  The existing low-memory BLS implementation proposed recently avoids the need
for storing and inverting large matrices, to achieve efficient usage of
memories. However, the existing low-memory BLS implementation sacrifices the
testing accuracy as a price for efficient usage of memories, since it can no
longer obtain the generalized inverse or ridge solution for the output weights
during incremental learning, and it cannot work under the very small ridge
parameter that is utilized in the original BLS. Accordingly, it is required to
develop the low-memory BLS implementations, which can work under very small
ridge parameters and compute the generalized inverse or ridge solution for the
output weights in the process of incremental learning. In this paper, firstly
we propose the low-memory implementations for the recently proposed recursive
and square-root BLS algorithms on added inputs and the recently proposed
squareroot BLS algorithm on added nodes, by simply processing a batch of inputs
or nodes in each recursion. Since the recursive BLS implementation includes the
recursive updates of the inverse matrix that may introduce numerical
instabilities after a large number of iterations, and needs the extra
computational load to decompose the inverse matrix into the Cholesky factor
when cooperating with the proposed low-memory implementation of the square-root
BLS algorithm on added nodes, we only improve the low-memory implementations of
the square-root BLS algorithms on added inputs and nodes, to propose the full
lowmemory implementation of the square-root BLS algorithm. All the proposed
low-memory BLS implementations compute the ridge solution for the output
weights in the process of incremental learning, and most of them can work under
very small ridge parameters.
