CuBICA: Independent component analysis by simultaneous third- and fourth-order cumulant diagonalization. CuBICA, which is an improved method for independent component analysis (ICA) based on the diagonalization of cumulant tensors is proposed. It is based on Comon’s algorithm, but it takes third- and fourth-order cumulant tensors into account simultaneously. The underlying contrast function is also mathematically much simpler and has a more intuitive interpretation. It is therefore easier to optimize and approximate. A comparison with Comon’s and three other ICA algorithms on different data sets demonstrates its performance.