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The localization of nodes on a network is a challenging research topic. It arises in a variety of applications such as communications and sensor network analysis. We propose a computational approach to recovering the positions of network nodes given partial and corrupted distance measurements, and the positions of a small subset of anchor nodes. First, we show how to derive geometrically adaptive diffusion bases defined over the entire network, given only partial distance measurements. Second, we propose to utilize several diffusion bases simultaneously to derive multiscale diffusion frames. Last, we utilize the diffusion frames to formulate a L1 regression based extension of the anchor points coordinates to the entire network. We experimentally show that under a wide range of conditions our method compares favorably with state-of-the-art approaches.