It is of soaring demand to develop statistical analysis tools that are robust against contamination as well as preserving individual data owners' privacy. In spite of the fact that both topics host a rich body of literature, to the best of our knowledge, we are the first to systematically study the connections between the optimality under Huber's contamination model and the local differential privacy (LDP) constraints. We start with a general minimax lower bound result, which disentangles the costs of being robust against Huber's contamination and preserving LDP. We further study four concrete examples: a two-point testing problem, a potentially-diverging mean estimation problem, a nonparametric density estimation problem and a univariate median estimation problem. For each problem, we demonstrate procedures that are optimal in the presence of both contamination and LDP constraints, comment on the connections with the state-of-the-art methods that are only studied under either contamination or privacy constraints, and unveil the connections between robustness and LDP via partially answering whether LDP procedures are robust and whether robust procedures can be efficiently privatised. Overall, our work showcases a promising prospect of joint study for robustness and local differential privacy.
This is joint work with Mengchu Li and Yi Yu.[-]

The advent of large scale inference has spurred reexamination of conventional statistical thinking. In a series of highly original articles, Efron showed in some examples that the ensemble of the null distributed test statistics grossly deviated from the theoretical null distribution, and Efron persuasively illustrated the danger in assuming the theoretical null's veracity for downstream inference. Though intimidating in other contexts, the large scale setting is to the statistician's benefit here. There is now potential to estimate, rather than assume, the null distribution.
In a model for n many z-scores with at most k nonnulls, we adopt Efron's suggestion and consider estimation of location and scale parameters for a Gaussian null distribution. Placing no assumptions on the nonnull effects, we consider rate-optimal estimation in the entire regime k < n/2, that is, precisely the regime in which the null parameters are identifiable. The minimax upper bound is obtained by considering estimators based on the empirical characteristic function and the classical kernel mode estimator. Faster rates than those in Huber's contamination model are achievable by exploiting the Gaussian character of the data. As a consequence, it is shown that consistent estimation is indeed possible in the practically relevant regime k ≍ n. In a certain regime, the minimax lower bound involves constructing two marginal distributions whose characteristic functions match on a wide interval containing zero. The construction notably differs from those in the literature by sharply capturing a second-order scaling of n/2 − k in the minimax rate.[-]

I will discuss a model of interacting particles in continuous space which is reversible with respect to Poisson point measures with constant density. Similar discrete models are known to ”homogenize”, in the sense that the evolution of the particle density can be approximated by the solution to a partial differential equation over large scales. The goal of the talk is to present some results that make this approximation quantitative.
Based on joint works with Arianna Giunti, Chenlin Gu and Maximilian Nitzschner.[-]

We consider the assignment (or bipartite matching) problem between $n$ source points and $n$ target points on the real line, where the assignment cost is a concave power of the distance, i.e. |x − y|p, for 0 < p < 1. It is known that, differently from the convex case (p > 1) where the solution is rigid, i.e. it does not depend on p, in the concave case it may varies with p and exhibit interesting long-range connections, making it more appropriate to model realistic situations, e.g. in economics and biology. In the random version of the problem, the points are samples of i.i.d. random variables, and one is interested in typical properties as the sample size n grows. Barthe and Bordenave in 2013 proved asymptotic upper and lower bounds in the range 0 < p < 1/2, which they conjectured to be sharp. Bobkov and Ledoux, in 2020, using optimal transport and Fourier-analytic tools, determined explicit upper bounds for the average assignment cost in the full range 0 < p < 1, naturally yielding to the conjecture that a “phase transition” occurs at p = 1/2. We settle affirmatively both conjectures. The novel mathematical tool that we develop, and may be of independent interest, is a formulation of Kantorovich problem based on Young integration theory, where the difference between two measures is replaced by the weak derivative of a function with finite q-variation.
Joint work with M. Goldman (arXiv:2305.09234).[-]

Traditionally, theories of “Sobolev” spaces on metric spaces have used local Lipschitz constants as a substitute for the gradient of functions. However, a recent study by Kajino and Murugan revealed that such an idea does not work for a class of self-similar sets including the planar Sierpinski carpet. The notion of conductive homogeneity was proposed to construct a counterpart of Sobolev spaces and Sobolev p-energy even for such cases. In this talk, I will review the method of construction of Sobolev spaces under the conductive homogeneity and give a class of regular polygon-based self-similar sets having the conductive homogeneity. Our condition is the local symmetry of the space with some (or no) global symmetry. In particular, we show that any locally symmetric triangle-based self-similar sets possess the conductive homogeneity. This is joint work with Y. Ota.[-]

Genetic differences are a critical driver of disease risk and healthy variation, across the tree of life. Mutations arise and spread in our distant, genealogical ancestors, and so genetic variation data can provide a window into our evolutionary past, allowing us to understand processes such as population size changes, admixture, natural selection, and even evolution of the mutation and recombination processes that generate the variation itself. It has long been recognised that knowledge of genealogical relationships among individuals would allow us to capture almost all the information available from such data. However, only in recent years has it become computationally feasible to infer such genealogies, genome-wide, from variation patterns. One such method, Relate, developed in our lab, allows approximate inference of genealogical trees under coalescent-like models, for up to tens of thousands of samples. Here, we will show that a powerful approach for inference is to identify and characterise departures from the relatively simple models used to build these trees. By defining a 'population' as a set of coalescence rates between labelled individuals backwards in time, we can uncover variability in these rates, and use a single collection of trees to identify ancient mixing events among populations - including 'ghost' groups we have never sampled - natural selection favouring the descendents of particular branches of the genealogy, and departures from mathematical expectations under clock-like behaviour, indicating disruption of recombination or mutation.[-]

Consider two ancestral lineages sampled from a system of two-dimensional branching random walks with logistic regulation in the stationary regime. We study the asymptotics of their coalescence time for large initial separation and find that it agrees with well known results for a suitably scaled two-dimensional stepping stone model and also with Malécot's continuous-space approximation for the probability of identity by descent as a function of sampling distance.
This can be viewed as a justification for the replacement of locally fluctuating population sizes by fixed effective sizes. Our main tool is a joint regeneration construction for the spatial embeddings of the two ancestral lineages.[-]

We investigate the spread of a population of pathogens infecting a spatially distributed host population with immunity. We model this situation by placing susceptible immobile hosts on the vertices of $\mathbb{Z}$. Pathogens diffuse in space according to symmetric simple random walks on $\mathbb{Z}$ and attempt an infection when they meet a host. As hosts often have an immune response against infections, we assume that each host needs to be attacked a random number of times, according to some distribution $I$, before it will be infected. Otherwise parasite reproduction is prevented and the parasite gets killed. In case of an successful infection the parasite kills the host and sets free a random number of offspring, according to some distribution $A$. We characterize the survival probability of the pathogen population depending on the initial distribution of pathogens and show under some relatively mild conditions on $I$ and $A$, that conditioned on survival of the parasite population, the infection spreads a.s. asymptotically at least and at most linearly fast. This talk is based on joint work in progress with Sascha Franck.[-]

We consider a population model in which the season alternates between winter and summer, and individuals can acquire mutations either that are advantageous in the summer and disadvantageous in the winter, or vice versa. Also, we assume that individuals in the population can either be active or dormant, and that individuals can move between these two states. Dormant individuals do not reproduce but do not experience selective pressures. We show that, under certain conditions, over time we see two waves of adaptation. Some individuals repeatedly acquire mutations that are beneficial in the summer, while others repeatedly acquire mutations that are beneficial in the winter. Individuals can survive the season during which they are less fit by entering a dormant state. This result suggests that, for populations in fluctuating environments, dormancy may have the potential to induce speciation. This is joint work with Fernando Cordero and Adrian Gonzalez Casanova.[-]