24 research outputs found
Hearing the clusters in a graph: A distributed algorithm
We propose a novel distributed algorithm to cluster graphs. The algorithm
recovers the solution obtained from spectral clustering without the need for
expensive eigenvalue/vector computations. We prove that, by propagating waves
through the graph, a local fast Fourier transform yields the local component of
every eigenvector of the Laplacian matrix, thus providing clustering
information. For large graphs, the proposed algorithm is orders of magnitude
faster than random walk based approaches. We prove the equivalence of the
proposed algorithm to spectral clustering and derive convergence rates. We
demonstrate the benefit of using this decentralized clustering algorithm for
community detection in social graphs, accelerating distributed estimation in
sensor networks and efficient computation of distributed multi-agent search
strategies
A Spectral Assignment Approach for the Graph Isomorphism Problem
In this paper, we propose algorithms for the graph isomorphism (GI) problem
that are based on the eigendecompositions of the adjacency matrices. The
eigenvalues of isomorphic graphs are identical. However, two graphs and
can be isospectral but non-isomorphic. We first construct a graph
isomorphism testing algorithm for friendly graphs and then extend it to
unambiguous graphs. We show that isomorphisms can be detected by solving a
linear assignment problem. If the graphs possess repeated eigenvalues, which
typically correspond to graph symmetries, finding isomorphisms is much harder.
By repeatedly perturbing the adjacency matrices and by using properties of
eigenpolytopes, it is possible to break symmetries of the graphs and
iteratively assign vertices of to vertices of , provided that an
admissible assignment exists. This heuristic approach can be used to construct
a permutation which transforms into if the graphs are
isomorphic. The methods will be illustrated with several guiding examples
Efficient Quantum Algorithms for Nonlinear Stochastic Dynamical Systems
In this paper, we propose efficient quantum algorithms for solving nonlinear
stochastic differential equations (SDE) via the associated Fokker-Planck
equation (FPE). We discretize the FPE in space and time using two well-known
numerical schemes, namely Chang-Cooper and implicit finite difference. We then
compute the solution of the resulting system of linear equations using the
quantum linear systems algorithm. We present detailed error and complexity
analyses for both these schemes and demonstrate that our proposed algorithms,
under certain conditions, provably compute the solution to the FPE within
prescribed error bounds with polynomial dependence on state
dimension . Classical numerical methods scale exponentially with dimension,
thus, our approach, under the aforementioned conditions, provides an
\emph{exponential speed-up} over traditional approaches.Comment: IEEE International Conference on Quantum Computing and Engineering
(QCE23
A Koopman framework for rare event simulation in stochastic differential equations
We exploit the relationship between the stochastic Koopman operator and the
Kolmogorov backward equation to construct importance sampling schemes for
stochastic differential equations. Specifically, we propose using
eigenfunctions of the stochastic Koopman operator to approximate the Doob
transform for an observable of interest (e.g., associated with a rare event)
which in turn yields an approximation of the corresponding zero-variance
importance sampling estimator. Our approach is broadly applicable and
systematic, treating non-normal systems, non-gradient systems, and systems with
oscillatory dynamics or rank-deficient noise in a common framework. In
nonlinear settings where the stochastic Koopman eigenfunctions cannot be
derived analytically, we use dynamic mode decomposition (DMD) methods to
compute them numerically, but the framework is agnostic to the particular
numerical method employed. Numerical experiments demonstrate that even coarse
approximations of a few eigenfunctions, where the latter are built from
non-rare trajectories, can produce effective importance sampling schemes for
rare events
math-PVS: A Large Language Model Framework to Map Scientific Publications to PVS Theories
As artificial intelligence (AI) gains greater adoption in a wide variety of
applications, it has immense potential to contribute to mathematical discovery,
by guiding conjecture generation, constructing counterexamples, assisting in
formalizing mathematics, and discovering connections between different
mathematical areas, to name a few.
While prior work has leveraged computers for exhaustive mathematical proof
search, recent efforts based on large language models (LLMs) aspire to position
computing platforms as co-contributors in the mathematical research process.
Despite their current limitations in logic and mathematical tasks, there is
growing interest in melding theorem proving systems with foundation models.
This work investigates the applicability of LLMs in formalizing advanced
mathematical concepts and proposes a framework that can critically review and
check mathematical reasoning in research papers. Given the noted reasoning
shortcomings of LLMs, our approach synergizes the capabilities of proof
assistants, specifically PVS, with LLMs, enabling a bridge between textual
descriptions in academic papers and formal specifications in PVS. By harnessing
the PVS environment, coupled with data ingestion and conversion mechanisms, we
envision an automated process, called \emph{math-PVS}, to extract and formalize
mathematical theorems from research papers, offering an innovative tool for
academic review and discovery
Spectral Complexity of Directed Graphs and Application to Structural Decomposition
We introduce a new measure of complexity (called spectral complexity) for
directed graphs. We start with splitting of the directed graph into its
recurrent and non-recurrent parts. We define the spectral complexity metric in
terms of the spectrum of the recurrence matrix (associated with the reccurent
part of the graph) and the Wasserstein distance. We show that the total
complexity of the graph can then be defined in terms of the spectral
complexity, complexities of individual components and edge weights. The
essential property of the spectral complexity metric is that it accounts for
directed cycles in the graph. In engineered and software systems, such cycles
give rise to sub-system interdependencies and increase risk for unintended
consequences through positive feedback loops, instabilities, and infinite
execution loops in software. In addition, we present a structural decomposition
technique that identifies such cycles using a spectral technique. We show that
this decomposition complements the well-known spectral decomposition analysis
based on the Fiedler vector. We provide several examples of computation of
spectral and total complexities, including the demonstration that the
complexity increases monotonically with the average degree of a random graph.
We also provide an example of spectral complexity computation for the
architecture of a realistic fixed wing aircraft system.Comment: We added new theoretical results in Section 2 and introduced a new
section 2.2 devoted to intuitive and physical explanations of the concepts
from the pape