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Neural abstractions have been recently introduced as formal approximations of complex, nonlinear dynamical models. They comprise a neural ODE and a certified upper bound on the error between the abstract neural network and the concrete dynamical model. So far neural abstractions have exclusively been obtained as neural networks consisting entirely of $ReLU$ activation functions, resulting in neural ODE models that have piecewise affine dynamics, and which can be equivalently interpreted as linear hybrid automata. In this work, we observe that the utility of an abstraction depends on its use: some scenarios might require coarse abstractions that are easier to analyse, whereas others might require more complex, refined abstractions. We therefore consider neural abstractions of alternative shapes, namely either piecewise constant or nonlinear non-polynomial (specifically, obtained via sigmoidal activations). We employ formal inductive synthesis procedures to generate neural abstractions that result in dynamical models with these semantics. Empirically, we demonstrate the trade-off that these different neural abstraction templates have vis-a-vis their precision and synthesis time, as well as the time required for their safety verification (done via reachability computation). We improve existing synthesis techniques to enable abstraction of higher-dimensional models, and additionally discuss the abstraction of complex neural ODEs to improve the efficiency of reachability analysis for these models.

Performance analysis is carried out in a near-field multiple-input multiple-output (MIMO) system for both discrete and continuous aperture antennas. The effective degrees of freedom (EDoF) is first derived. It is shown that near-field MIMO systems have a higher EDoF than free-space far-field ones. Additionally, the near-field EDoF further depends on the communication distance. Based on the derived EDoF, closed-form expressions of channel capacity with a fixed distance are obtained. As a further advance, with randomly deployed receivers, ergodic capacity is derived. Simulation results reveal that near-field MIMO has an enhanced multiplexing gain even under line-of-sight transmissions. In addition, the performance of discrete MIMO converges to that of continuous aperture MIMO.

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.

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 study functional and concurrent calculi with non-determinism, along with type systems to control resources based on linearity. The interplay between non-determinism and linearity is delicate: careless handling of branches can discard resources meant to be used exactly once. Here we go beyond prior work by considering non-determinism in its standard sense: once a branch is selected, the rest are discarded. Our technical contributions are three-fold. First, we introduce a $\pi$-calculus with non-deterministic choice, governed by session types. Second, we introduce a resource $\lambda$-calculus, governed by intersection types, in which non-determinism concerns fetching of resources from bags. Finally, we connect our two typed non-deterministic calculi via a correct translation.

Originally introduced in the context of the algebraic approach to term graph rewriting, the notion of gs-monoidal category has surfaced a few times under different monikers in the last decades. They can be thought of as symmetric monoidal categories whose arrows are generalised relations, with enough structure to talk about domains and partial functions, but less structure than cartesian bicategories. The aim of this paper is threefold. The first goal is to extend the original definition of gs-monoidality by enriching it with a preorder on arrows, giving rise to what we call oplax cartesian categories. Second, we show that (preorder-enriched) gs-monoidal categories naturally arise both as Kleisli categories and as span categories, and the relation between the resulting formalisms is explored. Finally, we present two theorems concerning Yoneda embeddings on the one hand and functorial completeness on the other, the latter inducing a completeness result also for lax functors from oplax cartesian categories to $\mathbf{Rel}$.

We develop lower bounds on communication in the memory hierarchy or between processors for nested bilinear algorithms, such as Strassen's algorithm for matrix multiplication. We build on a previous framework that establishes communication lower bounds by use of the rank expansion, or the minimum rank of any fixed size subset of columns of a matrix, for each of the three matrices encoding a bilinear algorithm. This framework provides lower bounds for a class of dependency directed acyclic graphs (DAGs) corresponding to the execution of a given bilinear algorithm, in contrast to other approaches that yield bounds for specific DAGs. However, our lower bounds only apply to executions that do not compute the same DAG node multiple times. Two bilinear algorithms can be nested by taking Kronecker products between their encoding matrices. Our main result is a lower bound on the rank expansion of a matrix constructed by a Kronecker product derived from lower bounds on the rank expansion of the Kronecker product's operands. We apply the rank expansion lower bounds to obtain novel communication lower bounds for nested Toom-Cook convolution, Strassen's algorithm, and fast algorithms for contraction of partially symmetric tensors.

Langevin dynamics are widely used in sampling high-dimensional, non-Gaussian distributions whose densities are known up to a normalizing constant. In particular, there is strong interest in unadjusted Langevin algorithms (ULA), which directly discretize Langevin dynamics to estimate expectations over the target distribution. We study the use of transport maps that approximately normalize a target distribution as a way to precondition and accelerate the convergence of Langevin dynamics. We show that in continuous time, when a transport map is applied to Langevin dynamics, the result is a Riemannian manifold Langevin dynamics (RMLD) with metric defined by the transport map. We also show that applying a transport map to an irreversibly-perturbed ULA results in a geometry-informed irreversible perturbation (GiIrr) of the original dynamics. These connections suggest more systematic ways of learning metrics and perturbations, and also yield alternative discretizations of the RMLD described by the map, which we study. Under appropriate conditions, these discretized processes can be endowed with non-asymptotic bounds describing convergence to the target distribution in 2-Wasserstein distance. Illustrative numerical results complement our theoretical claims.

Due to their inherent capability in semantic alignment of aspects and their context words, attention mechanism and Convolutional Neural Networks (CNNs) are widely applied for aspect-based sentiment classification. However, these models lack a mechanism to account for relevant syntactical constraints and long-range word dependencies, and hence may mistakenly recognize syntactically irrelevant contextual words as clues for judging aspect sentiment. To tackle this problem, we propose to build a Graph Convolutional Network (GCN) over the dependency tree of a sentence to exploit syntactical information and word dependencies. Based on it, a novel aspect-specific sentiment classification framework is raised. Experiments on three benchmarking collections illustrate that our proposed model has comparable effectiveness to a range of state-of-the-art models, and further demonstrate that both syntactical information and long-range word dependencies are properly captured by the graph convolution structure.

We introduce a multi-task setup of identifying and classifying entities, relations, and coreference clusters in scientific articles. We create SciERC, a dataset that includes annotations for all three tasks and develop a unified framework called Scientific Information Extractor (SciIE) for with shared span representations. The multi-task setup reduces cascading errors between tasks and leverages cross-sentence relations through coreference links. Experiments show that our multi-task model outperforms previous models in scientific information extraction without using any domain-specific features. We further show that the framework supports construction of a scientific knowledge graph, which we use to analyze information in scientific literature.