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的过程，旨在直接从示例数据中学习合适的变换，使数据获得稀疏表示。本文研究在预指定变换群作用下保持不变的字典学习问题。该问题的典型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的角度审视该问题，利用紧群上函数的非阿贝尔傅里叶分析工具，提出了一种能实现此类不变性字典学习的算法框架。我们将物理域（自然建模为无穷维）的字典学习问题与计算域（必然为有限维）的相关计算问题建立联系，证明字典学习可有效理解为特定矩阵轨道流形上的优化问题——该流形具有由对称群不可约表示决定的特殊块对角结构。这一视角使我们能够引入带限处理流程，在实际应用中实现降维。我们为所提出的计算方案提供了能获得理想字典学习效果的保证。采用该框架分别研究了SO(2)群和SO(3)群的字典学习问题：虽然SO(2)轨道流形具有精确的谱面体描述，但对SO(3)轨道流形的认知尚不充分。我们提出了SO(3)轨道流形的一种可计算谱面体外近似，并构建了交替最小化框架进行优化。通过在合成数据与真实数据上的数值实验，验证了该方法在学习SO(3)不变字典方面的有效性。