Ioannis Z. Emiris

Ioannis Z. Emiris

Sparse (or toric) elimination exploits the structure of polynomials by measuring their complexity in terms of Newton polytopes instead of total degree. The sparse, or Newton, resultant generalizes the classical homogeneous resultant and its degree is a function of the mixed volumes of the Newton polytopes. We sketch the sparse resultant constructions of Canny and Emiris and show how they reduce the problem of root-finding to an eigenproblem. A novel method for achieving this reduction is presented which does not increase the dimension of the problem. Together with an implementation of the sparse resultant construction, it provides a general solver for polynomial systems. We discuss the overall implementation and illustrate its use by applying it to concrete problems from vision, robotics and structural biology. The high efficiency and accuracy of the solutions suggest that sparse elimination may be the method of choice for systems of moderate size.

Panagiotis Rigas, Panagiotis Drivas, Charalambos Tzamos et al.

We present \textbf{GARLIC}, a representation learning approach for Euclidean approximate nearest neighbor (ANN) search in high dimensions. Existing partitions tend to rely on isotropic cells, fixed global resolution, or balanced constraints, which fragment dense regions and merge unrelated points in sparse ones, thereby increasing the candidate count when probing only a few cells. Our method instead partitions \(\mathbb{R}^d\) into anisotropic Gaussian cells whose shapes align with local geometry and sizes adapt to data density. Information-theoretic objectives balance coverage, overlap, and geometric alignment, while split/clone refinement introduces Gaussians only where needed. At query time, Mahalanobis distance selects relevant cells and localized quantization prunes candidates. This yields partitions that reduce cross-cell neighbor splits and candidate counts under small probe budgets, while remaining robust even when trained on only a small fraction of the dataset. Overall, GARLIC introduces a geometry-aware space-partitioning paradigm that combines information-theoretic objectives with adaptive density refinement, offering competitive recall--efficiency trade-offs for Euclidean ANN search.