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I propose an alternative algorithm to compute the MMS voting rule. Instead of using linear programming, in this new algorithm the maximin support value of a committee is computed using a sequence of maximum flow problems.

Problems from metric graph theory such as Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had a strong impact on the field of parameterized complexity by being the first problems in NP to admit double-exponential lower bounds in the treewidth, and even in the vertex cover number for the latter. We initiate the study of enumerating minimal solution sets for these problems and show that they are also of great interest in enumeration. More specifically, we show that enumerating minimal resolving sets in graphs and minimal geodetic sets in split graphs are equivalent to hypergraph dualization, arguably one of the most important open problems in algorithmic enumeration. This provides two new natural examples to a question that emerged in different works this last decade: for which vertex (or edge) set graph property $\Pi$ is the enumeration of minimal (or maximal) subsets satisfying $\Pi$ equivalent to hypergraph dualization? As only very few properties are known to fit within this context -- namely, properties related to minimal domination -- our results make significant progress in characterizing such properties, and provide new angles of approach for tackling hypergraph dualization. In a second step, we consider cases where our reductions do not apply, namely graphs with no long induced paths, and show these cases to be mainly tractable.

We introduce a flexible method to simultaneously infer both the drift and volatility functions of a discretely observed scalar diffusion. We introduce spline bases to represent these functions and develop a Markov chain Monte Carlo algorithm to infer, a posteriori, the coefficients of these functions in the spline basis. A key innovation is that we use spline bases to model transformed versions of the drift and volatility functions rather than the functions themselves. The output of the algorithm is a posterior sample of plausible drift and volatility functions that are not constrained to any particular parametric family. The flexibility of this approach provides practitioners a powerful investigative tool, allowing them to posit a variety of parametric models to better capture the underlying dynamics of their processes of interest. We illustrate the versatility of our method by applying it to challenging datasets from finance, paleoclimatology, and astrophysics. In view of the parametric diffusion models widely employed in the literature for those examples, some of our results are surprising since they call into question some aspects of these models.

We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of non-local in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at $t = 0$ which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.

Auxiliary data sources have become increasingly important in epidemiological surveillance, as they are often available at a finer spatial and temporal resolution, larger coverage, and lower latency than traditional surveillance signals. We describe the problem of spatial and temporal heterogeneity in these signals derived from these data sources, where spatial and/or temporal biases are present. We present a method to use a ``guiding'' signal to correct for these biases and produce a more reliable signal that can be used for modeling and forecasting. The method assumes that the heterogeneity can be approximated by a low-rank matrix and that the temporal heterogeneity is smooth over time. We also present a hyperparameter selection algorithm to choose the parameters representing the matrix rank and degree of temporal smoothness of the corrections. In the absence of ground truth, we use maps and plots to argue that this method does indeed reduce heterogeneity. Reducing heterogeneity from auxiliary data sources greatly increases their utility in modeling and forecasting epidemics.

We investigate the frequentist guarantees of the variational sparse Gaussian process regression model. In the theoretical analysis, we focus on the variational approach with spectral features as inducing variables. We derive guarantees and limitations for the frequentist coverage of the resulting variational credible sets. We also derive sufficient and necessary lower bounds for the number of inducing variables required to achieve minimax posterior contraction rates. The implications of these results are demonstrated for different choices of priors. In a numerical analysis we consider a wider range of inducing variable methods and observe similar phenomena beyond the scope of our theoretical findings.

Flexible estimation of the mean outcome under a treatment regimen (i.e., value function) is the key step toward personalized medicine. We define our target parameter as a conditional value function given a set of baseline covariates which we refer to as a stratum based value function. We focus on semiparametric class of decision rules and propose a sieve based nonparametric covariate adjusted regimen-response curve estimator within that class. Our work contributes in several ways. First, we propose an inverse probability weighted nonparametrically efficient estimator of the smoothed regimen-response curve function. We show that asymptotic linearity is achieved when the nuisance functions are undersmoothed sufficiently. Asymptotic and finite sample criteria for undersmoothing are proposed. Second, using Gaussian process theory, we propose simultaneous confidence intervals for the smoothed regimen-response curve function. Third, we provide consistency and convergence rate for the optimizer of the regimen-response curve estimator; this enables us to estimate an optimal semiparametric rule. The latter is important as the optimizer corresponds with the optimal dynamic treatment regimen. Some finite-sample properties are explored with simulations.

It often happens that free algebras for a given theory satisfy useful reasoning principles that are not preserved under homomorphisms of algebras, and hence need not hold in an arbitrary algebra. For instance, if $M$ is the free monoid on a set $A$, then the scalar multiplication function $A\times M \to M$ is injective. Therefore, when reasoning in the formal theory of monoids under $A$, it is possible to use this injectivity law to make sound deductions even about monoids under $A$ for which scalar multiplication is not injective -- a principle known in algebra as the permanence of identity. Properties of this kind are of fundamental practical importance to the logicians and computer scientists who design and implement computerized proof assistants like Lean and Coq, as they enable the formal reductions of equational problems that make type checking tractable. As type theories have become increasingly more sophisticated, it has become more and more difficult to establish the useful properties of their free models that enable effective implementation. These obstructions have facilitated a fruitful return to foundational work in type theory, which has taken on a more geometrical flavor than ever before. Here we expose a modern way to prove a highly non-trivial injectivity law for free models of Martin-L\"of type theory, paying special attention to the ways that contemporary methods in type theory have been influenced by three important ideas of the Grothendieck school: the relative point of view, the language of universes, and the recollement of generalized spaces.

We study the multivariate deconvolution problem of recovering the distribution of a signal from independent and identically distributed observations additively contaminated with random errors (noise) from a known distribution. For errors with independent coordinates having ordinary smooth densities, we derive an inversion inequality relating the $L^1$-Wasserstein distance between two distributions of the signal to the $L^1$-distance between the corresponding mixture densities of the observations. This smoothing inequality outperforms existing inversion inequalities. As an application of the inversion inequality to the Bayesian framework, we consider $1$-Wasserstein deconvolution with Laplace noise in dimension one using a Dirichlet process mixture of normal densities as a prior measure on the mixing distribution (or distribution of the signal). We construct an adaptive approximation of the sampling density by convolving the Laplace density with a well-chosen mixture of normal densities and show that the posterior measure concentrates around the sampling density at a nearly minimax rate, up to a log-factor, in the $L^1$-distance. The same posterior law is also shown to automatically adapt to the unknown Sobolev regularity of the mixing density, thus leading to a new Bayesian adaptive estimation procedure for mixing distributions with regular densities under the $L^1$-Wasserstein metric. We illustrate utility of the inversion inequality also in a frequentist setting by showing that an appropriate isotone approximation of the classical kernel deconvolution estimator attains the minimax rate of convergence for $1$-Wasserstein deconvolution in any dimension $d\geq 1$, when only a tail condition is required on the latent mixing density and we derive sharp lower bounds for these problems

This study focuses on how different modalities of human communication can be used to distinguish between healthy controls and subjects with schizophrenia who exhibit strong positive symptoms. We developed a multi-modal schizophrenia classification system using audio, video, and text. Facial action units and vocal tract variables were extracted as low-level features from video and audio respectively, which were then used to compute high-level coordination features that served as the inputs to the audio and video modalities. Context-independent text embeddings extracted from transcriptions of speech were used as the input for the text modality. The multi-modal system is developed by fusing a segment-to-session-level classifier for video and audio modalities with a text model based on a Hierarchical Attention Network (HAN) with cross-modal attention. The proposed multi-modal system outperforms the previous state-of-the-art multi-modal system by 8.53% in the weighted average F1 score.