Related papers: Computational topology and normal surfaces: Theore…

One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are…

Many proposed applications of neural networks in machine learning, cognitive/brain science, and society hinge on the feasibility of inner interpretability via circuit discovery. This calls for empirical and theoretical explorations 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…

The notion of weak truth-table reducibility plays an important role in recursion theory. In this paper, we introduce an elaboration of this notion, where a computable bound on the use function is explicitly specified. This elaboration…

Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by…

In this paper, we explore statistical versus computational trade-off to address a basic question in the application of a distributed algorithm: what is the minimal computational cost in obtaining statistical optimality? In smoothing spline…

In this paper we provide, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e. Gauss curvature and mean curvature. In practice, the…

Confined electronic states and optical transitions in 3D topological insulator nanoparticles have been studied in the literature, assuming idealized geometries such as spheres or infinitely long cylinders, that allow to obtain analytical…

A central question in computer science and statistics is whether efficient algorithms can achieve the information-theoretic limits of statistical problems. Many computational-statistical tradeoffs have been shown under average-case…

We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions and genus-surfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most…

We show that there are a finite number of possible pictures for a surface in a tetrahedron with local index $n$. Combined with previous results, this establishes that any topologically minimal surface can be transformed into one with a…

1) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with a certain number of critical cells. Our…

In quantum chemistry, the price paid by all known efficient model chemistries is either the truncation of the Hilbert space or uncontrolled approximations. Theoretical computer science suggests that these restrictions are not mere…

The operation of crushing a normal surface has proven to be a powerful tool in computational $3$-manifold topology, with applications both to triangulation complexity and to algorithms. The main difficulty with crushing is that it can…

Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time…

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but…

We consider the problem of measuring the margin of robust feasibility of solutions to a system of nonlinear equations. We study the special case of a system of quadratic equations, which shows up in many practical applications such as the…

In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have…

The dynamics of symbolic systems, such as multidimensional subshifts of finite type or cellular automata, are known to be closely related to computability theory. In particular, the appropriate tools to describe and classify topological…

We effectively bound T-singularities on non-rational projective surfaces with an arbitrary amount of T-singularities and ample canonical class. This fully generalizes the previous work for the case of one singularity, and illustrates the…