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In-Depth Information
Table 8.5 Comparison of prediction qualities and error norms for different regularization param-
eter values and with variable rank
λ ¼
0.1
λ ¼
0.01
λ ¼
0.001
k
p
1
p
3
e
F
e
Ω
p
1
p
3
e
F
e
Ω
p
1
p
3
e
F
e
Ω
2
1.18
2.11
13.22
3.62
0.20
1.22
34.52
3.29
0.02
0.32
55.98
3.28
5
2.66
4.83
11.78
3.33
0.96
3.66
29.77
2.93
2.05
3.66
41.52
2.89
50
5.74
9.16
8.13
1.94
5.77
8.21
20.47
0.63
5.41
7.84
27.02
0.55
100
6.13
9.75
7.32
1.80
6.29
9.84
17.62
0.28
6.15
9.12
21.82
0.12
200
6.09
9.86
7.11
1.79
6.32
10.02
16.64
0.28
6.32
9.93
18.06
0.12
previous approach. But, on the contrary, we may also argue that there is too little
statistical volume and the user cannot view all products that he/she is potentially
interested in, simply because there are too many of them. This would suggest the
matrix completion approach. So there are pros and cons for both assumptions.
Example 8.8
We next repeat the test of Example 8.6 with the factorization
according to formulation (
8.37
). Instead of a gradient descent algorithm, we used
an ALS algorithm as described in [ZWSP08] which is more robust.
The results are contained in Table
8.5
whose structure basically corresponds to
that of Table
8.3
. Instead of the time, we have included the error norm
e
Ω
which
corresponds to the Frobenius norm
e
F
but is calculated only on the given entries
(
i
,
j
)
∈
Ω
. Thus,
e
Ω
is equal to the root-mean-square error (RMSE) multiplied by
the square root of the number of entries
j
p
. Additionally, we compare different
values of the regularization parameter
.
From Table
8.5
we see that for increasing rank the RMSE is strongly declining,
and we capture the given probabilities on
λ
almost perfectly. A rank of 50-100 is
already sufficient to bring the RMSE close to zero, and higher ranks also do not
improve the prediction rate significantly. Unlike in Table
8.3
, where we needed
almost full rank to zero the approximation error, this is because here we just have to
approximate the unknown entries.
In contrast to Table
8.3
, the overall error
e
F
is slowly decreasing and remains
very high. This is again because we do not approximate the zero values outside
Ω
.
The prediction rate is comparable to Table
8.3
. This indicates that for the proba-
bility matrix
P
, the approach to consider all non-visited entries to be zero is equally
reasonable like assuming them to be unknown.
Ω
■
The result of Example 8.8 does not mean that the matrix completion approach
is outright useless for the recommendation engine task. In fact, it could be, e.g.,
used to complete the matrix of transactions or transitions before it is further
processed.
