Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Compact Linear Programming Relaxation for Binary Sub-modular MRF
158
0
0.0
(
0
)
Added by
Junyan Wang
Publication date
2014
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We propose a novel compact linear programming (LP) relaxation for binary sub-modular MRF in the context of object segmentation. Our model is obtained by linearizing an $l_1^+$-norm derived from the quadratic programming (QP) form of the MRF energy. The resultant LP model contains significantly fewer variables and constraints compared to the conventional LP relaxation of the MRF energy. In addition, unlike QP which can produce ambiguous labels, our model can be viewed as a quasi-total-variation minimization problem, and it can therefore preserve the discontinuities in the labels. We further establish a relaxation bound between our LP model and the conventional LP model. In the experiments, we demonstrate our method for the task of interactive object segmentation. Our LP model outperforms QP when converting the continuous labels to binary labels using different threshold values on the entire Oxford interactive segmentation dataset. The computational complexity of our LP is of the same order as that of the QP, and it is significantly lower than the conventional LP relaxation.
rate research
Read More
Superpixels have become prevalent in computer vision. They have been used to achieve satisfactory performance at a significantly smaller computational cost for various tasks. People have also combined superpixels with Markov random field (MRF) models. However, it often takes additional effort to formulate MRF on superpixel-level, and to the best of our knowledge there exists no principled approach to obtain this formulation. In this paper, we show how generic pixel-level binary MRF model can be solved in the superpixel space. As the main contribution of this paper, we show that a superpixel-level MRF can be derived from the pixel-level MRF by substituting the superpixel representation of the pixelwise label into the original pixel-level MRF energy. The resultant superpixel-level MRF energy also remains submodular for a submodular pixel-level MRF. The derived formula hence gives us a handy way to formulate MRF energy in superpixel-level. In the experiments, we demonstrate the efficacy of our approach on several computer vision problems.
We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In contrast to the situation in dimensions 8 and 24, where the linear programming bound is sharp, we show that it comes nowhere near the best packing densities known in dimensions 12, 16, 20, 28, and 32. More generally, we provide a systematic technique for proving separations of this sort.
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.
We give a characterization result for the integrality gap of the natural linear programming relaxation for the vertex cover problem. We show that integrality gap of the standard linear programming relaxation for any graph G equals $left(2-frac{2}{chi^f(G)}right)$ where $chi^f(G)$ denotes the fractional chromatic number of G.
Goldreich suggested candidates of one-way functions and pseudorandom generators included in $mathsf{NC}^0$. It is known that randomly generated Goldreichs generator using $(r-1)$-wise independent predicates with $n$ input variables and $m=C n^{r/2}$ output variables is not pseudorandom generator with high probability for sufficiently large constant $C$. Most of the previous works assume that the alphabet is binary and use techniques available only for the binary alphabet. In this paper, we deal with non-binary generalization of Goldreichs generator and derives the tight threshold for linear programming relaxation attack using local marginal polytope for randomly generated Goldreichs generators. We assume that $u(n)in omega(1)cap o(n)$ input variables are known. In that case, we show that when $rge 3$, there is an exact threshold $mu_mathrm{c}(k,r):=binom{k}{r}^{-1}frac{(r-2)^{r-2}}{r(r-1)^{r-1}}$ such that for $m=mufrac{n^{r-1}}{u(n)^{r-2}}$, the LP relaxation can determine linearly many input variables of Goldreichs generator if $mu>mu_mathrm{c}(k,r)$, and that the LP relaxation cannot determine $frac1{r-2} u(n)$ input variables of Goldreichs generator if $mu<mu_mathrm{c}(k,r)$. This paper uses characterization of LP solutions by combinatorial structures called stopping sets on a bipartite graph, which is related to a simple algorithm called peeling algorithm.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
