Related papers: Beyond the Laurent phenomenon
This paper addresses the problem of unsupervised clustering which remains one of the most fundamental challenges in machine learning and artificial intelligence. We propose the clustered generator model for clustering which contains both…
Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine learning. Network…
We study a two-species bidirectional exclusion process, and a single species variant, which is motivated by the motion of organelles and vesicles along microtubules. Specifically, we are interested in the clustering of the particles and…
We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving…
Clustering is a widely used technique in data mining applications for discovering patterns in underlying data. Most traditional clustering algorithms are limited to handling datasets that contain either numeric or categorical attributes.…
We propose a new unsupervised learning method for clustering a large number of time series based on a latent factor structure. Each cluster is characterized by its own cluster-specific factors in addition to some common factors which impact…
Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural…
In an age of increasingly large data sets, investigators in many different disciplines have turned to clustering as a tool for data analysis and exploration. Existing clustering methods, however, typically depend on several nontrivial…
We initiate a study of the dependence on the choice of ground ring on the question of whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is equality of the cluster algebra and upper cluster algebra…
By studying the structures of clusters bound by a model potential that favours polytetrahedral order, we find a previously unknown series of `magic numbers' (i.e. sizes of special stability) whose polytetrahedral structures are…
We show that modularity, a quantity introduced in the study of networked systems, can be generalized and used in the clustering problem as an indicator for the quality of the solution. The introduction of this measure arises very naturally…
Let G be any additive abelian group with cyclic torsion subgroup, and let A, B and C be finite subsets of G with cardinality n>0. We show that there is a numbering {a_i}_{i=1}^n of the elements of A, a numbering {b_i}_{i=1}^n of the…
We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation…
We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the…
In this paper, we study some properties of Euler polynomials arising from umbral calculus. Finally, we give some interesting identities of Euler polynomials using our results. Recently, Dere and Simsek have studied umbral calculus related…
In 2015, Phulara established a generalization of the famous central set theorem by an original idea. Roughly speaking, this idea extends a combinatorial result from one large subset of the given semigroup to countably many. In this paper,…
A clustering algorithm partitions a set of data points into smaller sets (clusters) such that each subset is more tightly packed than the whole. Many approaches to clustering translate the vector data into a graph with edges reflecting a…
We introduce admissible group actions on cluster algebras, cluster categories and quivers with potential and study the resulting orbit spaces. The orbit space of the cluster algebra has the structure of a generalized cluster algebra. This…
We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…
We study the problem of explainability-first clustering where explainability becomes a first-class citizen for clustering. Previous clustering approaches use decision trees for explanation, but only after the clustering is completed. In…