I have softmax as follows: $$\sigma(\mathbf{z})_i = \frac{e^{z_i}}{\sum_{j=1}^K e^{z_j}} \text{ for } i = 1, \dotsc , K \text{ and } \mathbf z=(z_1,\dotsc,z_K) \in\mathbb{R}^K$$

As borrowed from Wikipedia.

I have a tensor of dimension $100 \times 100 \times 256$, let's call this $\mathcal{X}$. I have a variable that I have to denote just as $\mathbf{x}_{i,j} \in \mathcal{X}$ - I can't use triple indices. I am doing a channel-wise softmax using python. However, I'd also like to write this mathematically but unsure how to represent this along the depth dimension. Here's my attempt at it:

$$\sigma(\mathbf{x}_{i,j}) = \frac{e^{\mathbf{x}_{i,j,k'}}}{\sum_{k=1}^{256}{e^{\mathbf{x}_{i,j,k}}}}$$ where $1 \leq k' \leq 256$.

Is this proper notation or does it not have the intended understanding?



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