Toward a Quantitative Survey of Dimension Reduction Techniques.

Dimensionality reduction methods, also known as projections, are frequently used in multidimensional data exploration in machine learning, data science, and information visualization. Tens of such techniques have been proposed, aiming to address a wide set of requirements, such as ability to show the high-dimensional data structure, distance or neighborhood preservation, computational scalability, stability to data noise and/or outliers, and practical ease of use. However, it is far from clear for practitioners how to choose the best technique for a given use context. We present a survey of a wide body of projection techniques that helps answering this question. For this, we characterize the input data space, projection techniques, and the quality of projections, by several quantitative metrics. We sample these three spaces according to these metrics, aiming at good coverage with bounded effort. We describe our measurements and outline observed dependencies of the measured variables. Based on these results, we draw several conclusions that help comparing projection techniques, explain their results for different types of data, and ultimately help practitioners when choosing a projection for a given context. Our methodology, datasets, projection implementations, metrics, visualizations, and results are publicly open, so interested stakeholders can examine and/or extend this benchmark.

Multilevel training of binary morphological operators.

The design of binary morphological operators that are translation-invariant and locally defined by a finite neighborhood window corresponds to the problem of designing Boolean functions. As in any supervised classification problem, morphological operators designed from training sample also suffer from overfitting. Large neighborhood tends to lead to performance degradation of the designed operator. This work proposes a multi-level design approach to deal with the issue of designing large neighborhood based operators. The main idea is inspired from stacked generalization (a multi-level classifier design approach) and consists in, at each training level, combining the outcomes of the previous level operators. The final operator is a multi-level operator that ultimately depends on a larger neighborhood than of the individual operators that have been combined. Experimental results show that two-level operators obtained by combining operators designed on subwindows of a large window consistently outperforms the single-level operators designed on the full window. They also show that iterating two-level operators is an effective multi-level approach to obtain better results.

An exact algorithm for optimal MAE stack filter design.

We propose a new algorithm for optimal MAE stack filter design. It is based on three main ingredients. First, we show that the dual of the integer programming formulation of the filter design problem is a minimum cost network flow problem. Next, we present a decomposition principle that can be used to break this dual problem into smaller subproblems. Finally, we propose a specialization of the network Simplex algorithm based on column generation to solve these smaller subproblems. Using our method, we were able to efficiently solve instances of the filter problem with window size up to 25 pixels. To the best of our knowledge, this is the largest dimension for which this problem was ever solved exactly.