Interactively visualizing pattern-based summaries of fuzzy tensors

Many datasets are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> -way fuzzy tensors, that is, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> -dimensional tables where every cell grades in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:math> the truth of a statement instantiated by the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> elements indexing the cell. In the special case where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , fuzzy matrices encode to what extent objects (the rows) have attributes (the columns). Boolean tensors are another special case, in which the statement is always either false ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ) or true ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ). Theoretically simple pattern-based models exist for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> -way Boolean or fuzzy tensors, and so do algorithms to discover sets of patterns well summarizing the tensors, according to these models. However, intuitively understanding such summaries has remained difficult. Tools specifically assisting data scientists in that endeavor are lacking. This article addresses that issue. It proposes the first interactive visualization for pattern-based summaries of fuzzy tensors. Pieces of information that are relevant to the interpretation of a summary are extracted from the study of the disjunctive box cluster model. They are turned into visual objects, attributes or interaction techniques. Special attention is given to allowing the exploration of summaries with many large patterns, as shown with the analysis of 55 patterns involving thousands of elements and summarizing a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mn>12</mml:mn> <mml:mo>×</mml:mo> <mml:mn>170</mml:mn> <mml:mspace width="0.25em"/> <mml:mn>670</mml:mn> <mml:mo>×</mml:mo> <mml:mn>29</mml:mn> </mml:mrow> </mml:math> real-world tensor.