Given sparse observations of buoy velocities, oceanographers are interested in reconstructing ocean currents away from the buoys and identifying divergences in a current vector field. As a first and modular step, we focus on the time-stationary case - for instance, by restricting to short time periods. Since we expect current velocity to be a continuous but highly non-linear function of spatial location, Gaussian processes (GPs) offer an attractive model. But we show that applying a GP with a standard stationary kernel directly to buoy data can struggle at both current reconstruction and divergence identification, due to some physically unrealistic prior assumptions. To better reflect known physical properties of currents, we propose to instead put a standard stationary kernel on the divergence and curl-free components of a vector field obtained through a Helmholtz decomposition. We show that, because this decomposition relates to the original vector field just via mixed partial derivatives, we can still perform inference given the original data with only a small constant multiple of additional computational expense. We illustrate the benefits of our method with theory and experiments on synthetic and real ocean data.
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Characterizing shapes of high-dimensional objects via Ricci curvatures plays a critical role in many research areas in mathematics and physics. However, even though several discretizations of Ricci curvatures for discrete combinatorial objects such as networks have been proposed and studied by mathematicians, the computational complexity aspects of these discretizations have escaped the attention of theoretical computer scientists to a large extent. In this paper, we study one such discretization, namely the Ollivier-Ricci curvature, from the perspective of efficient computation by fine-grained reductions and local query-based algorithms. Our main contributions are the following. (a) We relate our curvature computation problem to minimum weight perfect matching problem on complete bipartite graphs via fine-grained reduction. (b) We formalize the computational aspects of the curvature computation problems in suitable frameworks so that they can be studied by researchers in local algorithms. (c) We provide the first known lower and upper bounds on queries for query-based algorithms for the curvature computation problems in our local algorithms framework. En route, we also illustrate a localized version of our fine-grained reduction. We believe that our results bring forth an intriguing set of research questions, motivated both in theory and practice, regarding designing efficient algorithms for curvatures of objects.
Strong stability is a property of time integration schemes for ODEs that preserve temporal monotonicity of solutions in arbitrary (inner product) norms. It is proved that explicit Runge--Kutta schemes of order $p\in 4\mathbb{N}$ with $s=p$ stages for linear autonomous ODE systems are not strongly stable, closing an open stability question from [Z.~Sun and C.-W.~Shu, SIAM J. Numer. Anal. 57 (2019), 1158--1182]. Furthermore, for explicit Runge--Kutta methods of order $p\in\mathbb{N}$ and $s>p$ stages, we prove several sufficient as well as necessary conditions for strong stability. These conditions involve both the stability function and the hypocoercivity index of the ODE system matrix. This index is a structural property combining the Hermitian and skew-Hermitian part of the system matrix.
Autonomous experimentation has emerged as an efficient approach to accelerate the pace of materials discovery. Although instruments for autonomous synthesis have become popular in molecular and polymer science, solution processing of hybrid materials and nanoparticles, examples of autonomous tools for physical vapor deposition are scarce yet important for the semiconductor industry. Here, we report the design and implementation of an autonomous workflow for sputter deposition of thin films with controlled composition, leveraging a highly automated sputtering reactor custom-controlled by Python, optical emission spectroscopy (OES), and a Bayesian optimization algorithm. We modeled film composition, measured by x-ray fluorescence, as a linear function of emission lines monitored during the co-sputtering from elemental Zn and Ti targets in N$_2$ atmosphere. A Bayesian control algorithm, informed by OES, navigates the space of sputtering power to fabricate films with user-defined composition, by minimizing the absolute error between desired and measured emission signals. We validated our approach by autonomously fabricating Zn$_x$Ti$_{1-x}$N$_y$ films with deviations from the targeted cation composition within relative 3.5 %, even for 15 nm thin films, demonstrating that the proposed approach can reliably synthesize thin films with specific composition and minimal human interference. Moreover, the proposed method can be extended to more difficult synthesis experiments where plasma intensity depends non-linearly on pressure, or the elemental sticking coefficients strongly depend on the substrate temperature.
Turbulent fluctuations of the atmospheric refraction index, so-called optical turbulence, can significantly distort propagating laser beams. Therefore, modeling the strength of these fluctuations ($C_n^2$) is highly relevant for the successful development and deployment of future free-space optical communication links. In this letter, we propose a physics-informed machine learning (ML) methodology, $\Pi$-ML, based on dimensional analysis and gradient boosting to estimate $C_n^2$. Through a systematic feature importance analysis, we identify the normalized variance of potential temperature as the dominating feature for predicting $C_n^2$. For statistical robustness, we train an ensemble of models which yields high performance on the out-of-sample data of $R^2=0.958\pm0.001$.
This study performs an ablation analysis of Vector Quantized Generative Adversarial Networks (VQGANs), concentrating on image-to-image synthesis utilizing a single NVIDIA A100 GPU. The current work explores the nuanced effects of varying critical parameters including the number of epochs, image count, and attributes of codebook vectors and latent dimensions, specifically within the constraint of limited resources. Notably, our focus is pinpointed on the vector quantization loss, keeping other hyperparameters and loss components (GAN loss) fixed. This was done to delve into a deeper understanding of the discrete latent space, and to explore how varying its size affects the reconstruction. Though, our results do not surpass the existing benchmarks, however, our findings shed significant light on VQGAN's behaviour for a smaller dataset, particularly concerning artifacts, codebook size optimization, and comparative analysis with Principal Component Analysis (PCA). The study also uncovers the promising direction by introducing 2D positional encodings, revealing a marked reduction in artifacts and insights into balancing clarity and overfitting.
We consider the problem of estimating the roughness of the volatility in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. To this end, we introduce a new estimator that measures the so-called roughness exponent of a continuous trajectory, based on discrete observations of its antiderivative. We provide conditions on the underlying trajectory under which our estimator converges in a strictly pathwise sense. Then we verify that these conditions are satisfied by almost every sample path of fractional Brownian motion (with drift). As a consequence, we obtain strong consistency theorems in the context of a large class of rough volatility models. Numerical simulations show that our estimation procedure performs well after passing to a scale-invariant modification of our estimator.
Very distinct strategies can be deployed to recognize and characterize an unknown environment or a shape. A recent and promising approach, especially in robotics, is to reduce the complexity of the exploratory units to a minimum. Here, we show that this frugal strategy can be taken to the extreme by exploiting the power of statistical geometry and introducing new invariant features. We show that an elementary robot devoid of any orientation or observation system, exploring randomly, can access global information about an environment such as the values of the explored area and perimeter. The explored shapes are of arbitrary geometry and may even non-connected. From a dictionary, this most simple robot can thus identify various shapes such as famous monuments and even read a text.
Ghost, or fictitious points allow to capture boundary conditions that are not located on the finite difference grid discretization. We explore in this paper the impact of ghost points on the stability of the explicit Euler finite difference scheme in the context of the diffusion equation. In particular, we consider the case of a one-touch option under the Black-Scholes model. The observations and results are however valid for a much wider range of financial contracts and models.
Knowledge graphs (KGs) of real-world facts about entities and their relationships are useful resources for a variety of natural language processing tasks. However, because knowledge graphs are typically incomplete, it is useful to perform knowledge graph completion or link prediction, i.e. predict whether a relationship not in the knowledge graph is likely to be true. This paper serves as a comprehensive survey of embedding models of entities and relationships for knowledge graph completion, summarizing up-to-date experimental results on standard benchmark datasets and pointing out potential future research directions.
Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.
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