Grouping high dimension Y-space to lower dimensions

I have a ML problem with 300 variables to predict, Meaning I have a multi-label binary classification problem with a Y space of 300 and with about 2000 rows only.
Thus we assume that there are 2^300 permutations. However, it is not so.

Let me explain with an example.
X1,….X400 | Y1,Y2,Y3………..Y299,Y300
‘healthy’,..’sad’| 0,1..1,0
‘unhealthy’…’sad’|1,0,….0

However when i aggregate across the rows,there are maybe about 100 combinations far less than 2^300 combinations such as this.
so Y1,Y2,Y100 => C1 OR Y33,Y44,Y291,Y299,Y300 =>C2 … till C100
Thus, I can change it to a multi-class problem to predict these 100 classes instead.
However, there are a lot of classes with count of 1 or 2. More than 50%. Thus, this approach doesnt work so well.

Is there an algorithmic way to still do the multi-label but by grouping the variables in the most optimal way like from Y space to Z space with dimension 5 for example.
{Y1,Y10,Y222,Y232}=> Z1 can be grouped together
{Y2…} => Z2 can be grouped together
Thus,we can predict on the Z space instead. At this point, I am not sure whether the Y variables need to be exclusive to one Z group or not.

But I am open to all suggestions at this moment ..

Cheers