Corpus ID: 166228623

Fast Decomposable Submodular Function Minimization using Constrained Total Variation

  title={Fast Decomposable Submodular Function Minimization using Constrained Total Variation},
  author={Senanayak Sesh Kumar Karri and Francis R. Bach and Thomas Pock},
We consider the problem of minimizing the sum of submodular set functions assuming minimization oracles of each summand function. Most existing approaches reformulate the problem as the convex minimization of the sum of the corresponding Lovasz extensions and the squared Euclidean norm, leading to algorithms requiring total variation oracles of the summand functions; without further assumptions, these more complex oracles require many calls to the simpler minimization oracles often available in… Expand
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This paper develops an algorithm, SLG, that can efficiently minimize decomposable submodular functions with tens of thousands of variables, and applies it to synthetic benchmarks and a joint classification-and-segmentation task, and shows that it outperforms the state-of-the-art general purpose sub modular minimization algorithms by several orders of magnitude. Expand
On the Convergence Rate of Decomposable Submodular Function Minimization
It is shown that the algorithm converges linearly, and the upper and lower bounds on the rate of convergence are provided, which relies on the geometry of submodular polyhedra and draws on results from spectral graph theory. Expand
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This paper uses random coordinate descent methods to obtain algorithms with faster linear convergence rates and cheaper iteration costs, and their algorithms converge in significantly fewer iterations. Expand
Minimizing a sum of submodular functions
  • V. Kolmogorov
  • Computer Science, Mathematics
  • Discret. Appl. Math.
  • 2012
This work casts the problem of minimizing a function represented as a sum of submodular terms in an auxiliary graph in such a way that applying most existing SF algorithms would rely only on these subroutines, and shows how to improve its complexity in the case when the function contains cardinality-dependent terms. Expand
Active-set Methods for Submodular Minimization Problems
A new active-set algorithm for total variation denoising with the assumption of an oracle that solves the corresponding SFM problem is proposed, which performs local descent over ordered partitions and its ability to warm start considerably improves the performance of the algorithm. Expand
Convex Analysis for Minimizing and Learning Submodular Set Functions
The connections between convexity and submodularity are explored, for purposes of minimizing and learning submodular set functions. First, we develop a novel method for minimizing a particular classExpand
Submodular functions: from discrete to continuous domains
  • F. Bach
  • Computer Science, Mathematics
  • Math. Program.
  • 2019
This paper shows that most results relating submodularity and convexity for set-functions can be extended to all submodular functions, and provides practical algorithms which are based on function evaluations, to minimize generic sub modular functions on discrete domains, with associated convergence rates. Expand
Decomposable Submodular Function Minimization: Discrete and Continuous
Improved running time estimates are provided for the state-of-the-art continuous algorithms for the decomposable submodular function minimization problem using combinatorial arguments and a systematic experimental comparison of the two types of methods is provided. Expand
Geometric Rescaling Algorithms for Submodular Function Minimization
A new class of polynomial-time algorithms for submodular function minimization (SFM), as well as a unified framework to obtain stronglyPolynomial SFM algorithms, which can be applied to a wide range of combinatorial and continuous algorithms, including pseudo-polynomial ones. Expand
Learning with Submodular Functions: A Convex Optimization Perspective
  • F. Bach
  • Computer Science, Mathematics
  • Found. Trends Mach. Learn.
  • 2013
In Learning with Submodular Functions: A Convex Optimization Perspective, the theory of submodular functions is presented in a self-contained way from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems. Expand