Approximate Decomposable Submodular Function Minimization for Cardinality-Based Components

1 Introduction . Given a ground set V , a function f : 2V → R is submodular if for every A , B ⊆ V it satisfies f ( A ) + f ( B ) ≥ f ( A ∩B ) + f ( A ∪B ) . Submodular functions are ubiquitous in combinatorial optimization and machine learning , arising , e.g. , in image segmentation [ 22 ] , hypergraph clustering [ 30 ] , data subset selection [ 44 ] , document summarization [ 31 ] , and dense subgraph discovery [ 43 ] . Algorithms for minimizing submodular functions are well studied . Strongly-polynomial time algorithms for general submodular minimization exist [ 36 , 17 , 18 ] , but their runtimes are impractical in most cases . The past decade has witnessed several advances in faster algorithms for decomposable submodular function minimization ( DSFM ) , i.e. , minimizing a sum of simpler submodular functions [ 9 , 24 , 39 , 29 , 35 , 10 , 21 , 19 ] . This problem is defined by identifying a set E ⊆ 2V of subsets of the ground set . The goal is then to solve minimizeS⊆V f ( S ) = ∑ e∈E fe ( S ∩ e ) , ( 1 ) where for each e ∈ E , fe is a submodular function supported on a subset e ⊆ V . Functions of this form often arise as energy functions in computer vision [ 22 , 23 , 13 ] , and generalized cut functions for hypergraph clustering and learning [ 12 , 28 , 30 , 32 , 41 , 42 ] , among other applications . This paper focuses on cardinality-based DSFM ( Card-DSFM ) , where every component fe in the sum is a concave cardinality function , i.e. , fe ( A ) = ge ( |A| ) for some concave function ge . A single concave cardinality function is effectively a function of one variable and is trivial to minimize . However , set functions obtained via sums of these functions are much more complex and have broad modeling power , making this one of the most widely studied and applied variants since the earliest ∗This research was performed while the author was a postdoctoral associate at Cornell University . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . work on DSFM [ 24 , 39 ] . In terms of theory , previous research has addressed specialized runtimes and solution techniques [ 24 , 39 , 22 , 41 ] . In practice , cardinality-based decomposable submodular functions frequently arise as higher-order energy functions in computer vision [ 22 ] and set cover functions [ 39 ] . The most widely used generalized hypergraph cut functions are also cardinality based [ 30 , 32 , 41 , 42 , 1 ] . Even previous research on algorithms for the more general DSFM problem tends to focus on cardinality-based examples in experimental results [ 9 , 39 , 19 , 29 ] . We present the first approximation algorithms for Card-DSFM , using a purely combinatorial approach that relies on approximately reducing ( 1 ) to a sparse graph cut problem . The fact that concave cardinality functions are representable by graph cuts has previously been noted in different contexts [ 22 , 39 , 41 ] —this can be accomplished by combining simple graph gadgets , whose cut properties together model the function . However , previous techniques are limited in that they ( i ) focus only on exact reduction techniques , ( ii ) do not consider the density of the resulting graph , and ( iii ) do not address any questions regarding the optimality of such a reduction . Here we develop new approximate reduction methods leading to a sparse graph , which we show is optimally sparse in terms of the standard gadget reduction strategy . We show that representing a concave cardinality function with a sparse graph is equivalent to approximating a concave function with a piecewise linear curve , where the number of linear pieces determines the sparsity of the reduced graph . We develop a new algorithm for the resulting piecewise linear approximation problem , and prove new bounds on the number of edges needed to model different concave functions that arise in practice . Combining our reduction strategy with state of the art algorithms for maximum flow [ 15 , 26 , 40 , 14 ] leads to fast runtime guarantees for finding approximate solutions to Card-DSFM . Our algorithm is also easy to implement and achieves substantial practical improvements on benchmark image segmentation experiments [ 19 , 9 , 29 ] and algorithms for hypergraph clustering [ 42 ] . 2 Background on Graph Reductions . A nonnegative set function f : 2V → R+ on a ground set V is graph reducible if there exists a directed graph G = ( Vf , Ef ) on a larger node set Vf = V ∪ A ∪ { s , t } , which includes an auxiliary node setA as well as source and sink nodes s and t , whose s-t cut structure models f . More precisely , f is graph reducible if for every S ⊆ V we have f ( S ) = minimize T⊆A cutG ( { s } ∪ S ∪ T ) , ( 2 ) where cutG is the cut function on the graph G. In words , this says that evaluating f ( S ) is equivalent to finding the minimum s-t cut penalty that could be obtained by placing S on the s-side of the cut and V \S on the t-side . Given such a graph , f can be minimized by finding the minimum s-t cut set U∗ ⊆ V ∪ A in G and taking S∗ = V ∩ U∗ . When ( 2 ) holds , we say that G models f . Graph reduction strategy for Card-DSFM . In general , the function f ( S ) = ∑ e∈E fe ( S ∩ e ) is not a concave cardinality function , even if individual components fe in the sum are . However , in order to show f is graph reducible , it suffices to find a small reduced graph Ge for each e ∈ E that models fe in the sense of ( 2 ) , and combine these together into a larger graph . A number of related approaches for this task have been developed [ 39 , 22 , 41 ] . We review our own previous reduction strategy [ 41 ] , which we build on in this paper . Modeling fe can be accomplished by combining a set of cardinality-based ( CB ) gadgets [ 41 ] . For a set e ⊆ V of size k = |e| , a CB-gadget is a small graph parameterized by positive weights ( a , b ) . Each v ∈ e defines a node , and the gadget also involves a single auxiliary node ve . For each v ∈ e , there is a directed edge ( v , ve ) with weight a · ( k − b ) , and a directed edge ( ve , v ) with weight a · b . The resulting graph gadget models the function fa , b ( S ) = a ·min { |S| · ( k − b ) , ( k − |S| ) · b } . ( 3 ) To see why , consider where we must place the auxiliary node ve when solving a minimum s-t cut problem involving this small graph gadget . If we place i = |S| nodes on the s-side , then placing ve on the s-side has a cut penalty of ab ( k − i ) , whereas placing ve on the t-side gives a penalty of ai ( k − b ) . To minimize the cut as in ( 2 ) for a fixed S ⊆ e , we choose the smaller of the two cut penalties . We previously showed that any cardinality-based submodular function fe on a ground set e can be modeled by combining |e| − 1 CB-gadgets [ 41 ] . Analogously , Stobbe and Krause [ 39 ] showed that a concave cardinality function on a ground set e can be decomposed into a sum of |e| − 1 threshold potentials [ 39 ] and can be represented by graph cuts , though they provided no explicit reduction strategy . Priori to this , Kohli et al . [ 22 ] highlighted a similar type of gadget for modeling a class of robust potential functions for image segmentation , noting that multiple gadgets could be combined to model arbitrary concave functions . Using any of these techniques , the function fe can be modeled in the sense of ( 2 ) using a graph G with O ( |e| ) nodes and O ( |e|2 ) edges . 3 Sparse Reductions via Piecewise Linear Approximation . We now turn to a new and refined problem of finding sparse and approximate reduction strategies for Card-DSFM , which we prove can be cast as approximating a concave function with a piecewise linear curve . All proofs are given in the supplement , where we also derive a similar graph reduction scheme that is slightly more efficient for modeling symmetric functions , i.e. , fe ( A ) = fe ( e\A ) . 3.1 The sparse approximate reduction problem . We first introduce a special parameterized function with broad modeling power . Definition 1 A k-node combined gadget function ( CGF ) of order J is a function of the form : ` ( x ) = z0 · ( k − x ) + zk · x+ ∑J j=1 aj min { x · ( k − bj ) , ( k − x ) · bj } , ( 4 ) parameterized by scalars z0 ≥ 0 , zk ≥ 0 , and vectors a = ( aj ) ∈ RJ > 0 and b = ( bj ) ∈ RJ > 0 , where bj < bj+1 for j ∈ { 1 , 2 , . . . , J − 1 } and bJ < k. If we define a set function on a ground set e of size k by fe ( S ) = ` ( |S| ) , then fe is exactly the function that is modeled by combining J CB-gadgets with parameters ( aj , bj ) Jj=1 , and additionally placing a directed edge from a source node s to each v ∈ e of weight z0 , and an edge from v ∈ e to a sink node t with weight zk . Previous reduction techniques amount to proving that any concave function g on an interval [ 0 , k ] ( representing a submodular function fe ( S ) = g ( |S| ) ) matches some k-node CGF of order k − 1 at integer inputs [ 41 , 39 ] . We focus on a new sparse reduction problem . Definition 2 Let g : [ 0 , k ] → R+ be a nonnegative concave function and fix ε ≥ 0 . The sparsest approximate reduction ( SpAR ) problem seeks a k-node CGF ` with minimum order J so that g ( i ) ≤ ` ( i ) ≤ ( 1 + ε ) g ( i ) for all i ∈ { 0 , 1 , 2 , . . . k } . ( 5 ) This is equivalent to finding the minimum number of CB-gadgets needed to approximately model a concave cardinality function . Thus , solving this problem provides the sparsest reduction in terms of a standard reduction strategy . Without loss of generality , Definition 2 is restricted to considering functions that upper bound g. Any other approximating function could be scaled to produce an upper bound leading to the same guarantees for Card-DSFM . Importantly , we care about approximating the concave function g only at integer inputs , since we are ultimately concerned with modeling the set function fe ( S ) = g ( |S| ) . Satisfying g ( x ) ≤ ` ( x ) ≤ ( 1 + ε ) g ( x ) for all x ∈ [ 0 , k ] is a much stronger requirement and may require more CB-gadgets than is actually necessary to approximate fe . Connection to piecewise linear approximation Our first result establishes a precise one-to-one correspondence between a certain class of piecewise linear functions and combined gadget functions . Lemma 1 The k-node CGF in ( 4 ) is nonnegative , piecewise linear , concave , and has exactly J + 1 linear pieces . Conversely , let ` ′ : [ 0 , k ] → R+ be concave and piecewise linear with J + 1 linear pieces , and let mi be the slope of the ith linear piece and Bi be the ith breakpoint . Then ` ′ is uniquely characterized as the k-node CGF parameterized by ai = 1k ( mi −mi+1 ) and bi = Bi for i ∈ { 1 , 2 , . . . J } , z0 = ` ′ ( 0 ) /k , and zk = ` ′ ( k ) /k . This result tells us that solving SpAR ( Def 2 ) for g is equivalent to finding a concave piecewise linear function with the smallest number of linear pieces approximating g at integer points . Although some techniques for approximating a concave function with a piecewise linear curve already exist [ 34 ] and could be used as heuristics , these are ultimately unable to find optimal solutions for SpAR . First of all , these methods focus on approximating concave functions over continuous intervals rather than just at integer inputs . More importantly , they do not provide instance optimal approximations , but only give upper bounds on the number of linear pieces needed . We therefore turn to new methods that will allow us to exactly solve our sparse approximation problem . Algorithm 1 GREEDYPLCOVER ( g , ε ) Input : ε ≥ 0 , concave function g Output : piecewise linear ` with fewest linear pieces such that g ( i ) ≤ ` ( i ) ≤ ( 1 + ε ) g ( i ) L ← ∅ , u← 0 //u = smallest integer not covered by approximating line while u ≤ k do u′ , L← NEXTLINE ( g , u , ε ) //line covering widest range [ u , u ’ -1 ] L ← L ∪ { L } , u← u′ end while Return ` ( x ) = minL∈L L ( x )

Summary: This paper presents an algorithm for minimizing submodular functions that are the sum of cardinality-based components. This is a special case of decomposable submodular function minimization. Formally, the problem considered is to minimize $f(S) = \sum\limits_{e\in E} g_e(|S \cap e|)$, where $E\subseteq 2^V$ and $g_e$ are 1-dimensional concave functions. One instance of such functions are the region potentials used in image segmentation, where $g_e(x) = c_e \cdot x (|e| - x)$ for some constant $c_e \geq 0$. Such minimization problems are also considered in hypergraph clustering applications.  The authors propose an approximation algorithm that works by approximately reducing their problem to a graph min cut problem. The fact that concave functions can be _exactly_ represented by graphs is known, and the resulting number of edges in the graph is $O\left(\sum\limits_{e\in E} |e|^2\right)$. The main result of the paper is to show how to reduce the number of edges by allowing _approximate_ reductions, leading to a graph with $\widetilde{O}\left(\frac{1}{\epsilon} \sum\limits_{e\in E} |e|\right)$ edges while incurring an $(1+\epsilon)$-multiplicative error in the objective.  The idea is to simplify each concave function before converting it to a graph. Viewing each concave function as a piecewise linear function, the size of the resulting graph depends on the number of pieces. The authors present an algorithm that is able to find the _best_ (in terms of the number of pieces) $(1+\epsilon)$-approximation to this function. By known results, the optimal number of pieces is $\widetilde{O}\left(\frac{1}{\epsilon}\right)$, thus follows the $\widetilde{O}\left(\frac{1}{\epsilon} \sum\limits_{e\in E} |e|\right)$ bound on the number of edges.  For the special case of the region potentials $g_e(x) = c_e \cdot x (|e| - x)$, the authors are able to get a better bound of $\widetilde{O}\left(\frac{1}{\sqrt{\epsilon}} \sum\limits_{e\in E} |e|\right)$ on the number of edges.  The theoretical results are accompanied by experimental results in the problems of image segmentation and hypergraph clustering. For image segmentation, the results show a speedup of at least 3x in 3 out of 4 experiments compared to previous methods, while it is slower in 1 of the 4 experiments. For hypergraph clustering, the F1 score is improved by 3 percentage points at the expense of a 30% runtime increase.