The dictionary learning problem concerns the task of representing data as sparse linear sums drawn from a smaller collection of basic building blocks. In application domains where such techniques are deployed, we frequently encounter datasets where some form of symmetry or invariance is present. Motivated by this observation, we develop a framework for learning dictionaries for data under the constraint that the collection of basic building blocks remains invariant under such symmetries. Our procedure for learning such dictionaries relies on representing the symmetry as the action of a matrix group acting on the data, and subsequently introducing a convex penalty function so as to induce sparsity with respect to the collection of matrix group elements. Our framework specializes to the convolutional dictionary learning problem when we consider integer shifts. Using properties of positive semidefinite Hermitian Toeplitz matrices, we develop an extension that learns dictionaries that are invariant under continuous shifts. Our numerical experiments on synthetic data and ECG data show that the incorporation of such symmetries as priors are most valuable when the dataset has few data-points, or when the full range of symmetries is inadequately expressed in the dataset.

中文翻译：

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群不变字典学习


字典学习问题涉及将数据表示为从较小的基本构建块集合中提取的稀疏线性和的任务。在部署此类技术的应用程序领域中，我们经常遇到存在某种形式的对称性或不变性的数据集。受这一观察的启发，我们开发了一个框架，用于在基本构建块的集合在这种对称性下保持不变的约束下学习数据字典。我们学习此类字典的过程依赖于将对称性表示为矩阵组作用于数据的作用，然后引入凸罚函数，以引起矩阵组元素集合的稀疏性。当我们考虑整数移位时，我们的框架专门解决卷积字典学习问题。利用正半定 Hermitian Toeplitz 矩阵的特性，我们开发了一种扩展，可以学习在连续移位下不变的字典。我们对合成数据和心电图数据的数值实验表明，当数据集数据点很少，或者当数据集中没有充分表达全部对称性时，将此类对称性合并为先验是最有价值的。