Research Statement Noah Forman 2018

Transcription

1 Research Statement Noah Forman 018 I am a probability theorist focusing on random combinatorial structures and their continuum limits or analogues. Consider the simple random walk on Z, where each increment is ±1 with equal probability. This process converges in a scaling limit to a self-similar continuous process, Brownian motion. Thus, combinatorial study of the simple random walk is used to prove properties of the continuous Brownian motion, and conversely, analytical methods applied to Brownian motion are used to study random walks. In the same spirit, David Aldous conjectured in 1999 that a certain natural random walk on the space of binary (combinatorial) trees has a continuum analogue, a continuum-tree-valued diffusion [1,, 6]. I have spent the past several years solving this conjecture in [31, 3, 33, 3, 3]. Aldous s conjecture connects to central topics in probability theory. Continuum random trees can be mapped to a wide range of other important, complex random objects, so our continuumtree-valued diffusion can be mapped to processes on various other spaces. These include biological models such as coagulation and fragmentation processes [11, 1], branching processes, and exchangeable random partitions, compositions [31, 39, 6], and hierarchies [37], as well as random geometries such as the Brownian map [7, 8] and Liouville quantum gravity [0, 1, ]. For example, in [31] our study of this conjecture led to a method that resolves the first cases of a well-studied [17,, 6] 010 conjecture of Feng and Sun [3, ] on the existence of measure-valued diffusions as continuum analogues to a two-parameter family of random walks on a space of discrete partitions. Processes in this family are used to model fluctuating allele frequencies in a population. The analogous random walks on partitions may also be of interest in Bayesian clustering models. My collaborators Pal, Rizzolo and Winkel and I will use our method to resolve Feng and Sun s conjecture in full generality in forthcoming work. In addition to my work on these conjectures, I study the exchangeable structures mentioned above. Exchangeable random hierarchies, which I classified in [7, 30], are of interest for Bayesian non-parametric nested clustering [1, 3], as well as Bayesian phylogenetic inference [60, 61]. I also study excursion theory, fluctuation theory, local times, and path transformations and decompositions of R-valued random walks and Lévy processes [8, 8, 36]. I have harnessed and further cultivated this background in my study of Aldous s and Feng and Sun s conjectures. The Aldous diffusion. A cladogram is a rooted binary tree with labeled leaves and unlabeled internal vertices. We consider a Markov chain on the space of cladograms in which each transition comprises a down-move followed by an up-move, as follows. Down-move: a uniform random leaf is deleted, and its parent branch point is contracted away. Up-move: a uniform random edge is selected, a branch point is inserted into the middle of the edge, and the removed leaf is re-attached at this branch point. See Fig. 1. This process is used in Markov chain Monte Carlo algorithms for phylogenetic inference [9]. We call it the Aldous chain. A cousin of this chain in which the transitions consist only of up-moves, with no down-move component, is called Rémy growth [9]. If T 1 is the unique 1-leaf cladogram and (T n, n 1) is Rémy growth then T n is a uniform random n-leaf cladogram for every n 1. It follows that the Aldous chain is stationary under the uniform distribution on n-leaf cladograms, for any n 1. The uniform random cladograms that arise in Rémy growth converge to a scaling limit. We view the trees as metric spaces equipped with graph distance the distance between two vertices is the number of edges along the shortest path between them. If we scale each edge to have length 1/ n Figure 1. One down-up move of the Aldous chain. 1

2 Figure. Above: the Harris path [] corresponding to a rooted plane tree, with nodes colored to illustrate the correspondence. Below: Brownian excursion and a BCRT conformally embedded in R, the latter courtesy of Igor Kortchemski [6] (the particular excursion and CRT do not correspond). instead of 1, then the resulting trees T n / n converge as metric spaces to the Brownian continuum random tree (BCRT) []. The original development of the BCRT by Aldous [, 3, ] was based on the following well-known bijection. A rooted plane tree a tree in which the children of each branch point are ordered from left to right can be mapped to a path by considering an ant walking around the perimeter of the tree at unit speed and plotting the distance from the ant to the root as a function of time. See Fig.. To invert this map, imagine taking the graph of this path, applying glue to the bottom of each increment, and then squeezing the graph together laterally. E.g. in Fig., the 1 st increment would be glued to the 1 th to form the edge between red and green nodes in the tree, while the th and th increments would not be glued to each other, as the tops, rather than the bottoms, of those increments face each other. This map can be extended to apply not just to discrete (Dyck) paths, but to continuous, nonnegative excursion functions: functions e: [0, ) [0, ) with e(t) > 0 if and only if t (0, ζ(e)), for some ζ(e) > 0. In this manner, a standard Brownian excursion informally, Brownian motion conditioned to escape upwards from zero at time 0 and not return until time 1 can be mapped to a continuum random tree (CRT), called the BCRT. See the lower part of Fig.. Formally, a continuum tree or R-tree is a bounded metric space (T, d) with the tree-like property that, between any two points x, y T, there exists a unique non-self-intersecting path, and this is homeomorphic to the interval [0, d(x, y)]. See [1]. Recall the convergence of simple random walks to Brownian motion, mentioned at the outset of this document. Key in proving that convergence is the scaling convergence of the Binomial ( j, 1 ) probability distribution, which describes the walk at a fixed time j, to the Normal(0, t) probability distribution of the Brownian motion evaluated at a fixed time t. The Aldous chain evaluated at a fixed time is distributed as a uniform random cladogram. Thus, the scaling convergence of these random cladograms to the BCRT suggests the following. Conjecture 1 (Aldous, 99 [1,, 6]). The Aldous chain has a continuum analogue that is a continuum-tree-valued diffusion, stationary with the law of the BCRT. Theorem 1 (F-Pal-Rizzolo-Winkel, 18 [3]). The Aldous chain has a continuum-tree-valued analogous process, stationary with the law of the BCRT.

3 3 Σ Σ Σ leaf leaf leaf Σ 3 leaf 3 leaf 1 Σ 1 ρ Figure 3. -tree multi-spinal projection of a BCRT. (a) (b) (e) (c) (d) Figure. Stable ( 3 ) Lévy process, with jumps marked by BESQ( 1) excursions, depicted here as laterally symmetric colored spindle shapes. Form an interval partition by drawing a horizontal skewer through the picture, taking slices of spindles. The partition evolves as the skewer line moves upwards. Simulation by G. Brito, R. Chou, A. Forney, and C. Li, supervised by the author [9]. We prove this with a pathwise construction. In forthcoming work, we will establish further properties of this process, including the Markov property and continuity of its paths. Our construction is the first known (non-constant) continuous, stationary R-tree-valued Markov process. As the Aldous chain is a natural random walk on binary trees, this might be viewed as a canonical diffusion, like Brownian motion, on the space of binary R-trees. Recently, using a very different approach, Lohr, Mytnick and Winter [9] proved existence of an algebraic-tree-valued continuum limit of the Aldous chain, but this process lacks the continuously evolving distances of the conjectured diffusion. Our construction goes by building a tower of projectively consistent Markov processes on spaces of discrete trees with internal edges marked by interval partitions, with each interval representing the mass of a subtree branching off of a path within the CRT. See Fig. 3. The dynamics on the interval partitions are given by a novel construction [31, 33] depicted in Fig.. A discrete analogue to Fig. can be used to describe dynamics of a projection of the Aldous chain. Impacts of Aldous diffusion. The Aldous chain is used in Markov chain Monte Carlo (MCMC) algorithms for phylogenetic inference, such as LVB [9]. In studying this problem, we have gained a better understanding of this chain and, in [3], introduced a novel, related Markov chain on cladograms, that tends to leave low labels in the same region of the tree. Such new chains may also be of use in MCMC algorithms. The methods developed in this study might also shed light on other such Markov chains, such as the branch rotation chain, which is also used in LVB, and about which extremely little is known. A Markov chain on quadrangulations that is related to the branch rotation chain on trees was recently studied in [16], giving the first polynomial bound on its mixing times.

4 Figure. An approximation of the Brownian map. Simulation by Nicolas Curien [18]. As mentioned at the top of this statement, the BCRT can be mapped to many other objects. One notable example is a canonical random surface called the Brownian map, though sending the CRT to the map takes some additional random data [7, 8]. See Fig.. There is no known nonconstant diffusion on the space of surfaces that is stationary with the law of the Brownian map; it would be interesting to use the Aldous diffusion to construct one. The most immediate impact of this work, however, is in its connection to a conjecture of Feng and Sun, described below. Labeled two-parameter Poisson-Dirichlet diffusions. Dubins and Pitman s two-parameter Chinese restaurant processes (CRP(α, θ)) are a family of Markov processes to describe growing partitions of {1,,..., n}, n 1 []. Customers, representing the natural numbers, enter an infinitely large restaurant with infinitely large tables one-by-one, choosing randomly whether to join one of the tables with other patrons or to sit alone, with a bias to join a table that already holds more other customers. In particular, they join a table with m others with probability proportional to m α, or sit alone with probability proportional to Kα + θ, where K is the number of tables already occupied, with α [0, 1), θ > α. The random partitions thus formed turn out to be exchangeable: they are distributionally invariant under permutations of labels, meaning that the probability of any given partition arising depends only on the multiset of block sizes. Exchangeable structures are natural models on which one can do Bayesian inference. In particular, the CRPs are used in Bayesian non-parametric clustering [1, 38, 3], in which a collection of data is partitioned into an indeterminate number of subsets. The classic application is topic modeling, in which a collection of documents are partitioned by topic. In the CRP, the proportion of all customers that join any given table converges to a positive limit. These limiting frequencies, in ranked order from largest on down, have an important and well-studied probability distribution: the two-parameter Poisson-Dirichlet distribution (PD(α, θ)) [7]. This is a two-parameter family of laws on the Kingman simplex, := { (x i, i 1) [0, ) N i x i = 1 and j 1, x j x j+1 }. The one-parameter family with α = 0 was originally conceived of by Kingman [] and used as a model for allele frequencies in a population with infinitely many genetic types. In 1981, Ethier and Kurtz [0] introduced the infinitely-many-neutral-alleles models to describe such type frequencies fluctuating over time due to genetic drift, when there is no natural selection preference for any of the alleles. This is a one-parameter family of purely-atomic-measure-valued diffusions in which the atoms locations in type space represent distinct genetic types and masses of atoms represent type frequencies. The image of this diffusion on the Kingman simplex under the

5 map sending a measure to its ranked sequence of atom masses is also a diffusion, and is stationary under PD(0, θ). This diffusion is a continuum analogue to the reseating chain on the Chinese restaurant, in which at each step, a uniform random customer leaves and then selects a new seat as if entering the restaurant for the first time. This type of chain can be used for MCMC algorithms on clusterings. In 009, Petrov [] generalized Either and Kurtz s ranked masses diffusion to a two-parameter family of ranked allele frequency diffusions on, stationary under PD(α, θ). Petrov s diffusions are closely related to the work of Borodin and Olshanki [13, 1] regarding harmonic analysis on the infinite symmetric group. These diffusions have attracted considerable interest in the probability community [17, 19,,, 6, 63, 6]. The big question about these diffusions, which arguably predates Petrov s work (see [6]), is the following. Conjecture (Feng-Sun, 10 [3, ]). There exists a two-parameter family of Fleming-Viot processes (measure-valued diffusions) on purely atomic measures, with ranked atom masses evolving as Petrov s two-parameter diffusions. Theorem (F-Pal-Rizzolo-Winkel, 16 [31]). Such diffusions exist, for (α, θ) = ( 1, 0) and ( 1, 1 ). These processes arise from the pathwise construction illustrated in Fig., which was discovered in our work on Aldous s conjecture, with minimal modification to give measure-valued processes rather than interval partitions. A straightforward generalization of this construction solves the problem in the cases θ = α (0, 1) and θ = 0. In forthcoming work, we generalize this approach to give pathwise constructions of the conjectured diffusions for all (α, θ) (0, 1) [0, ). Our novel construction reveals different properties of Petrov s diffusions than those accessible via Petrov s generator methods or Feng and Sun s Dirichlet form methods. For example, we find that the biodiversity of the system evolves continuously in time, but that it is not, in itself, a Markov process. Ruggiero [6] had speculated about Markovian diversity evolutions related to Petrov s models, but he seems not to have reached a conclusion about the actual diversity derived from the models. As we complete our construction of the full two-parameter family of Feng and Sun s conjecture, we look forward to exploring other properties of these and Petrov s diffusions that are made accessible by our pathwise construction. Exchangeable hierarchies. A hierarchy on N is a collection H of subsets of N with: (i), N H, (ii) {j} H for every j N, and (iii) for every A, B H, the intersection A B equals either A, B, or (omitting some technicalities). Exchangeable random hierarchies arise in algorithms for nested clustering. Nested topic models sort collections of documents by topic and subtopic, as in the nested Chinese restuarant process [1]; or as mixtures of subtopics, as in the nested hierarchical Dirichlet process [3]. Such structures are also of interest in Bayesian phylogenetic inference [60, 61]. A rooted, weighted R-tree is a quadruple (T, d, r, p) where (T, d) is R-tree as discussed in the Aldous diffusion section, r T is a distinguished point called the root, and p is a probability distribution on T. For such a quadruple, let (t i, i 1) be i.i.d. with law p. We can use this to define a hierarchy: (1) G := { {j N: t j F (y)}: y T } where F (y) := {x T : y is between x and r}. I.e. the elements (called blocks ) of G correspond to the sets of samples falling within fringe subtrees F (y) cut off above points y T. The quadruple (T, d, r, p) is an interval partition (IP) tree [7] if: (i) p is supported out to the leaves of T and (ii) for x T, if x is a branch point or is in the closed support of p, then d(r, x) + p(f (x)) = 1. An IP tree can be thought of as a R-tree in which the metric is specified by the weight and the underlying branching structure. Theorem 3 (F-Haulk-Pitman, 18 [30]; F, 18 [7]). Given an exchangeable random hierarchy H on N, there exists an a.s. unique random (isomorphism class of) IP tree(s) (T, d, r, p) measurable in

6 6 Figure 6. Simulation of an IP tree representation of a hierarchy with continuous erosion (colored shading on the skeleton) and Cantor-like sets of branch points on each path. the tail σ-algebra associated with H such that, given the tail σ-algebra, the hierarchy is conditionally distributed as in (1). This theorem is in the spirit of de Finetti s characterization of exchangeable sequences or Kingman s descriptions of exchangeable partitions [7, ]. The idea is that, given the tail behavior of the hierarchy, the blocks arise as if by i.i.d. sampling of points from a CRT. In fact, the (isomorphism classes of) IP trees are in bijective correspondence with exchangeable, independently generated (analogous to i.i.d.) laws for hierarchies [7]. Related, recent work on CRT constructions and representations can be found in [, 0, 8]. Theorem (F, 18 [7]). The weight p on an IP tree (T, d, r, p) can be uniquely decomposed into the sum of a purely atomic measure, a continuous measure on the leaves of T, and the restriction of the length measure to a subset of the skeleton (non-leaves) of T. This result gives a hierarchies analogue to the property of self-similar fragmentation processes [10] that components break down via three mechanisms: repeated splitting into macroscopic blocks (corresponding to branch points and continuous mass on leaves), continuous erosion of dust out of large blocks (length measure on the skeleton), and instantaneous explosions of all or part of a large block into dust (atoms). Thus, an IP tree can be understood as a recipe for interspersing these three mechanisms. Directions for further research. (1) Construct a Markov process on the space of partitions of N, stationary with the law of the CRP(α, θ), in which blocks evolve like those of the labeled PD(α, θ) diffusions. () Apply the ideas of [7, 30] to devise new families of exchangeable hierarchies, combining explosions, erosion, and macroscopic splitting, that may be useful for nested topic models or phylogenetic inference. Hierarchies arising from simple iterative growth models, akin to the Chinese restaurant process, are of particular interest. Also in this vein... (3) Study bridges of sticky Brownian motion, with the aim of defining a natural sticky BCRT, combining splitting and erosion, akin to the IP tree in Fig. 6. () Use the Aldous diffusion to construct a diffusion on a space of surfaces, stationary with the law of the Brownian map of Fig.. () Study other combinatorial-tree-valued Markov chains, such as the branch rotation chain mentioned earlier, that are of interest for phylogenetic inference, and that could give rise to different diffusion dynamics on continuum trees or surfaces. In particular, nothing is known about the mixing time of branch rotation.

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