Subspace procrustes analysis

Postprint (author's final draft

Continuous optimization methods for the graph isomorphism problem

The graph isomorphism problem looks deceptively simple, but although polynomial-time algorithms exist for certain types of graphs such as planar graphs and graphs with bounded degree or eigenvalue multiplicity, its complexity class is still unknown. Information about potential isomorphisms between two graphs is contained in the eigenvalues and eigenvectors of their adjacency matrices. However, symmetries of graphs often lead to repeated eigenvalues so that associated eigenvectors are determined only up to basis rotations, which complicates graph isomorphism testing. We consider orthogonal and doubly stochastic relaxations of the graph isomorphism problem, analyze the geometric properties of the resulting solution spaces, and show that their complexity increases significantly if repeated eigenvalues exist. By restricting the search space to suitable subspaces, we derive an efficient Frank-Wolfe based continuous optimization approach for detecting isomorphisms. We illustrate the efficacy of the algorithm with the aid of various highly symmetric graphs

Sparse cointegration

Cointegration analysis is used to estimate the long-run equilibrium relations between several time series. The coefficients of these long-run equilibrium relations are the cointegrating vectors. In this paper, we provide a sparse estimator of the cointegrating vectors. The estimation technique is sparse in the sense that some elements of the cointegrating vectors will be estimated as zero. For this purpose, we combine a penalized estimation procedure for vector autoregressive models with sparse reduced rank regression. The sparse cointegration procedure achieves a higher estimation accuracy than the traditional Johansen cointegration approach in settings where the true cointegrating vectors have a sparse structure, and/or when the sample size is low compared to the number of time series. We also discuss a criterion to determine the cointegration rank and we illustrate its good performance in several simulation settings. In a first empirical application we investigate whether the expectations hypothesis of the term structure of interest rates, implying sparse cointegrating vectors, holds in practice. In a second empirical application we show that forecast performance in high-dimensional systems can be improved by sparsely estimating the cointegration relations