SEKE 2012 Proceedings - Knowledge Systems Institute

Where each
is the influence vector
for the -th node. We take the -norm on each so that it
will encourage all the elements in to go to zero together.
With the estimated , we can directly identify those most
influential nodes. Let us denote the set of hub nodes by :
FISTA Algorithm for SLIM
Input: ,
Initialization:
Iterate for
Step 1:
, until convergence
The cardinality of (i.e., the number of hub nodes) is
controlled by the regularization parameter . When ,
then all will become zero and none of the nodes will be
selected as hub node (i.e., ). On the othe r hand, if
, . Put anothe r way, the sm aller is, the
more nodes will be selected as hub nodes. In practice, we
could tune the regularizatio n parameter to achieve the
desirable number of hub nodes.
Although the formulation for SLIM is a co nvex
optimization problem, the n on-smoothness arising from
and the additional non-negativity constraint make th e
computation quite challenging. In the next section, w e
present an efficient and scalable solver for SLIM which
could be applied to solve web-scale problems.
Step 2:
Step 3:
Output:
Since (4) is separable in ter ms of
independently, i.e.,
, we solve it by e ach
IV. OPTIMIZATION FOR SLIM
In this sectio n, we present an efficient optimization
algorithm for solving SLIM in (3). We first introduce some
necessary notations. Let
be the
smooth part of the objective in (3). The gradient of over
takes the following form:
In addition, is Lipschitz con tinuous with the
constants , which is the maximum
eigenvalue of . In other word, we have for any ,
With these notations in place, we use the fast accelerated
gradient method framework. In particular, we adopt the fastiterative
shrinkage-thresholding algorithm (FISTA) [9] as
shown in the next Algorithm.
It is known that this algorithm has the optim al
convergence rate of .
To apply this algorithm, the main difficulty is how to
compute the first step. Firstly, we observe Step 1 can be
written as:
The closed form solution of the above minimization
problem can be characterized by the following Theorem.
Theorem 1. The optimal solution for the following
optimization can be represented as following,
Let
be indices for the positive values
in and let be the complement of . Then the optimal
is,
Proof. Firstly, we utilize the fact that the dual norm of
-norm is -norm, so that we could present as
Therefore, (6) can be reformulated as:
where .
Interchange the min and max, we have,
3