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能够使数据获得稀疏表示的合适变换。本文研究在预定义变换群作用下保持不变的字典学习问题，其自然应用场景包括冷冻电镜、多目标跟踪、同步问题、姿态估计等。我们特别从数学表示理论的视角探讨该问题。通过利用紧群上函数的非阿贝尔傅里叶分析工具，我们提出了一种学习具有此类不变性字典的算法框架。我们将物理域中自然建模为无限维的字典学习问题，与必然有限维的相关计算问题建立联系。我们证明字典学习问题可被有效理解为在特定矩阵轨道体上的优化实例，这些轨道体具有由对称群的不可约表示决定的块对角结构。该视角使我们能够引入带限处理程序，在应用中实现降维。我们为计算方案提供了保证字典学习获得理想结果的理论依据。应用该范式，我们研究了SO(2)群和SO(3)群的字典学习问题。虽然SO(2)轨道体存在精确的谱面体描述，但对SO(3)轨道体的理解仍存在显著不足。我们提出了SO(3)轨道体的可处理谱面体外逼近方法，并构建了在此设定下执行优化的交替最小化范式。通过合成数据与真实数据的数值实验，我们验证了所提方法在学习SO(3)不变字典方面的有效性。