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...r guarantees. 5 Experiments In this section, we empirically compare the performance of the various algorithms discussed in this paper. We are motivated by the speech data subset selection application =-=[30, 33, 21]-=- with the submodular function f encouraging limited vocabulary while g tries to achieve acoustic variability. A natural choice of the function f is a function of the form |Γ(X)|, where Γ(X) is the nei...

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...) can be posed as submodular function minimization subject to a size constraint. Alternatively, cardinality can be treated as a penalty, reducing the problem to unconstrained submodular minimization (=-=Jegelka et al., 2011-=-). Minimum Power Assignment: In wireless networks, one seeks a connectivity structure that maintains connectivity at a minimum energy consumption. This problem is equivalent to finding a suitable stru...

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...ence, but can be quite slow in practice (Section 5); or the minimum-normpoint/Fujishige-Wolfe algorithm [20] that has expensive iterations but finite convergence. Other recent methods are approximate =-=[24]-=-. In contrast to several iterative methods based on convex relaxations, we seek to obtain exact discrete solutions. To the best of our knowledge, all generic algorithms that use only submodularity are...

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...graph-cut problem which is relatively easy. If S = 2V above, we get a form of the approximate submodular minimization algorithm suggested for arbitrary (non-graph representable) submodular functions (=-=[18]-=-). The proximal minimization algorithm also generalizes three submodular function minimization algorithms IMA-I, II and III, described in detail in [15] again with λ = 1, S = 2V and d(X, Xt ) = d f 1 ...

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...ning, has inspired a large body of work on practically fast procedures (see [1] for a survey). But most of these procedures focus either on special submodular functions such as decomposable functions =-=[16, 20]-=- or on constrained SFM problems [13, 12, 15, 14]. Fujishige-Wolfe’s Algorithm for SFM: For any submodular function f , the base polytope Bf of f is defined as follows: Bf = {x ∈ Rn : x(A) ≤ f(A), ∀A ⊂...

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..., made possible by the fact that the supermodularity is restricted to a tree. 3 Optimization and Bounds Estimation The four approaches to problem (1) are described in the following subsection: alternating minimization (AM) in subsection 3.1; dual decomposition (DD) in subsection 3.2, continuous relaxation (CR) in subsection 3.3; and submodular-supermodular procedure (SSP) in subsection 3.4. Our algorithms require submodular minimization as a subroutine, which can be performed efficiently in general and even more efficiently for sub-classes of submodular functions (e.g., generalized graph-cuts [11, 16]). Exact inference on trees is always efficient using dynamic programming. 2 Algorithm 1 Alternating minimization (AM). Input: α and y0. Output: xt−1 and yt−1. 1: Set t← 1. 2: while not converged do 3: xt ← argminx∈{0,1}V Eα(x,yt−1). 4: yt ← argminy∈{0,1}V Eα(xt,y). 5: t← t+ 1. 6: end while Algorithm 2 AM with greedy scheduling (AM-Greedy). Input: y0. Output: xt−1. 1: Set α0 ← 0 and t← 1. 2: repeat 3: αt←argmaxα[Eα(xt−1,yt−1)−Eα(x∗α,yt−1 ,y ∗ α,yt−1 )] ((x∗α,yt−1 ,y ∗ α,yt−1 ) is the output of AM(α,yt−1)). 4: (xt,yt)← AM(αt,yt−1). 5: t← t+ 1. 6: until xt−1 = yt−1 holds 3.1 Alternating Minimiza...

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...ality matroid-intersection problem, as a SM problem (see [3]). SM is well known to be solvable in polynomial time using the ellipsoid method (see [7]). But the practicality of such an algorithm is very limited. Iwata, Fleischer, and Fujishige (see [8]) and Schrijver (see [14]) developed the first “combinatorial” polynomial-time algorithms for SM, however, again the practical use of such algorithms is quite limited. An algorithm that has had the most success seeks the minimum-norm point of the “base polyhedron” (see [5]), but even that algorithm has been found to be slow and/or inaccurate (see [9]). So we regard the challenge of developing practically-efficient approaches for SM as open. The aforementioned algorithms for SM take advantage of the Lovasz extension of a submodular function. This function is an extension of a submodular 2 Submodular Minimization via LP and MILP function, viewed as defined on the vertices of the unit hypercube [0, 1]E , to the entire hypercube, as a piecewise-linear convex function (see [11]). Using the Lovasz extension, one can derive an equivalent linear-programing (LP) problem with a very large number of columns (see [13], for example). Solving this LP...

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... function can be mapped to a form that permits the use of nice numerical optimization techniques; however, many submodular functions cannot be minimized using this technique, and it can also be slow [=-=Jegelka et al., 2011-=-]. Along with analyzing the deficiencies of existing methods, Jegelka, Lin, and Bilmes [2011] propose a powerful approach that relies on approximating the submodular functions with a sequence of graph...