Joint sparse optimization: Lower-order regularization method and application in cell fate conversion

Abstract

Multiple measurement signals are commonly collected in practical applications, and joint sparse optimization adopts the synchronous effect within multiple measurement signals to improve model analysis and sparse recovery capability. In this paper, we investigate the joint sparse optimization problem via $\ell_{p,q}$ regularization ($0\le q \le 1 \le p$) in three aspects: theory, algorithm and application. In the theoretical aspect, we introduce a weak notion of joint restricted Frobenius norm condition associated with the $\ell_{p,q}$ regularization, and apply it to establish an oracle property and a recovery bound for the $\ell_{p,q}$ regularization of joint sparse optimization problem. In the algorithmic aspect, we apply the well-known proximal gradient algorithm to solve the $\ell_{p,q}$ regularization problems, provide analytical formulas for proximal subproblems of certain specific $\ell_{p,q}$ regularizations, and establish the global convergence and linear convergence rate of the proximal gradient algorithm under some mild conditions. More importantly, we propose two types of proximal gradient algorithms with the truncation technique and the continuation technique, respectively, and establish their convergence to the ground true joint sparse solution within a tolerance relevant to the noise level and the recovery bound under the assumption of restricted isometry property. In the aspect of application, we develop a novel method, based on joint sparse optimization with lower-order regularization and proximal gradient algorithm, to infer the master transcription factors for cell fate conversion, which is a powerful tool in developmental biology and regenerative medicine. Numerical results indicate that the novel method facilitates fast identification of master transcription factors, give raise to the possibility of higher successful conversion rate and in the hope of reducing biological experimental cost.