To work with categorical features, machine learning systems employ embedding tables. These tables can become exceedingly large in modern recommendation systems, necessitating the development of new methods for fitting them in memory, even during training. Some of the most successful methods for table compression are Product-and Residual Vector Quantization (Gray & Neuhoff, 1998). These methods replace table rows with references to k-means clustered "codewords." Unfortunately, this means they must first know the table before compressing it, so they can only save memory during inference, not training. Recent work has used hashing-based approaches to minimize memory usage during training, but the compression obtained is inferior to that obtained by "post-training" quantization. We show that the best of both worlds may be obtained by combining techniques based on hashing and clustering. By first training a hashing-based "sketch", then clustering it, and then training the clustered quantization, our method achieves compression ratios close to those of post-training quantization with the training time memory reductions of hashing-based methods. We show experimentally that our method provides better compression and/or accuracy that previous methods, and we prove that our method always converges to the optimal embedding table for least-squares training.