There are plenty of results for muli-class learning with with fixed discrete labels:$$\text{Standard multi-class classification:} \begin{cases}f: X \rightarrow Y_{index} = \{1, 2, 3, ..., k \}, \\ s.t. X \subset \mathbb{R}^{d_x}\end{cases}$$I was wondering if there are any results which study multi-class learning with (countable, possibly finite) vector-valued outputs. $$\text{Vector-output multi-class classification:}  \begin{cases}f: X \rightarrow Y_{vector} = \{y_1, y_2, y_3, ..., y_k \}, \\ s.t. X \subset \mathbb{R}^{d_x}, y_i \in  \mathbb{R}^{d_y}\end{cases}$$One may suggest that this is another problem: metric-learning between $X$ and $Y$ space:$$\text{Standard metric-learning problem:}  \begin{cases}f: X, Y \rightarrow [0,1], \\ s.t. X \subset \mathbb{R}^{d_x}, Y \subset  \mathbb{R}^{d_y}\end{cases}$$However all the results I came across in the metric-learning study the functions on one single vector space (as in unsupervised metric learning). In addition, vector-output multi-class classification is much more restricted (easier to learn?) than metric learning, since in what I am asking, the number of possible output vectors is fixed (unlike metric learning). 
Generalization bounds for multiclass learning when the output is vector space?
machine learning