Transformation of a random variable is a common need in a design of many algorithms in signal processing, automatic control, and fault detection. Typically, the design is tied to an assumption on a probability density function of the random variable, often in the form of the Gaussian distribution. The assumption may be, however, difficult to be met in algorithms involving non-linear transformation of the random variable. This paper focuses on techniques capable to ensure validity of the Gaussian assumption of the non-linearly transformed Gaussian variable by approximating the to-be-transformed random variable distribution by a Gaussian mixture (GM) distribution. The stress is laid on an analysis and selection of design parameters of the approximate GM distribution to minimise the error imposed by the non-linear transformation such as the location and number of the GM terms. A special attention is devoted to the definition of the novel GM splitting directions based on the measures of non-Gaussianity. The proposed splitting directions are analysed and illustrated in numerical simulations.

Recently, a tight frame, called convolution framelets (CFs) and constructed by convolving local and non-local bases, is proposed to provide valuable insights in understanding the patch-based processing approaches in the viewpoint of sparse representation (SR). However, it is still unclear how to represent signals with energy concentration guarantee in its lifted space and how to optimise the local base for a given non-local base. To address these issues, the equivalence between the signal space and its lifted space is established by the Hankel operator. In the lifted space, the energy concentration property of signals is measured by the sparsity instead of the Euclidean norm and the rank. With the new objective function, an optimisation model is built to train the optimal local base for a given nonlocal base from the training samples, motivating us to propose the optimal matching convolution framelets (OMCFs). In addition, a numerical algorithm is also designed to solve the proposed model using alternating optimisation strategy. The OMCF for SR is tested on the speech signals, and the comparisons with traditional popular SR tools, such as discrete cosine transform (DCT) and Haar wavelets, demonstrate its better performance.

In sparse signal recovery, to overcome the -norm sparse regularisation's disadvantages tendency of uniformly penalise the signal amplitude and underestimate the high-amplitude components, a new algorithm based on a non-convex minimax-concave penalty is proposed, which can approximate the -norm more accurately. Moreover, the authors employ the -norm loss function instead of the -norm for the residual error, as the -loss is less sensitive to the outliers in the measurements. To rise to the challenges introduced by the non-convex non-smooth problem, they first employ a smoothed strategy to approximate the -norm loss function, and then use the difference-of-convex algorithm framework to solve the non-convex problem. They also show that any cluster point of the sequence generated by the proposed algorithm converges to a stationary point. The simulation result demonstrates the authors’ conclusions and indicates that the algorithm proposed in this study can obviously improve the reconstruction quality.