So we have $A_{m\times n}\approx P_{m\times k}Q_{k\times n}.$ Using the multiplicative update rule due to Lee we, in general, write that: $$P\leftarrow P \circ \frac{AQ^T}{PQQ^T}$$ and $$Q\leftarrow Q \circ \frac{P^TA}{P^TPQ}.$$ Since I am working with very large matrices, I don't have access to the whole matrix in memory. Give $(i,j,A[i][j])$, I can fetch $P[i][:]$ and $Q[:][j],$ where $P[i][:]$ is the $i'$th row vector of the matrix $P$ and $Q[:][j]$ is the $j$'th column vector of the matrix $Q.$ I therefore intend to write a method which does the following:
def update(i, j, matrix_value_at_ij):
// Goal is to update the vector P[i][:] and Q[:][j]
p_vect = get_vector_from_memory(i) // returns P[i][:] as a column vector
q_vect = get_vector_from_memory(j) // return Q[:][j] as a column vector
for c in range(k):
p_vect[k] = p_vect[k]* (coeff_AQ_t)/(coeff_PQQ_t)
q_vect[k] = q_vect[k]* (coeff_P_tA)/(coeff_P_tPQ)
I don't know how to compute (coeff_AQ_t)/(coeff_PQQ_t) and (coeff_P_tA)/(coeff_P_tPQ) under this constraint. Any ideas would be much appreciated.