Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning

The paper introduces the application of information geometry to describe the ground states of Ising models by utilizing parity-check matrices of cyclic and quasi-cyclic codes on toric and spherical topologies. The approach establishes a connection between machine learning and error-correcting coding. This proposed approach has implications for the development of new embedding methods based on trapping sets. Statistical physics and number geometry applied for optimize error-correcting codes, leading to these embedding and sparse factorization methods. The paper establishes a direct connection between DNN architecture and error-correcting coding by demonstrating how state-of-the-art architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range arena can be equivalent to of block and convolutional LDPC codes (Cage-graph, Repeat Accumulate). QC codes correspond to certain types of chemical elements, with the carbon element being represented by the mixed automorphism Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix are elaborated upon in detail. The Quantum Approximate Optimization Algorithm (QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous to the back-propagation loss function landscape in training DNNs. This similarity creates a comparable problem with TS pseudo-codeword, resembling the belief propagation method. Additionally, the layer depth in QAOA correlates to the number of decoding belief propagation iterations in the Wiberg decoding tree. Overall, this work has the potential to advance multiple fields, from Information Theory, DNN architecture design (sparse and structured prior graph topology), efficient hardware design for Quantum and Classical DPU/TPU (graph, quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text overlap with arXiv:2109.08184 by other author

State Transfer & Strong Cospectrality in Cayley Graphs

This thesis is a study of two graph properties that arise from quantum walks: strong cospectrality of vertices and perfect state transfer. We prove various results about these properties in Cayley graphs. We consider how big a set of pairwise strongly cospectral vertices can be in a graph. We prove an upper bound on the size of such a set in normal Cayley graphs in terms of the multiplicities of the eigenvalues of the graph. We then use this to prove an explicit bound in cubelike graphs and more generally, Cayley graphs of $Z_2^{d_1} \times Z_4^{d_2}$. We further provide an infinite family of examples of cubelike graphs (Cayley graphs of $Z_2^d$ ) in which this set has size at least four, covering all possible values of $d$. We then look at perfect state transfer in Cayley graphs of abelian groups having a cyclic Sylow-2-subgroup. Given such a group, G, we provide a complete characterization of connection sets C such that the corresponding Cayley graph for G admits perfect state transfer. This is a generalization of a theorem of Ba\v{s}i\'{c} from 2013, where he proved a similar characterization for Cayley graphs of cyclic groups

Walks and games on graphs

Herrman, Rebekah Ph.D. The University of Memphis, May 2020. Walks and Games on Graphs. Major Professor: B\\u27ela Bollob\\u27as, Ph.D.Chapter 1 is joint work with Dr. Travis Humble and appears in the journal Physical Review A. In this work, we consider continuous-time quantum walks on dynamic graphs. Continuous-time quantum walks have been well studied on graphs that do not change as a function of time. We offer a mathematical formulation for how to express continuous-time quantum walks on graphs that can change in time, find a universal set of walks that can perform any operation, and use them to simulate basic quantum circuits. This work was supported in part by the Department of Energy Student Undergraduate Laboratory Internship and the National Science Foundation Mathematical Sciences Graduate Internship programs.The $(t,r)$ broadcast domination number of a graph $G$, $\gamma_{t,r}(G)$, is a generalization of the domination number of a graph. In Chapter 2, we consider the $(t,r)$ broadcast domination number on graphs, specifically powers of cycles, powers of paths, and infinite grids. This work is joint with Peter van Hintum and has been submitted to the journal Discrete Applied Mathematics.Bridge-burning cops and robbers is a variant of the cops and robbers game on graphs in which the robber removes an edge from the graph once it is traversed. In Chapter 3, we study the maximum time it takes the cops to capture the robber in this variant. This is joint with Peter van Hintum and Dr. Stephen Smith.In Chapter 4, we study a variant of the chip-firing game called the \emph{diffusion game}. In the diffusion game, we begin with some integer labelling of the vertices of a graph, interpreted as a number of chips on each vertex, and for each subsequent step every vertex simultaneously fires a chip to each neighbour with fewer chips. In general, this could result in negative vertex labels. Long and Narayanan asked whether there exists an $f(n)$ for each $n$, such that whenever we have a graph on $n$ vertices and an initial allocation with at least $f(n)$ chips on each vertex, then the number of chips on each vertex will remain non-negative. We answer their question in the affirmative, showing further that $f(n)=n-2$ is the best possible bound. We also consider the existence of a similar bound $g(d)$ for each $d$, where $d$ is the maximum degree of the graph. This work is joint with Andrew Carlotti and has been submitted to the journal Discrete Mathematics.In Chapter 5, we consider the eternal game chromatic number of random graphs. The eternal graph colouring problem, recently introduced by Klostermeyer and Mendoza \cite{klostermeyer}, is a version of the graph colouring game, where two players take turns properly colouring a graph. In this chapter, we show that with high probability $\chi_{g}^{\infty}(G_{n,p}) = (\frac{p}{2} + o(1))n$ for odd $n$, and also for even $n$ when $p=\frac{1}{k}$ for some $k \in \N$. This work is joint with Vojt\u{e}ch Dvo\u{r}\\u27ak and Peter van Hintum, and has been submitted to the European Journal of Combinatorics

Graph clustering by flow simulation

Dit proefschrift heeft als onderwerp het clusteren van grafen door middel van simulatie van stroming, een probleem dat in zijn algemeenheid behoort tot het gebied der clusteranalyse. In deze tak van wetenschap ontwerpt en onderzoekt men methoden die gegeven bepaalde data een onderverdeling in groepen genereren, waarbij het oogmerk is een onderverdeling in groepen te vinden die natuurlijk is. Dat wil zeggen dat verschillende data-elementen in dezelfde groep idealiter veel op elkaar lijken, en dat data-elementen uit verschillende groepen idealiter veel van elkaar verschillen. Soms ontbreken zulke groepjes helemaal; dan is er weinig patroon te herkennen in de data. Het idee is dat de aanwezigheid van natuurlijke groepjes het mogelijk maakt de data te categoriseren. Een voorbeeld is het clusteren van gegevens (over symptomen of lichaamskarakteristieken) van patienten die aan dezelfde ziekte lijden. Als er duidelijke groepjes bestaan in die gegevens, kan dit tot extra inzicht leiden in de ziekte. Clusteranalyse kan aldus gebruikt worden voor exploratief onderzoek. Verdere voorbeelden komen uit de scheikunde, taxonomie, psychiatrie, archeologie, marktonderzoek en nog vele andere disicplines. Taxonomie, de studie van de classificatie van organismen, heeft een rijke geschiedenis beginnend bij Aristoteles en culminerend in de werken van Linnaeus. In feite kan de clusteranalyse gezien worden als het resultaat van een steeds meer systematische en abstracte studie van de diverse methoden ontworpen in verschillende toepassingsgebieden, waarbij methode zowel wordt gescheiden van data en toepassingsgebied als van berekeningswijze. In de cluster analyse kunnen grofweg twee richtingen onderscheiden worden, naar gelang het type data dat geclassificeerd moet worden. De data-elementen in het voorbeeld hierboven worden beschreven door vectoren (lijstjes van scores of metingen), en het verschil tussen twee elementen wordt bepaald door het verschil van de vectoren. Deze dissertatie betreft cluster analyse toegepast op data van het type `graaf'. Voorbeelden komen uit de patroonherkenning, het computer ondersteund ontwerpen, databases voorzien van hyperlinks en het World Wide Web. In al deze gevallen is er sprake van `punten' die verbonden zijn of niet. Een stelsel van punten samen met hun verbindingen heet een graaf. Een goede clustering van een graaf deelt de punten op in groepjes zodanig dat er weinig verbindingen lopen tussen (punten uit) verschillende groepjes en er veel verbindingen zijn in elk groepje afzonderlijk

Consensus problems and the effects of graph topology in collaborative control

In this dissertation, several aspects of design for networked systems are addressed. The main focus is on combining approaches from system theory and graph theory to characterize graph topologies that result in efficient decision making and control. In this framework, modelling and design of sparse graphs that are robust to failures and provide high connectivity are considered. A decentralized approach to path generation in a collaborative system is modelled using potential functions. Taking inspiration from natural swarms, various behaviors of the system such as target following, moving in cohesion and obstacle avoidance are addressed by appropriate encoding of the corresponding costs in the potential function and using gradient descent for minimizing the energy function. Different emergent behaviors emerge as a result of varying the weights attributed with different components of the potential function. Consensus problems are addressed as a unifying theme in many collaborative control problems and their robustness and convergence properties are studied. Implications of the continuous convergence property of consensus problems on their reachability and robustness are studied. The effects of link and agent faults on consensus problems are also investigated. In particular the concept of invariant nodes has been introduced to model the effect of nodes with different behaviors from regular nodes. A fundamental association is established between the structural properties of a graph and the performance of consensus algorithms running on them. This leads to development of a rigorous evaluation of the topology effects and determination of efficient graph topologies. It is well known that graphs with large diameter are not efficient as far as the speed of convergence of distributed algorithms is concerned. A challenging problem is to determine a minimum number of long range links (shortcuts), which guarantees a level of enhanced performance. This problem is investigated here in a stochastic framework. Specifically, the small world model of Watts and Strogatz is studied and it is shown that adding a few long range edges to certain graph topologies can significantly increase both the rate of convergence for consensus algorithms and the number of spanning trees in the graph. The simulations are supported by analytical stochastic methods inspired from perturbations of Markov chains. This approach is further extended to a probabilistic framework for understanding and quantifying the small world effect on consensus convergence rates: Time varying topologies, in which each agent nominally communicates according to a predefined topology, and switching with non-neighboring agents occur with small probability is studied. A probabilistic framework is provided along with fundamental bounds on the convergence speed of consensus problems with probabilistic switching. The results are also extended to the design of robust topologies for distributed algorithms. The design of a semi-distributed two-level hierarchical network is also studied, leading to improvement in the performance of distributed algorithms. The scheme is based on the concept of social degree and local leader selection and the use of consensus-type algorithms for locally determining topology information. Future suggestions include adjusting our algorithm towards a fully distributed implementation. Another important aspect of performance in collaborative systems is for the agents to send and receive information in a manner that minimizes process costs, such as estimation error and the cost of control. An instance of this problem is addressed by considering a collaborative sensor scheduling problem. It is shown that in finding the optimal joint estimates, the general tree-search solution can be efficiently solved by devising a method that utilizes the limited processing capabilities of agents to significantly decrease the number of search hypotheses

Structured Compressed Sensing Using Deterministic Sequences

The problem of estimating sparse signals based on incomplete set of noiseless or noisy measurements has been investigated for a long time from different perspec- tives. In this dissertation, after the review of the theory of compressed sensing (CS) and existing structured sensing matrices, a new class of convolutional sensing matri- ces based on deterministic sequences are developed in the first part. The proposed matrices can achieve a near optimal bound with O(K log(N)) measurements for non-uniform recovery. Not only are they able to approximate compressible signals in the time domain, but they can also recover sparse signals in the frequency and discrete cosine transform domain. The candidates of the deterministic sequences include maximum length sequence (or called m-sequence), Golay's complementary sequence and Legendre sequence etc., which will be investigated respectively. In the second part, Golay-paired Hadamard matrices are introduced as structured sensing matrices, which are constructed from the Hadamard matrix, followed by diagonal Golay sequences. The properties and performances are analyzed in the following. Their strong structures ensure special isometry properties, and make them be easier applicable to hardware potentially. Finally, we exploit novel CS principles successfully in a few real applications, including radar imaging and dis- tributed source coding. The performance and the effectiveness of each scenario are verified in both theory and simulations

Application of constrained optimisation techniques in electrical impedance tomography

A Constrained Optimisation technique is described for the reconstruction of temporal resistivity images. The approach solves the Inverse problem by optimising a cost function under constraints, in the form of normalised boundary potentials. Mathematical models have been developed for two different data collection methods for the chosen criterion. Both of these models express the reconstructed image in terms of one dimensional (I-D) Lagrange multiplier functions. The reconstruction problem becomes one of estimating these 1-D functions from the normalised boundary potentials. These models are based on a cost criterion of the minimisation of the variance between the reconstructed resistivity distribution and the true resistivity distribution. The methods presented In this research extend the algorithms previously developed for X-ray systems. Computational efficiency is enhanced by exploiting the structure of the associated system matrices. The structure of the system matrices was preserved in the Electrical Impedance Tomography (EIT) implementations by applying a weighting due to non-linear current distribution during the backprojection of the Lagrange multiplier functions. In order to obtain the best possible reconstruction it is important to consider the effects of noise in the boundary data. This is achieved by using a fast algorithm which matches the statistics of the error in the approximate inverse of the associated system matrix with the statistics of the noise error in the boundary data. This yields the optimum solution with the available boundary data. Novel approaches have been developed to produce the Lagrange multiplier functions. Two alternative methods are given for the design of VLSI implementations of hardware accelerators to improve computational efficiencies. These accelerators are designed to implement parallel geometries and are modelled using a verification description language to assess their performance capabilities

Projections et distances discrètes

Le travail se situe dans le domaine de la géométrie discrète. La tomographie discrète sera abordée sous l'angle de ses liens avec la théorie de l'information, illustrés par l'application de la transformation Mojette et de la "Finite Radon Transform" au codage redondant d'information pour la transmission et le stockage distribué. Les distances discrètes seront exposées selon les points de vue théorique (avec une nouvelle classe de distances construites par des chemins à poids variables) et algorithmique (transformation en distance, axe médian, granulométrie) en particulier par des méthodes en un balayage d'image (en "streaming"). Le lien avec les séquences d'entiers non-décroissantes et l'inverse de Lambek-Moser sera mis en avant