In this section, we discuss an algorithm that can be used to minimize the LP relax- ation of a multi-label MRF over the local polytope. In the remainder of this section, we use the termLP relaxationto denote the LP over the local polytope,
min τ∈Λ
hθ,τi. (2.78)
To this end, let us first briefly describe the well-knownmin-sum message passing (or min-sum belief propagation) [Pearl, 1988].
Min-Sum Message Passing. Min-sum message passing is an approximate algo- rithm for minimizing the multi-label energy function (2.60). This algorithm main- tains amessage vector mij:µ; µ ∈ L for each directed edge9 (i,j) ∈ E
+. The notation
mij:µ denotes the message from i → j indexed byµ. The basic operation of this al-
gorithm is passing a message from i → j for each directed edge (i,j) ∈ E+. This
operation can be written as
mij:µ←min λ∈L θij:λµ+θi:λ+
∑
(k,i)∈E+ k6=j mki:λ , (2.79)for all µ ∈ L. A message from i → j is valid if this update does not change the message vector mij. The idea of this algorithm is to keep passing messages of this form, in some particular order, until convergence, i.e., until all messages are valid. Note that the time complexity of one message passing step is O(`2), where` is the number of labels. Upon convergence, the algorithm provides beliefs (approximate min-marginals) for all random variables,
θi0:λ = θi:λ+
∑
(k,i)∈E+
mki:λ , (2.80)
9E+ denotes the set of directed edges between the vertices in the MRF,i.e., if(i,j)∈ E then,(i,j)∈
Figure 2.18: An elementary reparametrization. Subtract αfrom all incoming edges of node Xi:λ and add it to the node potential. This corresponds to the message mji:λ =α. The reverse
operation is also a valid reparametrization.
for all i ∈ V and λ ∈ L. Then, the approximate labelling can be obtained from the beliefs,
x0i:λ =argmin
λ∈L
θi0:λ . (2.81)
It will be seen later, that min-sum message passing is optimal on tree structured MRFs. However, for a graph with loops, the convergence of this algorithm is not guaranteed. Despite of this shortcoming, the idea of min-sum message passing will be useful in developing algorithms to minimize the LP relaxation.
Reparametrization. A given multi-label energy function can be written in different ways as a sum of unary and pairwise terms. In particular, there may exist a different set of parametersθ0 such thatEθ(x) =Eθ0(x)for allx, denoted asEθ≡Eθ0.
Lemma 2.3.4. Two energy functions Eθ and Eθ0 are equivalent if and only if there exist
values mji:λand mij:µfor(i,j)∈ E andλ,µ∈ Lsuch that
θij0:λµ=θij:λµ−mji:λ−mij:µ , (2.82)
θi0:λ =θi:λ+
∑
(k,i)∈E+
mki:λ .
Proof. This result is well known, and we refer the interested reader to [Kolmogorov, 2006; Werner, 2007].
The values ofmij:µconstitute amessagemij passed from the edge(i,j)to the node
j10; it may be thought of as a message vector (indexed by µ). A messagem
ij causes valuesmij:µto beswept outof all the edgesθij:λµand added to the nodesθj:µ. Messages
are passed in both directions from an edge(i,j). An elementary reparametrization is illustrated in Figure 2.18.
10This is the same as themessage vectorpassed fromi→jin the min-sum message passing algorithm.
In fact, reparametrization provides an alternative way of implementing the min- sum message passing algorithm, where the parametersθare updated directly based
on the messages. Similarly to Eq. (2.81), the labelling can be obtained as x0i:λ=argmin
λ∈L
θi0:λ , (2.83)
whereEθ0 ≡Eθ. This suggests a lower bound ofEθ(x),i.e.,
∑
i∈V min
λ∈L
θ0i:λ≤ Eθ(x). (2.84)
Now, one can try to maximize this lower bound over all possible reparametrizations, while keeping the parameters non-negative (the parameters θ are non-negative to
start with). This leads to the following optimization problem, max θ0 i
∑
∈V min λ∈L θ0i:λ , (2.85) s.t. Eθ0 ≡Eθ, θ0ij:λµ≥0 ∀(i,j)∈ E, λ,µ∈ L.We refer to this problem as the optimal message passing problem. The interesting relationship between the above problem and the LP relaxation is characterized by the following theorem [Werner, 2010].
Theorem 2.3.2. The LP relaxation over the local polytope(2.78) and the optimal message passing(2.85)are dual LP problems.
Proof. The proof of this theorem is a direct application of Lagrange duality to the LP relaxation over the local polytope. See [Werner, 2010] for details.
This means that the lower bound computed by optimizing either of these prob- lems yields the same value. Therefore, reparametrization (or message passing) is a good candidate to optimize the LP relaxation. However, the order in which the messages should be passed is still not clear. In particular, an algorithm based on message passing that would optimize the LP relaxation would be ideal. To this end, let us first give an algorithm for MRFs defined on trees and then turn to the more general algorithm.