An embedding mapping $$m$$ is defined as the pointwise distance between a target frame embedding and a source frame embedding. A conceptual mapping is then calculated by adding the embedding of the target verb to the mapping $$m$$, and selecting the most similar word to the resulting vector.

Benefits of this approach include relying only on embeddings of input words and frames, and requiring no labeled metaphor data (though FrameNet-tagged corpora are still 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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