The Paulsen problem is a basic open problem in operator theory: Given vectors @math that are @math -nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors @math that exactly satisfy the Parseval's condition and the equal norm condition? Given @math , the squared distance (to the set of exact solutions) is defined as @math where the infimum is over the set of exact solutions. Previous results show that the squared distance of any @math -nearly solution is at most @math and there are @math -nearly solutions with squared distance at least @math . The fundamental open question is whether the squared distance can be independent of the number of vectors @math . We answer this question affirmatively by proving that the squared distance of any @math -nearly solution is @math . Our approach is based on a continuous version of the operator scaling algorithm and consists of two parts. First, we define a dynamical system based on operator scaling and use it to prove that the squared distance of any @math -nearly solution is @math . Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an @math -nearly solution is @math when @math is large enough and @math is small enough. To analyze the convergence of the dynamical system, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyze the operator scaling algorithm.
We consider a fundamental problem in unsupervised learning called : given a collection of @math points in @math , if many but not necessarily all of these points are contained in a @math -dimensional subspace @math can we find it? The points contained in @math are called inliers and the remaining points are outliers . This problem has received considerable attention in computer science and in statistics. Yet efficient algorithms from computer science are not robust to adversarial outliers, and the estimators from robust statistics are hard to compute in high dimensions. Are there algorithms for subspace recovery that are both robust to outliers and efficient? We give an algorithm that finds @math when it contains more than a @math fraction of the points. Hence, for say @math this estimator is both easy to compute and well-behaved when there are a constant fraction of outliers. We prove that it is Small Set Expansion hard to find @math when the fraction of errors is any larger, thus giving evidence that our estimator is an optimal compromise between efficiency and robustness. As it turns out, this basic problem has a surprising number of connections to other areas including small set expansion, matroid theory and functional analysis that we make use of here. We prove a reverse form of the multidimensional Brascamp-Lieb inequality. Our method also gives a new way to derive the Brascamp-Lieb inequality and is rather convenient for the study of equality cases.  We prove a general lower bound on the complexity of unbounded error probabilistic communication protocols. This result improves on a lower bound for bounded error protocols from Krause (1996). As a simple consequence we get the, to our knowledge, first linear lower bound on the complexity of unbounded error probabilistic communication protocols for the functions defined by Hadamard matrices. We also give an upper bound on the margin of any embedding of a concept class in half spaces.