Adaptive Transition Probability Matrix Learning for Multiview Spectral Clustering.

Multiview clustering as an important unsupervised method has been gathering a great deal of attention. However, most multiview clustering methods exploit the self-representation property to capture the relationship among data, resulting in high computation cost in calculating the self-representation coefficients. In addition, they usually employ different regularizers to learn the representation tensor or matrix from which a transition probability matrix is constructed in a separate step, such as the one proposed by Wu et al.. Thus, an optimal transition probability matrix cannot be guaranteed. To solve these issues, we propose a unified model for multiview spectral clustering by directly learning an adaptive transition probability matrix (MCA2M), rather than an individual representation matrix of each view. Different from the one proposed by Wu et al., MCA2M utilizes the one-step strategy to directly learn the transition probability matrix under the robust principal component analysis framework. Unlike existing methods using the absolute symmetrization operation to guarantee the nonnegativity and symmetry of the affinity matrix, the transition probability matrix learned from MCA2M is nonnegative and symmetric without any postprocessing. An alternating optimization algorithm is designed based on the efficient alternating direction method of multipliers. Extensive experiments on several real-world databases demonstrate that the proposed method outperforms the state-of-the-art methods.

Generalized Nonconvex Low-Rank Tensor Approximation for Multi-View Subspace Clustering.

The low-rank tensor representation (LRTR) has become an emerging research direction to boost the multi-view clustering performance. This is because LRTR utilizes not only the pairwise relation between data points, but also the view relation of multiple views. However, there is one significant challenge: LRTR uses the tensor nuclear norm as the convex approximation but provides a biased estimation of the tensor rank function. To address this limitation, we propose the generalized nonconvex low-rank tensor approximation (GNLTA) for multi-view subspace clustering. Instead of the pairwise correlation, GNLTA adopts the low-rank tensor approximation to capture the high-order correlation among multiple views and proposes the generalized nonconvex low-rank tensor norm to well consider the physical meanings of different singular values. We develop a unified solver to solve the GNLTA model and prove that under mild conditions, any accumulation point is a stationary point of GNLTA. Extensive experiments on seven commonly used benchmark databases have demonstrated that the proposed GNLTA achieves better clustering performance over state-of-the-art methods.

Designing Hyperchaotic Cat Maps With Any Desired Number of Positive Lyapunov Exponents.

Generating chaotic maps with expected dynamics of users is a challenging topic. Utilizing the inherent relation between the Lyapunov exponents (LEs) of the Cat map and its associated Cat matrix, this paper proposes a simple but efficient method to construct an -dimensional ( -D) hyperchaotic Cat map (HCM) with any desired number of positive LEs. The method first generates two basic -D Cat matrices iteratively and then constructs the final -D Cat matrix by performing similarity transformation on one basic -D Cat matrix by the other. Given any number of positive LEs, it can generate an -D HCM with desired hyperchaotic complexity. Two illustrative examples of -D HCMs were constructed to show the effectiveness of the proposed method, and to verify the inherent relation between the LEs and Cat matrix. Theoretical analysis proves that the parameter space of the generated HCM is very large. Performance evaluations show that, compared with existing methods, the proposed method can construct -D HCMs with lower computation complexity and their outputs demonstrate strong randomness and complex ergodicity.

$n$ -Dimensional Discrete Cat Map Generation Using Laplace Expansions.

Different from existing methods that use matrix multiplications and have high computation complexity, this paper proposes an efficient generation method of n -dimensional ( [Formula: see text]) Cat maps using Laplace expansions. New parameters are also introduced to control the spatial configurations of the [Formula: see text] Cat matrix. Thus, the proposed method provides an efficient way to mix dynamics of all dimensions at one time. To investigate its implementations and applications, we further introduce a fast implementation algorithm of the proposed method with time complexity O(n4) and a pseudorandom number generator using the Cat map generated by the proposed method. The experimental results show that, compared with existing generation methods, the proposed method has a larger parameter space and simpler algorithm complexity, generates [Formula: see text] Cat matrices with a lower inner correlation, and thus yields more random and unpredictable outputs of [Formula: see text] Cat maps.

Dynamic Parameter-Control Chaotic System.

This paper proposes a general framework of 1-D chaotic maps called the dynamic parameter-control chaotic system (DPCCS). It has a simple but effective structure that uses the outputs of a chaotic map (control map) to dynamically control the parameter of another chaotic map (seed map). Using any existing 1-D chaotic map as the control/seed map (or both), DPCCS is able to produce a huge number of new chaotic maps. Evaluations and comparisons show that chaotic maps generated by DPCCS are very sensitive to their initial states, and have wider chaotic ranges, better unpredictability and more complex chaotic behaviors than their seed maps. Using a chaotic map of DPCCS as an example, we provide a field-programmable gate array design of this chaotic map to show the simplicity of DPCCS in hardware implementation, and introduce a new pseudo-random number generator (PRNG) to investigate the applications of DPCCS. Analysis and testing results demonstrate the excellent randomness of the proposed PRNG.

Cascade Chaotic System With Applications.

Chaotic maps are widely used in different applications. Motivated by the cascade structure in electronic circuits, this paper introduces a general chaotic framework called the cascade chaotic system (CCS). Using two 1-D chaotic maps as seed maps, CCS is able to generate a huge number of new chaotic maps. Examples and evaluations show the CCS's robustness. Compared with corresponding seed maps, newly generated chaotic maps are more unpredictable and have better chaotic performance, more parameters, and complex chaotic properties. To investigate applications of CCS, we introduce a pseudo-random number generator (PRNG) and a data encryption system using a chaotic map generated by CCS. Simulation and analysis demonstrate that the proposed PRNG has high quality of randomness and that the data encryption system is able to protect different types of data with a high-security level.