Related papers: Learning to Compute Gr\"obner Bases
This paper studies the concept and the computation of approximately vanishing ideals of a finite set of data points. By data points, we mean that the points contain some uncertainty, which is a key motivation for the approximate treatment.…
The majority of model-based learned image reconstruction methods in medical imaging have been limited to uniform domains, such as pixelated images. If the underlying model is solved on nonuniform meshes, arising from a finite element method…
We present a formalization of Gr\"obner basis theory in Lean 4, built on top of Mathlib's infrastructure for multivariate polynomials and monomial orders. Our development covers the core foundations of Gr\"obner basis theory, including…
Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper…
Many real-world problems can be naturally described by mathematical formulas. The task of finding formulas from a set of observed inputs and outputs is called symbolic regression. Recently, neural networks have been applied to symbolic…
We consider solving ill-posed imaging inverse problems without access to an explicit image prior or ground-truth examples. An overarching challenge in inverse problems is that there are many undesired images that fit to the observed…
Border basis schemes are open subschemes of Hilbert schemes parametrizing 0-dimensional subschemes of $\mathbb{P}^n$ of given length. They yield open coverings and are easy to describe and to compute with. Our topic is to find re-embeddings…
We propose a novel hybrid quantum-classical approach to calculate Graver bases, which have the potential to solve a variety of hard linear and non-linear integer programs, as they form a test set (optimality certificate) with very appealing…
Loop invariants are properties of a program loop that hold both before and after each iteration of the loop. They are often used to verify programs and ensure that algorithms consistently produce correct results during execution.…
Inverse problems correspond to a certain type of optimization problems formulated over appropriate input distributions. Recently, there has been a growing interest in understanding the computational hardness of these optimization problems,…
Estimating camera geometry typically involves solving minimal problems formulated as systems of multivariate polynomial equations, which often pose computational challenges when using existing Gr\"obner-basis or resultant-based methods due…
While Bayesian inference provides a principled framework for reasoning under uncertainty, its widespread adoption is limited by the intractability of exact posterior computation, necessitating the use of approximate inference. However,…
Quantum neural networks (QNNs) provide expressive probabilistic models by leveraging quantum superposition and entanglement, yet their practical training remains challenging due to highly oscillatory loss landscapes and noise inherent to…
The so called generalized down-up algebras are revisited from a viewpoint of Gr\"obner basis theory. Particularly it is shown explicitly that generalized down-up algebras are solvable polynomial algebras (provided $\lambda\omega\ne 0$), and…
The redundancy allocation problem is formulated minimizing the design cost for a series-parallel system with multiple component choices whereas ensuring a given system reliability level. The obtained model is a nonlinear integer programming…
We provide a new complexity bound for the computation of grevlex Gr\"obner bases in the generic zero-dimensional case, relying on Moreno-Soc\'ias' conjecture. We first formalize a property of regular sequences that implies a well-known…
The algebraic method provides useful techniques to identify models in designs and to understand aliasing of polynomial models. The present note surveys the topic of Gr\"obner bases in experimental design and then describes the notion of…
Incorporating a deep generative model as the prior distribution in inverse problems has established substantial success in reconstructing images from corrupted observations. Notwithstanding, the existing optimization approaches use gradient…
We present a new algorithm for computing a truncated Markov basis of a lattice. In general, this new algorithm is faster than existing methods. We then extend this new algorithm so that it solves the linear integer feasibility problem with…
These lecture notes evolve around mathematical concepts arising in inverse problems. We start by introducing inverse problems through examples such as differentiation, deconvolution, computed tomography and phase retrieval. This then leads…