose that H is a complex Hilbert space and that ℬ(H) denotes the bounded linear operators on H. We show that every abelian, amenable operator algebra is similar to a C∗-algebra. We do this by showing that if 𝒜⊆ℬ(H) is an abelian algebra with the property that given any bounded representation ϱ:𝒜→ℬ(Hϱ) of 𝒜 on a Hilbert space Hϱ, every invariant subspace of ϱ(𝒜) is topologically complemented by another invariant subspace of ϱ(𝒜), then 𝒜 is similar to an abelian C∗-algebra.
Abstract and Explicit Constructions of Jacobian Varieties
(University of Waterloo, 2018-08-10) Urbanik, David; Satriano, Matthew; Jao, David
Abelian varieties, in particular Jacobian varieties, have long attracted interest in mathematics. Their influence pervades arithmetic geometry and number theory, and understanding their construction was a primary motivator for Weil in his work on developing new foundations for algebraic geometry in the 1930s and 1940s. Today, these exotic mathematical objects find applications in cryptography and computer science, where they can be used to secure confidential communications and factor integers in subexponential time. Although in many respects well-studied, working in concrete, explicit ways with abelian varieties continues to be difficult. The issue is that, aside from the case of elliptic curves, it is often difficult to find ways of modelling and understanding these objects in ways amenable to computation. Often, the approach taken is to work ``indirectly'' with abelian varieties, in particular with Jacobians, by working instead with divisors on their associated curves to simplify computations. However, properly understanding the mathematics underlying the direct approach --- why, for instance, one can view the degree zero divisor classes on a curve as being points of a variety --- requires sophisticated mathematics beyond what is usually understood by algorithms designers and even experts in computational number theory. A direct approach, where explicit polynomial and rational functions are given that define both the abelian variety and its group law, cannot be found in the literature for dimensions greater than two. In this thesis, we make two principal contributions. In the first, we survey the mathematics necessary to understand the construction of the Jacobian of a smooth algebraic curve as a group variety. In the second, we present original work with gives the first instance of explicit rational functions defining the group law of an abelian variety of dimension greater than two. In particular, we derive explicit formulas for the group addition on the Jacobians of hyperelliptic curves of every genus g, and so give examples of explicit rational formulas for the group law in every positive dimension.
Accelerating and Privatizing Diffusion Models
(University of Waterloo, 2023-08-17) Dockhorn, Tim; Yu, Yaoliang
Diffusion models (DMs) have emerged as a powerful class of generative models. DMs offer both state-of-the-art synthesis quality and sample diversity in combination with a robust and scalable learning objective. DMs rely on a diffusion process that gradually perturbs the data towards a normal distribution, while the neural network learns to denoise. Formally, the problem reduces to learning the score function, i.e., the gradient of the log-density of the perturbed data. The reverse of the diffusion process can be approximated by a differential equation, defined by the learned score function, and can therefore be used for generation when starting from random noise. In this thesis, we give a thorough and beginner-friendly introduction to DMs and discuss their history starting from early work on score-based generative models. Furthermore, we discuss connections to other statistical models and lay out applications of DMs, with a focus on image generative modeling. We then present CLD: a new DM based on critically-damped Langevin dynamics. CLD can be interpreted as running a joint diffusion in an extended space, where the auxiliary variables can be considered "velocities" that are coupled to the data variables as in Hamiltonian dynamics. We derive a novel score matching objective for CLD-based DMs and introduce a fast solver for the reverse diffusion process which is inspired by methods from the statistical mechanics literature. The CLD framework provides new insights into DMs and generalizes many existing DMs which are based on overdamped Langevin dynamics. Next, we present GENIE, a novel higher-order numerical solver for DMs. Many existing higher-order solvers for DMs built on finite difference schemes which break down in the large step size limit as approximations become too crude. GENIE, on the other hand, learns neural network-based models for higher-order derivatives whose precision do not depend on the step size. The additional networks in GENIE are implemented as small output heads on top of the neural backbone of the original DM, keeping the computational overhead minimal. Unlike recent sampling distillation methods that fundamentally alter the generation process in DMs, GENIE still solves the true generative differential equation, and therefore naturally enables applications such as encoding and guided sampling. The fourth chapter presents differentially private diffusion models (DPDMs), DMs trained with strict differential privacy guarantees. While modern machine learning models rely on increasingly large training datasets, data is often limited in privacy-sensitive domains. Generative models trained on sensitive data with differential privacy guarantees can sidestep this challenge, providing access to synthetic data instead. DPDMs enforce privacy by using differentially private stochastic gradient descent for training. We thoroughly study the design space of DPDMs and propose noise multiplicity, a simple yet powerful modification of the DM training objective tailored to the differential privacy setting. We motivate and show numerically why DMs are better suited for differentially private generative modeling than one-shot generators such as generative adversarial networks or normalizing flows. Finally, we propose to distill the knowledge of large pre-trained DMs into smaller student DMs. Large-scale DMs have achieved unprecedented results across several domains, however, they generally require a large amount of GPU memory and are slow at inference time, making it difficult to deploy them in real-time or on resource-limited devices. In particular, we propose an approximate score matching objective that regresses the student model towards predictions of the teacher DM rather than the clean data as is done in standard DM training. We show that student models outperform the larger teacher model for a variety of compute budgets. Additionally, the student models may also be deployed on GPUs with significantly less memory than was required for the original teacher model.

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