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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. [ABSTRACT FROM AUTHOR]

Dimensionality reduction methods, also known as projections, are often used to explore multidimensional data in machine learning, data science, and information visualization. However, several such methods, such as the well-known t-distributed stochastic neighbor embedding and its variants, are computationally expensive for large datasets, suffer from stability problems, and cannot directly handle out-of-sample data. We propose a learning approach to construct any such projections. We train a deep neural network based on sample set drawn from a given data universe, and their corresponding two-dimensional projections, compute with any user-chosen technique. Next, we use the network to infer projections of any dataset from the same universe. Our approach generates projections with similar characteristics as the learned ones, is computationally two to four orders of magnitude faster than existing projection methods, has no complex-to-set user parameters, handles out-of-sample data in a stable manner, and can be used to learn any projection technique. We demonstrate our proposal on several real-world high-dimensional datasets from machine learning. [ABSTRACT FROM AUTHOR]

We revisit graph grammar and graph parsing as tools for recognizing graphics. A top-down approach for parsing families of handwritten graphics containing different kinds of symbols and of structural relations is proposed. It has been tested on two distinct domains, namely the recognition of handwritten mathematical expressions and of handwritten flowcharts. In the proposed approach, a graphic is considered as a labeled graph generated by a graph grammar. The recognition problem is translated into a graph parsing problem: Given a set of strokes (input data), a parse tree which represents the best interpretation is extracted. The graph parsing algorithm generates multiple interpretations (consistent with the grammar) that can be ranked according to a global cost function that takes into account the likelihood of symbols and structures. The parsing algorithm consists in recursively partitioning the stroke set according to rules defined in the graph grammar. To constrain the number of partitions to be evaluated, we propose the use of a hypothesis graph, built from data-driven machine learning techniques, to encode the most likely symbol and relation hypotheses. Within this approach, it is easy to relax the stroke ordering constraint allowing interspersed symbols, as opposed to some previous works. Experiments show that our method obtains accuracy comparable to methods specifically developed to recognize domain-dependent data. [ABSTRACT FROM AUTHOR]

Abstract: The design of translation invariant and locally defined binary image operators over large windows is made difficult by decreased statistical precision and increased training time. We present a complete framework for the application of stacked design, a recently proposed technique to create two-stage operators that circumvents that difficulty. We propose a novel algorithm, based on Information Theory, to find groups of pixels that should be used together to predict the output value. We employ this algorithm to automate the process of creating a set of first-level operators that are later combined in a global operator. We also propose a principled way to guide this combination, by using feature selection and model comparison. Experimental results show that the proposed framework leads to better results than single stage design. [Copyright &y& Elsevier]

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 a training sample also suffer from overfitting. Large neighborhood tends to lead to performance degradation of the designed operator. This work proposes a multilevel design approach to deal with the issue of designing large neighborhood- based operators. The main idea is inspired by stacked generalization (a multilevel classifier design approach) and consists of, at each training level, combining the outcomes of the previous level operators. The final operator is a multilevel 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 outperform the single-level operators designed on the full window. They also show that iterating two-level operators is an effective multilevel approach to obtain better results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Morphological operators are nonlinear transformations commonly used in image processing. Their theoretical foundation is based on lattice theory, and it is a well-known result that a large class of image operators can be expressed in terms of two basic ones, the erosions and the dilations. In practice, useful operators can be built by combining these two operators, and the new operators can be further combined to implement more complex transformations. The possibility of implementing a compact combination that performs a complex transformation of images is particularly appealing in resource-constrained hardware scenarios. However, finding a proper combination may require a considerable trial-and-error effort. This difficulty has motivated the development of machine-learning-based approaches for designing morphological image operators. In this work, we present an overview of this topic, divided in three parts. First, we review and discuss the representation structure of morphological image operators. Then we address the problem of learning morphological image operators from data, and how representation manifests in the formulation of this problem as well as in the learned operators. In the last part we focus on recent morphological image operator learning methods that take advantage of deep-learning frameworks. We close with discussions and a list of prospective future research directions. [ABSTRACT FROM AUTHOR]