Title: Towards Understanding and Mitigating Dimensional Collapse in Heterogeneous Federated Learning. Abstract: Federated learning aims to train models collaboratively across different clients without sharing data for privacy considerations. However, one major challenge for this learning paradigm is the data heterogeneity problem, which refers to the discrepancies between the local data distributions among various clients. To tackle this problem, we first study how data heterogeneity affects the representations of the globally aggregated models. Interestingly, we find that heterogeneous data results in the global model suffering from severe dimensional collapse, in which representations tend to reside in a lower-dimensional space instead of the ambient space. Moreover, we observe a similar phenomenon on models locally trained on each client and deduce that the dimensional collapse on the global model is inherited from local models. In addition, we theoretically analyze the gradient flow dynamics to shed light on how data heterogeneity result in dimensional collapse for local models. To remedy this problem caused by the data heterogeneity, we propose FedDecorr, a novel method that can effectively mitigate dimensional collapse in federated learning. Specifically, FedDecorr applies a regularization term during local training that encourages different dimensions of representations to be uncorrelated. FedDecorr, which is implementation-friendly and computationally-efficient, yields consistent improvements over baselines on standard benchmark datasets. Code: https://github.com/bytedance/FedDecorr.