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Given high-dimensional, graph-smooth and grossly-corrupted signals, this paper presents the algorithm to simultaneously estimate the intrinsic low-rank components and the underlying graph from high-dimensional noisy graph signal. We assume that the perturbation on low-rank components is sparse and signals are smooth on unknown underlying graph. The proposed algorithm learns the low-rank components by exploiting estimate of the graph, and refines the graph estimation with learned low-rank components. We propose two solutions to this problem: One applies alternating optimization to solve the subproblems. The other solves the problem directly. Furthermore, we analyze the impact of inexact graph on low-rank components estimation to justify our approach. We conduct extensive experiments on the proposed algorithm with synthetic data and real brain imaging data, Magnetoencephalography (MEG) and compare it with state-of-the-art methods. In particular, our algorithm is applied to estimate the low-rank components for classifying MEG signals evoked when a subject views a face or non-face image. We observe that our proposed algorithm is competitive in estimating low-rank components, adequately capturing intrinsic task-related information in lower-dimensional representation. This leads to improved classification performance. Furthermore, we notice that our estimated graph indicates brain active regions for the visual activity that are consistent with neuroscientific findings.