# Linearly-scalable learning of smooth low-dimensional patterns with permutation-aided entropic dimension reduction

arXiv: https://arxiv.org/abs/2306.10287 · HTML: https://mlsanity.com/paper/2306.10287/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 28 references that are in the map.
- First posted on arXiv 17 Jun 2023.

## Builds on (most cited first, 3 of 3)

- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Robust learning of data anomalies with analytically-solvable entropic outlier sparsification](https://mlsanity.com/paper/2112.11768/) (2021)
- [Permute Me Softly: Learning Soft Permutations for Graph Representations](https://mlsanity.com/paper/2110.01872/) (2021)

## Abstract (arXiv)

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 O(DT2), 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.