Among the recent work on designing algorithms for selecting citizens' assembly participants, one key property of these algorithms has not yet been studied: their manipulability. Strategic manipulation is a concern because these algorithms must satisfy representation constraints according to volunteers' self-reported features; misreporting these features could thereby increase a volunteer's chance of being selected, decrease someone else's chance, and/or increase the expected number of seats given to their group. Strikingly, we show that Leximin — an algorithm that is widely used for its fairness — is highly manipulable in this way. We then introduce a new class of selection algorithms that use Lp norms as objective functions. We show that the manipulability of the Lp-based algorithm decreases in O(1/n^(1-1/p)) as the number of volunteers n grows, approaching the optimal rate of O(1/n) as p approaches infinity. These theoretical results are confirmed via experiments in eight real-world datasets.

Ordinary differential equations (ODEs), via their induced flow maps, provide a powerful framework to parameterize invertible transformations for representing complex probability distributions. While such models have achieved enormous success in machine learning, little is known about their statistical properties. This work establishes the first general nonparametric statistical convergence analysis for distribution learning via ODE models trained through likelihood maximization. We first prove a convergence theorem applicable to arbitrary velocity field classes $\mathcal{F}$ satisfying certain simple boundary constraints. This general result captures the trade-off between the approximation error and complexity of the ODE model. We show that the latter can be quantified via the $C^1$-metric entropy of the class $\mathcal{F}$. We then apply this general framework to the setting of $C^k$-smooth target densities, and establish nearly minimax-optimal convergence rates for two relevant velocity field classes $\mathcal{F}$: $C^k$ functions and neural networks. The latter is the practically important case of neural ODEs. Our results also provide insight on how the choice of velocity field class, and the dependence of this choice on sample size (e.g., the scaling of neural network classes), impact statistical performance.

Diffusion models have had a profound impact on many application areas, including those where data are intrinsically infinite-dimensional, such as images or time series. The standard approach is first to discretize and then to apply diffusion models to the discretized data. While such approaches are practically appealing, the performance of the resulting algorithms typically deteriorates as discretization parameters are refined. In this paper, we instead directly formulate diffusion-based generative models in infinite dimensions and apply them to the generative modelling of functions. We prove that our formulations are well posed in the infinite-dimensional setting and provide dimension-independent distance bounds from the sample to the target measure. Using our theory, we also develop guidelines for the design of infinite-dimensional diffusion models. For image distributions, these guidelines are in line with current canonical choices. For other distributions, however, we can improve upon these canonical choices. We demonstrate these results both theoretically and empirically, by applying the algorithms to data distributions on manifolds and to distributions arising in Bayesian inverse problems or simulation-based inference.

A key promise of voting is that, by accounting for all constituents' preferences, it produces decisions that benefit society overall. It is alarming, then, that all deterministic voting rules have unbounded distortion (i.e., arbitrarily suboptimal social welfare). Existing work usually circumvents this stark impossibility by assuming restrictions on voters' possible latent utilities [Anshelevich et al. 2021]; however, it is not clear whether such assumptions reliably hold in practice, because voters' utilities are, to a large degree, inherent.