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Linear compatibility functions can be part of structured prediction work, where a machine learning program (or similar technology) attempts to solve for an identity in a classification problem by examining training inputs.
This sort of construct makes sense within the general framework of the neural network model that is innovating artificial intelligence at a fast clip.
Linear compatibility functions may be useful in joint feature representation of input/output pairs where the system encodes combined properties of these inputs and outputs in order to achieve structured production task. The system may predict a most compatible result for a given input or set of inputs.
These types of algorithms and mathematical constructs can be applied to parse trees or decision trees or other models, in order to come up with structured prediction outcomes in supervised machine learning, where typically, labels help the program to achieve the identification result.
Many experts talk about how supervised ML is generally more easily implemented than unsupervised ML; the utility of labels seems particularly clear in application to linear compatibility functions and other aspects of structured prediction.