arXiv research

Study on optimal partitions and nodal solutions for the Yamabe equation.

New framework links fractal complexity to separation dimension.

The study limits how many parts regular simplicial partitions can overlap.

This thesis classifies pseudo-Anosov homeomorphisms using geometric Markov partitions.

Although consistency is a minimum requirement of any estimator, little is known about consistency of the mean partition approach in consensus clustering. This contribution studies the asymptotic behavior of mean partitions. We show that under normal assumptions, the mean partition approach is consistent and asymptotic …

A graph clustering method that moves nodes to highest-degree neighbors.

Minimal partitions with minimal perimeter found in metric spaces.

We explore the geometrical interpretation of the PCA based clustering algorithm Principal Direction Divisive Partitioning (PDDP). We give several examples where this algorithm breaks down, and suggest a new method, gap partitioning, which takes into account natural gaps in the data between clusters. Geometric features …

The paper studies partition functions of point processes on Kähler manifolds, generalizing geometric functionals and relating to QHE.

Standard bubbles and partitions are stable in various model spaces.

Region-specific linear models are widely used in practical applications because of their non-linear but highly interpretable model representations. One of the key challenges in their use is non-convexity in simultaneous optimization of regions and region-specific models. This paper proposes novel convex region-specific…

Study optimal partitions on spheres using fractional Q-curvature and variational methods.

We calculate the free energy of Coulomb gas systems on Riemann surfaces.

A new method for multi-task learning improves performance without weakening inductive bias.

We propose the use of the functional determinant of geometric operators in constructing an entropy functional associated to geometric flows. Our approach is based on the direct computation of the partition function, with a well-defined set of microstates and macrostates in the canonical ensemble. The approach is motiva…

Study optimizes perimeter in convex domains with anisotropic constraints.

A new algorithm, Regular Tree Search, tackles non-convex simulation optimization problems.

New spectral clustering method using LASSO regularization for robust graph partitioning.

Distributed sparse learning with a cluster of multiple machines has attracted much attention in machine learning, especially for large-scale applications with high-dimensional data. One popular way to implement sparse learning is to use $L_1$ regularization. In this paper, we propose a novel method, called proximal \mb…

We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in $\mathbb{R}^n$. Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity…

Geometrically proves Zabrodin-Wiegmann conjecture for integer QH states.

POTATOES improves autoencoder UOD accuracy without tuning.

Personalization of cardiac models involves the optimization of organ tissue properties that vary spatially over the non-Euclidean geometry model of the heart. To represent the high-dimensional (HD) unknown of tissue properties, most existing works rely on a low-dimensional (LD) partitioning of the geometrical model. Wh…

This paper develops a geometric framework for Wilson surfaces in higher gauge theory.

Study introduces indecomposability for varifolds, leading to geometric consequences.

The paper calculates a formula for knot complements using holomorphic curves.

This paper solves a problem in 3D geometry by defining a canonical partition for certain manifolds.

Definition of the partition function of U(1) gauge theory is extended to a class of four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) ones including the multi-Taub--NUT spaces. The partition function is calculated via zeta-function regularization with special attention to its mo…

Motivated by a geometric problem, we introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreas…

Paper recovers lattice signal partitions efficiently.

Min-cut clustering, based on minimizing one of two heuristic cost-functions proposed by Shi and Malik, has spawned tremendous research, both analytic and algorithmic, in the graph partitioning and image segmentation communities over the last decade. It is however unclear if these heuristics can be derived from a more g…

We propose an algorithm, HPREF (Hierarchical Partitioning by Repeated Features), that produces a hierarchical partition of a set of clusterings of a fixed dataset, such as sets of clusterings produced by running a clustering algorithm with a range of parameters. This gives geometric structure to such sets of clustering…

ALMA improves clustering of multilayer networks.

This paper proves properties of uniformly hyperbolic sets and constructs Markov partitions.

Anisotropic curvature flow studied for planar networks.

A new method for high-dimensional functional regression reduces multicollinearity and improves interpretability.

This work proposes a method to optimize hyperparameters without validation data.

In this paper, we consider unsupervised partitioning problems, such as clustering, image segmentation, video segmentation and other change-point detection problems. We focus on partitioning problems based explicitly or implicitly on the minimization of Euclidean distortions, which include mean-based change-point detect…

We define an invariant of graphs embedded in a three-manifold and a partition function for 2-complexes embedded in a triangulated four-manifold by specifying the values of variables in the Turaev-Viro and Crane-Yetter state sum models. In the case of the three-dimensional invariant, we prove a duality formula relating …

New proof associates partitions to isotopic pseudo-Anosov homeomorphisms.

We study 4-dimensional higher-derivative conformal higher spin (CHS) fields generalising Weyl graviton and conformal gravitino. They appear, in particular, as "induced" theories in the AdS/CFT context. We consider their partition function on curved Einstein-space backgrounds like (A)dS or sphere and Ricci-flat spaces. …

Rational maps structure theorem with geometric decomposition and realizability proof.

The paper studies Teichmüller TQFT for hyperbolic knots, proving exponential decay of partition functions.

BN refines local partition geometry in piecewise-affine networks during training.

We prove that the open topological string partition function on a D-brane configuration in a Calabi-Yau manifold X takes the form of a closed topological string partition function on a different Calabi-Yau manifold X_b. This identification shows that the physics of D-branes in an arbitrary background X of topological s…

A Dirichlet $k$-partition of a domain $U \subseteq \mathbb{R}^d$ is a collection of $k$ pairwise disjoint open subsets such that the sum of their first Laplace-Dirichlet eigenvalues is minimal. A discrete version of Dirichlet partitions has been posed on graphs with applications in data analysis. Both versions admit va…

We propose a scalable Gromov-Wasserstein learning (S-GWL) method and establish a novel and theoretically-supported paradigm for large-scale graph analysis. The proposed method is based on the fact that Gromov-Wasserstein discrepancy is a pseudometric on graphs. Given two graphs, the optimal transport associated with th…

Enhanced spectral clustering for geometric graphs improves clustering accuracy.