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We present an entirely new geometric and probabilistic approach to synchronization of correspondences across multiple sets of objects or images. In particular, we present two algorithms: (1) Birkhoff-Riemannian L-BFGS for optimizing the…
Let B be an n by n doubly substochastic matrix. We show that B can be written as a convex combination of no more than {\sigma}(B)+t subpermutation matrices, where {\sigma}(B) is the number of nonzero elements in B and t is the number of…
The Birkhoff's theorem states that any doubly stochastic matrix lies inside a convex polytope with the permutation matrices at the corners. It can be proven that a similar theorem holds for unitary matrices with equal line sums for prime…
We consider quadratic optimization in variables $(x,y)$ where $0\le x\le y$, and $y\in\{0,1\}^n$. Such binary $y$ are commonly refered to as "indicator" or "switching" variables and occur commonly in applications. One approach to such…
We consider the problem of computing a sequence of rankings that maximizes consumer-side utility while minimizing producer-side individual unfairness of exposure. While prior work has addressed this problem using linear or quadratic…
Over the last two decades, pseudospectral methods based on Lagrange interpolants have flourished in solving trajectory optimization problems and their flight implementations. In a seemingly unjustified departure from these highly successful…
Convex optimization is a well-established research area with applications in almost all fields. Over the decades, multiple approaches have been proposed to solve convex programs. The development of interior-point methods allowed solving a…
This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of…
We propose and develop a novel framework for analyzing permutation-based combinatorial optimization problems, which could eventually be extended to other types of problems. Our approach is based on the decomposition of the objective…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
The still-unsolved problem of determining the set of eigenvalues realized by $n$-by-$n$ doubly stochastic matrices, those matrices with row sums and column sums equal to $1$, has attracted much attention in the last century. This problem is…
In this paper, we consider convex quadratic optimization problems with indicators on the continuous variables. In particular, we assume that the Hessian of the quadratic term is a Stieltjes matrix, which naturally appears in sparse…
It was shown recently that the constraints on the initial data for Einstein's equations may be posed as an evolutionary problem [9]. In one of the proposed two methods the constraints can be replaced by a first order symmetrizable…
The standard simplex in R^n, also known as the probability simplex, is the set of nonnegative vectors whose entries sum up to 1. They frequently appear as constraints in optimization problems that arise in machine learning, statistics, data…
Continuously extending combinatorial optimization objectives is a powerful technique commonly applied to the optimization of set functions. However, few such methods exist for extending functions on permutations, despite the fact that many…
We investigate how sorting algorithms efficiently overcome the exponential size of the permutation space. Our main contribution is a new continuous-time formulation of sorting as a gradient flow on the permutohedron, yielding an independent…
Recently, Sinkhorn's algorithm was applied for approximately solving linear programs emerging from optimal transport very efficiently. This was accomplished by formulating a regularized version of the linear program as Bregman projection…
It is well known that the permutahedron Pi_n has 2^n-2 facets. The Birkhoff polytope provides a symmetric extended formulation of Pi_n of size Theta(n^2). Recently, Goemans described a non-symmetric extended formulation of Pi_n of size…
We present an explicit closed-form formula for the vertices of the classical cut polytope $\operatorname{CUT}(n)$, defined as the convex hull of cut vectors of the complete graph $K_n$. Our derivation proceeds via a related polytope,…
Conical zeta values associated with rational convex polyhedral cones generalise multiple zeta values. We renormalise conical zeta values at poles by means of a generalisation of Connes and Kreimer's Algebraic Birkhoff Factorisation. This…