In many data science applications, the objective is to extract
appropriately-ordered smooth low-dimensional data patterns from
high-dimensional data sets. This is challenging since common sorting algorithms
are primarily aiming at finding monotonic orderings in low-dimensional data,
whereas typical dimension reduction and feature extraction algorithms are not
primarily designed for extracting smooth low-dimensional data patterns. We show
that when selecting the Euclidean smoothness as a pattern quality criterium,
both of these problems (finding the optimal 'crisp' data permutation and
extracting the sparse set of permuted low-dimensional smooth patterns) can be
efficiently solved numerically as one unsupervised entropy-regularized
iterative optimization problem. We formulate and prove the conditions for
monotonicity and convergence of this linearly-scalable (in dimension) numerical
procedure, with the iteration cost scaling of $\mathcal{O}(DT^2)$, where $T$ is
the size of the data statistics and $D$ is a feature space dimension. The
efficacy of the proposed method is demonstrated through the examination of
synthetic examples as well as a real-world application involving the
identification of smooth bankruptcy risk minimizing transition patterns from
high-dimensional economical data. The results showcase that the statistical
properties of the overall time complexity of the method exhibit linear scaling
in the dimensionality $D$ within the specified confidence intervals.

该研究介绍了一种基于欧式平滑度作为模式质量标准的无监督熵正则化迭代优化问题，能够高效地从高维数据中提取出稀疏的、经过排列的低维平滑模式，有效地实现了降维和特征提取，且在实际应用中能够识别同时最小化破产风险的平滑转换模式。