Section: New Results

Analysis of tree data

Participants : Romain Azaïs, Christophe Godin, Florian Ingels, Clément Legrand.

• Related Research Axes: RW1 (Representations of forms in silico)

• Related Key Modeling Challenges: KMC1 (A new paradigm for modeling tree structures in biology)

Tree-structured data naturally appear at different scales and in various fields of biology where plants and blood vessels may be described by trees. In the team, we aim to investigate a new paradigm for modeling tree structures in biology in particular to solve complex problems related to the representation of biological organisms and their forms in silico.

In 2018, we investigated the following questions linked to the analysis of tree data. (i) How to control the complexity of the algorithms used to solve queries on tree structures? For example, computing the edit distance matrix of a dataset of large trees is numerically expensive. (ii) How to estimate the parameters within a stochastic model of trees? And finally, (iii) how to develop statistical learning algorithms adapted to tree data? In general, trees do not admit a Euclidean representation, while most of classification algorithms are only adapted to Euclidean data. Consequently, we need to study methods that are specific to tree data.

Approximation of trees by self-nested trees. Complex queries on tree structures (e.g., computation of edit distance, finding common substructures, compression) are required to handle tree objects. A critical question is to control the complexity of the algorithms implemented to solve these queries. One way to address this issue is to approximate the original trees by simplified structures that achieve good algorithmic properties. One can expect good algorithmic properties from structures that present a high level of redundancy in their substructures. Indeed, one can take account these repetitions to avoid redundant computations on the whole structure. In the team, we think that the class of self-nested trees, that are the most compressed trees by DAG compression scheme, is a good candidate to be such an approximation class.

In [7], we have proved the algorithmic efficiency of self-nested trees through different questions (compression, evaluation of recursive functions, evaluation of edit distance) and studied their combinatorics. In particular, we have established that self-nested trees are roughly exponentially less frequent than general trees. This combinatorics can be an asset in exhaustive search problems. Nevertheless, this result also says that one can not always take advantage of the remarkable algorithmic properties of self-nested trees when working with general trees. Consequently, our aim is to investigate how general trees can be approximated by simplified trees in the class of self-nested trees from both theoretical and numerical perspectives.

We conjecture that the problem of optimal approximation by a self-nested tree is NP-hard. Despite a substantial work in 2018 (internship of Clément Legrand), this remains an open question. Consequently, we have developed a suboptimal approximation algorithm based on the height profile of a tree that can be used to very rapidly predict the edit distance between two trees, which is a usual but costly operation for comparing tree data in computational biology [7]. Another algorithm based on the simulation of Gibbs measures on the space of trees is currently under development. This work should result in a publication next year.

Statistical inference. The main objective of statistical inference is to retrieve the unknown parameters of a stochastic model from observations. A Galton-Watson tree is the genealogical tree of a population starting from one initial ancestor in which each individual gives birth to a random number of children according to the same probability distribution, independently of each other. In a recent work [12], we have focused on Galton-Watson trees conditional on their number of nodes. Several main classes of random trees can be seen as conditioned Galton-Watson trees. For instance, an ordered tree picked uniformly at random in the set of all ordered trees of a given size is a conditioned Galton-Watson tree with offspring distribution the geometric law with parameter $1/2$. Statistical methods were developed for conditioned Galton-Watson trees in [19]. We have introduced new estimators and stated their consistency. Our techniques improve the existing results both theoretically and numerically. A simulation study shows the good behavior of our procedure on finite-sample sizes and from missing or noisy data.

In a very different context, a substantial work has been made on statistical inference for piecewise-deterministic processes [2], [9], [8].

Kernel methods for tree data. In statistical learning, one aims to build a decision rule of a qualitative variable $Y$ as a function of a feature $X$ (typically a vector of ${ℝ}^d$) from a training dataset ${(X_i,Y_i)}_{1\le i\le n}$. We assume that $X$ is a tree, ordered or not, with or without labels. This framework is quite original since the state space of $X$ is not endowed with a canonical inner product. Kernel methods are particularly adapted to this setting since they enable to transform the raw data into a Hilbert space. In this context, the main issue is related to the construction of a good kernel. A kernel function adapted to trees is the subtree kernel introduced [24]. While the literature has never been focused on the weight function involved in the subtree kernel, we have shown that this function is crucial in prediction problems. We have proposed a new algorithm for computing the subtree kernel. It has been designed to allow learning the weight function directly from the data. On some difficult datasets, the prediction error is dramatically decreased from $>50\%$ to $3\%$.

This work is part of the ROMI project, that aims to develop an open and lightweight robotics platform for microfarms. This project requires to investigate advanced analysis and modeling techniques for plant structures. A main issue that arises in this context is to predict a feature of the plant (species, health status, etc) from its topology.

Invited talk on tree structures and algorithms Christophe Godin gave a invited talk entitled Can we manipulate tree forms like numbers? that was prepared with Romain Azaïs at the workshop on Mathematics for Developmental Biology organized at the Banff International Research Station for Mathematical Innovation and Discovery, organized by P. Prusinkiewicz and E. Mjolsness (Banff, Canada, December 2017).

Abstract: Tree-forms are ubiquitous in nature and recent observation technologies make it increasingly easy to capture their details, as well as the dynamics of their development, in 3 dimensions with unprecedented accuracy. These massive and complex structural data raise new conceptual and computational issues related to their analysis and to the quantification of their variability. Mathematical and computational techniques that usually successfully apply to traditional scalar or vectorial datasets fail to apply to such structural objects: How to define the average form of a set of tree-forms? How to compare and classify tree-forms? Can we solve efficiently optimization problems in tree-form spaces? How to approximate tree-forms? Can their intrinsic exponential computational curse be circumvented? In this talk, we presented a recent work to approach these questions from a new perspective, in which tree-forms show properties similar to that of numbers or real functions: they can be decomposed, approximated, averaged, and transformed in dual spaces where specific computations can be carried out more efficiently. We will discuss how these first results can be applied to the analysis and simulation of tree-forms in developmental biology (https://www.birs.ca/events/2017/5-day-workshops/17w5164).