Accelerated iterative hard thresholding

The iterative hard thresholding algorithm (IHT) is a powerful and versatile algorithm for compressed sensing and other sparse inverse problems. The standard IHT implementation faces several challenges when applied to practical problems. The step-size and sparsity parameters have to be chosen appropriately and, as IHT is based on a gradient descend strategy, convergence is only linear. Whilst the choice of the step-size can be done adaptively as suggested previously, this letter studies the use of acceleration methods to improve convergence speed. Based on recent suggestions in the literature, we show that a host of acceleration methods are also applicable to IHT. Importantly, we show that these modifications not only significantly increase the observed speed of the method, but also satisfy the same strong performance guarantees enjoyed by the original IHT method.