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Fully homomorphic encryption (FHE) allows direct computations over the encrypted data without access to the decryption. Hence multi-key FHE is well suitable for secure multiparty computation. Recently, Brakerski et al. (TCC 2019 and EUROCRYPT 2020) utilized additively homomorphic encryption to construct FHE schemes with different properties. Motivated by their work, we are attempting to construct multi-key FHE schemes via additively homomorphic encryption. In this paper, we propose a general framework of constructing multi-key FHE, combining the additively homomorphic encryption with specific multiparty computation protocols constructed from encryption switching protocol. Concretely, every involved party encrypts his plaintexts with an additively homomorphic encryption under his own public key. Then the ciphertexts are evaluated by suitable multiparty computation protocols performed by two cooperative servers without collusion. Furthermore, an instantiation with an ElGamal variant scheme is presented. Performance comparisons show that our multi-key FHE from additively homomorphic encryption is more efficient and practical. [ABSTRACT FROM AUTHOR]

Traditional manifold learning algorithms, such as ISOMAP, LLE, and Laplacian Eigenmap, mainly focus on uncovering the latent low-dimensional geometry structure of the training samples in an unsupervised manner where useful class information is ignored. Therefore, the derived low-dimensional representations are not necessarily optimal in discriminative capability. In this paper, we study the discriminant analysis problem by considering the nonlinear manifold structure of data space. To this end, firstly, a new clustering algorithm, called Intra-Cluster Balanced K-Means (ICBKM), is proposed to partition the samples into multiple clusters while ensure that there are balanced samples for the classes within each cluster; approximately, each cluster can be considered as a local patch on the embedded manifold. Then, the local discriminative projections for different clusters are simultaneously calculated by optimizing the global Fisher Criterion based on the cluster weighted data representation. Compared with traditional linear/kernel discriminant analysis (KDA) algorithms, our proposed algorithm has the following characteristics: 1) it essentially is a KDA algorithm with specific geometry-adaptive-kernel tailored to the specific data structure, in contrast to traditional KDA in which the kernel is fixed and independent to the data set; 2) it is approximately a locally linear while globally nonlinear discriminant analyzer; 3) it does not need to store the original samples for computing the low-dimensional representation of a new data; and 4) it is computationally efficient compared with traditional KDA when the sample number is large. The toy problem on artificial data demonstrates the effectiveness of our proposed algorithm in deriving discriminative representations for problems with nonlinear classification hyperplane. The face recognition experiments on YALE and CMU PIE databases show that our proposed algorithm significantly outperforms linear discriminant analysis (LDA) as well as Mixture LDA, and has higher accuracy than KDA with traditional kernels. Index Terms--Kernel design, kernel machine, kernel selection, linear discriminant analysis (LDA), manifold learning, principal component analysis (PCA), subspace learning.

Over the past few decades, a large family of algorithms--supervised or unsupervised; stemming from statistics or geometry theory--has been designed to provide different solutions to the problem of dimensionality reduction. Despite the different motivations of these algorithms, we present in this paper a general formulation known as graph embedding to unify them within a common framework. In graph embedding, each algorithm can be considered as the direct graph embedding or its linear/kernel/tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set, with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided. Furthermore, the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms. By utilizing this framework as a tool, we propose a new supervised dimensionality reduction algorithm called Marginal Fisher Analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class, while the penalty graph connects the marginal points and characterizes the interclass separability. We show that MFA effectively overcomes the limitations of the traditional Linear Discriminant Analysis algorithm due to data distribution assumptions and available projection directions. Real face recognition experiments show the superiority of our proposed MFA in comparison to LDA, also for corresponding kernel and tensor extensions. Index Terms--Dimensionality reduction, manifold learning, subspace learning, graph embedding framework.

Dimensionality reduction is an essential data preprocessing technique for large-scale and streaming data classification tasks. It can be used to improve both the efficiency and the effectiveness of classifiers. Traditional dimensionality reduction approaches fall into two categories: Feature Extraction and Feature Selection. Techniques in the feature extraction category are typically more effective than those in feature selection category. However, they may break down when processing large-scale data sets or data streams due to their high computational complexities. Similarly, the solutions provided by the feature selection approaches are mostly solved by greedy strategies and, hence, are not ensured to be optimal according to optimized criteria. In this paper, we give an overview of the popularly used feature extraction and selection algorithms under a unified framework. Moreover, we propose two novel dimensionality reduction algorithms based on the Orthogonal Centroid algorithm (OC). The first is an Incremental OC (IOC) algorithm for feature extraction. The second algorithm is an Orthogonal Centroid Feature Selection (OCFS) method which can provide optimal solutions according to the OC criterion. Both are designed under the same optimization criterion. Experiments on Reuters Corpus Volume-1 data set and some public large-scale text data sets indicate that the two algorithms are favorable in terms of their effectiveness and efficiency when compared with other state-of-the-art algorithms. Index Terms--Feature extraction, feature selection, orthogonal centroid algorithm.