Diffusion models are one of the key architectures of generative AI. Their main drawback, however, is the high computational cost. The first part of the talk introduces diffusion models, outlining their main ideas and their role in modern generative AI. The second part of the talk will focus on our recent research showing how the concept of sparsity, well known especially in statistics, can provide a pathway to more efficient diffusion pipelines. Our mathematical guarantees prove that sparsity can reduce the influence of the input dimension on computational complexity to that of a much smaller intrinsic dimension of the data. Our empirical findings further confirm that inducing sparsity can indeed lead to better samples at a lower cost.