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We provide a theoretical study of the iterative hard thresholding with partially known support set (IHT-PKS) algorithm when used to solve the compressed sensing recovery problem. Recent work has shown that IHT-PKS performs better than the traditional IHT in reconstructing sparse or compressible signals. However, less work has been done on analyzing the performance guarantees of IHT-PKS. In this paper, we improve the current RIP-based bound of IHT-PKS algorithm from ${\delta }_{3s-2k}<\frac{1}{\sqrt{32}}\approx 0.1768$ to ${\delta }_{3s-2k}<\frac{\sqrt{5}-1}{4}\approx 0.309$, where ${\delta }_{3s-2k}$ is the restricted...