arXiv research

Paper addresses limitations of traditional hierarchical clustering methods.

Constructs tri-Hamiltonian structure and Frobenius manifold for asymmetric gAL hierarchy

We prove that the extended Toda hierarchy of \cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators $L_m$, $m\geq -1$ of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau fu…

Local tri-Hamiltonian structure for Ablowitz-Ladik hierarchy established.

HLTF generates chemically valid 3D molecules with improved topology control.

Enhances image classification by integrating semantic hierarchy into CNN models.

Proposes DCV-ROOD framework for robust OOD detection evaluation.

In a polynomial regression model, the divisibility conditions implicit in polynomial hierarchy give way to a natural construction of constraints for the model parameters. We use this principle to derive versions of strong and weak hierarchy and to extend existing work in the literature, which at the moment is only conc…

Generates infinite-depth hierarchical clusters from few examples.

We study the problem of topic modeling in corpora whose documents are organized in a multi-level hierarchy. We explore a parametric approach to this problem, assuming that the number of topics is known or can be estimated by cross-validation. The models we consider can be viewed as special (finite-dimensional) instance…

This paper examines limitations of machine learning models in social systems.

CAF-HFCM automatically forms a cluster hierarchy and optimizes the number of clusters without trial-and-validation.

Kernels for structured data are commonly obtained by decomposing objects into their parts and adding up the similarities between all pairs of parts measured by a base kernel. Assignment kernels are based on an optimal bijection between the parts and have proven to be an effective alternative to the established convolut…

Constructs integrable hierarchies for generalized Frobenius manifolds with non-flat unity.

Graph neural networks (GNNs) are a powerful tool to learn representations on graphs by iteratively aggregating features from node neighbourhoods. Many variant models have been proposed, but there is limited understanding on both how to compare different architectures and how to construct GNNs systematically. Here, we p…

Study identifies pitfalls in assessing hierarchies for multi-class classification.

Super tau-covers extend bihamiltonian hierarchies' symmetries.

Hydrodynamic hierarchy deformed using conservation laws.

Compact reachability embeddings for hierarchical data

Actors in realistic social networks play not one but a number of diverse roles depending on whom they interact with, and a large number of such role-specific interactions collectively determine social communities and their organizations. Methods for analyzing social networks should capture these multi-faceted role-spec…

Legendre transformations link related integrable hierarchies.

Twisted $U$- and twisted $U/K$-hierarchies are soliton hierarchies introduced by Terng to find higher flows of the generalized sine-Gordon equation. Twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are among the most important classes of twisted hierarchies. In this paper, interesting first and higher flows of twi…

New method controls false discoveries in structured hypothesis spaces.

Derive bihamiltonian structure for rational reduction of 2D-Toda hierarchy

We propose a formally completely integrable extension of heat hierarchy based on the space of symmetries isomorphic to the Weyl algebra $\mathcal{A}_1$. The extended heat hierarchy will be the basic model for the analysis of the extension of KP hierarchy, and other integrable equations.

Wise's Quasiconvex Hierarchy Theorem classifying hyperbolic virtually compact special groups in terms of quasiconvex hierarchies played an essential role in Agol's proof of the Virtual Haken Conjecture. Answering a question of Wise, we construct a new virtual quasiconvex hierarchy for relatively hyperbolic virtually co…

New hierarchies derived from KP hierarchy using non-formal operators and Yang-Mills action.

Symmetry reduction of Painlevé IV to Flaschka-Newell Painlevé II

Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.

A relation between the Goldstein-Petrich hierarchy for plane curves and the Toda lattice hierarchy is investigated. A representation formula for plane curves is given in terms of a special class of $τ$-functions of the Toda lattice hierarchy. A representation formula for discretized plane curves is also discussed.

New integrable deformations for topological hierarchies from Frobenius manifolds.

We introduce two families of soliton hierarchies: the twisted hierarchies associated to symmetric spaces. The Lax pairs of these two hierarchies are Laurent polynomials in the spectral variable. Our constructions gives a hierarchy of commuting flows for the generalized sine-Gordon equation (GSGE), which is the Gauss-Co…

Adversarial MoE learns category-specific models for product search.

Paper investigates methods to improve classification by inducing a hierarchy from flat labels.

Hierarchical Latent Attribute Models (HLAMs) are a family of discrete latent variable models that are attracting increasing attention in educational, psychological, and behavioral sciences. The key ingredients of an HLAM include a binary structural matrix and a directed acyclic graph specifying hierarchical constraints…

We compute the central invariants of the bihamiltonian structures of the constrained KP hierarchies, and show that these integrable hierarchies are topological deformations of their hydrodynamic limits.

We present the Lax pair formalism for certain extension of the continuous limit of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the $CP^1$ topological sigma model and the extended Toda hierarchy. We al…

Proposes a neural network for accurate and reconciled hierarchical time series forecasting.

New algorithms for hierarchical classification using conformal prediction.

New diffiety theory leads to a well-posed KP hierarchy.

In this work, we introduce a novel probabilistic representation of deep learning, which provides an explicit explanation for the Deep Neural Networks (DNNs) in three aspects: (i) neurons define the energy of a Gibbs distribution; (ii) the hidden layers of DNNs formulate Gibbs distributions; and (iii) the whole architec…

We propose an extension of the differential system for constant mean curvature (CMC) surfaces in a three dimensional space form to an associated hierarchy of evolution equations by the higher-order commuting symmetries. The infinite sequence of higher-order conservation laws of CMC surfaces admit the corresponding exte…

This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group $L$ as positive and negative subgroups L_+, L_- and a commuting sequence in the Lie algebr…

Study bihamiltonian structures and Frobenius manifolds for specific Toda hierarchies.

We observe that the modular class of a Poisson-Nijhenhuis manifold has a canonical representative and that, under a cohomological assumption, this vector field is bi-hamiltonian. In many examples the associated hierarchy of flows reproduces classical integrable hierarchies.

ProHOC detects OOD samples in class hierarchies, predicting them to correct internal nodes.

We construct integrable hierarchies of flows for curves in centroaffine ${\mathbb R}^3$ through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and …

The hierarchy structure associated with a (2+1)-dimensional Nonlinear Schroedinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using …