Algorithms for learning a dictionary to sparsely represent a given dataset typically alternate between sparse coding and dictionary update stages. Methods for dictionary update aim to minimise expansion error by updating dictionary vectors and expansion coefficients given patterns of non-zero coefficients obtained in the sparse coding stage. We propose a block total least squares (BLOTLESS) algorithm for dictionary update. BLOTLESS updates a block of dictionary elements and the corresponding sparse coefficients simultaneously. In the error free case, three necessary conditions for exact recovery are identified. Lower bounds on the number of training data are established so that the necessary conditions hold with high probability. Numerical simulations show that the bounds approximate well the number of training data needed for exact dictionary recovery. Numerical experiments further demonstrate several benefits of dictionary learning with BLOTLESS update compared with state-of-the-art algorithms especially when the amount of training data is small.

中文翻译：

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使用 BLOTLESS 更新的字典学习

用于学习字典以稀疏表示给定数据集的算法通常在稀疏编码和字典更新阶段之间交替。字典更新的方法旨在通过在稀疏编码阶段获得的非零系数模式更新字典向量和扩展系数来最小化扩展误差。我们提出了一种用于字典更新的块总最小二乘 (BLOTLESS) 算法。BLOTLESS 同时更新一个字典元素块和相应的稀疏系数。在无差错的情况下，确定了准确恢复的三个必要条件。建立了训练数据数量的下限，以便以高概率满足必要条件。数值模拟表明，边界很好地近似了精确字典恢复所需的训练数据数量。数值实验进一步证明了与最先进的算法相比，使用 BLOTLESS 更新进行字典学习的几个好处，尤其是在训练数据量很小的情况下。