2,032 research outputs found
MAP inference via Block-Coordinate Frank-Wolfe Algorithm
We present a new proximal bundle method for Maximum-A-Posteriori (MAP)
inference in structured energy minimization problems. The method optimizes a
Lagrangean relaxation of the original energy minimization problem using a multi
plane block-coordinate Frank-Wolfe method that takes advantage of the specific
structure of the Lagrangean decomposition. We show empirically that our method
outperforms state-of-the-art Lagrangean decomposition based algorithms on some
challenging Markov Random Field, multi-label discrete tomography and graph
matching problems
Efficient Relaxations for Dense CRFs with Sparse Higher Order Potentials
Dense conditional random fields (CRFs) have become a popular framework for
modelling several problems in computer vision such as stereo correspondence and
multi-class semantic segmentation. By modelling long-range interactions, dense
CRFs provide a labelling that captures finer detail than their sparse
counterparts. Currently, the state-of-the-art algorithm performs mean-field
inference using a filter-based method but fails to provide a strong theoretical
guarantee on the quality of the solution. A question naturally arises as to
whether it is possible to obtain a maximum a posteriori (MAP) estimate of a
dense CRF using a principled method. Within this paper, we show that this is
indeed possible. We will show that, by using a filter-based method, continuous
relaxations of the MAP problem can be optimised efficiently using
state-of-the-art algorithms. Specifically, we will solve a quadratic
programming (QP) relaxation using the Frank-Wolfe algorithm and a linear
programming (LP) relaxation by developing a proximal minimisation framework. By
exploiting labelling consistency in the higher-order potentials and utilising
the filter-based method, we are able to formulate the above algorithms such
that each iteration has a complexity linear in the number of classes and random
variables. The presented algorithms can be applied to any labelling problem
using a dense CRF with sparse higher-order potentials. In this paper, we use
semantic segmentation as an example application as it demonstrates the ability
of the algorithm to scale to dense CRFs with large dimensions. We perform
experiments on the Pascal dataset to indicate that the presented algorithms are
able to attain lower energies than the mean-field inference method
Efficient Linear Programming for Dense CRFs
The fully connected conditional random field (CRF) with Gaussian pairwise
potentials has proven popular and effective for multi-class semantic
segmentation. While the energy of a dense CRF can be minimized accurately using
a linear programming (LP) relaxation, the state-of-the-art algorithm is too
slow to be useful in practice. To alleviate this deficiency, we introduce an
efficient LP minimization algorithm for dense CRFs. To this end, we develop a
proximal minimization framework, where the dual of each proximal problem is
optimized via block coordinate descent. We show that each block of variables
can be efficiently optimized. Specifically, for one block, the problem
decomposes into significantly smaller subproblems, each of which is defined
over a single pixel. For the other block, the problem is optimized via
conditional gradient descent. This has two advantages: 1) the conditional
gradient can be computed in a time linear in the number of pixels and labels;
and 2) the optimal step size can be computed analytically. Our experiments on
standard datasets provide compelling evidence that our approach outperforms all
existing baselines including the previous LP based approach for dense CRFs.Comment: 24 pages, 10 figures and 4 table
Barrier Frank-Wolfe for Marginal Inference
We introduce a globally-convergent algorithm for optimizing the
tree-reweighted (TRW) variational objective over the marginal polytope. The
algorithm is based on the conditional gradient method (Frank-Wolfe) and moves
pseudomarginals within the marginal polytope through repeated maximum a
posteriori (MAP) calls. This modular structure enables us to leverage black-box
MAP solvers (both exact and approximate) for variational inference, and obtains
more accurate results than tree-reweighted algorithms that optimize over the
local consistency relaxation. Theoretically, we bound the sub-optimality for
the proposed algorithm despite the TRW objective having unbounded gradients at
the boundary of the marginal polytope. Empirically, we demonstrate the
increased quality of results found by tightening the relaxation over the
marginal polytope as well as the spanning tree polytope on synthetic and
real-world instances.Comment: 25 pages, 12 figures, To appear in Neural Information Processing
Systems (NIPS) 2015, Corrected reference and cleaned up bibliograph
A Multi-Plane Block-Coordinate Frank-Wolfe Algorithm for Training Structural SVMs with a Costly max-Oracle
Structural support vector machines (SSVMs) are amongst the best performing
models for structured computer vision tasks, such as semantic image
segmentation or human pose estimation. Training SSVMs, however, is
computationally costly, because it requires repeated calls to a structured
prediction subroutine (called \emph{max-oracle}), which has to solve an
optimization problem itself, e.g. a graph cut.
In this work, we introduce a new algorithm for SSVM training that is more
efficient than earlier techniques when the max-oracle is computationally
expensive, as it is frequently the case in computer vision tasks. The main idea
is to (i) combine the recent stochastic Block-Coordinate Frank-Wolfe algorithm
with efficient hyperplane caching, and (ii) use an automatic selection rule for
deciding whether to call the exact max-oracle or to rely on an approximate one
based on the cached hyperplanes.
We show experimentally that this strategy leads to faster convergence to the
optimum with respect to the number of requires oracle calls, and that this
translates into faster convergence with respect to the total runtime when the
max-oracle is slow compared to the other steps of the algorithm.
A publicly available C++ implementation is provided at
http://pub.ist.ac.at/~vnk/papers/SVM.html
Reflection methods for user-friendly submodular optimization
Recently, it has become evident that submodularity naturally captures widely
occurring concepts in machine learning, signal processing and computer vision.
Consequently, there is need for efficient optimization procedures for
submodular functions, especially for minimization problems. While general
submodular minimization is challenging, we propose a new method that exploits
existing decomposability of submodular functions. In contrast to previous
approaches, our method is neither approximate, nor impractical, nor does it
need any cumbersome parameter tuning. Moreover, it is easy to implement and
parallelize. A key component of our method is a formulation of the discrete
submodular minimization problem as a continuous best approximation problem that
is solved through a sequence of reflections, and its solution can be easily
thresholded to obtain an optimal discrete solution. This method solves both the
continuous and discrete formulations of the problem, and therefore has
applications in learning, inference, and reconstruction. In our experiments, we
illustrate the benefits of our method on two image segmentation tasks.Comment: Neural Information Processing Systems (NIPS), \'Etats-Unis (2013
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