Multiple kernel clustering (MKC) is an important research topic that has been widely studied for decades. However, current methods still face two problems: inefficient when handling out-of-sample data points and lack of theoretical study of the stability and generalization of clustering. In this paper, we propose a novel method that can efficiently compute the embedding of out-of-sample data with a solid generalization guarantee. Specifically, we approximate the eigen functions of the integral operator associated with the linear combination of base kernel functions to construct low-dimensional embeddings of out-of-sample points for efficient multiple kernel clustering. In addition, we, for the first time, theoretically study the stability of clustering algorithms and prove that the single-view version of the proposed method has uniform stability as $\mathcal{O}\left(Kn^{-3/2}\right)$ and establish an upper bound of excess risk as $\widetilde{\mathcal{O}}\left(Kn^{-3/2}+n^{-1/2}\right)$, where $K$ is the cluster number and $n$ is the number of samples. We then extend the theoretical results to multiple kernel scenarios and find that the stability of MKC depends on kernel weights. As an example, we apply our method to a novel MKC algorithm termed SimpleMKKM and derive the upper bound of its excess clustering risk, which is tighter than the current results. Extensive experimental results validate the effectiveness and efficiency of the proposed method.

Kernel k-means has been widely studied in machine learning. However, existing kernel k-means methods often ignore the \textit{fairness} issue, which may cause discrimination. To address this issue, in this paper, we propose a novel Fair Kernel K-Means (FKKM) framework. In this framework, we first propose a new fairness regularization term that can lead to a fair partition of data. The carefully designed fairness regularization term has a similar form to the kernel k-means which can be seamlessly integrated into the kernel k-means framework.

Multiple kernel clustering (MKC) is an important research topic that has been widely studied for decades. However, current methods still face two problems: inefficient when handling out-of-sample data points and lack of theoretical study of the stability and generalization of clustering. In this paper, we propose a novel method that can efficiently compute the embedding of out-of-sample data with a solid generalization guarantee. Specifically, we approximate the eigen functions of the integral operator associated with the linear combination of base kernel functions to construct low-dimensional embeddings of out-of-sample points for efficient multiple kernel clustering. In addition, we, for the first time, theoretically study the stability of clustering algorithms and prove that the single-view version of the proposed method has uniform stability as \mathcal{O}\left(Kn {-3/2}\right) and establish an upper bound of excess risk as \widetilde{\mathcal{O}}\left(Kn {-3/2} n {-1/2}\right), where K is the cluster number and n is the number of samples. We then extend the theoretical results to multiple kernel scenarios and find that the stability of MKC depends on kernel weights.

As the analytic tools become more powerful, and more data are generated on a daily basis, the issue of data privacy arises. This leads to the study of the design of privacy-preserving machine learning algorithms. Given two objectives, namely, utility maximization and privacy-loss minimization, this work is based on two previously non-intersecting regimes -- Compressive Privacy and multi-kernel method. Compressive Privacy is a privacy framework that employs utility-preserving lossy-encoding scheme to protect the privacy of the data, while multi-kernel method is a kernel based machine learning regime that explores the idea of using multiple kernels for building better predictors. The compressive multi-kernel method proposed consists of two stages -- the compression stage and the multi-kernel stage. The compression stage follows the Compressive Privacy paradigm to provide the desired privacy protection. Each kernel matrix is compressed with a lossy projection matrix derived from the Discriminant Component Analysis (DCA). The multi-kernel stage uses the signal-to-noise ratio (SNR) score of each kernel to non-uniformly combine multiple compressive kernels. The proposed method is evaluated on two mobile-sensing datasets -- MHEALTH and HAR -- where activity recognition is defined as utility and person identification is defined as privacy. The results show that the compression regime is successful in privacy preservation as the privacy classification accuracies are almost at the random-guess level in all experiments. On the other hand, the novel SNR-based multi-kernel shows utility classification accuracy improvement upon the state-of-the-art in both datasets. These results indicate a promising direction for research in privacy-preserving machine learning.

For supervised and unsupervised learning, positive definite kernels allow to use large and potentially infinite dimensional feature spaces with a computational cost that only depends on the number of observations. This is usually done through the penalization of predictor functions by Euclidean or Hilbertian norms. In this paper, we explore penalizing by sparsity-inducing norms such as the L1-norm or the block L1-norm. We assume that the kernel decomposes into a large sum of individual basis kernels which can be embedded in a directed acyclic graph; we show that it is then possible to perform kernel selection through a hierarchical multiple kernel learning framework, in polynomial time in the number of selected kernels. This framework is naturally applied to non linear variable selection; our extensive simulations on synthetic datasets and datasets from the UCI repository show that efficiently exploring the large feature space through sparsity-inducing norms leads to state-of-the-art predictive performance.

For supervised and unsupervised learning, positive definite kernels allow to use large and potentially infinite dimensional feature spaces with a computational cost that only depends on the number of observations. This is usually done through the penalization of predictor functions by Euclidean or Hilbertian norms. In this paper, we explore penalizing by sparsity-inducing norms such as the l1-norm or the block l1-norm. We assume that the kernel decomposes into a large sum of individual basis kernels which can be embedded in a directed acyclic graph; we show that it is then possible to perform kernel selection through a hierarchical multiple kernel learning framework, in polynomial time in the number of selected kernels. This framework is naturally applied to non linear variable selection; our extensive simulations on synthetic datasets and datasets from the UCI repository show that efficiently exploring the large feature space through sparsity-inducing norms leads to state-of-the-art predictive performance.