Related papers: Phase constants in the Fock-Goncharov quantization…
Spectral clustering is a powerful unsupervised machine learning algorithm for clustering data with non convex or nested structures. With roots in graph theory, it uses the spectral properties of the Laplacian matrix to project the data in a…
The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted…
For any point $\mathrm{X}$ in the cluster complex $\mathrm{Cpx}(\mathcal{C})$ of a 2-Calabi-Yau category $\mathcal{C}$, we introduce $\mathrm{X}$-evolution flow on $\mathrm{Cpx}(\mathcal{C})$. We show that such a flow induces a piecewise…
We construct symplectic groupoids integrating log-canonical Poisson structures on cluster varieties of type $\mathcal{A}$ and $\mathcal{X}$ over both the real and complex numbers. Extensions of these groupoids to the completions of the…
We study the problem of graph clustering under a broad class of objectives in which the quality of a cluster is defined based on the ratio between the number of edges in the cluster, and the total weight of vertices in the cluster. We show…
We propose an extension of the Goncharov-Kenyon class of cluster integrable systems by their Hamiltonian reductions. This extension allows us to fill in the gap in cluster construction of the $q$-difference Painlev\'e equations, showing…
In this article, we introduce the notion of cluster automorphism of a given cluster algebra as a $\ZZ$-automorphism of the cluster algebra that sends a cluster to another and commutes with mutations. We study the group of cluster…
We prove a conjecture of Fock and Goncharov which provides a birational equivalence of a cluster variety called the cluster symplectic double and a certain moduli space of local systems associated to a surface.
It is known that the homogeneous coordinate ring of a Grassmannian has a cluster structure, which is induced from the combinatorial structure of a plabic graph. A plabic graph is a certain bipartite graph described on the disk, and there is…
For a fixed seed $(X, Q)$, a \emph{rooted mutation loop} is a sequence of mutations that preserves $(X, Q)$. The group generated by all rooted mutation loops is called \emph{rooted mutation group} and will be denoted by $\mathcal{M}(Q)$.…
Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the associated preprojective algebra. To each terminal representation M of Q (these are certain preinjective representations), we attach a natural subcategory $C_M$ of…
Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of…
The tetrad constraint is a condition of which the satisfaction signals a rank reduction of a covariance submatrix and is used to design causal discovery algorithms that detects the existence of latent (unmeasured) variables, such as FOFC.…
We develop (quantum) cluster algebra structures over arbitrary commutative unital rings $\Bbbk$ and prove that the (quantized) coordinate rings of connected simply-connected complex simple algebraic groups $G$ over $\Bbbk$ admit such…
Okounkov [Oko03] conjectured the log-concavity about the structure constants for many interesting basis from representation theory. For the cluster algebra, Gross, Hacking, Keel, Kontsevich [GHKK18] introduced the atomic theta basis. We…
We present a categorification of four mutation finite cluster algebras by the cluster category of the category of coherent sheaves over a weighted projective line of tubular weight type. Each of these cluster algebras which we call tubular…
Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional…
Bayesian clustering typically relies on mixture models, with each component interpreted as a different cluster. After defining a prior for the component parameters and weights, Markov chain Monte Carlo (MCMC) algorithms are commonly used to…
In this paper, we prove Conjecture 4.8 of "Cluster algebras IV" by S. Fomin and A. Zelevinsky, stating that the mutation classes of rectangular matrices associated with cluster algebras of finite type are precisely those classes which are…
In 2003, Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by…