arXiv research

Edge subdivision affects the Perron eigenvalue of tree Ricci matrices.

Study of Ricci flow on trees, focusing on edge weights and curvatures.

Study of discrete period matrices on embedded graphs, relating to Riemann surfaces.

Existence and uniqueness of discrete Einstein metrics on trees proven.

The paper introduces a new type of Ricci flow on graphs to study their curvature.

Proves convergence of gradient Ricci shrinkers with uniform bounds.

Mathematical foundation for phylogenetic tree uncertainty quantification.

Researchers construct explicit bundles for ALF metrics, revealing rational patching matrices for gravitational instantons.

Positive-curvature metrics on trees identified for specific configurations.

Decision Machines embeds decision trees into vector spaces for improved optimization.

We discuss some methods to quantitatively investigate the properties of correlation matrices. Correlation matrices play an important role in portfolio optimization and in several other quantitative descriptions of asset price dynamics in financial markets. Specifically, we discuss how to define and obtain hierarchical …

Local existence and uniqueness of Bakry-Émery Ricci flow solutions on finite graphs.

Sparse oblique decision tree improves security rules for renewable power systems.

This paper uses rank correlation methods to construct MSTs from financial returns, finding them more stable and robust.

Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.

Proposes a new phylogenetic tree space with biologically principled geometry.

Correlation matrices of foreign exchange rate time series are investigated for 60 world currencies. Minimal Spanning Tree (MST) graphs for the gold, silver and platinum are presented. Inverse power like scaling is discussed for these graphs as well as for four distinct currency groups (major, liquid, less liquid and no…

The paper characterizes Kenmotsu metrics as almost $*$-Ricci solitons.

The spectral geometry of mesh matrices of graphs is explored, leading to new formulas and eigenvalue estimates.

Proves a conjecture about 3-manifold invariants using trees and asymptotic formulas.

Deviance-style normalization for sparse, jointly overdispersed count matrices

SNJ recovers latent tree models from similarity matrices.

We consider Ricci flow on two classes of nilpotent Lie groups that generalize the three-dimensional Heisenberg group: the higher-dimensional classical Heisenberg groups, and the groups of real unitriangular matrices. Each group is known to admit a Ricci soliton, but we construct them \textit{explicitly} on each group. …

The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we d…

Decision trees are a popular technique in statistical data classification. They recursively partition the feature space into disjoint sub-regions until each sub-region becomes homogeneous with respect to a particular class. The basic Classification and Regression Tree (CART) algorithm partitions the feature space using…

In this article we review several techniques to extract information from stock market data. We discuss recurrence analysis of time series, decomposition of aggregate correlation matrices to study co-movements in financial data, stock level partial correlations with market indices, multidimensional scaling and minimum s…

Many state-of-the-art results obtained with deep networks are achieved with the largest models that could be trained, and if more computation power was available, we might be able to exploit much larger datasets in order to improve generalization ability. Whereas in learning algorithms such as decision trees the ratio …

Study finds Calabi-Yau models' operator spectra match random matrix theory.

The topology of $SU(3)$-representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to $-1$ are explored, and compared to data coming from spider evaluation of the webs. Corresponding to an evaluation of a web as a spider is a rooted tree. We associate t…

We prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolves under the Ricci flow. These upper bounds supplement the well-known global lower bound.

The paper sets bounds on how much regret is unavoidable in adaptive LQR with unknown B-matrix.

New Lie algebras from quivers lead to rigid Ricci solitons.

This paper studies how adding leaves to a tree affects its spectral properties.

Gaussian latent tree models, or more generally, Gaussian latent forest models have Fisher-information matrices that become singular along interesting submodels, namely, models that correspond to subforests. For these singularities, we compute the real log-canonical thresholds (also known as stochastic complexities or l…

The paper proves continuity of Morse index for Ricci shrinkers.

We prove that the Ricci flow equation for left invariant metrics on Lie groups reduces to a first order ordinary differential equation for a map $Q : (-a,a) \to UT$, where $UT$ is the group of upper triangular matrices. We decompose the matrix $R_{ij}$ of Ricci tensor coordinates with respect to an orthonormal frame fi…

This paper presents an improvement to model learning when using multi-class LogitBoost for classification. Motivated by the statistical view, LogitBoost can be seen as additive tree regression. Two important factors in this setting are: 1) coupled classifier output due to a sum-to-zero constraint, and 2) the dense Hess…

We present a technique for clustering categorical data by generating many dissimilarity matrices and averaging over them. We begin by demonstrating our technique on low dimensional categorical data and comparing it to several other techniques that have been proposed. Then we give conditions under which our method shoul…

The eigendeomposition of nearest-neighbor (NN) graph Laplacian matrices is the main computational bottleneck in spectral clustering. In this work, we introduce a highly-scalable, spectrum-preserving graph sparsification algorithm that enables to build ultra-sparse NN (u-NN) graphs with guaranteed preservation of the or…

Paper proves conditions for estimating precision matrices with Laplacian constraints.

This work employs some techniques in order to filter random noise from the information provided by minimum spanning trees obtained from the correlation matrices of international stock market indices prior to and during times of crisis. The first technique establishes a threshold above which connections are considered a…

Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and struc…

Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further s…

xRFM improves tabular data inference with better accuracy and scalability.

Decision forests are widely used for classification and regression tasks. A lesser known property of tree-based methods is that one can construct a proximity matrix from the tree(s), and these proximity matrices are induced kernels. While there has been extensive research on the applications and properties of kernels, …

With the advent of massive data sets much of the computational science and engineering community has moved toward data-intensive approaches in regression and classification. However, these present significant challenges due to increasing size, complexity and dimensionality of the problems. In particular, covariance mat…

Classifies cobounded hyperbolic actions of metabelian groups.

Simple matrix formulas for Grassmannian curvatures.