Element-wise matrix multiplication (Hadamard product) and traditional matrix multiplication serve different purposes. Hadamard product multiplies corresponding elements of two matrices, while traditional matrix multiplication computes the dot product of rows and columns.
Trade-offs:
1. Computational complexity: Traditional matrix multiplication has higher complexity (O(n^3)) compared to element-wise multiplication (O(n^2)).
2. Matrix dimensions: Element-wise requires equal dimensions, whereas traditional allows for different inner dimensions (A[m x n] * B[n x p]).
3. Resultant matrix size: Element-wise results in a matrix of the same size as input matrices, while traditional produces a matrix with outer dimensions (A[m x n] * B[n x p] = C[m x p]).
Scenarios favoring element-wise multiplication:
– When performing element-wise operations like scaling or masking.
– When both matrices have the same dimensions and no need for linear transformation.
Scenarios favoring traditional multiplication:
– When transforming data between spaces (e.g., rotation, scaling).
– When combining features from multiple sources (e.g., neural networks).
