## Abstract

We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation methods for finite-dimensional problems. While for the latter several optimality bounds are known, to our knowledge no such bounds exist in the infinite-dimensional setting. We provide such a bound by analyzing a probabilistic rounding method, showing that it is possible to obtain an integral solution of the original partitioning problem from a solution of the relaxed problem with an *a priori* upper bound on the objective. The approach has a natural interpretation as an approximate, multiclass variant of the celebrated coarea formula.

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## Acknowledgements

This publication is partly based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).

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## Appendix

### Appendix

### Proof of Proposition 3

In order to prove the first assertion (88), note that the mapping *w*↦*Ψ*(*νw*
^{⊤}) is convex, therefore it must assume its maximum on the polytope Δ_{
l
}−Δ_{
l
}:={*z*
^{1}−*z*
^{2}|*z*
^{1},*z*
^{2}∈Δ_{
l
}} in a vertex of the polytope. Since the polytope Δ_{
l
}−Δ_{
l
} is the difference of two polytopes, its vertex set is at most the difference of their vertex sets, *V*:={*e*
^{i}−*e*
^{j}|*i*,*j*∈{1,…,*l*}}. On this set, the bound *Ψ*(*νw*
^{⊤})⩽*λ*
_{
u
} holds for *w*∈*V* due to the upper-boundedness condition (25), which proves (88).

The second equality (90) follows from the fact that *G*:={*b*
^{ik}:=*e*
^{k}(*e*
^{i}−*e*
^{i+1})^{⊤}∣1⩽*k*⩽*d*,1⩽*i*⩽*l*−1} is a basis of the linear subspace *W*, satisfying *Ψ*(*b*
^{ik})⩽*λ*
_{
u
}, and *Ψ* is positively homogeneous and convex, and thus subadditive. Specifically, there is a linear transform *T*:*W*→ℝ^{d×(l−1)} such that *w*=∑_{
i,k
}
*b*
^{ik}
*α*
_{
ik
} for *α*=*T*(*w*). Then

Since (25) ensures *Ψ*(±*b*
^{ik})⩽*λ*
_{
u
}, we obtain

for any suitable operator norm ∥⋅∥ and any *w*∈*W*. □

### Proof of Proposition 4

Denote \(\mathcal{B}_{\delta} :=\mathcal{B}_{\delta}(x)\). We prove mutual inclusion:

“⊆”: From the definition of the measure-theoretic interior,

Since \(|\mathcal{B}_{\delta} | \geqslant|\mathcal{B}_{\delta} \cap E| \geqslant|\mathcal{B}_{\delta} \cap E \cap F|\) (and vice versa for \(|\mathcal{B}_{\delta} \cap F|\)), it follows by the “sandwich” criterion that both \(\lim_{\delta\searrow0} |\mathcal{B}_{\delta} \cap E| / |\mathcal{B}_{\delta} |\) and \(\lim_{\delta\searrow0} |\mathcal{B}_{\delta} \cap F| / |\mathcal{B}_{\delta} |\) exist and are equal to 1, which shows *x*∈*E*
^{1}∩*F*
^{1}.

“⊇”: Assume that *x*∈*E*
^{1}∩*F*
^{1}. Then

We obtain equality,

from which we conclude that

i.e., *x*∈(*E*∩*F*)^{1}. □

### Proof of Proposition 5

First note that

The inequality (∗) is a consequence of the definition of \(w^{\pm}_{\mathcal{F}E}\) and [2, Theorem 3.77], and (∗∗) follows directly from *w*(*x*),*w*(*y*)∈Δ_{
l
} a.e. on *Ω*. The upper bound (187) permits applying [2, Theorem 3.84] on *w*, which provides \(w \in\operatorname{BV} (\varOmega)^{l}\) and (94). Due to [2, Proposition 3.61], the sets (*E*)^{0},(*E*)^{1} and \(\mathcal{F}E\) form a (pairwise disjoint) partition of *Ω*, up to an \(\mathcal{H}^{d - 1}\)-zero set. Therefore, since \(\varPsi(D u) \ll|D u| \ll\mathcal{H}^{d - 1}\) by construction, from [2, Theorem 2.37, 3.84] we obtain, for any Borel set *A*,

Since *w*(*x*)∈Δ_{
l
} a.e. by assumption, we conclude that \(w^{+}_{\mathcal{F}E}\) and \(w^{-}_{\mathcal{F}E}\) must have values in Δ_{
l
} as well, see [2, Theorem 3.77]. Therefore we can apply Proposition 3 to obtain

We rewrite *Ψ*(*Dw*) using (94),

From [2, Proposition 2.37] we obtain that *Ψ* is additive on mutually singular Radon measures *μ*,*ν*, i.e., if |*μ*|⊥|*ν*|, then

for any Borel set *B*⊆*Ω*. This holds in particular for the three measures in (193), therefore

Since *Du*⌞(*E*)^{1}≪|*Du*⌞(*E*)^{1}|=|*Du*|⌞(*E*)^{1}, we conclude *Ψ*(*Dw*)⌞(*E*)^{1}=*Ψ*(*Du*)⌞(*E*)^{1} and *Ψ*(*Dw*)⌞(*E*)^{0}=*Ψ*(*Dv*)⌞(*E*)^{0}. Substitution into (192) proves the remaining assertion,

□

### Proof of Proposition 6

We first show (98). It suffices to show that

This can be seen by considering the precise representative \(\widetilde{1_{E}}\) of 1_{
E
} [2, Definition 3.63]: Starting with the definition,

the fact that \(\lim_{\delta\searrow0} \frac{| \varOmega\cap\mathcal {B}_{\delta} (x) |}{|\mathcal{B}_{\delta} (x) |} = 1\) implies

Substituting *E* by *Ω*∖*E*, the same equivalence shows that \(x \in(E)^{0} \Leftrightarrow\widetilde{1_{\varOmega\setminus E}} (x) = 1 \Leftrightarrow\widetilde{1_{E}} (x) = 0\). As \(\mathcal{L}^{d} (\varOmega \setminus((E)^{0} \cup(E)^{1})) = 0\), this shows that \(1_{E^{1}} = \widetilde{1_{E}}\)
\(\mathcal{L}^{d}\)-a.e. Using the fact that \(\widetilde {1_{E}} = 1_{E}\) [2, Proposition 3.64], we conclude that \(1_{(E)^{1}} = 1_{E}\)
\(\mathcal{L}^{d}\)-a.e., which proves (197) and therefore the assertion (98).

Since the measure-theoretic interior (*E*)^{1} is defined over \(\mathcal{L}^{d}\)-integrals, it is invariant under \(\mathcal{L}^{d}\)-negligible modifications of *E*. Together with (197) this implies

To show the relation (*Du*)⌞(*E*)^{1}=(*Dv*)⌞(*E*)^{1}, consider

The equality (∗) holds due to the assumption (96), and due to the fact that *Df*=*Dg* if *f*=*g*
\(\mathcal{L}^{d}\)-a.e. (see, e.g., [2, Proposition 3.2]). We continue from (204) via

Therefore *Du*⌞(*E*)^{1}=*Dv*⌞(*E*)^{1}. Then,

In the equality (∗) we used the additivity of *Ψ* on mutually singular Radon measures [2, Proposition 2.37]. By definition of the total variation, |*μ*⌞*A*|=|*μ*|⌞*A* holds for any measure *μ*, therefore |*Du*⌞(*Ω*∖(*E*)^{1})|=|*Du*|⌞(*Ω*∖(*E*)^{1}) and |*Du*⌞(*Ω*∖(*E*)^{1})|((*E*)^{1})=0, which together with (again by definition) *Ψ*(*μ*)≪|*μ*| implies that the second term in (211) vanishes. Since all observations equally hold for *v* instead of *u*, we conclude

Equation (97) follows immediately. □

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Lellmann, J., Lenzen, F. & Schnörr, C. Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem.
*J Math Imaging Vis* **47**, 239–257 (2013). https://doi.org/10.1007/s10851-012-0390-7

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DOI: https://doi.org/10.1007/s10851-012-0390-7