No abstract

In this article, we consider the manifold learning problem when the data set is invariant under the action of a compact Lie group K. Our approach consists in augmenting the data-induced graph Laplacian by integrating over the K-orbits of the existing data points, which yields a K-invariant graph Laplacian L. We prove that L can be diagonalized by using the unitary irreducible representation matrices of K, and we provide an explicit formula for computing its eigenvalues and eigenfunctions. In addition, we show that the normalized Laplacian operator LN converges to the Laplace-Beltrami operator of the data manifold with an improved convergence rate, where the improvement grows with the dimension of the symmetry group K. This work extends the steerable graph Laplacian framework of Landa and Shkolnisky from the case of SO(2) to arbitrary compact Lie groups.

In this article we develop an algorithm which computes a divisor of an integer N , which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If N has m distinct prime factors, then the probability that our algorithm runs successfully is at least p(m) = 1 − (m + 1)/2 m . We compute the computational complexity of the algorithm in terms of classical, or digital, steps and in terms of diffusion steps, which is a concept that we define here. As we will discuss below, we assert that a diffusion step can and should be considered as being comparable to a quantum step for an algorithm which runs on a quantum computer. With this, we prove that our factorization algorithm uses at most O((log N ) 2 ) deterministic steps and at most O((log N ) 2 ) diffusion steps with an implied constant which is effective. By comparison, Shor's algorithm is known to use at most O((log N ) 2 log(log N ) log(log log N )) quantum steps on a quantum computer.As an example of our algorithm, we simulate the diffusion computer algorithm on a desktop computer and obtain factorizations of N = 33 and N = 1363.

In this article we develop a general method by which one can explicitly evaluate certain sums of n-th powers of products of d ≥ 1 elementary trigonometric functions evaluated at m = (m 1 , . . . , m d )-th roots of unity. Our approach is to first identify the individual terms in the expression under consideration as eigenvalues of a discrete Laplace operator associated to a graph whose vertices form a d-dimensional discrete torus G m which depends on m. The sums in question are then related to the n-th step of a Markov chain on G m . The Markov chain admits the interpretation as a particular random walk, also viewed as a discrete time and discrete space heat diffusion, so then the sum in question is related to special values of the associated heat kernel. Our evaluation follows by deriving a combinatorial expression for the heat kernel, which is obtained by periodizing the heat kernel on the infinite lattice Z d which covers G m .