Corpus ID: 31861010

Active-set Methods for Submodular Minimization Problems

  title={Active-set Methods for Submodular Minimization Problems},
  author={K. S. Sesh Kumar and Francis R. Bach},
  journal={J. Mach. Learn. Res.},
We consider the submodular function minimization (SFM) and the quadratic minimization problemsregularized by the Lov'asz extension of the submodular function. These optimization problemsare intimately related; for example,min-cut problems and total variation denoising problems, wherethe cut function is submodular and its Lov'asz extension is given by the associated total variation.When a quadratic loss is regularized by the total variation of a cut function, it thus becomes atotal variation… Expand
Fast Decomposable Submodular Function Minimization using Constrained Total Variation
A modified convex problem requiring constrained version of the total variation oracles that can be solved with significantly fewer calls to the simple minimization oracles is considered. Expand
Stochastic Submodular Maximization: The Case of Coverage Functions
This model captures situations where the discrete objective arises as an empirical risk, or is given as an explicit stochastic model, and yields solutions that are guaranteed to match the optimal approximation guarantees, while reducing the computational cost by several orders of magnitude, as demonstrated empirically. Expand
Optimization of Graph Total Variation via Active-Set-based Combinatorial Reconditioning
This work proposes a novel adaptive preconditioner driven by a sharp analysis of the local linear convergence rate depending on the "active set" at the current iterate, and shows that nested-forest decomposition of the inactive edges yields a guaranteed locallinear convergence rate. Expand
Differentiable Learning of Submodular Models
Can we incorporate discrete optimization algorithms within modern machine learning models? For example, is it possible to use in deep architectures a layer whose output is the minimal cut of aExpand
Submodular Kernels for Efficient Rankings
This work exploits geometric structure of ranked data and additional available information about the objects to derive a submodular kernel that drastically reduces the computational cost compared to state-of-the-art kernels and scales well to large datasets while attaining good empirical performance. Expand
Novel applications of intelligent computing paradigms for the analysis of nonlinear reactive transport model of the fluid in soft tissues and microvessels
The methodology integrates the artificial neural network, genetic algorithms, and pattern search aided by active-set technique (AST) and interior-point technique (IPT) to solve a one-dimensional steady-state nonlinear reactive transport model (RTM) that is meant for fluid and solute transport model of soft tissues and microvessels. 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
Efficient Minimization of Decomposable Submodular Functions
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
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
Reflection methods for user-friendly submodular optimization
This work proposes a new method that exploits existing decomposability of submodular functions, and solves both the continuous and discrete formulations of the problem, and therefore has applications in learning, inference, and reconstruction. Expand
Provable Submodular Minimization using Wolfe's Algorithm
A maiden convergence analysis of Wolfe's algorithm is given and a robust version of Fujishige's theorem is proved which shows that an O(1/n2)-approximate solution to the min-norm point on the base polytope implies exact submodular minimization. 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
MRF Energy Minimization and Beyond via Dual Decomposition
It is shown that by appropriately choosing what subproblems to use, one can design novel and very powerful MRF optimization algorithms, which are able to derive algorithms that generalize and extend state-of-the-art message-passing methods, and take full advantage of the special structure that may exist in particular MRFs. Expand
Convex Optimization for Parallel Energy Minimization
This work reformulates the quadratic energy minimization problem as a total variation denoising problem, which, when viewed geometrically, enables the use of projection and reflection based convex methods and performs an extensive empirical evaluation comparing state-of-the-art combinatorial algorithms and convex optimization techniques. Expand
Submodular systems and related topics
Let Open image in new window be a distributive lattice formed by subsets of a finite set with set union and intersection as the lattice operations, and let f be a submodular function on Open imageExpand
Active set algorithms for isotonic regression; A unifying framework
The active set approach provides a unifying framework for studying algorithms for isotonic regression, simplifies the exposition of existing algorithms and leads to several new efficient algorithms, including a new O(n) primal feasible active set algorithm. Expand