Understanding how neural systems efficiently process information through distributed representations is a fundamental challenge at the interface of neuroscience and machine learning. Recent approaches analyze the statistical and geometrical attributes of neural representations as population-level mechanistic descriptors of task implementation. In particular, manifold capacity has emerged as a promising framework linking population geometry to the separability of neural manifolds. However, this metric has been limited to linear readouts. To address this limitation, we introduce a theoretical framework that leverages latent directions in input space, which can be related to contextual information. We derive an exact formula for the context-dependent manifold capacity that depends on manifold geometry and context correlations, and validate it on synthetic and real data. Our framework's increased expressivity captures representation reformatting in deep networks at early stages of the layer hierarchy, previously inaccessible to analysis. As context-dependent nonlinearity is ubiquitous in neural systems, our data-driven and theoretically grounded approach promises to elucidate context-dependent computation across scales, datasets, and models.

翻译：理解神经系统如何通过分布式表示高效处理信息，是神经科学与机器学习交叉领域的一个基本挑战。近期研究通过分析神经表征的统计与几何属性，将其作为任务实现的群体水平机制描述符。特别是流形容量的提出，为连接群体几何与神经流形的可分性提供了一个有前景的框架。然而，该指标此前仅限于线性读出。为突破这一限制，我们引入了一个理论框架，该框架利用输入空间中的潜在方向，这些方向可与上下文信息相关联。我们推导了上下文依赖流形容量的精确公式，该公式取决于流形几何与上下文相关性，并在合成数据与真实数据上进行了验证。我们框架更强的表达能力捕捉到了深度网络在层级早期阶段表征重组的过程，这是以往分析方法无法触及的。鉴于上下文依赖的非线性在神经系统中普遍存在，我们这种数据驱动且理论扎实的方法，有望阐明跨尺度、数据集和模型的上下文依赖计算。