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:
$$str(f) = \sum_{n\in N}\frac{cos(E_n, E_f)}{|N|} - \sum_{n\notin N}^{k}\frac{cos(E_n, E_f)}{|k|}$$

Learning Frame Embeddings

**CM-Lex** is an approach for metaphor generation that learns joint embedded representations for domains and lexical items. It uses the linear transformation between two domains as a mapping. This process relies on:

- **Frames (Fillmore, 1982):** Related systems of concepts, which are exchangeable with the term "domain" used in conceptual metaphor theory.
- **FrameNet (Baker et al., 1998):** Semantic frames along with the set of lexical units that evoke them.

Thus, CM-Lex considers the transformation from one frame to another as a proxy for a conceptual metaphoric mapping.

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Conceptual metaphor theory (CMT): uses conceptual mappings between domains (conceptual structures that group related concepts) to generate linguistic metaphors.

Framework: Given a literal input sentence --> evokes a target domain --> generates metaphoric sentences that evoke desired corresponding source domain.

Eg. Three inputs: 
1. a literal input sentence (They argued against the contract)
2. the target domain evoked by the literal input (ARGUMENT)
3. the desired source domain (WAR) for the metaphorical sentence

The output is a metaphorical sentence that evokes the desired corresponding source domain (They fought against the contract).

How? Using an intended conceptual mapping (ARGUMENT - WAR)

Introduction (Metaphor Generation with Conceptual Mappings)

To automatically evaluate metaphor generation models using Sentence-BERT (SBERT), sentence embeddings are generated for:

1. The literal input $$L$$
2. The original gold metaphoric expression $$M$$
3. The generated output $$G$$

Using these embeddings, the following metrics are calculated:
- **Distance from Gold Metaphor (dis)**: $$\cos(M, G)$$
- **Relational distance (rel)**: $$\cos(L, M) - \cos(L, G)$$
- **Exact Match**: The percentage of times the model produced the exact gold output.

SBERT-based Evaluation Metrics for Metaphor Generation

CM-BART is a metaphor generation model that relies on parallel training data. Creating this data involves three steps: 1. Identify and mask metaphoric verbs using a BERT-based metaphor classification model (e.g., masking "died" in "The house where love had died"), and replace them with infillings from a pre-trained BERT language model. 2. Filter for quality by ensuring that at least 4 out of 5 symbolic beams returned by COMET (Bosselut et al., 2019) are identical. 3. Tag each sentence with FrameNet frames, assuming that the relationship between the frames for the source and target domains reflects a metaphoric mapping.

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