The three-way-in and three-way-out framework to treat and exploit ambiguity in data

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Abstract

In this paper, we address ambiguity, intended as a characteristic of any data expression for which a unique meaning cannot be associated by the computational agent for either lack of information or multiple interpretations of the same configuration. In particular, we will propose and discuss ways in which a decision-support classifier can accept ambiguous data and make some (informative) value out of them for the decision maker. Towards this goal we propose a set of learning algorithms within what we call the three-way-in and three-way-out approach, that is, respectively, learning from partially labeled data and learning classifiers that can abstain. This approach is based on orthopartitions, as a common representation framework, and on three-way decisions and evidence theory, as tools to enable uncertain and approximate reasoning and inference. For both the above learning settings, we provide experimental results and comparisons with standard Machine Learning techniques, and show the advantages and promising performances of the proposed approaches on a collection of benchmarks, including a real-world medical dataset.

Section snippets

Introduction and related works

Recently Machine Learning (ML) has continuously attracted the interest of the research community, both from a mathematical-theoretical point of view and, more predominantly, from an application point of view. This interest has been stimulated by the fact that different research communities (e.g. health-care and medicine, finance and economics, …) have acknowledged the ubiquity of uncertainty, in different forms, e.g. vagueness, randomness, ambiguity, as an intrinsic part of their practice [14],

Basic notions

In this section, we give the mathematical background on decision tables, orthopairs and orthopartions that will be used in the following.

Definition 2.1

A multi–observer decision table is a tuple U,A,t,D where

  • U is a universe of objects of interest;

  • A is a set of attributes (or features) that we use to represent objects in U. In particular, we define each attribute as a function a:UVa where Va is the domain of values that the attribute a can assume;

  • tAD is a distinguished decision attribute, that we assume

Three–way output

In this section, we will describe in greater detail the Three-way Out (TWO) learning setting. In this context, the chosen data representation, i.e., the selected features and/or their level of granularity, is such that a form of c-ambiguity arises. Thus, the chosen data representation does not allow us to distinguish different objects that are either identical or “too near” in the sample space, similarly to the concept of indiscernibility in standard Pawlak's rough sets [36] or generalized

Three–way input

While Three–Way Output denotes a single phenomenon, i.e., a classifier emitting a set–valued classification, with Three–Way Input (TWI) we denote two different phenomena leading to a form of r-ambiguity, i.e., the presence of set–valued values in the training set (which is the input to the learning process):

  • 1.

    The target attribute d is set–valued, this setting is also called learning from imprecise/partial labels [9], [25]: a set-valued classification can be seen as a partial abstention of the

Experimental validation and discussion

In order to assess the validity and efficacy of the proposed algorithms, in both the Three-way Out and Three-way In learning settings, we performed two sets of experimental validations: in Section 5.1 we report the experimental setting and obtained results in the Three-way Out case, while in Section 5.2 we report the same information for the Three-way In case.

Conclusions

In this article, we studied the ambiguity occurring in Machine Learning, from a twofold perspective: both as a problem affecting the input of the learning process, and as a potential resource to make the output of classifiers apter for sound human decision making.

In particular, we presented techniques to represent and manage this type of uncertainty in the training data that is fed into the learning algorithm, what we called Three-way In), and also techniques to represent ambiguity and

Declaration of Competing Interest

The authors confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

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