Database Reference
In-Depth Information
Fig. 8.1 The best
approximating
one-dimensional subspace
(
solid line
) to a set of data
residing in
x
u
2
u
3
2
3
. Projections
are indicated by
dotted lines
R
1
u
4
w
z
1
1
2
2
y
u
1
This manifold is chosen such that the mean-squared error resulting from the
projection is minimal among all possible choices. Mathematically, the problem
may be stated as follows:
X
n
s
2
,
min
a
j
Xy
j
b
ð
8
:
6
Þ
,
,
,
...
,
n
p
d
n
p
d
X
∈R
b
∈R
y
1
y
n
s
∈R
j¼
1
n
p
denote the given data. A straightforward argument
reveals that
b
is always given by the centroid of the data, i.e.,
b
where
a
1
,
...
,
a
n
s
∈ R
:¼ n
s
X
n
s
j¼
1
a
j
.
Hence, assuming without loss of generality that the data are mean centered (which
may always be achieved by replacing our data by
a
j
b
,
,
n
s
Þ
, the
translation
b
may always be taken to be 0. We may thus restrict ourselves to the
problem of finding the best approximating
subspace
to a set of mean-centered data:
j ¼
1,
...
X
n
s
2
min
a
j
Xy
j
:
ð
8
:
7
Þ
,
,
...
,
n
p
d
d
X
∈R
y
1
y
n
s
∈R
j¼
1
The
Frobenius norm
is defined as
m
,
n
:¼
X
2
,
A
2
F
mn
kk
a
ij
∈ R
:
i¼
1
,
j¼
1
Summarizing our data and intrinsic variables in matrices,
,
A
:¼ a
1
; ...;
½
a
n
s
,
Y
:¼ y
1
; ...;
y
n
s
we may cast (
8.7
) equivalently as the matrix factorization problem
2
F
min
k
A XY
k
:
ð
8
:
8
Þ
,
n
p
d
dn
s
X
∈R
Y
∈R
Recalling the general framework stipulated in (
8.1
), (
8.8
) may be stated in terms
of the former by assigning
fE
2
F
,
C
1
:¼
R
n
p
d
,
C
2
:¼
R
dn
s
ð
;
F
Þ :¼
k
E F
k
:
