Database Reference
In-Depth Information
x
fg
e
Therefore, if we now want to evaluate a newly given set of data points
i¼
1
(the test or evaluation set) by
,
M
,
ð
n
e
e
y
i
:¼ f
x
ð
,
i ¼
1,
...
we just form the combination of the associated values for
f
l
according to (
7.30
).
The learning on the training data is summarized in Algorithm 7.1. It consists of
assembling the matrices
C
l
and
B
l
for the different grids
Ω
l
and solving the
corresponding discrete systems (
7.26
). The main results are the coefficient vectors
α
l
for the grids. In the next section of adaptive learning, we will also need the
matrices
C
l
and
B
l
.
Algorithm 7.1: Computation of sparse grid classifier
Input: training data set {(x
i
,
y
i
)}
i¼
1
, regularization parameter
λ
Output: coefficients
α
l
(matrices
C
l
and
B
l
)
for
q ¼
0,
...
,
d
1 do
for
l
1
¼
1,
...
,
n q
do
for
l
2
¼
1,
,
n q
(
l
1
1) do
..
for
l
d
1
¼
1,
...
,
n q
(
l
1
1)
...
(
l
d
2
1) do
l
d
¼ n q
(
l
1
1)
...
(
l
d
2
1)
(
l
d
1
1)
assemble matrices
C
l
and
B
l
solve the linear system (
...
B
l
)
λ
C
l
+
B
l
α
l
¼ B
l
y
end for
...
end for
end for
end for
Algorithm 7.2 shows the application of the classifier (represented by the
α
l
) to the test data set x
fg
coefficients
i¼
1
as described above.
Algorithm 7.2: Evaluation of sparse grid classifier
Input: test data set x
fg
i¼
1
, coefficients
α
l
yfg
Output: set of score values
e
i¼
1
,
M
e
y
i
¼
0,
i ¼
1,
...
for
q ¼
0,
...
,
d
1 do
for
l
1
¼
1,
...
,
n q
do
for
l
2
¼
1,
...
,
n q
(
l
1
1) do
...
(continued)
