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We prove descent theorems for semiorthogonal decompositions using techniques from derived algebraic geometry. Our methods allow us to capture more general filtrations of derived categories and even marked filtrations, where one descends not…
We propose a procedure to build a decision tree which approximates the performance of complex machine learning models. This single approximation tree can be used to interpret and simplify the predicting pattern of random forests (RFs) and…
This paper is concerned with convergence estimates for fully discrete tree tensor network approximations of high-dimensional functions from several model classes. For functions having standard or mixed Sobolev regularity, new estimates…
The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths),…
Termination property of functions is an important issue in computability theory. In this paper, we show that repeated iterations of a function can induce an order amongst the elements of its domain set. Hasse diagram of the poset, thus…
In this paper we investigate the use of the concept of tree dimension in Horn clause analysis and verification. The dimension of a tree is a measure of its non-linearity - for example a list of any length has dimension zero while a complete…
The decomposition of a density function on a domain into a minimal sum of unimodal components is a fundamental problem in statistics, leading to the topological invariant of unimodal category of a density. This paper gives an efficient…
Contour trees describe the topology of level sets in scalar fields and are widely used in topological data analysis and visualization. A main challenge of utilizing contour trees for large-scale scientific data is their computation at scale…
The construction of a dendogram on a set of individuals is a key component of a genomewide association study. However even with modern sequencing technologies the distances on the individuals required for the construction of such a…
A general class of stochastic Runge-Kutta methods for the weak approximation of It\^o and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an…
We prove that functionality of compositions of top-down tree transducers is decidable by reducing the problem to the functionality of one top-down tree transducer with look-ahead.
For a graph G, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of…
The properties of randomly evolving special trees having defined and analyzed already in two earlier papers (arXiv:cond-mat/0205650 and arXiv:cond-mat/0211092) have been investigated in the case when the continuous time parameter converges…
Submodular function minimization is a fundamental optimization problem that arises in several applications in machine learning and computer vision. The problem is known to be solvable in polynomial time, but general purpose algorithms have…
We construct a small regular cellular decomposition of the Fulton MacPherson operad $FM_2$ that is compatible with the operad composition. The cells are indexed by trees with edges of two colors and vertices labelled by cells of the cacti…
We present a new method to count unrooted maps on the sphere up to orientation-preserving homeomorphisms. The principle, called tree-decomposition, is to deform a map into an arborescent structure whose nodes are occupied by constrained…
We propose a tree-based algorithm for classification and regression problems in the context of functional data analysis, which allows to leverage representation learning and multiple splitting rules at the node level, reducing…
We construct and study new generalisations to rooted trees and forests of some properties of shuffles of words. First, we build a coproduct on rooted trees which, together with their shuffle, endow them with bialgebra structure. We then…
This paper is concerned with the approximation of high-dimensional functions in a statistical learning setting, by empirical risk minimization over model classes of functions in tree-based tensor format. These are particular classes of…
Bicubic maps are in bijection with \beta(0,1)-trees. We introduce two new ways of decomposing \beta(0,1)-trees. Using this we define an endofunction on \beta(0,1)-trees, and thus also on bicubic maps. We show that this endofunction is in…