In this paper we present a novel method for efficient and effective 3D surface reconstruction in open scenes. Existing Neural Radiance Fields (NeRF) based works typically require extensive training and rendering time due to the adopted implicit representations. In contrast, 3D Gaussian splatting (3DGS) uses an explicit and discrete representation, hence the reconstructed surface is built by the huge number of Gaussian primitives, which leads to excessive memory consumption and rough surface details in sparse Gaussian areas. To address these issues, we propose Gaussian Voxel Kernel Functions (GVKF), which establish a continuous scene representation based on discrete 3DGS through kernel regression.

Kernel methods are widely utilized in machine learning field to learn, from training data, a latent function in a reproducing kernel Hilbert space. It is well known that the approximator thus obtained usually achieves a linear representation, which brings various computational benefits, while maintaining great representation power (i.e., universal approximation). However, when non-negativity constraints are imposed on the function's outputs, the literature usually takes the kernel method-based approximators as offering linear representations at the expense of limited model flexibility or good representation power by allowing for their nonlinear forms. The main contribution of this paper is to derive a sufficient condition for a positive definite kernel so that it may construct flexible and linear approximators of non-negative functions. We call a kernel function that offers these attributes an inverse M-kernel; it is a generalization of the inverse M-matrix. Furthermore, we show that for a one-dimensional input space, universal exponential/Abel kernels are inverse M-kernels and construct linear universal approximators of non-negative functions. To the best of our knowledge, it is the first time that the existence of linear universal approximators of non-negative functions has been elucidated. We confirm the effectiveness of our results by experiments on the problems of non-negativity-constrained regression, density estimation, and intensity estimation. Finally, we discuss issues and perspectives on multi-dimensional input settings.

Kernel methods are widely utilized in machine learning field to learn, from training data, a latent function in a reproducing kernel Hilbert space. It is well known that the approximator thus obtained usually achieves a linear representation, which brings various computational benefits, while maintaining great representation power (i.e., universal approximation). However, when non-negativity constraints are imposed on the function's outputs, the literature usually takes the kernel method-based approximators as offering linear representations at the expense of limited model flexibility or good representation power by allowing for their nonlinear forms. The main contribution of this paper is to derive a sufficient condition for a positive definite kernel so that it may construct flexible and linear approximators of non-negative functions. We call a kernel function that offers these attributes an inverse M-kernel; it is reminiscent of the inverse M-matrix.

For ease of reference, sections, propositions and equations that belong to this supplementary material are prefixed with an'S'. Additionally, labels in light blue refer to the main paper. Hyperlinks across documents should work if the two PDFs are placed in the same folder. Next, in a specific case, we show that the discrete problem (6) admits a closed-form solution that we discuss in Section S2. Noticeably, this special case allows the understanding of the behaviour of the estimated DPP kernel in both the small and large regularization (λ) limits.

The functional characterization of different neuronal types has been a longstanding and crucial challenge. With the advent of physical quantum computers, it has become possible to apply quantum machine learning algorithms to translate theoretical research into practical solutions. Previous studies have shown the advantages of quantum algorithms on artificially generated datasets, and initial experiments with small binary classification problems have yielded comparable outcomes to classical algorithms. However, it is essential to investigate the potential quantum advantage using realworld data. To the best of our knowledge, this study is the first to propose the utilization of quantum systems to classify neuron morphologies, thereby enhancing our understanding of the performance of automatic multiclass neuron classification using quantum kernel methods. We examined the influence of feature engineering on classification accuracy and found that quantum kernel methods achieved similar performance to classical methods, with certain advantages observed in various configurations. Furthermore, the advances in quantum computing systems have allowed a progress in the study of quantum ML algorithms, especially with kernel methods. The number of features determined the number of qubits, and a quantum circuit used to implement the feature map was of a depth that was a linear or polylogarithmic function of the dataset's size. Thus far, the studies that have been conducted to support the advantages of a quantum feature map have carefully selected synthetic datasets or applied it to small binary classification problems. Despite the fact that research on cortical circuits has been conducted for over a century, determining how many classes of cortical neurons exist remains an ongoing and uncompleted task. Moreover, the continuous development of techniques and the availability of an increasing number of phenotype datasets have not led to the maintenance of a unique classification system that is easy to update and can consider the different defining features of neurons specific to a given type. Despite the inherent complexity and challenges that neuroscientists must deal with while addressing neuronal classification, numerous reasons exist for interest in this topic.

The general perception is that kernel methods are not scalable, so neural nets become the choice for large-scale nonlinear learning problems. Have we tried hard enough for kernel methods? In this paper, we propose an approach that scales up kernel methods using a novel concept called "doubly stochastic functional gradients". Based on the fact that many kernel methods can be expressed as convex optimization problems, our approach solves the optimization problems by making two unbiased stochastic approximations to the functional gradient--one using random training points and another using random features associated with the kernel--and performing descent steps with this noisy functional gradient. Our algorithm is simple, need no commit to a preset number of random features, and allows the flexibility of the function class to grow as we see more incoming data in the streaming setting. We demonstrate that a function learned by this procedure after t iterations converges to the optimal function in the reproducing kernel Hilbert space in rate O(1/t), and achieves a generalization bound of O(1/ t). Our approach can readily scale kernel methods up to the regimes which are dominated by neural nets. We show competitive performances of our approach as compared to neural nets in datasets such as 2.3 million energy materials from MolecularSpace, 8 million handwritten digits from MNIST, and 1 million photos from ImageNet using convolution features.

This work investigates data-driven prediction and control of Hammerstein-Wiener systems using physics-informed Gaussian process models. Data-driven prediction algorithms have been developed for structured nonlinear systems based on Willems' fundamental lemma. However, existing frameworks cannot treat output nonlinearities and require a dictionary of basis functions for Hammerstein systems. In this work, an implicit predictor structure is considered, leveraging the multi-step-ahead ARX structure for the linear part of the model. This implicit function is learned by Gaussian process regression with kernel functions designed from Gaussian process priors for the nonlinearities. The linear model parameters are estimated as hyperparameters by assuming a stable spline hyperprior. The implicit Gaussian process model provides explicit output prediction by optimizing selected optimality criteria. The model is also applied to receding horizon control with the expected control cost and chance constraint satisfaction guarantee. Numerical results demonstrate that the proposed prediction and control algorithms are superior to black-box Gaussian process models.

Graph kernels are kernel methods measuring graph similarity and serve as a standard tool for graph classification. However, the use of kernel methods for node classification, which is a related problem to graph representation learning, is still ill-posed and the state-of-the-art methods are heavily based on heuristics. Here, we present a novel theoretical kernel-based framework for node classification that can bridge the gap between these two representation learning problems on graphs. Our approach is motivated by graph kernel methodology but extended to learn the node representations capturing the structural information in a graph. We theoretically show that our formulation is as powerful as any positive semidefinite kernels. To efficiently learn the kernel, we propose a novel mechanism for node feature aggregation and a data-driven similarity metric employed during the training phase. More importantly, our framework is flexible and complementary to other graph-based deep learning models, e.g., Graph Convolutional Networks (GCNs). We empirically evaluate our approach on a number of standard node classification benchmarks, and demonstrate that our model sets the new state of the art.

We would like to thank the reviewers for their comments. Below are our answers to the main questions/concerns. R1: How can runtime be independent of image size? We will add a clarification. R3: How are crops selected?

Each class contains 600 images of size 84 84. These classes are split into 64, 16, and 20 classes for meta-training, meta-validation, and meta-testing respectively [28]. CUB contains 200 classes with a total of 11,788 images of size 84 84. Following previous works [5], the base, validation, and novel split are 100, 50, and 50 classes respectively. CIFAR-FS is a variant of the CIFAR-100 dataset used for few-shot classification. It contains 100 classes, each with 600 images of 32 32 pixels. The classes are randomly split into 64, 16, and 20 for meta-training, meta-validation, and meta-testing respectively.