Null Space
In linear algebra, the null space of a matrix is the set of all vectors that the matrix maps to the zero vector, and it is orthogonal to the row space of the matrix.
Timeline (last 90 days)
Performing SVD on the projection matrix W yields a text subspace corresponding to 512 non-zero singular values and null-space directions corresponding to 256 zero singular values.
After audio features are projected onto the null-space directions, their dot product with all text vectors is zero, so they naturally do not interfere with text embeddings.
If actual training were restricted to the orthogonal complement of the null space (U_perp), performance could be expected to improve further.
All Facts (3)
Performing SVD on the projection matrix W yields a text subspace corresponding to 512 non-zero singular values and null-space directions corresponding to 256 zero singular values.
50%UnverifiedAfter audio features are projected onto the null-space directions, their dot product with all text vectors is zero, so they naturally do not interfere with text embeddings.
50%UnverifiedIf actual training were restricted to the orthogonal complement of the null space (U_perp), performance could be expected to improve further.
50%