🤖 AI Summary
This work addresses the ambiguity inherent in traditional scalar representations of angular data, which fail to distinguish between nearby angles differing by more than π due to periodicity. To resolve this, the authors propose a high-dimensional, real-valued distributed representation based on Fourier embeddings, integrated with spatial semantic pointers to enable neurally interpretable encoding of periodic signals. They formalize the Dirichlet kernel and the periodic Gaussian kernel within this framework, allowing flexible and theoretically grounded control over angular similarity measures. The resulting approach provides unambiguous representations for arbitrarily close angles and establishes a principled design framework for similarity functions with customizable kernel shapes and provable theoretical guarantees.
📝 Abstract
Periodic signals are critical for representing physical and perceptual phenomena. Scalar, real angular measures, e.g., radians and degrees, result in difficulty processing and distinguishing nearby angles, especially when their absolute difference exceeds pi. We can avoid this problem by using real-valued, periodic embeddings in high-dimensional space. These representations also allow us to control the nature of their dot product similarities, allowing us to construct a variety of different kernel shapes. In this work, we aim of highlight how these representations can be constructed and focus on the formalization of Dirichlet and periodic Gaussian kernels using the neurally-plausible representation scheme of Spatial Semantic Pointers.