Kernel (linear algebra)

About: Kernel (linear algebra) is a research topic. Over the lifetime, 5080 publications have been published within this topic receiving 105655 citations. The topic is also known as: null space & nullspace.

Learning Transferable Features with Deep Adaptation Networks

Abstract: Recent studies reveal that a deep neural network can learn transferable features which generalize well to novel tasks for domain adaptation. However, as deep features eventually transition from general to specific along the network, the feature transferability drops significantly in higher layers with increasing domain discrepancy. Hence, it is important to formally reduce the dataset bias and enhance the transferability in task-specific layers. In this paper, we propose a new Deep Adaptation Network (DAN) architecture, which generalizes deep convolutional neural network to the domain adaptation scenario. In DAN, hidden representations of all task-specific layers are embedded in a reproducing kernel Hilbert space where the mean embeddings of different domain distributions can be explicitly matched. The domain discrepancy is further reduced using an optimal multi-kernel selection method for mean embedding matching. DAN can learn transferable features with statistical guarantees, and can scale linearly by unbiased estimate of kernel embedding. Extensive empirical evidence shows that the proposed architecture yields state-of-the-art image classification error rates on standard domain adaptation benchmarks.

Exploiting the circulant structure of tracking-by-detection with kernels

Abstract: Recent years have seen greater interest in the use of discriminative classifiers in tracking systems, owing to their success in object detection. They are trained online with samples collected during tracking. Unfortunately, the potentially large number of samples becomes a computational burden, which directly conflicts with real-time requirements. On the other hand, limiting the samples may sacrifice performance. Interestingly, we observed that, as we add more and more samples, the problem acquires circulant structure. Using the well-established theory of Circulant matrices, we provide a link to Fourier analysis that opens up the possibility of extremely fast learning and detection with the Fast Fourier Transform. This can be done in the dual space of kernel machines as fast as with linear classifiers. We derive closed-form solutions for training and detection with several types of kernels, including the popular Gaussian and polynomial kernels. The resulting tracker achieves performance competitive with the state-of-the-art, can be implemented with only a few lines of code and runs at hundreds of frames-per-second. MATLAB code is provided in the paper (see Algorithm 1).

Support Vector Method for Novelty Detection

Abstract: Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S equals some a priori specified ν between 0 and 1. We propose a method to approach this problem by trying to estimate a function f which is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space. We provide a theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabelled data.

Introduction to Support Vector Machines

Abstract: Support Vector Machines (SVM’s) are a relatively new learning method used for binary classification. The basic idea is to find a hyperplane which separates the d-dimensional data perfectly into its two classes. However, since example data is often not linearly separable, SVM’s introduce the notion of a “kernel induced feature space” which casts the data into a higher dimensional space where the data is separable. Typically, casting into such a space would cause problems computationally, and with overfitting. The key insight used in SVM’s is that the higher-dimensional space doesn’t need to be dealt with directly (as it turns out, only the formula for the dot-product in that space is needed), which eliminates the above concerns. Furthermore, the VC-dimension (a measure of a system’s likelihood to perform well on unseen data) of SVM’s can be explicitly calculated, unlike other learning methods like neural networks, for which there is no measure. Overall, SVM’s are intuitive, theoretically wellfounded, and have shown to be practically successful. SVM’s have also been extended to solve regression tasks (where the system is trained to output a numerical value, rather than “yes/no” classification).

KPConv: Flexible and Deformable Convolution for Point Clouds

Abstract: We present Kernel Point Convolution (KPConv), a new design of point convolution, i.e. that operates on point clouds without any intermediate representation. The convolution weights of KPConv are located in Euclidean space by kernel points, and applied to the input points close to them. Its capacity to use any number of kernel points gives KPConv more flexibility than fixed grid convolutions. Furthermore, these locations are continuous in space and can be learned by the network. Therefore, KPConv can be extended to deformable convolutions that learn to adapt kernel points to local geometry. Thanks to a regular subsampling strategy, KPConv is also efficient and robust to varying densities. Whether they use deformable KPConv for complex tasks, or rigid KPconv for simpler tasks, our networks outperform state-of-the-art classification and segmentation approaches on several datasets. We also offer ablation studies and visualizations to provide understanding of what has been learned by KPConv and to validate the descriptive power of deformable KPConv.