  This paper develops a novel Continuous-time Accelerated Proximal Point
Algorithm (CAPPA) for $\ell_1$-minimization problems with provable fixed-time
convergence guarantees. The problem of $\ell_1$-minimization appears in several
contexts, such as sparse recovery (SR) in Compressed Sensing (CS) theory, and
sparse linear and logistic regressions in machine learning to name a few. Most
existing algorithms for solving $\ell_1$-minimization problems are
discrete-time, inefficient and require exhaustive computer-guided iterations.
CAPPA alleviates this problem on two fronts: (a) it encompasses a
continuous-time algorithm that can be implemented using analog circuits; (b) it
betters LCA and finite-time LCA (recently developed continuous-time dynamical
systems for solving SR problems) by exhibiting provable fixed-time convergence
to optimal solution. Consequently, CAPPA is better suited for fast and
efficient handling of SR problems. Simulation studies are presented that
corroborate computational advantages of CAPPA.
