# Hierarchical Concept Learning by Fuzzy Semantic Cells
## 3 Method
### Discussion  
In this part we discuss the similarity and differences between our proposed method and related supervised learning methods. The discussion is from four aspects: Prototype classification, hard/soft margin, limited samples and unsupervised learning.  
**Prototype classification**. Prototype classification has been a thriving area in artificial intelligence. Mean-of-class prototype classification is a typical method to learn prototypes based on which classification is done [28]. There are three kinds of mean-of-class prototype classification methods. Non-margin classifiers, such as linear discriminant analysis (LDA) [29; 30], form non-overlapping areas to represent concepts. Such methods use a similar strategy as our method to construct concepts. Other advanced works [31; 32] introduce deep models to extract features for learning prototypes. However, one concept may have multiple prototypes and a non-margin classifier assumes only one prototype, while our method assumes multiple prototypes intrinsically.  
**Hard/soft margin**. Hard margin classifiers include Support Vector Machine [21] and similar methods. They assume a hard hyper-plane or hyper-sphere to separate concepts. Such assumption does not consider the vagueness and uncertainty among samples, while our fuzzy semantic cells assume vagueness with an uncertain margin. Soft margin classifiers obtain prototypes by applying a regularized preprocessing and then classifying samples using hard margin classifiers [21]. This kind of methods also fail to consider multiple prototypes. Another typical soft margin classifier is mixture classification models [33; 34]. The main idea is to fit Gaussian mixture models on samples in each class respectively and then to classify new samples with linear discriminant analysis. The fitting process is similar to our prototype learning procedure. However, our method further considers the exclusion among concepts with conditional entropy.  
**Limited samples**. How to learn concepts with limited data is also a fundamental topic. As the number of samples is limited, prior knowledge [35], strong assumptions [36], appropriate augmentation [37; 38] or transferred knowledge [39] is important. They can serve as an important inductive bias for out-of-distribution samples or regularize the model to avoid potential overfitting. The virtue that learning concepts from data can prevent adversarial attacks is also discussed [40]. Our method is based on the assumption of prototype theory [11] and the hierarchical structure of prototypes and concepts [24; 25]. These two assumptions are from the perspective of how humans form concepts. Given the no-free lunch theorem, there is no universally applicable assumption. However, as our motivation is to model concepts, our assumptions of hierarchical prototypes are instructive.  
**Unsupervised learning**. Prototypes and concepts learning has also been an important topic in unsupervised learning. With the assumption of density peaks [41], message propagation [42], mixture of distributions [43] or disjunctive combination [44], the underlying concepts can also be learned. A recent work [45] also extracts deep features for clustering. The motivation of the introduced unsupervised learning methods is also to learn concepts from samples, but the learning procedures are not supervised by labels. In summary, how to learn fuzzy concepts unsupervisedly with a reasonable assumption is an important topic in our future work.