Lp Quasi-norm Minimization: Algorithm and Applications

Sparsity finds applications in areas as diverse as statistics, machine learning, and signal processing. Computations over sparse structures are less complex compared to their dense counterparts, and their storage consumes less space. This paper proposes a heuristic method for retrieving sparse approximate solutions of optimization problems via minimizing the $\ell_{p}$ quasi-norm, where $0<p<1$. An iterative two-block ADMM algorithm for minimizing the $\ell_{p}$ quasi-norm subject to convex constraints is proposed. For $p=s/q<1$, $s,q \in \mathbb{Z}_{+}$, the proposed algorithm requires solving for the roots of a scalar degree $2q$ polynomial as opposed to applying a soft thresholding operator in the case of $\ell_{1}$. The merit of that algorithm relies on its ability to solve the $\ell_{p}$ quasi-norm minimization subject to any convex set of constraints. However, it suffers from low speed, due to a convex projection step in each iteration, and the lack of mathematical convergence guarantee. We then aim to vanquish these shortcomings by relaxing the assumption on the constraints set to be the set formed due to convex and differentiable, with Lipschitz continuous gradient, functions, i.e. specifically, polytope sets. Using a proximal gradient step, we mitigate the convex projection step and hence enhance the algorithm speed while proving its convergence. We then present various applications where the proposed algorithm excels, namely, matrix rank minimization, sparse signal reconstruction from noisy measurements, sparse binary classification, and system identification. The results demonstrate the significant gains obtained by the proposed algorithm compared to those via $\ell_{1}$ minimization.

Comparison of Different Spike Sorting Subtechniques Based on Rat Brain Basolateral Amygdala Neuronal Activity

Developing electrophysiological recordings of brain neuronal activity and their analysis provide a basis for exploring the structure of brain function and nervous system investigation. The recorded signals are typically a combination of spikes and noise. High amounts of background noise and possibility of electric signaling recording from several neurons adjacent to the recording site have led scientists to develop neuronal signal processing tools such as spike sorting to facilitate brain data analysis. Spike sorting plays a pivotal role in understanding the electrophysiological activity of neuronal networks. This process prepares recorded data for interpretations of neurons interactions and understanding the overall structure of brain functions. Spike sorting consists of three steps: spike detection, feature extraction, and spike clustering. There are several methods to implement each of spike sorting steps. This paper provides a systematic comparison of various spike sorting sub-techniques applied to real extracellularly recorded data from a rat brain basolateral amygdala. An efficient sorted data resulted from careful choice of spike sorting sub-methods leads to better interpretation of the brain structures connectivity under different conditions, which is a very sensitive concept in diagnosis and treatment of neurological disorders. Here, spike detection is performed by appropriate choice of threshold level via three different approaches. Feature extraction is done through PCA and Kernel PCA methods, which Kernel PCA outperforms. We have applied four different algorithms for spike clustering including K-means, Fuzzy C-means, Bayesian and Fuzzy maximum likelihood estimation. As one requirement of most clustering algorithms, optimal number of clusters is achieved through validity indices for each method. Finally, the sorting results are evaluated using inter-spike interval histograms.