For cloud service providers, fine-grained packet loss detection across data centers is crucial in improving their service level and increasing business income. However, the inability to obtain sufficient measurements makes it difficult owing to the fundamental limit that the wide-area network links responsible for communication are not under their management. Moreover, millisecond-level delay jitter and clock synchronization errors in the WAN disable many tools that perform well in data center networks on this issue. Therefore, there is an urgent need to develop a new tool or method. In this work, we propose SketchDecomp, a novel loss detection method, from a mathematical perspective that has never been considered before. Its key is to decompose sketches upstream and downstream into several sub-sketches and builds a low-rank matrix optimization model to solve them. Extensive experiments on the test bed demonstrate its superiority.