Related papers: Sampling algebraic sets in local intrinsic coordin…

We present effective numerical algorithms for locally recovering unknown governing differential equations from measurement data. We employ a set of standard basis functions, e.g., polynomials, to approximate the governing equation with high…

We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree…

We introduce algorithms for robustly computing intrinsic coordinates on point clouds. Our approach relies on generating many candidate coordinates by subsampling the data and varying hyperparameters of the embedding algorithm (e.g.,…

In this article we provide examples, methods and algorithms to determine conditions on the parameters of certain type of parametric optimization problems, such that among the resulting local minima and maxima there is at least one which…

Representing token embeddings as probability distributions over learned manifolds allows for more flexible contextual inference, reducing representational rigidity while enhancing semantic granularity. Comparative evaluations demonstrate…

This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose…

Numerous interesting properties in nonlinear systems analysis can be written as polynomial optimization problems with nonconvex sum-of-squares problems. To solve those problems efficiently, we propose a sequential approach of local…

Many real-world physics and engineering problems arise in geometrically complex domains discretized by meshes for numerical simulations. The nodes of these potentially irregular meshes naturally form point clouds whose limited tractability…

We present certain techniques to find completely positive maps between matrix algebras that take prescribed values on given data. To this aim we describe a semidefinite programming approach and another convex minimization method supported…

In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems)…

Tabular datasets with low-sample-size or many variables are prevalent in biomedicine. Practitioners in this domain prefer linear or tree-based models over neural networks since the latter are harder to interpret and tend to overfit when…

This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix…

We propose a method called ideal regression for approximating an arbitrary system of polynomial equations by a system of a particular type. Using techniques from approximate computational algebraic geometry, we show how we can solve ideal…

Deep learning models have achieved significant success in various image related tasks. However, they often encounter challenges related to computational complexity and overfitting. In this paper, we propose an efficient approach that…

Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help reduce the problem dimension, cut…

Compressive sampling has become a widely used approach to construct polynomial chaos surrogates when the number of available simulation samples is limited. Originally, these expensive simulation samples would be obtained at random locations…

Designing experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. The local optimality approach is to study the regions in parameter space where a given design is optimal. In many…

In many applications it is important to understand the sensitivity of eigenvalues of a matrix polynomial to perturbations of the polynomial. The sensitivity commonly is described by condition numbers or pseudospectra. However, the…

Enumerative Geometry is concerned with the number of solutions to a structured system of polynomial equations, when the structure comes from geometry. Enumerative real algebraic geometry studies real solutions to such systems, particularly…

This paper reconsiders common benchmarking approaches to nearest neighbor search. It is shown that the concept of local intrinsic dimensionality (LID) allows to choose query sets of a wide range of difficulty for real-world datasets.…