Kernel means are frequently used to represent probability distributions in machine learning problems. In particular, the well known kernel density estimator and the kernel mean embedding both have the form of a kernel mean. Unfortunately, kernel means are faced with scalability issues. A single point evaluation of the kernel density estimator, for example, requires a computation time linear in the training sample size. To address this challenge, we present a method to efficiently construct a sparse approximation of a kernel mean. We do so by first establishing an incoherence-based bound on the approximation error, and then noticing that, for the case of radial kernels, the bound can be minimized by solving the $k$-center problem. The outcome is a linear time construction of a sparse kernel mean, which also lends itself naturally to an automatic sparsity selection scheme. We show the computational gains of our method by looking at three problems involving kernel means: Euclidean embedding of distributions, class proportion estimation, and clustering using the mean-shift algorithm.
In presence of sparse noise we propose kernel regression for predicting output vectors which are smooth over a given graph. Sparse noise models the training outputs being corrupted either with missing samples or large perturbations. The presence of sparse noise is handled using appropriate use of $\ell_1$-norm along-with use of $\ell_2$-norm in a convex cost function. For optimization of the cost function, we propose an iteratively reweighted least-squares (IRLS) approach that is suitable for kernel substitution or kernel trick due to availability of a closed form solution. Simulations using r…
Gaussian processes are powerful models for probabilistic machine learning, but are limited in application by their $O(N^3)$ inference complexity. We propose a method for deriving parametric families of kernel functions with compact spatial support, which yield naturally sparse kernel matrices and enable fast Gaussian process inference via sparse linear algebra. These families generalize known compactly-supported kernel functions, such as the Wendland polynomials. The parameters of this family of kernels can be learned from data using maximum likelihood estimation. Alternatively, we can quickly…
Kernel regression is a popular non-parametric fitting technique. It aims at learning a function which estimates the targets for test inputs as precise as possible. Generally, the function value for a test input is estimated by a weighted average of the surrounding training examples. The weights are typically computed by a distance-based kernel function and they strongly depend on the distances between examples. In this paper, we first review the latest developments of sparse metric learning and kernel regression. Then a novel kernel regression method involving sparse metric learning, which is …
Georgia Institute of Technology（佐治亚理工学院）；University of Michigan（密歇根大学）
Attention mechanisms take an expectation of a data representation with respect to probability weights. This creates summary statistics that focus on important features. Recently, (Martins et al. 2020, 2021) proposed continuous attention mechanisms, focusing on unimodal attention densities from the exponential and deformed exponential families: the latter has sparse support. (Farinhas et al. 2021) extended this to use Gaussian mixture attention densities, which are a flexible class with dense support. In this paper, we extend this to two general flexible classes: kernel exponential families and…
Non-Sparse Regularization for Multiple Kernel Learning
University of California（加利福尼亚大学）；Technische Universität Berlin（柏林工业大学）；Yahoo! Research（雅虎研究院）；Friedrich Miescher Laboratory（弗里德里希·米歇尔实验室）；Max Planck Society（马克斯·普朗克学会）；LIFE Biosystems GmbH（生命生物系统有限公司）
We consider the two-group classification problem and propose a kernel classifier based on the optimal scoring framework. Unlike previous approaches, we provide theoretical guarantees on the expected risk consistency of the method. We also allow for feature selection by imposing structured sparsity using weighted kernels. We propose fully-automated methods for selection of all tuning parameters, and in particular adapt kernel shrinkage ideas for ridge parameter selection. Numerical studies demonstrate the superior classification performance of the proposed approach compared to existing nonparam…
CITEC cluster of excellence（CITEC卓越集群）；Bielefeld University（比勒费尔德大学）
We are interested in the decomposition of motion data into a sparse linear combination of base functions which enable efficient data processing. We combine two prominent frameworks: dynamic time warping (DTW), which offers particularly successful pairwise motion data comparison, and sparse coding (SC), which enables an automatic decomposition of vectorial data into a sparse linear combination of base vectors. We enhance SC as follows: an efficient kernelization which extends its application domain to general similarity data such as offered by DTW, and its restriction to non-negative linear rep…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Imperial College London（帝国理工学院）；Zhejiang Zhelixin Credit Investigation Co., Ltd.（浙江浙里信征信有限公司）；Caltech（加州理工学院）；Johns Hopkins University（约翰斯·霍普金斯大学）
Regressing the vector field of a dynamical system from a finite number of observed states is a natural way to learn surrogate models for such systems. A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a data-adapted kernel which can be learned by using Kernel Flows. The method of Kernel Flows is a trainable machine learning method that learns the optimal parameters of a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used. The objective function could be a short-term …
Ulsan National Institute of Science and Technology（蔚山国立科学技术院）；VinAI Research（VinAI研究院）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；INEEJI
Choosing a proper set of kernel functions is an important problem in learning Gaussian Process (GP) models since each kernel structure has different model complexity and data fitness. Recently, automatic kernel composition methods provide not only accurate prediction but also attractive interpretability through search-based methods. However, existing methods suffer from slow kernel composition learning. To tackle large-scaled data, we propose a new sparse approximate posterior for GPs, MultiSVGP, constructed from groups of inducing points associated with individual additive kernels in composit…
We introduce a balloon estimator in a generalized expectation-maximization method for estimating all parameters of a Gaussian mixture model given one data sample per mixture component. Instead of limiting explicitly the model size, this regularization strategy yields low-complexity sparse models where the number of effective mixture components reduces with an increase of a smoothing probability parameter $\mathbf{P>0}$. This semi-parametric method bridges from non-parametric adaptive kernel density estimation (KDE) to parametric ordinary least-squares when $\mathbf{P=1}$. Experiments show that…
Despite their success, kernel methods suffer from a massive computational cost in practice. In this paper, in lieu of commonly used kernel expansion with respect to $N$ inputs, we develop a novel optimal design maximizing the entropy among kernel features. This procedure results in a kernel expansion with respect to entropic optimal features (EOF), improving the data representation dramatically due to features dissimilarity. Under mild technical assumptions, our generalization bound shows that with only $O(N^{\frac{1}{4}})$ features (disregarding logarithmic factors), we can achieve the optima…
MediaTek Research（联发科研究院）；National University of Singapore（新加坡国立大学）
Kernel-based models such as kernel ridge regression and Gaussian processes are ubiquitous in machine learning applications for regression and optimization. It is well known that a major downside for kernel-based models is the high computational cost; given a dataset of $n$ samples, the cost grows as $\mathcal{O}(n^3)$. Existing sparse approximation methods can yield a significant reduction in the computational cost, effectively reducing the actual cost down to as low as $\mathcal{O}(n)$ in certain cases. Despite this remarkable empirical success, significant gaps remain in the existing results…
Scientific workloads have traditionally exploited high levels of sparsity to accelerate computation and reduce memory requirements. While deep neural networks can be made sparse, achieving practical speedups on GPUs is difficult because these applications have relatively moderate levels of sparsity that are not sufficient for existing sparse kernels to outperform their dense counterparts. In this work, we study sparse matrices from deep learning applications and identify favorable properties that can be exploited to accelerate computation. Based on these insights, we develop high-performance G…
In this paper, we propose a new method to perform Sparse Kernel Principal Component Analysis (SKPCA) and also mathematically analyze the validity of SKPCA. We formulate SKPCA as a constrained optimization problem with elastic net regularization (Hastie et al.) in kernel feature space and solve it. We consider outlier detection (where KPCA is employed) as an application for SKPCA, using the RBF kernel. We test it on 5 real-world datasets and show that by using just 4% (or even less) of the principal components (PCs), where each PC has on average less than 12% non-zero elements in the worst case…
the University of Sydney（悉尼大学）；ARC Centre for Data Analytics for Resources and Environments（澳大利亚研究理事会资源与环境数据分析中心）；NVIDIA（英伟达）
We establish a general form of explicit, input-dependent, measure-valued warpings for learning nonstationary kernels. While stationary kernels are ubiquitous and simple to use, they struggle to adapt to functions that vary in smoothness with respect to the input. The proposed learning algorithm warps inputs as conditional Gaussian measures that control the smoothness of a standard stationary kernel. This construction allows us to capture non-stationary patterns in the data and provides intuitive inductive bias. The resulting method is based on sparse spectrum Gaussian processes, enabling close…
National University of Singapore（新加坡国立大学）；Shenzhen University（深圳大学）
This paper presents a variational Bayesian kernel selection (VBKS) algorithm for sparse Gaussian process regression (SGPR) models. In contrast to existing GP kernel selection algorithms that aim to select only one kernel with the highest model evidence, our proposed VBKS algorithm considers the kernel as a random variable and learns its belief from data such that the uncertainty of the kernel can be interpreted and exploited to avoid overconfident GP predictions. To achieve this, we represent the probabilistic kernel as an additional variational variable in a variational inference (VI) framewo…
The University of Sydney（悉尼大学）；University of Reading（雷丁大学）
A general framework of least squares support vector machine with low rank kernels, referred to as LR-LSSVM, is introduced in this paper. The special structure of low rank kernels with a controlled model size brings sparsity as well as computational efficiency to the proposed model. Meanwhile, a two-step optimization algorithm with three different criteria is proposed and various experiments are carried out using the example of the so-call robust RBF kernel to validate the model. The experiment results show that the performance of the proposed algorithm is comparable or superior to several exis…
Max Planck Institute for Astrophysics（马克斯·普朗克天体物理研究所）
Gaussian Processes (GPs) are highly expressive, probabilistic models. A major limitation is their computational complexity. Naively, exact GP inference requires $\mathcal{O}(N^3)$ computations with $N$ denoting the number of modeled points. Current approaches to overcome this limitation either rely on sparse, structured or stochastic representations of data or kernel respectively and usually involve nested optimizations to evaluate a GP. We present a new, generative method named Iterative Charted Refinement (ICR) to model GPs on nearly arbitrarily spaced points in $\mathcal{O}(N)$ time for dec…
Zero-inflated datasets, which have an excess of zero outputs, are commonly encountered in problems such as climate or rare event modelling. Conventional machine learning approaches tend to overestimate the non-zeros leading to poor performance. We propose a novel model family of zero-inflated Gaussian processes (ZiGP) for such zero-inflated datasets, produced by sparse kernels through learning a latent probit Gaussian process that can zero out kernel rows and columns whenever the signal is absent. The ZiGPs are particularly useful for making the powerful Gaussian process networks more interpre…
CITEC cluster of excellence（CITEC卓越集群）；Bielefeld University（比勒费尔德大学）
In recent years, kernel-based sparse coding (K-SRC) has received particular attention due to its efficient representation of nonlinear data structures in the feature space. Nevertheless, the existing K-SRC methods suffer from the lack of consistency between their training and test optimization frameworks. In this work, we propose a novel confident K-SRC and dictionary learning algorithm (CKSC) which focuses on the discriminative reconstruction of the data based on its representation in the kernel space. CKSC focuses on reconstructing each data sample via weighted contributions which are confid…
Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Gregorio Marañón Health Research Institute（格雷戈里奥·马拉尼翁健康研究所）
Multi-view problems can be faced with latent variable models since they are able to find low-dimensional projections that fairly capture the correlations among the multiple views that characterise each datum. On the other hand, high-dimensionality and non-linear issues are traditionally handled by kernel methods, inducing a (non)-linear function between the latent projection and the data itself. However, they usually come with scalability issues and exposition to overfitting. Here, we propose merging both approaches into single model so that we can exploit the best features of multi-view laten…
We provide a methodology for learning sparse statistical models that use as features all possible multiplicative interactions among an underlying atomic set of features. While the resulting optimization problems are exponentially sized, our methodology leads to algorithms that can often solve these problems exactly or provide approximate solutions based on combining highly correlated features. We also introduce an algorithmic paradigm, the Fine Control Kernel framework, so named because it is based on Fenchel Duality and is reminiscent of kernel methods. Its theory is tailored to large sparse …
Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data. In this paper we take the approximation theory point of view to explore various aspects of smooth kernels related to their inferential properties. We analyze eigenvalue decay of kernels operators and matrices, properties of eigenfunctions/eigenvectors and "Fourier" coefficients of functions in the kernel space restricted to a discrete set of data points. We also investigate the fitting capacity of kernels, giving expl…
Institute of Numerical Mathematics of the Russian Academy of Sciences（俄罗斯科学院数值数学研究所）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
We consider the problem of improving kernel approximation via randomized feature maps. These maps arise as Monte Carlo approximation to integral representations of kernel functions and scale up kernel methods for larger datasets. Based on an efficient numerical integration technique, we propose a unifying approach that reinterprets the previous random features methods and extends to better estimates of the kernel approximation. We derive the convergence behaviour and conduct an extensive empirical study that supports our hypothesis.
We develop a novel procedure for constructing confidence bands for components of a sparse additive model. Our procedure is based on a new kernel-sieve hybrid estimator that combines two most popular nonparametric estimation methods in the literature, the kernel regression and the spline method, and is of interest in its own right. Existing methods for fitting sparse additive model are primarily based on sieve estimators, while the literature on confidence bands for nonparametric models are primarily based upon kernel or local polynomial estimators. Our kernel-sieve hybrid estimator combines th…
Recently, classical kernel methods have been extended by the introduction of suitable tensor kernels so to promote sparsity in the solution of the underlying regression problem. Indeed, they solve an lp-norm regularization problem, with p=m/(m-1) and m even integer, which happens to be close to a lasso problem. However, a major drawback of the method is that storing tensors requires a considerable amount of memory, ultimately limiting its applicability. In this work we address this problem by proposing two advances. First, we directly reduce the memory requirement, by intriducing a new and mor…
We propose a generic spatiotemporal event forecasting method, which we developed for the National Institute of Justice's (NIJ) Real-Time Crime Forecasting Challenge. Our method is a spatiotemporal forecasting model combining scalable randomized Reproducing Kernel Hilbert Space (RKHS) methods for approximating Gaussian processes with autoregressive smoothing kernels in a regularized supervised learning framework. While the smoothing kernels capture the two main approaches in current use in the field of crime forecasting, kernel density estimation (KDE) and self-exciting point process (SEPP) mod…
Gaussian process regression generally does not scale to beyond a few thousands data points without applying some sort of kernel approximation method. Most approximations focus on the high eigenvalue part of the spectrum of the kernel matrix, $K$, which leads to bad performance when the length scale of the kernel is small. In this paper we introduce Multiresolution Kernel Approximation (MKA), the first true broad bandwidth kernel approximation algorithm. Important points about MKA are that it is memory efficient, and it is a direct method, which means that it also makes it easy to approximate $…
The kernel herding algorithm is used to construct quadrature rules in a reproducing kernel Hilbert space (RKHS). While the computational efficiency of the algorithm and stability of the output quadrature formulas are advantages of this method, the convergence speed of the integration error for a given number of nodes is slow compared to that of other quadrature methods. In this paper, we propose a modified kernel herding algorithm whose framework was introduced in a previous study and aim to obtain sparser solutions while preserving the advantages of standard kernel herding. In the proposed al…
Tohoku University（东北大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；Preferred Networks, Inc.（优选网络公司）；The Institute of Statistical Mathematics（统计数学研究所）
In this paper, we propose a new kernel-based co-occurrence measure that can be applied to sparse linguistic expressions (e.g., sentences) with a very short learning time, as an alternative to pointwise mutual information (PMI). As well as deriving PMI from mutual information, we derive this new measure from the Hilbert--Schmidt independence criterion (HSIC); thus, we call the new measure the pointwise HSIC (PHSIC). PHSIC can be interpreted as a smoothed variant of PMI that allows various similarity metrics (e.g., sentence embeddings) to be plugged in as kernels. Moreover, PHSIC can be estimate…
University of Oxford（牛津大学）；EURECOM（欧洲通信研究院）；University of Adelaide（阿德莱德大学）
We investigate the connections between sparse approximation methods for making kernel methods and Gaussian processes (GPs) scalable to large-scale data, focusing on the Nyström method and the Sparse Variational Gaussian Processes (SVGP). While sparse approximation methods for GPs and kernel methods share some algebraic similarities, the literature lacks a deep understanding of how and why they are related. This may pose an obstacle to the communications between the GP and kernel communities, making it difficult to transfer results from one side to the other. Our motivation is to remove this ob…
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University at Buffalo（布法罗大学）
A Gaussian Process (GP) is a prominent mathematical framework for stochastic function approximation in science and engineering applications. This success is largely attributed to the GP's analytical tractability, robustness, non-parametric structure, and natural inclusion of uncertainty quantification. Unfortunately, the use of exact GPs is prohibitively expensive for large datasets due to their unfavorable numerical complexity of $O(N^3)$ in computation and $O(N^2)$ in storage. All existing methods addressing this issue utilize some form of approximation -- usually considering subsets of the …
