arXiv subjects
Publications and source records attributed to Isabel Detherage.
We prove both lower and upper bounds on the Ollivier-Ricci curvature of the basis exchange walk on a matroid. We give several examples of non-negatively curved basis exchange walks and negatively curved basis exchange walks.
This paper highlights a formal connection between two families of widely used matrix factorization algorithms in numerical linear algebra. One family consists of the Jacobi eigenvalue algorithm and its variants for computing the Hermitian eigendecomposition and singular value decomposition. The other consists of Gaussian elimination and the Gram-Schmidt procedure with various pivoting rules for computing the Cholesky decomposition and QR decomposition respectively. Both families are cast as special cases of a more general class of factorization algorithms. We provide a randomized pivoting rule that applies to this general class (which differs substantially from the usual pivoting rules for Gaussian elimination / Gram-Schmidt) which admits a unified analysis of the entire class of algorithms. The result is the same linear rate of convergence for each algorithm, irrespective of which factorization it computes. One important consequence of this randomized pivoting rule is a provable polynomial bound on the numerical stability of the Jacobi eigenvalue algorithm without any preconditioning, which addresses a longstanding open problem of Demmel and Veseli\'c `92.
This paper asks if the following iterative procedure approximately orthogonalizes a set of $n$ linearly independent unit vectors while preserving their span: in each iteration, access a random pair of vectors and replace one with the component perpendicular to the other, renormalized to be a unit vector. We provide a positive answer: any given set of starting vectors converges almost surely to an orthonormal basis of their span. We specifically argue that the $n$-volume of the parallelepiped generated by the vectors approaches 1 (i.e. the parallelepiped approaches a hypercube). If $A$ is the matrix formed by taking these vectors as columns, this volume is simply $\det(|A|)$ where $|A|=(A^*A)^{1/2}$. We show that $O(n^2\log(1/(\det(|A|)\varepsilon)))$ iterations suffice to bring ${\det(|A|)}$ above $1-\varepsilon$ with constant probability.
We study the infimum of the spectrum, or ground state energy (g.s.e.), of a discrete Schr\"odinger operator on $\theta\mathbb{Z}^d$ parameterized by a potential $V:\mathbb{R}^d\rightarrow\mathbb{R}_{\ge 0}$ and a frequency parameter $\theta\in (0,1)$. We relate this g.s.e. to that of a corresponding continuous semiclassical Schr\"odinger operator on $\mathbb{R}^d$ with parameter $\theta$, arising from the same choice of potential. We show that: the discrete g.s.e. is at most the continuous one for continuous periodic $V$ and irrational $\theta$; the opposite inequality holds up to a factor of $1-o(1)$ as $\theta\rightarrow 0$ for sufficiently regular smooth periodic $V$; and the opposite inequality holds up to a constant factor for every bounded $V$ and $\theta$ with the property that discrete and continuous averages of $V$ on fundamental domains of $\theta \mathbb{Z}^d$ are comparable. Our proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on $\theta \mathbb{Z}^d$ to low-energy functions for the continuous operator on $\mathbb{R}^d$, and vice versa.