The dictionary learning problem can be viewed as a data-driven process to learn a suitable transformation so that data is sparsely represented directly from example data. In this paper, we examine the problem of learning a dictionary that is invariant under a pre-specified group of transformations. Natural settings include Cryo-EM, multi-object tracking, synchronization, pose estimation, etc. We specifically study this problem under the lens of mathematical representation theory. Leveraging the power of non-abelian Fourier analysis for functions over compact groups, we prescribe an algorithmic recipe for learning dictionaries that obey such invariances. We relate the dictionary learning problem in the physical domain, which is naturally modelled as being infinite dimensional, with the associated computational problem, which is necessarily finite dimensional. We establish that the dictionary learning problem can be effectively understood as an optimization instance over certain matrix orbitopes having a particular block-diagonal structure governed by the irreducible representations of the group of symmetries. This perspective enables us to introduce a band-limiting procedure which obtains dimensionality reduction in applications. We provide guarantees for our computational ansatz to provide a desirable dictionary learning outcome. We apply our paradigm to investigate the dictionary learning problem for the groups SO(2) and SO(3). While the SO(2) orbitope admits an exact spectrahedral description, substantially less is understood about the SO(3) orbitope. We describe a tractable spectrahedral outer approximation of the SO(3) orbitope, and contribute an alternating minimization paradigm to perform optimization in this setting. We provide numerical experiments to highlight the efficacy of our approach in learning SO(3) invariant dictionaries, both on synthetic and on real world data.

翻译：字典学习问题可视为一种数据驱动过程，旨在从示例数据中直接学习合适的变换，使得数据获得稀疏表示。本文研究在预指定变换群下具有不变性的字典学习问题。其自然应用场景包括冷冻电镜、多目标跟踪、同步定位、姿态估计等。我们特别从数学表示论视角研究该问题。通过利用紧群上函数的非阿贝尔傅里叶分析，我们提出了学习满足此类不变性的字典的算法框架。我们将物理域（天然建模为无限维）中的字典学习问题与计算域（必然有限维）中的关联计算问题相联系，证明字典学习问题可有效理解为特定矩阵轨道空间上的优化实例，这些轨道空间具有由对称群不可约表示支配的特定块对角结构。该视角使我们能够引入带限方法，在应用中实现降维。我们为计算方案提供了保证，确保获得理想的字典学习结果。我们将该范式应用于SO(2)群和SO(3)群的字典学习问题。尽管SO(2)轨道空间具有精确的谱面体描述，但对SO(3)轨道空间的认知尚不充分。我们描述了SO(3)轨道空间的可处理谱面体外逼近，并提出交替最小化范式以在此情境下执行优化。通过合成数据与真实数据的数值实验，验证了本方法在学习SO(3)不变字典方面的有效性。