Global nonlinear Dimensionality Reduction (DR) methods excel at capturing complex features of datasets while preserving their overall high-dimensional structure when projecting them into a lower-dimensional space. Force-Scheme (FS) is one such method, used in a variety of domains. However, its use is still hindered by distortions and high computational cost. In this paper, we introduce Enhanced Force-Scheme (EFS), a revisited approach to solve the optimization problem posed by FS. We build on the core ideas of the original FS algorithm and introduce a more advanced optimization framework grounded in gradient-based optimization, which yields higher-quality layouts. Additionally, we elaborate on multiple strategies to accelerate the computation of projections using EFS, thereby facilitating its use on large datasets. Finally, we compare it with FS and other popular DR techniques and show that, among the methods tested, EFS best captures global structure while still performing well on local metrics. • Presentation of a new model (Enhanced Force-Scheme, EFS) that corrects major artifacts in Force-Scheme (FS) layouts by rethinking the way in which points are moved during the optimization. • Introduction of gradient descent concepts to obtain more reliable convergence and more detail in the resulting layouts. • Introduction of multiple strategies for scaling EFS and enable the projection of large datasets.

Abstract As Deep Learning models become increasingly complex, Explainable AI becomes essential for deploying machine learning classifiers. Decision Boundary Mapping (DBM) is a technique for visualizing a classifier's global decision boundary. Despite their relative success, current DBM methods rely on global inverse multidimensional projections that map 2D points back to the input space. The resulting visualizations depend heavily on the chosen projection, distort elements of the structure and distribution of the input data, and contain limited classifier‐specific information, often leading to inaccurate boundaries. We propose Sampling‐based Precise Neighbourhood Estimation (SPINE), a novel DBM approach for differentiable classifiers that is projection‐agnostic and does not depend on inverse projections. SPINE generates additional high‐dimensional points near the decision boundary using counterfactuals and variational autoencoders, then applies local interpolation to build visual representations. Our results show that this strategy better captures the structure and distribution of the data and the classifier's predictions in boundary regions than the state‐of‐the‐art in DBMs, yielding higher‐resolution and more accurate representations of classifier behaviour. Beyond standard DBM representations, SPINE enables additional exploratory analyses, such as counterfactual feature changes and path visualizations, supporting richer insights into model decisions.

Dimensionality Reduction (DR) methods have become essential tools for the data analysis toolbox. Typically, DR methods combine features of a multivariate dataset to produce dimensions in a reduced space, preserving some data properties, usually pairwise distances or local neighborhoods. Preserving such properties makes DR methods attractive, but it is also one of their weaknesses. When calculating the embedded dimensions, usually through non-linear strategies, the original feature values are lost and not explicitly represented in the spatialization of the produced layouts, making it challenging to interpret the results and understand the features’ contributions to the attained representations. Some strategies have been proposed to tackle this issue, such as coloring the DR layouts or generating explanations. Still, they are post-processes, so specific features (values) are not guaranteed to be preserved or represented. This paper proposes DimenFix , a novel meta-DR strategy that explicitly preserves the values of a particular user-defined feature or external data (not used to generate a layout) in one of the embedded axes. DimenFix can be used to preserve ordinal (e.g., numerical measures) and nominal (e.g., labels) values and works with virtually any gradient-descent DR method. It requires minimum changes to the underlying DR technique, running in linear time considering the number of data instances. In our results, involving Force Scheme and t-SNE adaptations, DimenFix was capable of representing features without heavily impacting distance or neighborhood preservation, allowing for creating hybrid layouts that join characteristics of scatter plots and DR methods.

Abstract In the vast landscape of visualization research, Dimensionality Reduction (DR) and graph analysis are two popular subfields, often essential to most visual data analytics setups. DR aims to create representations to support neighborhood and similarity analysis on complex, large datasets. Graph analysis focuses on identifying the salient topological properties and key actors within network data, with specialized research investigating how such features could be presented to users to ease the comprehension of the underlying structure. Although these two disciplines are typically regarded as disjoint subfields, we argue that both fields share strong similarities and synergies that can potentially benefit both. Therefore, this paper discusses and introduces a unifying framework to help bridge the gap between DR and graph (drawing) theory. Our goal is to use the strongly math‐grounded graph theory to improve the overall process of creating DR visual representations. We propose how to break the DR process into well‐defined stages, discuss how to match some of the DR state‐of‐the‐art techniques to this framework, and present ideas on how graph drawing, topology features, and some popular algorithms and strategies used in graph analysis can be employed to improve DR topology extraction, embedding generation, and result validation. We also discuss the challenges and identify opportunities for implementing and using our framework, opening directions for future visualization research.

Dimensionality reduction (DR) techniques help analysts to understand patterns in high-dimensional spaces. These techniques, often represented by scatter plots, are employed in diverse science domains and facilitate similarity analysis among clusters and data samples. For datasets containing many granularities or when analysis follows the information visualization mantra, hierarchical DR techniques are the most suitable approach since they present major structures beforehand and details on demand. This work presents HUMAP, a novel hierarchical dimensionality reduction technique designed to be flexible on preserving local and global structures and preserve the mental map throughout hierarchical exploration. We provide empirical evidence of our technique's superiority compared with current hierarchical approaches and show a case study applying HUMAP for dataset labelling.

Line-chart visualizations of temporal data enable users to identify interesting patterns for the user to inquire about. Using Intelligent Agents (IA), Visual Analytic tools can automatically uncover explicit knowledge related information to said patterns. Yet, visualizing the association of data, patterns, and knowledge is not straightforward. In this paper, we present ChatKG , a novel visual analytics strategy that allows exploratory data analysis of a Knowledge Graph that associates temporal sequences, the patterns found in each sequence, the temporal overlap between patterns, the related knowledge of each given pattern gathered from a multi-agent IA, and the IA’s suggestions of related datasets for further analysis visualized as annotations. We exemplify and informally evaluate ChatKG by analyzing the world’s life expectancy. For this, we implement an oracle that automatically extracts relevant or interesting patterns, populates the Knowledge Graph to be visualized, and, during user interaction, inquires the multi-agent IA for related information and suggests related datasets to be displayed as visual annotations. Our tests and an interview conducted showed that ChatKG is well suited for temporal analysis of temporal patterns and their related knowledge when applied to history studies. • Novel VA strategy for intelligent agent-assisted analysis of temporal data. • Association between explicit knowledge from an intelligent agent to temporal patterns. • Visualization of Knowledge Graphs with multi-modal data. • Analysis of life expectancy indicators contextualized by an intelligent agent.

Dimensionality Reduction (DR) scatterplot layouts have become a ubiquitous visualization tool for analyzing multidimensional datasets. Despite their popularity, such scatterplots suffer from occlusion, especially when informative glyphs are used to represent data instances, potentially obfuscating critical information for the analysis under execution. Different strategies have been devised to address this issue, either producing overlap-free layouts that lack the powerful capabilities of contemporary DR techniques in uncovering interesting data patterns or eliminating overlaps as a post-processing strategy. Despite the good results of post-processing techniques, most of the best methods typically expand or distort the scatterplot area, thus reducing glyphs' size (sometimes) to unreadable dimensions, defeating the purpose of removing overlaps. This article presents Distance Grid (DGrid), a novel post-processing strategy to remove overlaps from DR layouts that faithfully preserves the original layout's characteristics and bounds the minimum glyph sizes. We show that DGrid surpasses the state-of-the-art in overlap removal (through an extensive comparative evaluation considering multiple different metrics) while also being one of the fastest techniques, especially for large datasets. A user study with 51 participants also shows that DGrid is consistently ranked among the top techniques for preserving the original scatterplots' visual characteristics and the aesthetics of the final results.

Multivariate or multidimensional visualization plays an essential role in exploratory data analysis by allowing users to derive insights and formulate hypotheses. Despite their popularity, it is usually users' responsibility to (visually) discover the data patterns, which can be cumbersome and time-consuming. Visual Analytics (VA) and machine learning techniques can be instrumental in mitigating this problem by automatically discovering and representing such patterns. One example is the integration of classification models with (visual) interpretability strategies, where models are used as surrogates for data patterns so that understanding a model enables understanding the phenomenon represented by the data. Although useful and inspiring, the few proposed solutions are based on visual representations of so-called black-box models, so the interpretation of the patterns captured by the models is not straightforward, requiring mechanisms to transform them into human-understandable pieces of information. This paper presents multiVariate dAta eXplanation (VAX), a new VA method to support identifying and visual interpreting patterns in multivariate datasets. Unlike the existing similar approaches, VAX uses the concept of Jumping Emerging Patterns, inherent interpretable logic statements representing class-variable relationships (patterns) derived from random Decision Trees. The potential of VAX is shown through use cases employing two real-world datasets covering different scenarios where intricate patterns are discovered and represented, something challenging to be done using usual exploratory approaches.

Abstract Despite the tremendous advances in machine learning (ML), training with imbalanced data still poses challenges in many real‐world applications. Among a series of diverse techniques to solve this problem, sampling algorithms are regarded as an efficient solution. However, the problem is more fundamental, with many works emphasizing the importance of instance hardness. This issue refers to the significance of managing unsafe or potentially noisy instances that are more likely to be misclassified and serve as the root cause of poor classification performance. This paper introduces HardVis, a visual analytics system designed to handle instance hardness mainly in imbalanced classification scenarios. Our proposed system assists users in visually comparing different distributions of data types, selecting types of instances based on local characteristics that will later be affected by the active sampling method, and validating which suggestions from undersampling or oversampling techniques are beneficial for the ML model. Additionally, rather than uniformly undersampling/oversampling a specific class, we allow users to find and sample easy and difficult to classify training instances from all classes. Users can explore subsets of data from different perspectives to decide all those parameters, while HardVis keeps track of their steps and evaluates the model's predictive performance in a test set separately. The end result is a well‐balanced data set that boosts the predictive power of the ML model. The efficacy and effectiveness of HardVis are demonstrated with a hypothetical usage scenario and a use case. Finally, we also look at how useful our system is based on feedback we received from ML experts.

Transitive Halifax is an activity-oriented mobility service that allows users to search for bus routes toward places where they can perform their desired activities. The service is based on the observation that individuals often go to a place to conduct an activity. Simultaneously, the activity is often not strictly related to a single place since one may go shopping or eating in different locations. Transitive Halifax has a web interface that helps the user find the most relevant bus routes and bus stops candidates that they could use to go to places where they can perform their intended activity. The system implements a search engine that ranks the bus stops candidates according to the user's preferences and desired activities.

Over the past decades, classification models have proven to be essential machine learning tools given their potential and applicability in various domains. In these years, the north of the majority of the researchers had been to improve quantitative metrics, notwithstanding the lack of information about models' decisions such metrics convey. This paradigm has recently shifted, and strategies beyond tables and numbers to assist in interpreting models' decisions are increasing in importance. Part of this trend, visualization techniques have been extensively used to support classification models' interpretability, with a significant focus on rule-based models. Despite the advances, the existing approaches present limitations in terms of visual scalability, and the visualization of large and complex models, such as the ones produced by the Random Forest (RF) technique, remains a challenge. In this paper, we propose Explainable Matrix (ExMatrix), a novel visualization method for RF interpretability that can handle models with massive quantities of rules. It employs a simple yet powerful matrix-like visual metaphor, where rows are rules, columns are features, and cells are rules predicates, enabling the analysis of entire models and auditing classification results. ExMatrix applicability is confirmed via different examples, showing how it can be used in practice to promote RF models interpretability.

Dimensionality Reduction (DR) techniques are widely used to analyze and make sense of high-dimensional data. Each method is geared towards preserving a different aspect of the data. For example, some techniques favor neighborhood preservation whereas others favor distance preservation. While these DR techniques help users to represent their data, it makes a complex task to select a suitable DR. Also, most DR techniques have additional parameters that affect the results, which make the task of choosing a technique more difficult. Existing methods compare DR techniques using some quality metrics, and some of them combine DR outputs by averaging projections. However, it does not yet provide enough mechanisms to create a new DR according to user requirements. In this paper, we present a way to analyze and compare different DR techniques. It is an interactive assessment method that allows a user to explore known DR techniques, identify the differences between them, and create a new DR technique that combines existing techniques to match user expectations.

A music piece contains a large amount of information represented as a series of instructions corresponding to notes that must be played at specific times. These simple notes are combined to form complex harmonic structures that can be difficult to identify and analyze. Due to its simplicity and straightforward interpretation, music sheets and piano rolls have been the visual metaphor employed by most music visualization tools to support interpretation. Albeit it can represent all necessary elements to perform a music piece, these metaphors do not explicitly show many of the patterns and structures inherent to music arrangements, such as rhythm progression and harmonic interactions, needing users to create a mental model of them. Moreover, comparing different pieces and visualizing how a particular instrument track relates to the others is an issue not only for music sheet-based techniques, but also for most existing music visualization methods. In this paper, we present a novel visualization framework, called Music Overview, Stability, and Harmony Visualization (MoshViz), which facilitates the visualization and understanding of music renditions, focusing mainly on the visual analysis of specific musical instruments. Our approach creates a high-level model of music data and highlights structures of interest, enabling a detail+overview visualization to assist users in the task of identifying harmonic and melodic patterns. The usefulness and representativeness of MoshViz are confirmed by a set of user tests which demonstrate that the proposed visual metaphor matches, with a high degree of accuracy, the mental model of different users regarding the recognizable patterns of sounds.

Kernel-based methods have experienced a substantial progress in the last years, tuning out an essential mechanism for data classification, clustering and pattern recognition. The effectiveness of kernel-based techniques, though, depends largely on the capability of the underlying kernel to properly embed data in the feature space associated to the kernel. However, visualizing how a kernel embeds the data in a feature space is not so straightforward, as the embedding map and the feature space are implicitly defined by the kernel. In this work, we present a novel technique to visualize the action of a kernel, that is, how the kernel embeds data into a high-dimensional feature space. The proposed methodology relies on a solid mathematical formulation to map kernelized data onto a visual space. Our approach is faster and more accurate than most existing methods while still allowing interactive manipulation of the projection layout, a game-changing trait that other kernel-based projection techniques do not have.

Space-filling techniques seek to use as much as possible the visual space to represent a dataset, splitting it into regions that represent the data elements. Amongst those techniques, Treemaps have received wide attention due to its simplicity, reduced visual complexity, and compact use of the available space. Several different Treemap algorithms have been proposed, however the core idea is the same, to divide the visual space into rectangles with areas proportional to some data attribute or weight. Although pleasant layouts can be effectively produced by the existing techniques, most of them do not take into account relationships that might exist between different data elements when partitioning the visual space. This violates the distance-similarity metaphor, that is, close rectangles do not necessarily represent similar data elements. In this paper, we propose a novel approach, called Neighborhood Treemap (Nmap), that seeks to solve this limitation by employing a slice and scale strategy where the visual space is successively bisected on the horizontal or vertical directions and the bisections are scaled until one rectangle is defined per data element. Compared to the current techniques with the same similarity preservation goal, our approach presents the best results while being two to three orders of magnitude faster. The usefulness of Nmap is shown by two applications involving the organization of document collections and the construction of cartograms illustrating its effectiveness on different scenarios.

Abstract Interactive visualization systems for exploring and manipulating high‐dimensional feature spaces have experienced a substantial progress in the last few years. State‐of‐art methods rely on solid mathematical and computational foundations that enable sophisticated and flexible interactive tools. Current methods are even capable of modifying data attributes during interaction, highlighting regions of potential interest in the feature space, and building visualizations that bring out the relevance of attributes. However, those methodologies rely on complex and non‐intuitive interfaces that hamper the free handling of the feature spaces. Moreover, visualizing how neighborhood structures are affected during the space manipulation is also an issue for existing methods. This paper presents a novel visualization‐assisted methodology for interacting and transforming data attributes embedded in feature spaces. The proposed approach relies on a combination of multidimensional projections and local transformations to provide an interactive mechanism for modifying attributes. Besides enabling a simple and intuitive visual layout, our approach allows the user to easily observe the changes in neighborhood structures during interaction. The usefulness of our methodology is shown in an application geared to image retrieval.

Abstract Word clouds have become one of the most widely accepted visual resources for document analysis and visualization, motivating the development of several methods for building layouts of keywords extracted from textual data. Existing methods are effective to demonstrate content, but are not capable of preserving semantic relationships among keywords while still linking the word cloud to the underlying document groups that generated them. Such representation is highly desirable for exploratory analysis of document collections. In this paper we present a novel approach to build document clouds, named ProjCloud that aim at solving both semantical layouts and linking with document sets. ProjCloud generates a semantically consistent layout from a set of documents. Through a multidimensional projection, it is possible to visualize the neighborhood relationship between highly related documents and their corresponding word clouds simultaneously. Additionally, we propose a new algorithm for building word clouds inside polygons, which employs spectral sorting to maintain the semantic relationship among words. The effectiveness and flexibility of our methodology is confirmed when comparisons are made to existing methods. The technique automatically constructs projection based layouts the user may choose to examine in the form of the point clouds or corresponding word clouds, allowing a high degree of control over the exploratory process.

Abstract Fiber tracts detection is an increasingly common technology for diagnosis and also understanding of brain function. Although tools for tracing and presenting brain fibers are advanced, it is still difficult for physicians or students to explore the dataset in 3D due to their intricate topology. In this work we present a visual exploration approach for fiber tracts data aimed at supporting exploration of such data. The work employs a local, precise and fast 2D multidimensional projection technique that allows a large number of fibers to be handled simultaneously and to select groups of bundled fibers for further exploration. In this approach, a DTI feature dataset, including curvature as well as spatial features, is projected on a 2D or 3D view. By handling groups formed in this view, exploration is linked to corresponding brain fibers in object space. The link exists in both directions and fibers selected in object space are also mapped to feature space. Our approach also allows users to modify the projection, controlling and improving, if necessary, the definition of groups of fibers for small and large datasets, due to the local nature of the projection. Compared to other related work, the method presented here is faster for creating visual representations, making it possible to explore complete sets of fibers tracts up to 250K fibers, which was not possible previously. Additionally, the ability to change configuration of the feature space representation adds a high degree of flexibility to the process.

Abstract Multidimensional projection has emerged as an important visualization tool in applications involving the visual analysis of high‐dimensional data. However, high precision projection methods are either computationally expensive or not flexible enough to enable feedback from user interaction into the projection process. A built‐in mechanism that dynamically adapts the projection based on direct user intervention would make the technique more useful for a larger range of applications and data sets. In this paper we propose the Piecewise Laplacian‐based Projection (PLP), a novel multidimensional projection technique, that, due to the local nature of its formulation, enables a versatile mechanism to interact with projected data and to allow interactive changes to alter the projection map dynamically, a capability unique of this technique. We exploit the flexibility provided by PLP in two interactive projection‐based applications, one designed to organize pictures visually and another to build music playlists. These applications illustrate the usefulness of PLP in handling high‐dimensional data in a flexible and highly visual way. We also compare PLP with the currently most promising projections in terms of precision and speed, showing that it performs very well also according to these quality criteria.

Abstract Visualization of high‐dimensional data requires a mapping to a visual space. Whenever the goal is to preserve similarity relations a frequent strategy is to use 2D projections, which afford intuitive interactive exploration, e.g., by users locating and selecting groups and gradually drilling down to individual objects. In this paper, we propose a framework for projecting high‐dimensional data to 3D visual spaces, based on a generalization of the Least‐Square Projection (LSP). We compare projections to 2D and 3D visual spaces both quantitatively and through a user study considering certain exploration tasks. The quantitative analysis confirms that 3D projections outperform 2D projections in terms of precision. The user study indicates that certain tasks can be more reliably and confidently answered with 3D projections. Nonetheless, as 3D projections are displayed on 2D screens, interaction is more difficult. Therefore, we incorporate suitable interaction functionalities into a framework that supports 3D transformations, predefined optimal 2D views, coordinated 2D and 3D views, and hierarchical 3D cluster definition and exploration. For visually encoding data clusters in a 3D setup, we employ color coding of projected data points as well as four types of surface renderings. A second user study evaluates the suitability of these visual encodings. Several examples illustrate the framework's applicability for both visual exploration of multidimensional abstract (non‐spatial) data as well as the feature space of multi‐variate spatial data.

In this paper, we present a novel approach for constructing bundled layouts of general graphs. As layout cues for bundles, we use medial axes, or skeletons, of edges which are similar in terms of position information. We combine edge clustering, distance fields, and 2D skeletonization to construct progressively bundled layouts for general graphs by iteratively attracting edges towards the centerlines of level sets of their distance fields. Apart from clustering, our entire pipeline is image-based with an efficient implementation in graphics hardware. Besides speed and implementation simplicity, our method allows explicit control of the emphasis on structure of the bundled layout, i.e. the creation of strongly branching (organic-like) or smooth bundles. We demonstrate our method on several large real-world graphs.

Multidimensional projection techniques have experienced many improvements lately, mainly regarding computational times and accuracy. However, existing methods do not yet provide flexible enough mechanisms for visualization-oriented fully interactive applications. This work presents a new multidimensional projection technique designed to be more flexible and versatile than other methods. This novel approach, called Local Affine Multidimensional Projection (LAMP), relies on orthogonal mapping theory to build accurate local transformations that can be dynamically modified according to user knowledge. The accuracy, flexibility and computational efficiency of LAMP is confirmed by a comprehensive set of comparisons. LAMP's versatility is exploited in an application which seeks to correlate data that, in principle, has no connection as well as in visual exploration of textual documents.

Most multidimensional projection techniques rely on distance (dissimilarity) information between data instances to embed high-dimensional data into a visual space. When data are endowed with Cartesian coordinates, an extra computational effort is necessary to compute the needed distances, making multidimensional projection prohibitive in applications dealing with interactivity and massive data. The novel multidimensional projection technique proposed in this work, called Part-Linear Multidimensional Projection (PLMP), has been tailored to handle multivariate data represented in Cartesian high-dimensional spaces, requiring only distance information between pairs of representative samples. This characteristic renders PLMP faster than previous methods when processing large data sets while still being competitive in terms of precision. Moreover, knowing the range of variation for data instances in the high-dimensional space, we can make PLMP a truly streaming data projection technique, a trait absent in previous methods.

Document interpretation is a crucial task in many visual analytics applications, made harder by the widespread availability of freely available textual files. In this paper we propose an approach based on topic detection coupled with multiple coordinated views to assist analysis of time varying document collections. Given multiple document maps built from a set of text files, we define a strategy to support users locating the evolution of topics addressed by the documents, along various time steps. The approach is supported by a new algorithm for topic extraction from texts, also introduced. Finally, we show several examples illustrating how the proposed strategy may be applied in the analysis of document collections.

Time series analysis poses many challenges to professionals in a wide range of domains. Several visualization solutions have been proposed for exploratory tasks on time series collections. For large data sets, however, current techniques fail to provide a global view that supports a good association between groups of similar time series. We employ fast multidimensional projection techniques to create concise visual representations of a collection of time series. The whole collection can be viewed in a two-dimensional graph-based representation that provides a starting point for further exploration and detailed analysis. The projections employ distance metrics to compare the series and generate a layout that attempts to group those with similar behavior. We illustrate the approach on a real data set containing streamflows describing the behavior of hydroelectric power plants in Brazil.

Multiple Views have been put forward as an alternative to assist exploration of evolving phenomena or associations between distinct data sets or distinct presentations of a single data set. Coordinating between views is a challenge that must be met to improve visualization support for exploratory tasks. This is particularly true for high-dimensional data, such as document collections. We introduce a coordination framework for multiple views of document collections created using projections and point placement visualizations. Coordination can occur between different views of a single data set or between views of multiple data sets. Multiple coordinations are also admitted. Three new types of coordination are presented to illustrate the framework; these have been implemented in a multipurpose multi-dimensional visualization system called PEx (Projection Explorer).

The efficacy of search engines has expanded the uses for the information available on the Web. An increasing number of applications make use of the WWW as a primary source of information. The usefulness of such applications is, however, impaired by the current styles of display of the Web search results. This paper presents a system that adapts two techniques to map and explore Web results visually in order to find relevant patterns and relationships amongst the resulting documents. The first technique creates a visual map of the search results using a content-based multidimensional projection. The second techniques is capable of identifying, labeling and displaying topics within sub-groups of documents on the map. The system (The Projection explorer for the WWW, or PEx-Web) implements these techniques and various additional tools as means to make better use of Web search results for exploratory applications.

Point placement strategies aim at mapping data points represented in higher dimensions to bi-dimensional spaces and are frequently used to visualize relationships amongst data instances.They have been valuable tools for analysis and exploration of datasets of various kinds. Many conventional techniques, however, do not behave well when the number of dimensions is high, such as in the case of documents collections. Later approaches handle that shortcoming, but may cause too much clutter to allow flexible exploration to take place. In this work we present a novel hierarchical point placement technique that is capable of dealing with these problems. While good grouping and separation of data with high similarity is maintained without increasing computation cost,its hierarchical structure lends itself both to exploration in various levels of detail and to handling data in subsets, improving analysis capability and also allowing manipulation of larger data sets.

The problem of projecting multidimensional data into lower dimensions has been pursued by many researchers due to its potential application to data analysis of various kinds. This paper presents a novel multidimensional projection technique based on least square approximations. The approximations compute the coordinates of a set of projected points based on the coordinates of a reduced number of control points with defined geometry. We name the technique Least Square Projections (LSP). From an initial projection of the control points, LSP defines the positioning of their neighboring points through a numerical solution that aims at preserving a similarity relationship between the points given by a metric in mD. In order to perform the projection, a small number of distance calculations is necessary and no repositioning of the points is required to obtain a final solution with satisfactory precision. The results show the capability of the technique to form groups of points by degree of similarity in 2D. We illustrate that capability through its application to mapping collections of textual documents from varied sources, a strategic yet difficult application. LSP is faster and more accurate than other existing high quality methods, particularly where it was mostly tested, that is, for mapping text sets.

This paper presents a tool, called text map explorer, which can be used to create and explore document maps (visual representations of document collections). This tool is capable of grouping (and separating) documents by their contents, revealing to the user relationships amongst them. This paper also presents a novel multi-dimensional projection technique for text that reduces the quadratic time complexity of our previous approach to O(N3/2), keeping the same quality of maps. The technique creates a surface that reveals intrinsic patterns and supports various kinds of exploration of a text collection

This paper introduces Least Square Projection (LSP), a fast technique for projection of multi-dimensional data onto lower dimensions developed and tested successfully in the context of creation of text maps based on their content. Current solutions are either based on computationally expensive dimension reduction with no proper guarantee of the outcome or on faster techniques that need some sort of post-processing for recovering information lost during the process. LSP is based on least square approximation, a technique originally employed for surface modeling and reconstruction. Least square approximations are capable of computing the coordinates of a set of projected points based on a reduced number of control points with defined geometry. We extend the concept for general data sets. In order to perform the projection, a small number of distance calculations is necessary and no repositioning of the final points is required to obtain a satisfactory precision of the final solution. Textual information is a typically difficult data type to handle, due to its intrinsic dimensionality. We employ document corpora as a benchmark to demonstrate the capabilities of the LSP to group and separate documents by their content with high precision.