Geometry of Lightning Self-Attention: Identifiability and Dimension

Nathan W. Henry, Giovanni Luca Marchetti, Kathlén Kohn
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Abstract

We consider function spaces defined by self-attention networks without normalization, and theoretically analyze their geometry. Since these networks are polynomial, we rely on tools from algebraic geometry. In particular, we study the identifiability of deep attention by providing a description of the generic fibers of the parametrization for an arbitrary number of layers and, as a consequence, compute the dimension of the function space. Additionally, for a single-layer model, we characterize the singular and boundary points. Finally, we formulate a conjectural extension of our results to normalized self-attention networks, prove it for a single layer, and numerically verify it in the deep case.
闪电自我关注的几何学:可识别性和维度
我们考虑了由无规范化的自注意网络定义的函数空间,并从理论上分析了它们的几何形状。由于这些网络是多项式的,我们依赖于代数几何的工具。特别是,我们通过对任意层数的参数化的一般纤维进行描述,研究了深度注意的可识别性,并由此计算了函数空间的维度。此外,对于单层模型,我们描述了奇异点和边界点的特征。最后,我们提出了将我们的结果扩展到归一化自我注意网络的猜想,证明了单层网络的结果,并在深层网络中进行了数值验证。
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