A simple choice of non-sparse gating function BIBREF17 is to multiply the input by a trainable weight matrix INLINEFORM0 and then apply the INLINEFORM1 function. DISPLAYFORM0

We add two components to the Softmax gating network: sparsity and noise. Before taking the softmax function, we add tunable Gaussian noise, then keep only the top k values, setting the rest to INLINEFORM0 (which causes the corresponding gate values to equal 0). The sparsity serves to save computation, as described above. While this form of sparsity creates some theoretically scary discontinuities in the output of gating function, we have not yet observed this to be a problem in practice. The noise term helps with load balancing, as will be discussed in Appendix SECREF51 . The amount of noise per component is controlled by a second trainable weight matrix INLINEFORM1 . DISPLAYFORM0 DISPLAYFORM1 A simple choice of non-sparse gating function BIBREF17 is to multiply the input by a trainable weight matrix INLINEFORM0 and then apply the INLINEFORM1 function. DISPLAYFORM0

We add two components to the Softmax gating network: sparsity and noise. Before taking the softmax function, we add tunable Gaussian noise, then keep only the top k values, setting the rest to INLINEFORM0 (which causes the corresponding gate values to equal 0). The sparsity serves to save computation, as described above. While this form of sparsity creates some theoretically scary discontinuities in the output of gating function, we have not yet observed this to be a problem in practice. The noise term helps with load balancing, as will be discussed in Appendix SECREF51 . The amount of noise per component is controlled by a second trainable weight matrix INLINEFORM1 . DISPLAYFORM0 DISPLAYFORM1 
 Question: What equations are used for the trainable gating network?
DISPLAYFORM0 DISPLAYFORM0 DISPLAYFORM1