arXiv research

The statistical leverage scores of a complex matrix $A\in\mathbb{C}^{n\times d}$ record the degree of alignment between col$(A)$ and the coordinate axes in $\mathbb{C}^n$. These score are used in random sampling algorithms for solving certain numerical linear algebra problems. In this paper we present a max-plus algebr…

New Max-Plus neural network exploits subgradient sparsity for efficient training.

We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any supermartingale of class $(\mathcal{D})$ as a conditional expectation of some running supremum process. As an application, we show how the Max-Plus supermartingale decomposition allows…

Estimates parameters in max-linear Bayesian networks with noise.

We consider deterministic Markov decision processes (MDPs) and apply max-plus algebra tools to approximate the value iteration algorithm by a smaller-dimensional iteration based on a representation on dictionaries of value functions. The setup naturally leads to novel theoretical results which are simply formulated due…

Following recent advances in morphological neural networks, we propose to study in more depth how Max-plus operators can be exploited to define morphological units and how they behave when incorporated in layers of conventional neural networks. Besides showing that they can be easily implemented with modern machine lea…

Efficiently finds sparse solutions to max-plus equations for convex regression.

The notion of max-plus convex subset of Euclidean space can be naturally extended to other linear spaces. The aim of this paper is to describe the topology of hyperspaces of max-plus convex subsets of Tychonov powers $\mathbb R^τ$ of the real line. We show that the corresponding spaces are AR's if and only if $τ\leω_1$…

TML package uses tropical geometry for machine learning tasks.

Tropical SVM tackles phylogenomics by classifying multi-locus data.

Tropical geometry and weighted lattices improve curve and surface fitting.

Tropical division approximates polynomial division for neural networks.

Agent maximizes utility with pathwise constraint on portfolio value.

Building on the line of work [DIRT15a], [DIRT15b], [NS17a], [DT17], [HLS18], [HS18] we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous conn…

New algebraic structure derived from Hopf algebra and Drinfel'd twist.

Study of cluster and skein algebras for surfaces, showing their connection.

Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.

Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.

We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we …

New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.

A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define…

Study biderivations in complete Leibniz algebras, extending Lie algebra results.

The paper extends a theorem to Lie-Rinehart algebras and provides new decompositions of universal enveloping algebras.

The paper classifies Lie algebras with special operators.

Symmetric spaces' connections form Lie admissible triple algebras.

The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.

The paper investigates gradings of complex simple Lie algebras, focusing on $|3|$-gradings and their algebraic structures.

New algebra pong algebra computed for knot Floer homology.

Study resolves conjecture linking two algebraic structures on surfaces.

Study on pre-Lie structures for semisimple Lie algebras over C.

Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.

Characterizes G2-structures on Lie algebras with non-trivial center.

Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms o…

This paper presents results on the framization of some knot algebras, defined by the authors. We explain the motivations of the concept of framization, coming from the Yokonuma--Hecke algebras, as well as recent results on the framization of the Temperley--Lieb algebra. Finally, we propose framizations for other knot a…

For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and non-existence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algeb…

Similarity algebra extends algebraic structures with quantitative bounds.

Extends Loday-Quillen-Tsygan theorem to bornological Lie algebra homology.

New algebra for twice-punctured torus curves.

We define a Poisson Algebra called the {\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functio…

New braided Frobenius algebras created from specific Hopf algebras.

Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables us to explore Poisson algebras in the realm of non-associative algebras. We stud…

The paper examines differential smoothness in specific algebra types.

Generalizes Hecke algebra for double torus, linking to skein algebra.

2-compatible Lie algebras are quadratic deformations of Lie algebras with specific constraints.

A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the categ…

In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…

We construct the Lie algebra of an n-Lie algebra and we also define the notion of cohomology of an n-Lie algebra.

The study describes the free Lie-Yamaguti algebra.