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The real-life data have a complex and non-linear structure due to their nature. These non-linearities and the large number of features can usually cause problems such as the empty-space phenomenon and the well-known curse of dimensionality.…

We introduce an adaptive scattered data fitting scheme as extension of local least squares approximations to hierarchical spline spaces. To efficiently deal with non-trivial data configurations, the local solutions are described in terms of…

Learning Spaces are certain set systems that are applied in the mathematical modeling of education. We propose a suitable compression (without loss of information) of such set systems to facilitate their logical and statistical analysis.…

In subset selection we search for the best linear predictor that involves a small subset of variables. From a computational complexity viewpoint, subset selection is NP-hard and few classes are known to be solvable in polynomial time. Using…

We show that one can enumerate the vertices of the convex hull of integer points in polytopes whose constraint matrices have bounded and nonzero subdeterminants, in time polynomial in the dimension and encoding size of the polytope. This…

Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling…

In this paper, we establish the local superlinear convergence property of some polynomial-time interior-point methods for an important family of conic optimization problems. The main structural property used in our analysis is the…

Large-scale optimization problems that seek sparse solutions have become ubiquitous. They are routinely solved with various specialized first-order methods. Although such methods are often fast, they usually struggle with not-so-well…

Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the…

Finding a reduction of complex, high-dimensional dynamics to its essential, low-dimensional "heart" remains a challenging yet necessary prerequisite for designing efficient numerical approaches. Machine learning methods have the potential…

We consider the problem of computing sample points in each connected component of a semi-algebraic set defined by the non-vanishing or the positivity of an n-variate polynomial of degree d, with rational coefficients of bit size bounded by…

Intrinsic computation refers to how dynamical systems store, structure, and transform historical and spatial information. By graphing a measure of structural complexity against a measure of randomness, complexity-entropy diagrams display…

Symbolic Computation algorithms and their implementation in computer algebra systems often contain choices which do not affect the correctness of the output but can significantly impact the resources required: such choices can benefit from…

This paper deals with the computation of polytopic invariant sets for polynomial dynamical systems. An invariant set of a dynamical system is a subset of the state space such that if the state of the system belongs to the set at a given…

Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic…

The Numerical Assembly Technique is extended to investigate arbitrary planar frame structures with the focus on the computation of natural frequencies. This allows us to obtain highly accurate results without resorting to spatial…

Estimating intrinsic dimensionality of data is a classic problem in pattern recognition and statistics. Principal Component Analysis (PCA) is a powerful tool in discovering dimensionality of data sets with a linear structure; it, however,…

In the present paper, some aspects of the finite-dimensional theory of set-inclusive generalized equations are studied. Set-inclusive generalized equations are problems arising in several contexts of optimization and variational analysis,…

It has been found that Transformer-based language models have the ability to perform basic quantitative reasoning. In this paper, we propose a method for studying how these models internally represent numerical data, and use our proposal to…

Covering and elimination inequalities are central to combinatorial optimization, yet their role has largely been studied in problem-specific settings or via no-good cuts. This paper introduces a unified perspective that treats these…