Sparse mixtures of experts combine multiple prediction models while evaluating only a small subset for each input. Although this architecture reduces computation, how expert separation and routing sparsity jointly govern expert recovery remains an open problem. In this paper, we address this question for location Gaussian mixtures of experts with a fixed number of experts by establishing how expert separation and spatial arrangement determine recovery rates. For dense models, our results reveal that when experts form well-separated clusters, the separation geometry is governed by the size of the largest cluster; without a prescribed cluster structure, this geometry is instead driven by the total number of experts. For sparse models, the separation geometry on retained routing regions is determined by the number of active experts, while the contribution of omitted regions is controlled by their total probability. Consequently, activating fewer experts reduces the separation complexity in these bounds. Furthermore, Top-1 routing yields a particularly striking result: experts can be recovered at a rate independent of the separation quantity. Our technical approach combines polynomial interpolation and Gaussian reproducing-kernel Hilbert space (RKHS) test functions with quantitative restriction inequalities that extend pointwise comparisons to expert functions.
When you read a paper about a new Vision-Language Model today, it can be easy to forget how strange this idea would have sounded not so long ago. Teaching machines to see was already hard. Teaching them to read and generate language was already hard. Asking them to do both at once - and then to reason, answer questions, follow instructions, and sometimes even surprise us - still carries a quiet trace of science fiction, even as it becomes routine. This book was born from a simple feeling: it is too easy to get lost. The field moves quickly, new model names appear constantly, and the gap between "I know the buzzwords" and "I actually understand how this works" can feel uncomfortably wide. I have felt that gap many times. If you are holding this book, you probably have too. My goal is not to provide an exhaustive catalog of every dataset, benchmark, and new model variant. Instead, I want to offer something more modest - and, I hope, more durable: a clear mental map of Vision-Language Models. Enough structure that you can read new papers with confidence; enough intuition that you can design your own systems without feeling as if you are assembling LEGO bricks blindly.