Dimensionality reduction-based dictionary learning methods in the literature have often used iterative random projections. The dimensionality of such a random projection matrix is a random number that might not lead to a separable subspace structure in the transformed space. The convergence of such methods highly depends on the initial seed values used. Also, gradient descent-based updates might result in local minima. This paper proposes a constructive approach to derandomize the projection matrix using the Johnson-Lindenstrauss lemma. Rather than reducing dimensionality via random projections, a projection matrix derived from the proposed Modified Supervised PC analysis is used. A heuristic is proposed to decide the data perturbation levels and the dictionary atom's corresponding suitable description length. The projection matrix is derived in a single step, provides maximum feature-label consistency of the transformed space, and preserves the geometry of the original data. The projection matrix thus constructed is proved to be a JL-embedding. Despite confusing classes in the OCR datasets, the dictionary trained in the transformed space generates discriminative sparse coefficients with reduced complexity. Empirical study demonstrates that the proposed method performs well even when the number of classes and dimensionality increase. Experimentation on OCR and face recognition datasets shows better classification performance than other algorithms.

翻译：现有文献中基于降维的字典学习方法常采用迭代随机投影。此类随机投影矩阵的维度为随机数值，可能导致变换后的空间无法形成可分离的子空间结构。此类方法的收敛性高度依赖于初始种子值的选择。此外，基于梯度下降的更新可能导致局部最优解。本文提出一种基于约翰逊-林登斯特劳斯引理的构造性方法来实现投影矩阵的去随机化。通过所提出的改进监督主成分分析导出投影矩阵，以替代随机投影降维方式。文中提出一种启发式方法来确定数据扰动水平及字典原子对应的合适描述长度。该投影矩阵通过单步推导获得，能最大化变换空间的特征-标签一致性，同时保持原始数据的几何结构。研究证明如此构建的投影矩阵满足约翰逊-林登斯特劳斯嵌入条件。即使在OCR数据集中存在类别混淆的情况下，于变换空间训练的字典仍能生成具有判别性的稀疏系数且复杂度更低。实证研究表明，即使类别数量和维度增加，所提方法仍能保持良好性能。在OCR和人脸识别数据集上的实验表明，该方法相比其他算法具有更优的分类性能。