Related papers: Unifying and generalizing known lower bounds via g…
The Gilbert-Varshamov (GV) lower bound on the maximum cardinality of a q-ary code of length n with minimum Hamming distance at least d can be obtained by application of Turan's theorem to the graph with vertex set {0,1,..,q-1}^n in which…
We argue that parameterized complexity is a useful tool with which to study global constraints. In particular, we show that many global constraints which are intractable to propagate completely have natural parameters which make them…
One goal of geometric measure theory is to understand how measures in the plane or higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises between the class of rectifiable measures,…
Motivated by the concept of Euclidean Distance Degree, which measures the complexity of finding the nearest point to an algebraic set in Euclidean space, we introduce the notion of Grassmann Distance Complexity (GDC). This concept…
Generalized matrix approximation plays a fundamental role in many machine learning problems, such as CUR decomposition, kernel approximation, and matrix low rank approximation. Especially with today's applications involved in larger and…
This study introduces a new unified structural framework for orbifold sigma models that incorporates twisted sectors, singularities, and smooth regions into a single algebraic object. Traditional approaches to orbifold theories often treat…
This paper presents and analyzes the first matrix optimization model which allows general coordinate and spectral constraints. The breadth of problems our model covers is exemplified by a lengthy list of examples from the literature,…
For a manifold embedded in an inner product space, we express geometric quantities such as {\it Hamilton vector fields, affine and Levi-Civita connections, curvature} in global coordinates. Instead of coordinate indices, the global formulas…
The article builds on several recent advances in the Monge-Kantorovich theory of mass transport which have -- among other things -- led to new and quite natural proofs for a wide range of geometric inequalities such as the ones formulated…
Composite minimization is a powerful framework in large-scale convex optimization, based on decoupling of the objective function into terms with structurally different properties and allowing for more flexible algorithmic design. We…
When the CG method for solving linear algebraic systems was formulated about 70 years ago by Lanczos, Hestenes, and Stiefel, it was considered an iterative process possessing a mathematical finite termination property. CG was placed into a…
We classify the allowed structures of the discrete 1-form gauge sector in six-dimensional supergravity theories realized as F-theory compactifications. This provides upper bounds on the 1-form gauge factors $\mathbb{Z}_m$ and in particular…
Nesterov's accelerated gradient (AG) method for minimizing a smooth strongly convex function $f$ is known to reduce $f({\bf x}_k)-f({\bf x}^*)$ by a factor of $\epsilon\in(0,1)$ after $k=O(\sqrt{L/\ell}\log(1/\epsilon))$ iterations, where…
We consider a general class of spin systems with potentially unbounded real-valued spins, defined via a single-site potential with super-Gaussian tails on general graphs, allowing for both short- and long-range interactions. This class…
We develop a novel approach to gravity that we call `matrix general relativity' (MGR) or `gravitational chromodynamics' (GCD or GQCD for quantum version). Gravity is described in this approach not by one Riemannian metric (i.e. a symmetric…
We discuss a link between graph theory and geometry that arises when considering graph dynamical systems with odd interactions. The equilibrium set in such systems is not a collection of isolated points, but rather a union of manifolds,…
This paper proves strong lower bounds for distributed computing in the CONGEST model, by presenting the bit-gadget: a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing…
Although machine learning is increasingly applied in control approaches, only few methods guarantee certifiable safety, which is necessary for real world applications. These approaches typically rely on well-understood learning algorithms,…
In this thesis we develop a novel framework to study smooth and strongly convex optimization algorithms, both deterministic and stochastic. Focusing on quadratic functions we are able to examine optimization algorithms as a recursive…
Let $H$, $T$ and $C_n$ be a graph, a tree and a cycle of order $n$, respectively. Let $H^{(i)}$ be the complete join of $H$ and an empty graph on $i$ vertices. Then the Cartesian product $H\Box T$ of $H$ and $T$ can be obtained by applying…