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We consider the problem of solving a family of parametric mixed-integer linear optimization problems where some entries in the input data change. We introduce the concept of cutting-plane layer (CPL), i.e., a differentiable cutting-plane generator mapping the problem data and previous iterates to cutting planes. We propose a CPL implementation to generate split cuts, and by combining several CPLs, we devise a differentiable cutting-plane algorithm that exploits the repeated nature of parametric instances. In an offline phase, we train our algorithm by updating the internal parameters controlling the CPLs, thus altering cut generation. Once trained, our algorithm computes, with predictable execution times and a fixed number of cuts, solutions with low integrality gaps. Preliminary computational tests show that our algorithm generalizes on unseen instances and captures underlying parametric structures.

The variational quantum eigensolver (VQE) is a hybrid algorithm that has the potential to provide a quantum advantage in practical chemistry problems that are currently intractable on classical computers. VQE trains parameterized quantum circuits using a classical optimizer to approximate the eigenvalues and eigenstates of a given Hamiltonian. However, VQE faces challenges in task-specific design and machine-specific architecture, particularly when running on noisy quantum devices. This can have a negative impact on its trainability, accuracy, and efficiency, resulting in noisy quantum data. We propose variational denoising, an unsupervised learning method that employs a parameterized quantum neural network to improve the solution of VQE by learning from noisy VQE outputs. Our approach can significantly decrease energy estimation errors and increase fidelities with ground states compared to noisy input data for the $\text{H}_2$, LiH, and $\text{BeH}_2$ molecular Hamiltonians, and the transverse field Ising model. Surprisingly, it only requires noisy data for training. Variational denoising can be integrated into quantum hardware, increasing its versatility as an end-to-end quantum processing for quantum data.

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there were finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, many real-world data are not naturally posed in the setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions, and then convert the GPs into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is further developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications.

This paper studies nested sequents for quantified modal logics. In particular, it considers extensions of the propositional modal logics definable by the axioms D, T, B, 4, and 5 with varying, increasing, decreasing, and constant domains. Each calculus is proved to have good structural properties: weakening and contraction are height-preserving admissible and cut is (syntactically) admissible. Each calculus is shown to be equivalent to the corresponding axiomatic system and, thus, to be sound and complete. Finally, it is argued that the calculi are internal -- i.e., each sequent has a formula interpretation -- whenever the existence predicate is expressible in the language.

Many important science and engineering problems can be converted into NP-complete problems which are of significant importance in computer science and mathematics. Currently, neither existing classical nor quantum algorithms can solve these problems in polynomial time. To address this difficulty, this paper proposes a quantum feasibility labeling (QFL) algorithm to label all possible solutions to the vertex coloring problem, which is a well-known NP-complete problem. The QFL algorithm converts the vertex coloring problem into the problem of searching an unstructured database where good and bad elements are labeled. The recently proposed variational quantum search (VQS) algorithm was demonstrated to achieve an exponential speedup, in circuit depth, up to 26 qubits in finding good element(s) from an unstructured database. Using the labels and the associated possible solutions as input, the VQS can find all feasible solutions to the vertex coloring problem. The number of qubits and the circuit depth required by the QFL each is a polynomial function of the number of vertices, the number of edges, and the number of colors of a vertex coloring problem. We have implemented the QFL on an IBM Qiskit simulator to solve a 4-colorable 4-vertex 3-edge coloring problem.

The log-conformation formulation, although highly successful, was from the beginning formulated as a partial differential equation that contains an, for PDEs unusual, eigenvalue decomposition of the unknown field. To this day, most numerical implementations have been based on this or a similar eigenvalue decomposition, with Knechtges et al. (2014) being the only notable exception for two-dimensional flows. In this paper, we present an eigenvalue-free algorithm to compute the constitutive equation of the log-conformation formulation that works for two- and three-dimensional flows. Therefore, we first prove that the challenging terms in the constitutive equations are representable as a matrix function of a slightly modified matrix of the log-conformation field. We give a proof of equivalence of this term to the more common log-conformation formulations. Based on this formulation, we develop an eigenvalue-free algorithm to evaluate this matrix function. The resulting full formulation is first discretized using a finite volume method, and then tested on the confined cylinder and sedimenting sphere benchmarks.

We introduce a novel unit-time ordinary differential equation (ODE) flow called the preconditioned F\"{o}llmer flow, which efficiently transforms a Gaussian measure into a desired target measure at time 1. To discretize the flow, we apply Euler's method, where the velocity field is calculated either analytically or through Monte Carlo approximation using Gaussian samples. Under reasonable conditions, we derive a non-asymptotic error bound in the Wasserstein distance between the sampling distribution and the target distribution. Through numerical experiments on mixture distributions in 1D, 2D, and high-dimensional spaces, we demonstrate that the samples generated by our proposed flow exhibit higher quality compared to those obtained by several existing methods. Furthermore, we propose leveraging the F\"{o}llmer flow as a warmstart strategy for existing Markov Chain Monte Carlo (MCMC) methods, aiming to mitigate mode collapses and enhance their performance. Finally, thanks to the deterministic nature of the F\"{o}llmer flow, we can leverage deep neural networks to fit the trajectory of sample evaluations. This allows us to obtain a generator for one-step sampling as a result.

Adjoint systems are widely used to inform control, optimization, and design in systems described by ordinary differential equations or differential-algebraic equations. In this paper, we explore the geometric properties and develop methods for such adjoint systems. In particular, we utilize symplectic and presymplectic geometry to investigate the properties of adjoint systems associated with ordinary differential equations and differential-algebraic equations, respectively. We show that the adjoint variational quadratic conservation laws, which are key to adjoint sensitivity analysis, arise from (pre)symplecticity of such adjoint systems. We discuss various additional geometric properties of adjoint systems, such as symmetries and variational characterizations. For adjoint systems associated with a differential-algebraic equation, we relate the index of the differential-algebraic equation to the presymplectic constraint algorithm of Gotay et al. [18]. As an application of this geometric framework, we discuss how the adjoint variational quadratic conservation laws can be used to compute sensitivities of terminal or running cost functions. Furthermore, we develop structure-preserving numerical methods for such systems using Galerkin Hamiltonian variational integrators (Leok and Zhang [23]) which admit discrete analogues of these quadratic conservation laws. We additionally show that such methods are natural, in the sense that reduction, forming the adjoint system, and discretization all commute, for suitable choices of these processes. We utilize this naturality to derive a variational error analysis result for the presymplectic variational integrator that we use to discretize the adjoint DAE system. Finally, we discuss the application of adjoint systems in the context of optimal control problems, where we prove a similar naturality result.

Humans perceive the world by concurrently processing and fusing high-dimensional inputs from multiple modalities such as vision and audio. Machine perception models, in stark contrast, are typically modality-specific and optimised for unimodal benchmarks, and hence late-stage fusion of final representations or predictions from each modality (`late-fusion') is still a dominant paradigm for multimodal video classification. Instead, we introduce a novel transformer based architecture that uses `fusion bottlenecks' for modality fusion at multiple layers. Compared to traditional pairwise self-attention, our model forces information between different modalities to pass through a small number of bottleneck latents, requiring the model to collate and condense the most relevant information in each modality and only share what is necessary. We find that such a strategy improves fusion performance, at the same time reducing computational cost. We conduct thorough ablation studies, and achieve state-of-the-art results on multiple audio-visual classification benchmarks including Audioset, Epic-Kitchens and VGGSound. All code and models will be released.

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.