Laurence Boulanger | Searcharxiv

arXiv2026

Introducing Feature-Based Trajectory Clustering, a clustering algorithm for longitudinal data

We present a new algorithm for clustering longitudinal data. Data of this type can be conceptualized as consisting of individuals and, for each such individual, observations of a time-dependent variable made at various times. Generically, the specific way in which this variable evolves with time is different from one individual to the next. However, there may also be commonalities; specific characteristic features of the time evolution shared by many individuals. The purpose of the method we put forward is to find clusters of individual whose underlying time-dependent variables share such characteristic features. This is done in two steps. The first step identifies each individual to a point in Euclidean space whose coordinates are determined by specific mathematical formulae meant to capture a variety of characteristic features. The second step finds the clusters by applying the Spectral Clustering algorithm to the resulting point cloud.

cs.LG↗

arXiv2015

Toric generalized Kähler structures

Given a compact symplectic toric manifold $(M,ω, \mathbb{T})$, we identify a class $DGK_ω^{\mathbb{T}}(M)$ of $\mathbb{T}$-invariant generalized Kähler structures for which a generalisation the Abreu-Guillemin theory of toric Kähler metrics holds. Specifically, elements of $DGK_ω^{\mathbb{T}}(M)$ are characterized by the data of a strictly convex function $τ$ on the moment polytope associated to $(M,ω, \mathbb{T})$ via the Delzant theorem, and an antisymmetric matrix $C$. For a given $C$, it is shown that a toric Kähler structure on $M$ can be explicitly deformed to a non-Kähler element of $DGK_ω^{\mathbb{T}}(M)$ by adding a small multiple of $C$. This constitutes an explicit realization of a recent unobstructedness theorem of R. Goto, where the choice of a matrix $C$ corresponds to choosing a holomorphic Poisson structure. Adapting methods from S. K. Donaldson, we compute the moment map for the action of $\mathrm{Ham}(M,ω)$ on $DGK_ω^{\mathbb{T}}(M)$. The result introduces a natural notion of "generalized Hermitian scalar curvature". In dimension 4, we find an expression for this generalized Hermitian scalar curvature in terms of the underlying bi-Hermitian structure in the sense of Apostolov-Gauduchon-Grantcharov.

math.DG↗