An embedding mapping $$m$$ is defined as the pointwise distance between a target frame embedding and a source frame embedding. A conceptual mapping is calculated by adding the target verb's embedding to $$m$$ and selecting the word most similar to the resulting vector. This approach relies only on embeddings of input words and frames, requiring no labeled metaphor data (though FrameNet-tagged corpora are needed). A drawback is that it operates without contextual information.

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To learn frame embeddings, replace a focus verb in the training data with its corresponding FrameNet frame label (either provided in gold-standard data or tagged via a semantic parser). Then, train word embedding models (e.g., word2vec skip-gram, GloVe, and FastText) on this modified data. This process yields a joint embedding space containing both common word embeddings and FrameNet frame embeddings.

The quality of these frame embeddings can be evaluated using two metrics:

1. Lexical Metric (lex): Evaluates whether a frame embedding is similar to words within its frame and dissimilar to those outside it:
$$lex(f) = \sum_{w\in f}\frac{cos(E_w, E_f)}{|f|} - \sum_{w\notin f}^{k}\frac{cos(E_w, E_f)}{|k|}$$

2. Structural Similarity Metric (str): Evaluates whether frame embeddings reflect the hierarchy of connected frames in FrameNet by checking similarity around their neighbors:
$$str(f) = \sum_{n\in N}\frac{cos(E_n, E_f)}{|N|} - \sum_{n\notin N}^{k}\frac{cos(E_n, E_f)}{|k|}$$

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