Related papers: An example driven introduction to probabilistic ro…
In line with the notion of probabilistic rough paths introduced in the previous contribution \cite{salkeld2021Probabilistic}, we address corresponding random controlled rough paths (first introduced in \cite{2019arXiv180205882.2B}), the…
Dynamic trees are mixtures of tree structured belief networks. They solve some of the problems of fixed tree networks at the cost of making exact inference intractable. For this reason approximate methods such as sampling or mean field…
A Bialgebra is a module over a ring that is both an associative algebra and a co-associative coalgebra with the product and coproduct additionally satisfying an appropriate commutative relationship. One application of Bialgebras is in the…
The purpose of this article is to solve rough differential equations with the theory of regularity structures. These new tools recently developed by Martin Hairer for solving semi-linear partial differential stochastic equations were…
Towards a future where machine learning systems will integrate into every aspect of people's lives, researching methods to interpret such systems is necessary, instead of focusing exclusively on enhancing their performance. Enriching the…
Rich out of equilibrium collective dynamics of strongly interacting large assemblies emerge in many areas of science. Some intriguing and not fully understood examples are the glassy arrest in atomic, molecular or colloidal systems,…
We introduce a new notion of "regularity structure" that provides an algebraic framework allowing to describe functions and / or distributions via a kind of "jet" or local Taylor expansion around each point. The main novel idea is to…
We generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise. As an illustration, we discuss a…
Mean field theory models of percolation on networks provide analytic estimates of network robustness under node or edge removal. We introduce a new mean field theory model based on generating functions that includes information about the…
We propose Partition Tree, a novel tree-based framework for conditional density estimation over general outcome spaces that supports both continuous and categorical variables within a unified formulation. Our approach models conditional…
We discuss several connections between discrete and continuous random trees. In the discrete setting, we focus on Galton-Watson trees under various conditionings. In particular, we present a simple approach to Aldous' theorem giving the…
Dynamic regression trees are an attractive option for automatic regression and classification with complicated response surfaces in on-line application settings. We create a sequential tree model whose state changes in time with the…
The potential lack of fairness in the outputs of machine learning algorithms has recently gained attention both within the research community as well as in society more broadly. Surprisingly, there is no prior work developing tree-induction…
We introduce a mean-field model of lattice trees based on embeddings into $\Z^d$ of abstract trees having a critical Poisson offspring distribution. This model provides a combinatorial interpretation for the self-consistent mean-field model…
In this paper we introduce a variation on the multidimensional segment tree, formed by unifying different interpretations of the dimensionalities of the data structure. We give some new definitions to previously well-defined concepts that…
In critical situations involving discrimination, gender inequality, economic damage, and even the possibility of casualties, machine learning models must be able to provide clear interpretations for their decisions. Otherwise, their obscure…
The theory of rough paths arose from a desire to establish continuity properties of ordinary differential equations involving terms of low regularity. While essentially an analytic theory, its main motivation and applications are in…
Transportation processes, which play a prominent role in the life and social sciences, are typically described by discrete models on lattices. For studying their dynamics a continuous formulation of the problem via partial differential…
In this paper we present a new interpretation of the Lions derivative as the Radon-Nikodym derivative of a vector measure, which provides a canonical extension of the Lions derivative for functions taking values in infinite dimensional…
To most mathematicians and computer scientists the word ``tree'' conjures up, in addition to the usual image, the image of a connected graph with no circuits. In the last few years various types of trees have been the subject of much…