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end, we resort to the matrix of Lanczos vectors
Q
k
and consider their left-projection
approximation
A
k
¼ Q
k
Q
k
A
:
Correspondingly, if using the cova
ria
nce (rather than the Gramian) matrix
C ¼
AA
T
and its matrix of Lanczos vectors
Q
k
, we obtain the associated right-projection
approximation
A
L
k
¼ AQ
k
Q
T
:
k
m
, the
matrix-vector operations
s
k
:
¼ A
k
b
and
t
k
:¼ A
k
b
provide good approximations to
A
k
b
, since their sequences {
s
i
} and {
t
i
} exhibit rapid convergence to
Ab
in terms of
the leading left singular directions of
A
. As the goal of data compression consists
precisely in preservation of accuracy with respect to these directions, the Lanczos
projection provides a good alternative to singular value projection.
As for computation of the updated session vector through left-projection approx-
imation by means of Lanczos vectors, we thus obtain
Now Chen and Saad were able to show that for arbitrary vectors
b
∈R
a
k
¼ Q
k
Q
k
a
:
ð
8
:
25
Þ
Compared with its counterpart for singular vectors (
8.21
), (
8.25
) is even easier
to compute, since we may abstain from a Ritz step, i.e., the solution of the
eigenproblem (
8.24
), outright. As we shall see in numerical tests, the practical
results, too, of approximation by means of Lanczos vector projection are hardly
worse than those of singular vector projection.
8.4.2 RE-Specific Requirements
As regards usage for reinforcement learning, of course, the factorization of transi-
tion probabilities
p
ðÞ
ss
0
is of particular interest. Here, we shall ignore the actions
a
by
considering either only the unconditional probabilities
p
ss
0
or, according to Assump-
tion 5.2, only the conditional probabilities of a transition to the recommended
product
p
ss
a
. For the sake of simplicity, we shall identify the two cases with
p
ss
0
.
The reason is that matrix factorization makes sense only for two indices
s
and
s
0
.
The case of an additional factorization with respect to the actions
a
, that is,
p
ss
0
,
will be treated later in the scope of tensor factorization.
Thus, we would like to factorize the matrix of
transition probabilities
s
,
s
0
∈
S
.
P ¼ P
ss
0
