5.3. METHODOLOGY 51
5.3.4 PROTOTYPE-GUIDED COMPATIBILITY MODELING
Intuitively, compatible fashion items can essentially follow several latent compatible attribute
interaction prototypes, while the incompatible ones would share certain unfavorable prototypes.
In a sense, each latent prototype can be characterized by a set of attribute elements. For example,
{jeans, boyfriend-style, ragged, street fashion} tends to form a harmonious prototype, while {office
lady, holed, cartoon, tiered skirt} is more likely to refer to an unfavorable one. Toward this end,
we assume that there is a set of latent compatible/incompatible attribute interaction prototypes.
Owing to its superior capability of latent factor modeling [66], we seek the latent attribute
interaction prototypes under the NMF. To derive the latent attribute interaction compatible
prototypes, it is natural to resort to the set of positive top-bottom pairs S. Here we define the
data matrix G
p
D Œg
1
; g
2
; : : : ; g
N
2 R
2M N
, where g
n
D Œf
t
i
n
I f
b
j
n
2 R
2M
denotes the semantic
attribute representation of the n-th positive top-bottom pair .t
i
n
; b
j
n
/.
According to NMF, we aim to solve the following objective:
min
P;H
p
G
p
PH
p
2
F
;
s.t. P > 0; H
p
> 0;
(5.5)
where P D Œp
1
; p
2
; ; p
L
p
2 R
2M L
p
refers to the latent basis matrix, each column of which
corresponds to a compatible prototype, and L
p
represents the total number of the latent proto-
types. H
p
2 R
L
p
N
corresponds to the latent representation matrix of the N top-bottom pairs
regarding the basis compatibility prototypes. In particular, p
l
2 R
2M
denotes the l-th latent
compatible prototype, which can be rewritten as follows:
p
l
D
"
p
t
l
p
b
l
#
; (5.6)
where p
t
l
2 R
M
and p
b
l
2 R
M
can be treated as the semantic attribute representations of the
prototype top and bottom for p
l
.
In the same manner, we can also derive the latent incompatible prototypes based on the set
of negative top-bottom pairs (t
i
, b
k
)’s, where the bottom b
k
… B
C
i
is randomly sampled for top t
i
.
Let
G
u
2 R
2M N
be the data matrix comprising semantic attribute representations of negative
top-bottom pairs and U D Œu
1
; u
2
; ; u
L
u
2 R
2M L
u
be the matrix of latent incompatible
prototypes, where L
u
is the total number of incompatible prototypes, and H
u
2 R
L
u
N
denotes
the latent representation matrix of the N negative top-bottom pairs in the prototype space.
Similarly, we represent the r-th latent incompatible prototype u
r
2 R
2M
as follows:
u
r
D
"
u
t
r
u
b
r
#
; (5.7)
where u
t
r
2 R
M
and u
b
r
2 R
M
denote the semantic attribute representations of the prototype
top and bottom of u
r
. Ultimately, we have the following NMF loss for the latent prototype
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