This Letter introduces sub-Nyquist sampling vertical scanning white light interferometry (SWLI) using deep learning. The method designs Envelope-Deep Residual Shrinkage Networks with channel-wise thresholds (E-DRSN-cw), a network model extracting oversampling envelopes from undersampled signals. The model improves the training efficiency, accuracy, and robustness by following the soft thresholding nonlinear layer approach, pre-padding undersampled interference signals with zeros, using LayerNorm for augmenting inputs and labels, and predicting regression envelopes. Simulation data train the network, and experiments demonstrate its superior performance over classical methods in the accuracy and the robustness. The E-DRSN-cw provides a swift measurement solution for SWLI, removing the need for prior knowledge.