In order to de emphasize the impact of very large

In order to de-emphasize the impact of very large absolute values of e
ij
, a new set of error
terms is defined:
ǫ
ij
=
braceleftBigg
e
ij
if | e
ij
| ≤ Δ
f ( | e
ij
| ) if | e
ij
| > Δ
(12.14)
Here Δ is a user-defined threshold, which defines the case when an entry becomes large.
f ( | e
ij
| ) is a damped (i.e., sublinear) function of | e
ij
| satisfying f (Δ) = Δ. This condition
ensures that ǫ
ij
is a continuous function of e
ij
at e
ij
= ± Δ. The damping ensures that large
values of the error are not given undue importance. An example of such a damped function
is as follows:
f ( | e
ij
| ) =
radicalBig
Δ(2 | e
ij
| − Δ) (12.15)
This type of damped function has been used in [ 428 ]. The objective function for robust
matrix factorization then replaces the error values e
ij
with the adjusted values ǫ
ij
as follows:
Minimize J
robust
=
1
2
summationdisplay
( i,j ) ∈ S
ǫ
2
ij
+
λ
2
m summationdisplay
i =1
k summationdisplay
s =1
u
2
is
+
λ
2
n summationdisplay
j =1
k summationdisplay
s =1
v
2
js
An iterative re-weighted least-squares algorithm, which is described in [ 426 ], is used for the
optimization process. Here, we describe a simplified algorithm. The first step is to compute
the gradient of the objective function J
robust
with respect to each of the decision variables:
∂J
robust
∂u
iq
=
1
2
summationdisplay
j :( i,j ) ∈ S
∂ǫ
2
ij
∂u
iq
+ λu
iq
, ∀ i ∈ { 1 . . . m } , ∀ q ∈ { 1 . . .k }
∂J
robust
∂v
jq
=
1
2
summationdisplay
i :( i,j ) ∈ S
∂ǫ 2
ij
∂v
jq
+ λv
jq
∀ j ∈ { 1 . . . n } , ∀ q ∈ { 1 . . .k }

12.5. STRATEGIES FOR ROBUST RECOMMENDER DESIGN 407
Note that the aforementioned gradients contain a number of partial derivatives with respect
to the decision variables. The value of
∂ǫ
2
ij
∂u
iq
can be computed as follows:
∂ǫ
2
ij
∂u
iq
=
braceleftBigg
2 · e
ij
( − v
jq
) if | e
ij
| ≤ Δ
2 · Δ · sign( e
ij
)( − v
jq
) if | e
ij
| > Δ
Here, the sign function takes on the value of +1 for positive quantities and − 1 for negative
quantities. The case-wise description of derivative can be consolidated to simplified form as
follows:
∂ǫ
2
ij
∂u
iq
= 2 · min {| e
ij
| , Δ } · sign( e
ij
) · ( − v
jq
)
It is noteworthy that the gradient is damped when the error is larger than Δ. This damping
of the gradient directly makes the approach more robust to a few large errors in the ratings
matrix. Similarly, we can compute the partial derivative with respect to v
jq
as follows:
∂ǫ
2
ij
∂v
jq
=
braceleftBigg
2 · e
ij
( − u
iq
) if | e
ij
| ≤ Δ
2 · Δ · sign( e
ij
)( − u
iq
) if | e
ij
| > Δ
As before, it is possible to consolidate this derivative as follows:
∂ǫ 2
ij
∂v
jq
= 2 · min {| e
ij
| , Δ } · sign( e
ij
) · ( − u
iq
)
One can now derive the update steps as follows, which need to be executed for each user i
and each item j :
u
iq
⇐ u
iq
+ α
⎛
⎝
summationdisplay
j :( i,j ) ∈ S
min {| e
ij
| , Δ } · sign( e
ij
) · v
jq
− λ · u
iq
⎞
⎠
∀ i, ∀ q ∈ { 1 . . .k }
v
jq
⇐ v
jq
+ α
⎛
⎝
summationdisplay
i :( i,j ) ∈ S
min {| e
ij
| , Δ } · sign( e
ij
) · u
iq
− λ · v
jq
⎞
⎠
∀ j, ∀ q ∈ { 1 . . .k }
These updates are performed to convergence. The aforementioned steps correspond to global
updates. These updates can be executed within the algorithmic framework of gradient
descent (cf. Figure 3.8 of Chapter 3 ).