In zero-shot learning, the model observes test samples from classes that were not observed during training, and needs to predict the category they belong to. Formally, the training data has a label space \(Y\) and the testing data has a different label space \(Y^{\prime}\), where \(Y \cap Y^{\prime}=\emptyset\). Given training data \(\left\{\left(\mathbf{x}_{i}, y_{i}ight) \mid \mathbf{x}_{i} \in \mathbb{R}^{n}, i=1, \ldots, Night\}\), zero-shot learning aims to learn a function \(f: \mathbb{R}^{n} ightarrow Y \cup Y^{\prime}\). Suppose for each label \(y \in Y \cup Y^{\prime}\), a label representation vector \(\mathbf{y}_{i} \in \mathbb{R}^{m}\) is also given.

a. Suppose we aim to learn a mapping function \(g: \mathbb{R}^{n} ightarrow \mathbb{R}^{m}\) (e.g., \(g(\mathbf{x})=\mathbf{W x}\) ) from the training data so that \(\mathbf{y}_{i}\) is close to \(g\left(\mathbf{x}_{i}ight)(i=1, \ldots, N)\). Use this idea to design a zero-shot learning algorithm.
b. Suppose we aim to learn a scoring function \(g: \mathbb{R}^{n+m} ightarrow \mathbb{R}\) (e.g., \(g(\mathbf{x}, \mathbf{y})=\mathbf{x}^{T} \mathbf{W y}\) ) from the training data so that \(g\left(\mathbf{x}_{i}, \mathbf{y}_{i}ight)\) is large \((i=1, \ldots, N)\). Use this idea to design a zero-shot learning algorithm.