Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference

Abstract

We propose a branch-and-cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for seeking violated constraints. In order to further speed up the computation, several strategies have been exploited, including model reduction, warm start and removal of inactive constraints. We analyze the performance of the proposed method under different settings, and demonstrate that our method either outperforms or performs on par with state-of-the-art approaches. Especially when the connectivities are dense or when the relative magnitudes of the unary costs are low, we achieve the best reported results. Experiments show that the proposed algorithm achieves better approximation than the state-of-the-art methods within a variety of time budgets on challenging non-submodular MAP-MRF inference problems.

Appendix

1.1 Appendix 1: Relationship between the Standard SDP Relaxation (11) and the Simplified Dual (21)

The Lagrangian dual of (11) can be expressed in the following general form:

$$\begin{aligned}&\min _{\mathbf {u}} \quad \mathbf {u}^{\top }\mathbf {b}\end{aligned}$$

(25a)

$$\begin{aligned}&\, \mathrm {s.t.}\,\quad \mathbf {Z}= \mathbf {A}+ \textstyle {\sum _{i=1}^m} u_i \mathbf {B}_i \succcurlyeq \mathbf {0}, \end{aligned}$$

(25b)

$$\begin{aligned}&\qquad \quad u_i \ge 0, \forall i \in \mathcal {I}_{in}. \end{aligned}$$

(25c)

The p.s.d. constraint (25b) can be replaced by a penalty function, which is considered as a measure of violation of this constraint. In our case, the penalty function is defined as $\mathrm {p}(\mathbf {u}) = ||\min (\mathbf {0}, {\varvec{\lambda }}) ||_2^2 = ||\varPi _{\mathcal {S}^{nh+1}_+} (\mathbf {C}(\mathbf {u})) ||_F^2$, where ${\varvec{\lambda }}$ is the vector of eigenvelues of $\mathbf {Z}$. We can find that if $\mathrm {p}(\mathbf {u}) = 0$, then $\mathbf {Z}\succcurlyeq \mathbf {0}$. Now the problem (25) can be transformed to

(26a)

$$\begin{aligned} \mathrm {s.t.}\,\,\quad u_i \ge 0, \forall i \in \mathcal {I}_{in}, \end{aligned}$$

(26b)

where $\gamma > 0$ serves as a penalty parameter. With the increase of $\gamma$, the solution to (26) converges to that of (25). It is clear that (26) is equivalent to (21).

1.2 Appendix 2: Proof of Propositions 1 and 2

Firstly, it is known (Malick 2007; Wang et al. 2013) that the set of p.s.d. matrices with fixed trace $\varTheta _\eta := \{ \mathbf {X}\succcurlyeq \mathbf {0} | \mathrm {trace}(\mathbf {X}) = \eta \}$, $\forall \eta > 0$ has the following property:

Theorem 2

(The spherical constraint). $\forall \eta >0, \forall \mathbf {X}\in \varTheta _\eta$, we have $||\mathbf {X}||_{F} \le \eta$, and $||\mathbf {X}||_{F} = \eta$ if and only if $\mathrm {rank}(\mathbf {X}) = 1$.

It is also shown in Wang et al. (2013) that the problem (21) is the Lagrangian dual of the following problem:

$$\begin{aligned}&\min _{\mathbf {y},\mathbf {Y}} \,\, \mathrm {E}(\mathbf {y},\mathbf {Y}) + \mathrm {g}_\gamma (\mathbf {y},\mathbf {Y}) \end{aligned}$$

(27a)

$$\begin{aligned}&\,\mathrm {s.t.}\,\, (12), (13), (14), (15), (16), (17), (18), (19),\end{aligned}$$

(27b)

$$\begin{aligned}&\,\qquad \varOmega (\mathbf {y}, \mathbf {Y}) \succcurlyeq \mathbf {0}, \end{aligned}$$

(27c)

where $\mathrm {g}_\gamma (\mathbf {y},\mathbf {Y}) = \frac{1}{2\gamma }(||\varOmega (\mathbf {y},\mathbf {Y}) ||^2_F - (n+1)^2)$.

Proof of Proposition 1

(i) $\forall \mathcal {D}_1 \subseteq \mathcal {D}_2 \subseteq \mathcal {Z}^n$, $\exists \mathcal {F}_{in}, \mathcal {F}_{eq} \in \{(p,i) \}_{p \in \mathcal {V}, i \in \mathcal {Z}}$ such that $\mathcal {D}_1 = \{ \mathbf { x}\in \mathcal {D}_2 \ | \ x_p \ne i, \forall (p,i) \in \mathcal {F}_{in}; x_p = i, \forall (p,i) \in \mathcal {F}_{eq} \}$. Consequently, the difference between the SDCut primal formulation (27) with respect to $\mathcal {D}_1$ and $\mathcal {D}_2$ is that the one with respect to $\mathcal {D}_1$ contains the following additional linear constraints:

$$\begin{aligned} \left\{ \begin{array}{ll} y_{p,i} = 0, Y_{pi,qj} = Y_{qj,pi} = 0, &{}\forall (p,i) \in \mathcal {F}_{in}, \\ y_{p,i} = 1, Y_{pi,qj} = Y_{qj,pi} = y_{q,j}, &{}\forall (p,i) \in \mathcal {F}_{eq}. \end{array} \right. \end{aligned}$$

(28)

Because of the strong duality, we know that $\mathrm {d}^\star _\gamma (\mathcal {D})$ equals to the optimal value of the corresponding primal problem (27). Then we have $\mathrm {d}^\star _\gamma (\mathcal {D}_1) \ge \mathrm {d}^\star _\gamma (\mathcal {D}_2)$, as the primal problem (27) with respect to $\mathcal {D}_1$ has more constraints than that with respect to $\mathcal {D}_2$.

(ii) This proof is simple. As $|{\mathcal {D}} |= 1$, there is only one point $\hat{\mathbf { x}}$ in the set $\mathcal {D}$ and $\mathrm {E}(\hat{\mathbf { x}}) = \min _{\mathbf { x}\in \mathcal {D}} \mathrm {E}(\mathbf { x})$. Then the feasible set of (27) also contains a single point $\{ \hat{\mathbf {y}}, \hat{\mathbf {Y}} \}$ corresponding to $\hat{\mathbf { x}}$ by applying constraints as (28). Because $||\varOmega (\hat{\mathbf {y}},\hat{\mathbf {Y}}) ||^2_F = (n+1)^2$, we have $\mathrm {d}^\star _\gamma (\mathcal {D}) = \mathrm {E}(\hat{\mathbf {y}},\hat{\mathbf {Y}}) = \min _{\mathbf { x}\in \mathcal {D}} \mathrm {E}(\mathbf { x})$. $\square$

Proof of Proposition 2

$\{ \mathbf {y}_\gamma ^\star , \mathbf {Y}_\gamma ^\star \}$ is the optimal solution of (27) based on the strong duality, and $\mathrm {d}_\gamma (\mathbf {u}^\star _\gamma )$ is the corresponding optimal objective value. Consider the following problem

$$\begin{aligned}&\min _{\mathbf {y},\mathbf {Y}} \,\, \mathrm {E}(\mathbf {y},\mathbf {Y}) + \mathrm {g}_\gamma (\mathbf {y},\mathbf {Y}) \end{aligned}$$

(29a)

$$\begin{aligned}&\mathrm {s.t.}\,\, \varOmega (\mathbf {y}, \mathbf {Y}) \succcurlyeq \mathbf {0}, \, \mathrm {rank}(\varOmega ({\mathbf {y}_\gamma ^\star , \mathbf {Y}_\gamma ^\star })) = 1, \end{aligned}$$

(29b)

$$\begin{aligned}&\qquad (12), (13), (14), (15), (16), (17), (18), (19), \end{aligned}$$

(29c)

which adds a rank-1 constraint to the problem (27). Then $\mathrm {d}_\gamma (\mathbf {u}^\star _\gamma )$ and $\{ \mathbf {y}_\gamma ^\star , \mathbf {Y}_\gamma ^\star \}$ are also optimal for the above problem. Note that the constraints (12), (13), $\varOmega ({\mathbf {y}_\gamma ^\star , \mathbf {Y}_\gamma ^\star }) \succcurlyeq \mathbf {0}$ and $\mathrm {rank}(\varOmega ({\mathbf {y}_\gamma ^\star , \mathbf {Y}_\gamma ^\star })) = 1$, force $\{ \mathbf {y}_\gamma ^\star , \mathbf {Y}_\gamma ^\star \}$ to be a vertex of $\mathcal {M}(\mathcal {G},\mathcal {Z})$. So the feasible set of (29) is $\mathcal {M}(\mathcal {G},\mathcal {Z})$. On the other hand, $\mathrm {g}_\gamma (\mathbf {y}_\gamma ^\star , \mathbf {Y}_\gamma ^\star ) = 0$ at $\mathrm {rank}(\varOmega ({\mathbf {y}_\gamma ^\star , \mathbf {Y}_\gamma ^\star })) = 1$ (Theorem 2), so the objective function of (29) is $\mathrm {E}(\mathbf {y},\mathbf {Y})$. In summary, the problem (29) is equivalent to the MAP problem $\displaystyle {\min _{\mathbf {y},\mathbf {Y}\in \mathcal {M}(\mathcal {G},\mathcal {Z})}} \mathrm {E}(\mathbf {y},\mathbf {Y})$. Then we have that $\mathbf {y}_\gamma ^\star , \mathbf {Y}_\gamma ^\star$ yield the exact MAP solution and $\mathrm {d}_\gamma (\mathbf {u}^\star _\gamma )$ is the minimum energy. The value of $\gamma$ does not affect the above results. $\square$