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Sparsification of Decomposable Submodular Functions

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Akbar Rafiey, Yuichi Yoshida

Data Structures and Algorithms · Submitted 18 Jan 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Our main result is a polynomial-time randomized sparsification algorithm such that the expected number of functions used in the output is independent of the number of underlying submodular functions in the original function.

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arXiv ID2201.07289 (version 1)
Pith IDVYTSFY6V, Pith's permanent record of this paper and its review
CitationAkbar Rafiey, Yuichi Yoshida. "Sparsification of Decomposable Submodular Functions." arXiv:2201.07289 (2022). https://pith.science/paper/VYTSFY6V
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,617 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.21 · human participants 0.02 · controlled comparison 0.05 · randomized assignment 0.05 · states quantitative result 0.30 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Submodular functions are at the core of many machine learning and data mining tasks. The underlying submodular functions for many of these tasks are decomposable, i.e., they are sum of several simple submodular functions. In many data intensive applications, however, the number of underlying submodular functions in the original function is so large that we need prohibitively large amount of time to process it and/or it does not even fit in the main memory. To overcome this issue, we introduce the notion of sparsification for decomposable submodular functions whose objective is to obtain an accurate approximation of the original function that is a (weighted) sum of only a few submodular functions. Our main result is a polynomial-time randomized sparsification algorithm such that the expected number of functions used in the output is independent of the number of underlying submodular functions in the original function. We also study the effectiveness of our algorithm under various constraints such as matroid and cardinality constraints. We complement our theoretical analysis with an empirical study of the performance of our algorithm.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Submodular functions are at the core of many machine learning and data mining tasks. background 1.00 0.04 0.01 asserted 1.00
s1 The underlying submodular functions for many of these tasks are decomposable, i.e., they are sum of several simple submodular functions. background 1.00 0.06 0.02 asserted 0.99
s2 In many data intensive applications, however, the number of underlying submodular functions in the original function is so large that we need prohibitively large amount of time to process it and/or it does not even fit in the main memory. background 0.78 0.09 0.02 asserted 0.95
s3 To overcome this issue, we introduce the notion of sparsification for decomposable submodular functions whose objective is to obtain an accurate approximation of the original function that is a (weighted) sum of only a few submodular functions. contribution 0.96 0.84 0.10 asserted 0.99
s4 Our main result is a polynomial-time randomized sparsification algorithm such that the expected number of functions used in the output is independent of the number of underlying submodular functions in the original function. result 0.82 0.96 0.97 asserted 1.00
s5 We also study the effectiveness of our algorithm under various constraints such as matroid and cardinality constraints. contribution 0.98 0.92 0.02 asserted 0.87
s6 We complement our theoretical analysis with an empirical study of the performance of our algorithm. contribution 0.89 0.95 0.03 asserted 0.94

What kind of paper the classifier reports this is: method or tool 0.99 · evidence from proof or derivation 0.90.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Submodular functions are at the core of many machine learning and data mining tasks. The underlying submodular functions for many of these tasks are decomposable, i.e., they are sum of several simple submodular functions. In many data intensive applications, however, the number of underlying submodular functions in the original function is so large that we need prohibitively large amount of time to process it and/or it does not even fit in the main memory. To overcome this issue, we introduce the notion of sparsification for decomposable submodular functions whose objective is to obtain an accurate approximation of the original function that is a (weighted) sum of only a few submodular functions. Our main result is a polynomial-time randomized sparsification algorithm such that the expected number of functions used in the output is independent of the number of underlying submodular functions in the original function. We also study the effectiveness of our algorithm under various constraints such as matroid and cardinality constraints. We complement our theoretical analysis with an empirical study of the performance of our algorithm.

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  • As a starting point, we observe in Section [ref] (and prove in Appendix [ref]) that the sampling algorithm from [ref] used to prove Theorem [ref] is largely independent of submodularity, leading to a more general sparsification algorithm for decomposable functions.

    Sparsification of Monotone $k$-Submodular Functions of Low Curvature · Introduction · 0.84

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  • This problem has been studied by [ref] and its special cases where submodular functions $f_e$ take particular forms have been studied with applications in semi-supervised learning, clustering, and rank learning (see [ref] for more details).

    Hypergraph Cuts with General Splitting Functions · Submodular function minimization · 0.96

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