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Hyperparameter tuning is an active area of research in machine learning, where the aim is to identify the optimal hyperparameters that provide the best performance on the validation set. Hyperparameter tuning is often achieved using naive…

We introduce a simple autoencoder based on hyperbolic geometry for solving standard collaborative filtering problem. In contrast to many modern deep learning techniques, we build our solution using only a single hidden layer. Remarkably,…

In this paper we describe an algorithm for implicitizing rational hypersurfaces in case there exists at most a finite number of base points. It is based on a technique exposed in math.AG/0210096, where implicit equations are obtained as…

Recent advances in feature learning have shown that self-supervised vision foundation models can capture semantic correspondences but often lack awareness of underlying 3D geometry. GECO addresses this gap by producing geometrically…

In this paper, we propose to study a new geometric optimization problem called "geometric prototype" in Euclidean space. Given a set of patterns, where each pattern is represented by a (weighted or unweighted) point set, the geometric…

Wireless localization of permanent magnets enables occlusion-free guidance for medical interventions, yet its practical accuracy is fundamentally limited by two coupled challenges: the poor observability of conventional planar sensor arrays…

In this paper, we apply the practical GADI-HS iteration as a smoother in algebraic multigrid (AMG) method for solving second-order non-selfadjoint elliptic problem. Additionally, we prove the convergence of the derived algorithm and…

In this paper, we introduce a Homogeneous Second-Order Descent Method (HSODM) using the homogenized quadratic approximation to the original function. The merit of homogenization is that only the leftmost eigenvector of a gradient-Hessian…

We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is `multi-valued'. This paper largely consisits of two…

We consider the problem of embedding the nodes of a hypergraph into Euclidean space under the assumption that the interactions arose through closeness to unknown hyperedge centres. In this way, we tackle the inverse problem associated with…

Automated algorithm selection for continuous black-box optimization depends on representing problem information under limited probing and selecting solvers under heavy-tailed performance distributions. This paper proposes a geometric…

Moving horizon estimation (MHE) is a widely studied state estimation approach in several practical applications. In the MHE problem, the state estimates are obtained via the solution of an approximated nonlinear optimization problem.…

In this paper, we develop a new parallel auxiliary grid algebraic multigrid (AMG) method to leverage the power of graphic processing units (GPUs). In the construction of the hierarchical coarse grid, we use a simple and fixed coarsening…

Euclidean geometry is among the earliest forms of mathematical thinking. While the geometric primitives underlying its constructions, such as perfect lines and circles, do not often occur in the natural world, humans rarely struggle to…

We revisit the task of learning a Euclidean metric from data. We approach this problem from first principles and formulate it as a surprisingly simple optimization problem. Indeed, our formulation even admits a closed form solution. This…

Information geometric optimization (IGO) is a general framework for stochastic optimization problems aiming at limiting the influence of arbitrary parametrization choices. The initial problem is transformed into the optimization of a smooth…

We primarily consider bilevel programs where the lower level is a convex quadratic minimization problem under integer constraints. We show that it is $\Sigma_2^p$-hard to decide if the optimal objective for the leader is lesser than a given…

Group Relative Policy Optimization (GRPO) significantly enhances the reasoning performance of Large Language Models (LLMs). However, this success heavily relies on expensive external verifiers or human rules. Such dependency not only leads…

Neural ODEs are a widely used, powerful machine learning technique in particular for physics. However, not every solution is physical in that it is an Euler-Lagrange equation. We present Helmholtz metrics to quantify this resemblance for a…

Black-box algorithms are designed to optimize functions without relying on their underlying analytical structure or gradient information, making them essential when gradients are inaccessible or difficult to compute. Traditional methods for…