We construct near-optimal coresets for kernel density estimates for points in $\mathbb{R}^d$ when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size $O(\sqrt{d}/\varepsilon\cdot \sqrt{\log 1/\varepsilon} )$, and we show a near-matching lower bound of size $Ω(\min\{\sqrt{d}/\varepsilon, 1/\varepsilon^2\})$. When $d\geq 1/\varepsilon^2$, it is known that the size of coreset can be $O(1/\varepsilon^2)$. The upper bound is a polynomial-in-$(1/\varepsilon)$ improvement when $d \in [3,1/\varepsilon^2)$ and the lower bound is the first known lower bound to depend on $d$ for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide-variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative.
We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For characteristic kernels (including Gaussian and Laplace kernels), our approximation preserves the $L_\infty$ error between kernel density estimates within error $ε$, with coreset size $2/ε^2$, but no other aspects of the data, including the dimension, the diameter of the point set, or the bandwidth of the kernel common to other approximations. When the dimension is…
Given a point set $P\subset \mathbb{R}^d$, the kernel density estimate of $P$ is defined as \[ \overline{\mathcal{G}}_P(x) = \frac{1}{\left|P\right|}\sum_{p\in P}e^{-\left\lVert x-p \right\rVert^2} \] for any $x\in\mathbb{R}^d$. We study how to construct a small subset $Q$ of $P$ such that the kernel density estimate of $P$ is approximated by the kernel density estimate of $Q$. This subset $Q$ is called a coreset. The main technique in this work is constructing a $\pm 1$ coloring on the point set $P$ by discrepancy theory and we leverage Banaszczyk's Theorem. When $d>1$ is a constant, our cons…
This paper defines the notion of class discrepancy for families of functions. It shows that low discrepancy classes admit small offline and streaming coresets. We provide general techniques for bounding the class discrepancy of machine learning problems. As corollaries of the general technique we bound the discrepancy (and therefore coreset complexity) of logistic regression, sigmoid activation loss, matrix covariance, kernel density and any analytic function of the dot product or the squared distance. Our results prove the existence of epsilon-approximation O(sqrt{d}/epsilon) sized coresets f…
Kernel regression is an essential and ubiquitous tool for non-parametric data analysis, particularly popular among time series and spatial data. However, the central operation which is performed many times, evaluating a kernel on the data set, takes linear time. This is impractical for modern large data sets. In this paper we describe coresets for kernel regression: compressed data sets which can be used as proxy for the original data and have provably bounded worst case error. The size of the coresets are independent of the raw number of data points, rather they only depend on the error guara…
Coreset (or core-set) is a small weighted \emph{subset} $Q$ of an input set $P$ with respect to a given \emph{monotonic} function $f:\mathbb{R}\to\mathbb{R}$ that \emph{provably} approximates its fitting loss $\sum_{p\in P}f(p\cdot x)$ to \emph{any} given $x\in\mathbb{R}^d$. Using $Q$ we can obtain approximation of $x^*$ that minimizes this loss, by running \emph{existing} optimization algorithms on $Q$. In this work we provide: (i) A lower bound which proves that there are sets with no coresets smaller than $n=|P|$ for general monotonic loss functions. (ii) A proof that, under a natural assum…
Coreset is usually a small weighted subset of $n$ input points in $\mathbb{R}^d$, that provably approximates their loss function for a given set of queries (models, classifiers, etc.). Coresets become increasingly common in machine learning since existing heuristics or inefficient algorithms may be improved by running them possibly many times on the small coreset that can be maintained for streaming distributed data. Coresets can be obtained by sensitivity (importance) sampling, where its size is proportional to the total sum of sensitivities. Unfortunately, computing the sensitivity of each p…
Massachusetts Institute of Technology（麻省理工学院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Coresets have emerged as a powerful tool to summarize data by selecting a small subset of the original observations while retaining most of its information. This approach has led to significant computational speedups but the performance of statistical procedures run on coresets is largely unexplored. In this work, we develop a statistical framework to study coresets and focus on the canonical task of nonparameteric density estimation. Our contributions are twofold. First, we establish the minimax rate of estimation achievable by coreset-based estimators. Second, we show that the practical core…
IBM T.J. Watson Research Center（IBM T.J. Watson研究中心）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
We study (constrained) least-squares regression as well as multiple response least-squares regression and ask the question of whether a subset of the data, a coreset, suffices to compute a good approximate solution to the regression. We give deterministic, low order polynomial-time algorithms to construct such coresets with approximation guarantees, together with lower bounds indicating that there is not much room for improvement upon our results.
An $\varepsilon$-coreset for a given set $D$ of $n$ points, is usually a small weighted set, such that querying the coreset \emph{provably} yields a $(1+\varepsilon)$-factor approximation to the original (full) dataset, for a given family of queries. Using existing techniques, coresets can be maintained for streaming, dynamic (insertion/deletions), and distributed data in parallel, e.g. on a network, GPU or cloud. We suggest the first coresets that approximate the negative log-likelihood for $k$-Gaussians Mixture Models (GMM) of arbitrary shapes (ratio between eigenvalues of their covariance m…
An $\varepsilon$-coreset for Least-Mean-Squares (LMS) of a matrix $A\in{\mathbb{R}}^{n\times d}$ is a small weighted subset of its rows that approximates the sum of squared distances from its rows to every affine $k$-dimensional subspace of ${\mathbb{R}}^d$, up to a factor of $1\pm\varepsilon$. Such coresets are useful for hyper-parameter tuning and solving many least-mean-squares problems such as low-rank approximation ($k$-SVD), $k$-PCA, Lassso/Ridge/Linear regression and many more. Coresets are also useful for handling streaming, dynamic and distributed big data in parallel. With high proba…
Quantum computing is anticipated to offer immense computational capabilities which could provide efficient solutions to many data science problems. However, the current generation of quantum devices are small and noisy, which makes it difficult to process large data sets relevant for practical problems. Coreset selection aims to circumvent this problem by reducing the size of input data without compromising the accuracy. Recent work has shown that coreset selection can help to implement quantum K-Means clustering problem. However, the impact of coreset selection on the performance of quantum K…
Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；Inception Institute of Artificial Intelligence（启智人工智能研究院）
In this paper, we study the statistical properties of kernel $k$-means and obtain a nearly optimal excess clustering risk bound, substantially improving the state-of-art bounds in the existing clustering risk analyses. We further analyze the statistical effect of computational approximations of the Nyström kernel $k$-means, and prove that it achieves the same statistical accuracy as the exact kernel $k$-means considering only $Ω(\sqrt{nk})$ Nyström landmark points. To the best of our knowledge, such sharp excess clustering risk bounds for kernel (or approximate kernel) $k$-means have never bee…
Heinz College of Information Systems and Public Policy, Carnegie Mellon University（卡内基梅隆大学海因茨信息系统与公共政策学院）
We establish the first nonasymptotic error bounds for Kaplan-Meier-based nearest neighbor and kernel survival probability estimators where feature vectors reside in metric spaces. Our bounds imply rates of strong consistency for these nonparametric estimators and, up to a log factor, match an existing lower bound for conditional CDF estimation. Our proof strategy also yields nonasymptotic guarantees for nearest neighbor and kernel variants of the Nelson-Aalen cumulative hazards estimator. We experimentally compare these methods on four datasets. We find that for the kernel survival estimator, …
Harvard-MIT Health Sciences and Technology, Massachusetts Institute of Technology（麻省理工学院哈佛-麻省理工健康科学与技术项目）；MIT Media Lab, Massachusetts Institute of Technology（麻省理工学院媒体实验室）；Harvard Medical School（哈佛医学院）；Boston Children’s Hospital（波士顿儿童医院）；Massachusetts General Hospital（麻省总医院）
Currently there is no validated objective measure of pain. Recent neuroimaging studies have explored the feasibility of using functional near-infrared spectroscopy (fNIRS) to measure alterations in brain function in evoked and ongoing pain. In this study, we applied multi-task machine learning methods to derive a practical algorithm for pain detection derived from fNIRS signals in healthy volunteers exposed to a painful stimulus. Especially, we employed multi-task multiple kernel learning to account for the inter-subject variability in pain response. Our results support the use of fNIRS and ma…
Developing efficient kernel methods for regression is very popular in the past decade. In this paper, utilizing boosting on kernel-based weaker learners, we propose a novel kernel-based learning algorithm called kernel-based re-scaled boosting with truncation, dubbed as KReBooT. The proposed KReBooT benefits in controlling the structure of estimators and producing sparse estimate, and is near overfitting resistant. We conduct both theoretical analysis and numerical simulations to illustrate the power of KReBooT. Theoretically, we prove that KReBooT can achieve the almost optimal numerical conv…
A Monte Carlo k-nearest neighbours (KNN) and a multi-resolution convolutional neural network (CNN) were developed to detect the presences of multiple gasses in near infrared (IR) spectrums. High Resolution Transmission database was used to synthesize the near IR spectrums. Monte Carlo KNN determined the optimal kernel sizes and the optimal number of channels. The multi-resolution CNN, composed of multiple different kernels, was created using the optimal kernel sizes and the optimal number of channels. The multi-resolution CNN outperforms the multilayer perceptron and the partial least squares.
Abstract We present PAC-Bayesian bounds for the generalisation error of the K-nearest-neighbour classifier (K-NN). This is achieved by casting the K-NN classifier into a kernel space framework in the limit of vanishing kernel bandwidth. We establish a relation between prior measures over the coefficients in the kernel expansion and the induced measure on the weight vectors in kernel space. Defining a sparse prior over the coefficients allows the application of a PAC-Bayesian folk theorem that leads to a generalisation bound that is a function of the number of redundant training examples: those…
University of Amsterdam（阿姆斯特丹大学）；Canadian Institute for Advanced Research（加拿大高等研究院）
A major challenge in Bayesian Optimization is the boundary issue (Swersky, 2017) where an algorithm spends too many evaluations near the boundary of its search space. In this paper, we propose BOCK, Bayesian Optimization with Cylindrical Kernels, whose basic idea is to transform the ball geometry of the search space using a cylindrical transformation. Because of the transformed geometry, the Gaussian Process-based surrogate model spends less budget searching near the boundary, while concentrating its efforts relatively more near the center of the search region, where we expect the solution to …
Freie Universität Berlin（柏林自由大学）；University of Cologne（科隆大学）
Kernel methods are a cornerstone of classical machine learning. The idea of using quantum computers to compute kernels has recently attracted attention. Quantum embedding kernels (QEKs) constructed by embedding data into the Hilbert space of a quantum computer are a particular quantum kernel technique that allows to gather insights into learning problems and that are particularly suitable for noisy intermediate-scale quantum devices. In this work, we first provide an accessible introduction to quantum embedding kernels and then analyze the practical issues arising when realizing them on a nois…
Max-Planck Institute for Informatics（马克斯·普朗克信息学研究所）
The Neural Tangent Kernel (NTK) characterizes the behavior of infinitely wide neural nets trained under least squares loss by gradient descent. However, despite its importance, the super-quadratic runtime of kernel methods limits the use of NTK in large-scale learning tasks. To accelerate kernel machines with NTK, we propose a near input sparsity time algorithm that maps the input data to a randomized low-dimensional feature space so that the inner product of the transformed data approximates their NTK evaluation. Our transformation works by sketching the polynomial expansions of arc-cosine ke…
Warrington College of Business, University of Florida（佛罗里达大学沃顿商学院）；Institute of Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）；Paul G. Allen School of Computer Science and Engineering, University of Washington（华盛顿大学保罗·G·艾伦计算机科学与工程学院）
Local polynomial regression (Fan and Gijbels 1996) is an important class of methods for nonparametric density estimation and regression problems. However, straightforward implementation of local polynomial regression has quadratic time complexity which hinders its applicability in large-scale data analysis. In this paper, we significantly accelerate the computation of local polynomial estimates by novel applications of multi-dimensional binary indexed trees (Fenwick 1994). Both time and space complexity of our proposed algorithm is nearly linear in the number of input data points. Simulation r…
LIGM, Université Gustave Eiffel, CNRS（古斯塔夫·埃菲尔大学 LIGM 实验室，法国国家科学研究中心）；INRIA Paris（法国国家信息与自动化研究所巴黎中心）；ENS - Département d’Informatique de l’École Normale Supérieure（巴黎高等师范学院计算机科学系）；PSL Research University（巴黎文理研究大学）
It was recently shown that under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds. However, rather than the distance itself, the object of interest for applications such as generative modeling is the underlying optimal transport map. Hence, computational and statistical guarantees need to be obtained for the estimated maps themselves. In this paper, we propose the first tractable algorithm for which the statistical $L^2$ error on the maps nearly matches the existing minimax lower-bounds…
IT University of Copenhagen（哥本哈根信息技术大学）；BARC；University of Copenhagen（哥本哈根大学）
Kernel Density Estimation (KDE) is a nonparametric method for estimating the shape of a density function, given a set of samples from the distribution. Recently, locality-sensitive hashing, originally proposed as a tool for nearest neighbor search, has been shown to enable fast KDE data structures. However, these approaches do not take advantage of the many other advances that have been made in algorithms for nearest neighbor algorithms. We present an algorithm called Density Estimation from Approximate Nearest Neighbors (DEANN) where we apply Approximate Nearest Neighbor (ANN) algorithms as a…
