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The growing demands on GPU memory posed by the increasing number of neural network parameters call for training approaches that are more memory-efficient. Previous memory reduction training techniques, such as Low-Rank Adaptation (LoRA) and ReLoRA, face challenges, with LoRA being constrained by its low-rank structure, particularly during intensive tasks like pre-training, and ReLoRA suffering from saddle point issues. In this paper, we propose **S**parse **S**pectral **T**raining **(SST)** to optimize memory usage for **pre-training**. SST **updates all singular values** and **selectively updates singular vectors** through a multinomial sampling method weighted by the magnitude of the singular values. Furthermore, SST employs **singular value decomposition to initialize and periodically reinitialize** low-rank parameters, reducing distortion relative to full-rank training compared to other low-rank methods. Through comprehensive testing on both Euclidean and hyperbolic neural networks across various tasks, SST demonstrates its ability to outperform existing memory reduction training methods and is comparable to full-rank training in various cases. On LLaMA-1.3B, with only 18.7\% of the parameters trainable compared to full-rank training (using a rank equivalent to 6\% of the embedding dimension), SST reduces the perplexity gap between other low-rank methods and full-rank training by **97.4\%**. This result highlights SST as an effective parameter-efficient technique for model pre-training.