Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space [Formula: see text]. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza’s formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

A random matrix analysis of random Fourier features: beyond the Gaussian kernel, a precise phase transition, and the corresponding double descent*

This article characterizes the exact asymptotics of random Fourier feature (RFF) regression, in the realistic setting where the number of data samples n, their dimension p, and the dimension of feature space N are all large and comparable. In this regime, the random RFF Gram matrix no longer converges to the well-known limiting Gaussian kernel matrix (as it does when N → ∞ alone), but it still has a tractable behavior that is captured by our analysis. This analysis also provides accurate estimates of training and test regression errors for large n, p, N. Based on these estimates, a precise characterization of two qualitatively different phases of learning, including the phase transition between them, is provided; and the corresponding double descent test error curve is derived from this phase transition behavior. These results do not depend on strong assumptions on the data distribution, and they perfectly match empirical results on real-world data sets.

A T-CNN Time Series Classification Method

Time series classification is a basic task in the field of streaming data event analysis and data mining. The existing time series classification methods have the problems of low classification accuracy and low efficiency. To solve these problems, this paper proposes a T-CNN time series classification method based on a Gram matrix. Specifically, we perform wavelet threshold denoising on time series to filter normal curve noise, and propose a lossless transformation method based on the Gram matrix, which converts the time series to the time domain image and retains all the information of events. Then, we propose an improved CNN time series classification method, which introduces the Toeplitz convolution kernel matrix into convolution layer calculation. Finally, we introduce a Triplet network to calculate the similarity between similar events and different classes of events, and optimize the squared loss function of CNN. The proposed T-CNN model can accelerate the convergence rate of gradient descent and improve classification accuracy. Experimental results show that, compared with the existing methods, our T-CNN time series classification method has great advantages in efficiency and accuracy.

Application of Image Style Transfer Technology in Interior Decoration Design Based on Ecological Environment

With the further development of the social economy, people pay more attention to spiritual and cultural needs. As the main place of people’s daily life, the family is very important to the creation of its cultural atmosphere. In fact, China has fully entered the era of interior decoration, and people are paying more and more attention to decorative effects and the comfort and individual characteristics of decoration. Therefore, it is of practical significance to develop the application of decorative art in interior space design. However, the transfer effect of current interior decoration art design tends to be artistic, which leads to image distortion, and image content transfer errors are easy to occur in the process of transfer. The application of image style transfer in interior decoration art can effectively solve such problems. This paper analyzes the basic theory of image style transfer through image style transfer technology, Gram matrix, and Poisson image editing technology and designs images from several aspects such as image segmentation, content loss, enhanced style loss, and Poisson image editing constrained image spatial gradient. The application process of style transfer in interior decoration art realizes the application of image style transfer in interior decoration art. The experimental results show that the application of image style transmission in interior decoration art design can effectively avoid the contents of the interior decoration errors and distortions and has a good style transfer effect.

Online Learning with Radial Basis Function Networks

We investigate the benefits of feature selection, nonlinear modelling and online learning when forecasting in financial time series. We consider the sequential and continual learning sub-genres of online learning. The experiments we conduct show that there is a benefit to online transfer learning, in the form of radial basis function networks, beyond the sequential updating of recursive least-squares models. We show that the radial basis function networks, which make use of clustering algorithms to construct a kernel Gram matrix, are more beneficial than treating each training vector as separate basis functions, as occurs with kernel Ridge regression. We demonstrate quantitative procedures to determine the very structure of the radial basis function networks. Finally, we conduct experiments on the log returns of financial time series and show that the online learning models, particularly the radial basis function networks, are able to outperform a random walk baseline, whereas the offline learning models struggle to do so.

Subsampling and Aggregation: A Solution to the Scalability Problem in Distance-Based Prediction for Mixed-Type Data

The distance-based linear model (DB-LM) extends the classical linear regression to the framework of mixed-type predictors or when the only available information is a distance matrix between regressors (as it sometimes happens with big data). The main drawback of these DB methods is their computational cost, particularly due to the eigendecomposition of the Gram matrix. In this context, ensemble regression techniques provide a useful alternative to fitting the model to the whole sample. This work analyzes the performance of three subsampling and aggregation techniques in DB regression on two specific large, real datasets. We also analyze, via simulations, the performance of bagging and DB logistic regression in the classification problem with mixed-type features and large sample sizes.

The Generalized Complex Kernel Affine Projection Algorithms

The complex kernel adaptive filter (CKAF) has been widely applied to the complex-valued nonlinear problem in signal processing and machine learning. However, most of the CKAF applications involve the complex kernel least mean square (CKLMS) algorithms, which work in a pure complex or complexified reproducing kernel Hilbert space (RKHS). In this paper, we propose the generalized complex kernel affine projection (GCKAP) algorithms in the widely linear complex-valued RKHS (WL-RKHS). The proposed algorithms have two main notable features. One is that they provide a complete solution for both circular and non-circular complex nonlinear problems and show many performance improvements over the CKAP algorithms. The other is that the GCKAP algorithms inherit the simplicity of the CKLMS algorithm while reducing its gradient noise and boosting its convergence. The second-order statistical characteristics of WL-RKHS have also been developed. An augmented Gram matrix consists of a standard Gram matrix and a pseudo-Gram matrix. This decomposition provides more underlying information when the real and imaginary parts of the signal are correlated and learning is independent. In addition, some online sparsification criteria are compared comprehensively in the GCKAP algorithms, including the novelty criterion, the coherence criterion, and the angle criterion. Finally, two nonlinear channel equalization experiments with non-circular complex inputs are presented to illustrate the performance improvements of the proposed algorithms.

A Saliency Detection and Gram Matrix Transform-Based Convolutional Neural Network for Image Emotion Classification

Using the convolutional neural network (CNN) method for image emotion recognition is a research hotspot of deep learning. Previous studies tend to use visual features obtained from a global perspective and ignore the role of local visual features in emotional arousal. Moreover, the CNN shallow feature maps contain image content information; such maps obtained from shallow layers directly to describe low-level visual features may lead to redundancy. In order to enhance image emotion recognition performance, an improved CNN is proposed in this work. Firstly, the saliency detection algorithm is used to locate the emotional region of the image, which is served as the supplementary information to conduct emotion recognition better. Secondly, the Gram matrix transform is performed on the CNN shallow feature maps to decrease the redundancy of image content information. Finally, a new loss function is designed by using hard labels and probability labels of image emotion category to reduce the influence of image emotion subjectivity. Extensive experiments have been conducted on benchmark datasets, including FI (Flickr and Instagram), IAPSsubset, ArtPhoto, and Abstract. The experimental results show that compared with the existing approaches, our method has a good application prospect.

Online Learning with Radial Basis Function Networks

We investigate the benefits of feature selection, nonlinear modelling and online learning when forecasting in financial time series. We consider the sequential and continual learning sub-genres of online learning. The experiments we conduct show that there is a benefit to online transfer learning, in the form of radial basis function networks, beyond the sequential updating of recursive least-squares models. We show that the radial basis function networks, which make use of clustering algorithms to construct a kernel Gram matrix, are more beneficial than treating each training vector as separate basis functions, as occurs with kernel Ridge regression. We demonstrate quantitative procedures to determine the very structure of the radial basis function networks. Finally, we conduct experiments on the log returns of financial time series and show that the online learning models, particularly the radial basis function networks, are able to outperform a random walk baseline, whereas the offline learning models struggle to do so.