We discuss three SYCL realisations of a simple Finite Volume scheme over multiple Cartesian patches. The realisation flavours differ in the way how they map the compute steps onto loops and tasks: We compare an implementation which is exclusively using a cascade of for-loops to a version which uses nested parallelism, and finally benchmark these against a version which models the calculations as task graph. Our work proposes realisation idioms to realise these flavours within SYCL. The idioms translate to some degree to other GPGPU programming techniques, too. Our preliminary results suggest that SYCL's capability to model calculations via tasks or nested parallelism does not yet allow such realisations to outperform their counterparts using exclusively data parallelism.
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Since Knowledge Graphs (KGs) contain rich semantic information, recently there has been an influx of KG-enhanced recommendation methods. Most of existing methods are entirely designed based on euclidean space without considering curvature. However, recent studies have revealed that a tremendous graph-structured data exhibits highly non-euclidean properties. Motivated by these observations, in this work, we propose a knowledge-based multiple adaptive spaces fusion method for recommendation, namely MCKG. Unlike existing methods that solely adopt a specific manifold, we introduce the unified space that is compatible with hyperbolic, euclidean and spherical spaces. Furthermore, we fuse the multiple unified spaces in an attention manner to obtain the high-quality embeddings for better knowledge propagation. In addition, we propose a geometry-aware optimization strategy which enables the pull and push processes benefited from both hyperbolic and spherical spaces. Specifically, in hyperbolic space, we set smaller margins in the area near to the origin, which is conducive to distinguishing between highly similar positive items and negative ones. At the same time, we set larger margins in the area far from the origin to ensure the model has sufficient error tolerance. The similar manner also applies to spherical spaces. Extensive experiments on three real-world datasets demonstrate that the MCKG has a significant improvement over state-of-the-art recommendation methods. Further ablation experiments verify the importance of multi-space fusion and geometry-aware optimization strategy, justifying the rationality and effectiveness of MCKG.
We propose and analyse an explicit boundary-preserving scheme for the strong approximations of some SDEs with non-globally Lipschitz drift and diffusion coefficients whose state-space is bounded. The scheme consists of a Lamperti transform followed by a Lie--Trotter splitting. We prove $L^{p}(\Omega)$-convergence of order $1$, for every $p \in \mathbb{N}$, of the scheme and exploit the Lamperti transform to confine the numerical approximations to the state-space of the considered SDE. We provide numerical experiments that confirm the theoretical results and compare the proposed Lamperti-splitting scheme to other numerical schemes for SDEs.
We study a family of distances between functions of a single variable. These distances are examples of integral probability metrics, and have been used previously for comparing probability measures. Special cases include the Earth Mover's Distance and the Kolmogorov Metric. We examine their properties for general signals, proving that they are robust to a broad class of perturbations and that the distance between one-dimensional tomographic projections of a two-dimensional function is bounded by the size of the difference in projection angles. We also establish error bounds for approximating the metric from finite samples, and prove that these approximations are robust to additive Gaussian noise. The results are illustrated in numerical experiments.
We introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for the dual-porosity-Stokes problem. This coupled problem describes the interaction between free flow in macrofractures/conduits, governed by the Stokes equations, and flow in microfractures/matrix, governed by a dual-porosity model. We prove that the HDG method is strongly conservative, well-posed, and give an a priori error analysis showing dependence on the problem parameters. Our theoretical findings are corroborated by numerical examples
Benefiting from the development of deep learning, text-to-speech (TTS) techniques using clean speech have achieved significant performance improvements. The data collected from real scenes often contain noise and generally needs to be denoised by speech enhancement models. Noise-robust TTS models are often trained using the enhanced speech, which thus suffer from speech distortion and background noise that affect the quality of the synthesized speech. Meanwhile, it was shown that self-supervised pre-trained models exhibit excellent noise robustness on many speech tasks, implying that the learned representation has a better tolerance for noise perturbations. In this work, we therefore explore pre-trained models to improve the noise robustness of TTS models. Based on HIFI-GAN we first propose a representation-to-waveform vocoder, which aims to learn to map the representation of pre-trained models to the waveform. We then propose a text-to-representation Fastspeech2 model, which aims to learn to map text to pre-trained model representations. Experimental results on the LJSpeech and LibriTTS datasets show that our method outperforms those using speech enhancement methods in both subjective and objective metrics. Audio samples are available at: //zqs01.github.io/rep2wav/.
Understanding the road genome is essential to realize autonomous driving. This highly intelligent problem contains two aspects - the connection relationship of lanes, and the assignment relationship between lanes and traffic elements, where a comprehensive topology reasoning method is vacant. On one hand, previous map learning techniques struggle in deriving lane connectivity with segmentation or laneline paradigms; or prior lane topology-oriented approaches focus on centerline detection and neglect the interaction modeling. On the other hand, the traffic element to lane assignment problem is limited in the image domain, leaving how to construct the correspondence from two views an unexplored challenge. To address these issues, we present TopoNet, the first end-to-end framework capable of abstracting traffic knowledge beyond conventional perception tasks. To capture the driving scene topology, we introduce three key designs: (1) an embedding module to incorporate semantic knowledge from 2D elements into a unified feature space; (2) a curated scene graph neural network to model relationships and enable feature interaction inside the network; (3) instead of transmitting messages arbitrarily, a scene knowledge graph is devised to differentiate prior knowledge from various types of the road genome. We evaluate TopoNet on the challenging scene understanding benchmark, OpenLane-V2, where our approach outperforms all previous works by a great margin on all perceptual and topological metrics. The code is released at //github.com/OpenDriveLab/TopoNet
We study a class of Gaussian processes for which the posterior mean, for a particular choice of data, replicates a truncated Taylor expansion of any order. The data consist of derivative evaluations at the expansion point and the prior covariance kernel belongs to the class of Taylor kernels, which can be written in a certain power series form. We discuss and prove some results on maximum likelihood estimation of parameters of Taylor kernels. The proposed framework is a special case of Gaussian process regression based on data that is orthogonal in the reproducing kernel Hilbert space of the covariance kernel.
This note shows how to compute, to high relative accuracy under mild assumptions, complex Jacobi rotations for diagonalization of Hermitian matrices of order two, using the correctly rounded functions $\mathtt{cr\_hypot}$ and $\mathtt{cr\_rsqrt}$, proposed for standardization in the C programming language as recommended by the IEEE-754 floating-point standard. The rounding to nearest (ties to even) and the non-stop arithmetic are assumed. The numerical examples compare the observed with theoretical bounds on the relative errors in the rotations' elements, and show that the maximal observed departure of the rotations' determinants from unity is smaller than that of the transformations computed by LAPACK.
We present a new framework for computing fine-scale solutions of multiscale Partial Differential Equations (PDEs) using operator learning tools. Obtaining fine-scale solutions of multiscale PDEs can be challenging, but there are many inexpensive computational methods for obtaining coarse-scale solutions. Additionally, in many real-world applications, fine-scale solutions can only be observed at a limited number of locations. In order to obtain approximations or predictions of fine-scale solutions over general regions of interest, we propose to learn the operator mapping from coarse-scale solutions to fine-scale solutions using a limited number (and possibly noisy) observations of the fine-scale solutions. The approach is to train multi-fidelity homogenization maps using mathematically motivated neural operators. The operator learning framework can efficiently obtain the solution of multiscale PDEs at any arbitrary point, making our proposed framework a mesh-free solver. We verify our results on multiple numerical examples showing that our approach is an efficient mesh-free solver for multiscale PDEs.
Recommender systems are widely used in big information-based companies such as Google, Twitter, LinkedIn, and Netflix. A recommender system deals with the problem of information overload by filtering important information fragments according to users' preferences. In light of the increasing success of deep learning, recent studies have proved the benefits of using deep learning in various recommendation tasks. However, most proposed techniques only aim to target individuals, which cannot be efficiently applied in group recommendation. In this paper, we propose a deep learning architecture to solve the group recommendation problem. On the one hand, as different individual preferences in a group necessitate preference trade-offs in making group recommendations, it is essential that the recommendation model can discover substitutes among user behaviors. On the other hand, it has been observed that a user as an individual and as a group member behaves differently. To tackle such problems, we propose using an attention mechanism to capture the impact of each user in a group. Specifically, our model automatically learns the influence weight of each user in a group and recommends items to the group based on its members' weighted preferences. We conduct extensive experiments on four datasets. Our model significantly outperforms baseline methods and shows promising results in applying deep learning to the group recommendation problem.
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