How would one solve length generalization with position embeddings?

L ength general ization is the problem of representing the relative position of an element in a sequence , regardless of its absolute position . One way to solve length general ization with position embed d ings is to use a technique called " pos itional encoding ". \n \n Pos itional encoding is a way to add information about the position of an element in a sequence to its embedding . There are several ways to implement position al encoding , but one common approach is to use sin us oid al functions . These functions encode the position of an element in a sequence using a combination of s ine and cos ine waves , with different frequencies for different positions . \n \n For example , consider a sequence of integers [ 1 , 2 , 3 , 4 , 5 ]. To represent the position of the element 3 in this sequence using a position embedding , we could use the following sin us oid al function : \n \n pe ( i ) = sin ( 2 \ * pi \ * ( i - 1 ) / ( 2 \ * seq \_ len )) \n \n where seq \_ len is the length of the sequence , and i is the position of the element in the sequence . This function enc odes the position of the element using a s ine wave with a frequency of 1 / seq \_ len , and sh ifts the wave by 2 \ * pi \ * ( i - 1 ) / ( 2 \ * seq \_ len ) rad ians to account for the position of the element in the sequence . \n \n By using position al encoding in this way , we can represent the relative position of an element in a sequence , regardless of its absolute position . This can help improve the performance of models that use position al information , such as language models and sequence - to - sequence models . </s>