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Algorithm 9.2 HOSVD update (continued)
ð
n k ;t k
Þ of principal left singular vectors of [ A ( k ) B ( k ) ], k ¼
Output: matrices U k R
T
A ðÞ
¼: A ¼
1,
...
, d 1, matrix U of principal right singular vectors of
h i , updated slice B according to updated HOSVD
1: Update U k , k ¼ 1,
T B ðÞ
A ðÞ
...
, d 1 according to the above-described incremental
SVD (Algorithm 8.1)
2: A : ¼ ( A ( d ) ) T , b : ¼ ( B ( d ) ) T
3: A
:¼ A½
4: z : ¼ U T b , c : ¼k b Uzk 2
5: Comput e U and U
6:
U
:¼ UU
z
c
:¼ U U
7: b
8: B ( d ) : ¼ b T
9: for k ¼ 1,
, d 1
10: B ( k ) : ¼ U k U k B ( k )
11: end for
...
We shall briefly explain Algorithm 9.2. It crucially relies on the decomposition
( 9.4 ). First, we update the frontal mode B U d U d ,
i.e., by virtue of
T
V V T e n . The latter is then projected along the other modes by means of
( 9.5 ). This relies on the insight that, for the left-projection, we only need the new
slice B to compute its approximation. Unfortunately, this does not hold for the right-
projection, which is needed in the frontal mode, and we must approximate the entire
tensor (even if only adaptively), to obtain the updated slice. This renders the
application to computing recommendations more complicated; we shall discuss
this in more detail in the next section.
Another drawback is the fact that due to the large number of rows, steps 5 and
6 are computationally expensive and, thus, the procedure scales poorly.
A ðÞ
9.1.5 Computing Recommendations
To compute recommendations, we might initially proceed along the lines of the
outset of Sect. 8.3.3 by adding the updated slice of the current session B to the
previous tensor A in each step of the session and carrying out the incremental
learning step according to Algorithm 9.2. The latter provides the approximated slice
B t , which we deploy to forecast the current session. When the session terminates,
we compute the HOSVD of A , i.e., we carry out a complete learning step.
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