Related papers: An Algebraic Theory of Complexity for Discrete Opt…
This paper studies the distributed optimization problem with possibly nonidentical local constraints, where its global objective function is composed of $N$ convex functions. The aim is to solve the considered optimization problem in a…
We propose a novel discretization procedure for the classical Euler equation based on the theory of Galois differential algebras and the finite operator calculus developed by G.C. Rota and collaborators. This procedure allows us to define…
In many contexts, simpler models are preferable to more complex models and the control of this model complexity is the goal for many methods in machine learning such as regularization, hyperparameter tuning and architecture design. In deep…
We first describe a general class of optimization problems that describe many natural, economic, and statistical phenomena. After noting the existence of a conserved quantity in a transformed coordinate system, we outline several instances…
Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings…
This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and…
Submodular optimization generalizes many classic problems in combinatorial optimization and has recently found a wide range of applications in machine learning (e.g., feature engineering and active learning). For many large-scale…
When analyzing complex networks a key target is to uncover their modular structure, which means searching for a family of modules, namely node subsets spanning each a subnetwork more densely connected than the average. This work proposes a…
We study the problem of quantization of discrete probability distributions, arising in universal coding, as well as other applications. We show, that in many situations this problem can be reduced to the covering problem for the unit…
In recent years, deep learning has been connected with optimal control as a way to define a notion of a continuous underlying learning problem. In this view, neural networks can be interpreted as a discretization of a parametric Ordinary…
Most classical results in circuit complexity theory concern circuits over the Boolean domain. Besides their simplicity and the ease of comparing different languages, the actual architecture of computers is also an important motivating…
We present theoretical and practical results on the order theory of lattices of functions, focusing on Galois connections that abstract (sets of) functions - a topic known as higher-order abstract interpretation. We are motivated by the…
Some sorts of generalized morphisms are defined from very basic mathematical objects such as sets, functions, and partial functions. A wide range of mathematical notions such as continuous functions between topological spaces, ring…
We define the algebra of Colombeau generalized functions on a subset A of the space of d-dimensional generalized points. If the domain A is open, such generalized functions can be identified with pointwise maps from A into the ring of…
A rational function $f(x)$ is rationally summable if there exists a rational function $g(x)$ such that $f(x)=g(x+1)-g(x)$. Detecting whether a given rational function is summable is an important and basic computational subproblem that…
Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the…
We consider general shape optimization problems governed by Dirichlet boundary value problems. The proposed approach may be extended to other boundary conditions as well. It is based on a recent representation result for implicitly defined…
Measuring software complexity plays an important role to meet the demands of complex software. The cyclomatic complexity is one of most used and renowned metric among the other three proposed and researched metrics that are namely: Line of…
We present a novel set of rigorous and computationally efficient topology-based complexity notions that exhibit a strong correlation with the generalization gap in modern deep neural networks (DNNs). DNNs show remarkable generalization…
Optimization is a key task in a number of applications. When the set of feasible solutions under consideration is of combinatorial nature and described in an implicit way as a set of constraints, optimization is typically NP-hard.…