# Understanding multiclass log-loss

I'm trying to understand the multiclass log-loss as described in sk-learns documentation.

The wording ' Let the true labels (Y) for a set of samples be encoded as a 1-of-K binary indicator matrix...', implies to me that we have K different matrices, one for each class, the elements of which take values of 0 or 1 indicating the presence of that class (a kind of one-hot coding?).

Or is it that Y is a K by N matrix, N the number of samples? This makes far more sense to me, maybe I'm just misinterpreting the wording.

Can anyone point me to a more thorough explanation of the multi-class case? I can find plenty on the binary case, not so much multi-class.

Let the true labels for a set of samples be encoded as a $$1$$-of-$$K$$ binary indicator matrix $$Y$$, i.e., $$y_{i,k}=1$$ if sample $$i$$ has label $$k$$ taken from a set of $$K$$ labels.
This is standard matrix nomenclature: $$Y$$ is a matrix with $$N$$ rows and $$K$$ columns (not $$K$$ by $$N$$ as you suspect), with the $$i$$-th row encoding the class membership of the $$i$$-th sample with a single $$1$$ at position $$k$$.
Incidentally, you could test your understanding by creating a small toy example, say with $$N=2$$ and $$K=3$$, and feeding it to the log_loss() function. It should quickly become apparent that $$Y\in\mathbb{R}^{N\times K}$$, not $$Y\in\mathbb{R}^{K\times N}$$ or something even stranger.