In classical pattern recognition tasks, neural networks are trained to recognise human-defined patterns for model inputs. Some Explainable AI (XAI) methods can explain other latent patterns that underlie the network's recognition of inputs as human-defined patterns; in this work, we call these latent patterns second-order patterns, and we propose to discover them. To this end, we apply a hierarchical clustering algorithm to analyse whether representations learned by a speaker recognition network from utterances naturally form hierarchical clusters. Each resulting cluster represents a second-order pattern that characterises how the network recognises some known utterances as speaker identities. All the resulting second-order patterns are then semantically interpreted using the existing Hierarchical Cluster-Class Matching (HCCM) method. Furthermore, we propose a new task, second-order pattern recognition, to identify which discovered second-order patterns characterising known utterances are exhibited by an unseen utterance. To achieve this, we design the Hierarchical Cluster Navigation and Assignment (HCNA) method. HCNA recognises a known second-order pattern as applying to an unseen utterance when the unseen utterance's network representation lies within the extrapolation space of the cluster regarded as that second-order pattern. Our experiments show that the extrapolation mechanism introduced by HCNA substantially improves performance on the second-order pattern recognition task.

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