Potts model, parametric maxflow and k-submodular functions

The problem of minimizing the Potts energy function frequently occurs in computer vision applications. One way to tackle this NP-hard problem was proposed by Kovtun [19,20]. It identifies a part of an optimal solution by running $k$ maxflow computations, where $k$ is the number of labels. The number of "labeled" pixels can be significant in some applications, e.g. 50-93% in our tests for stereo. We show how to reduce the runtime to $O(\log k)$ maxflow computations (or one {\em parametric maxflow} computation). Furthermore, the output of our algorithm allows to speed-up the subsequent alpha expansion for the unlabeled part, or can be used as it is for time-critical applications. To derive our technique, we generalize the algorithm of Felzenszwalb et al. [7] for {\em Tree Metrics}. We also show a connection to {\em $k$-submodular functions} from combinatorial optimization, and discuss {\em $k$-submodular relaxations} for general energy functions.Comment: Accepted to ICCV 201

A Unified View of Piecewise Linear Neural Network Verification

The success of Deep Learning and its potential use in many safety-critical applications has motivated research on formal verification of Neural Network (NN) models. Despite the reputation of learned NN models to behave as black boxes and the theoretical hardness of proving their properties, researchers have been successful in verifying some classes of models by exploiting their piecewise linear structure and taking insights from formal methods such as Satisifiability Modulo Theory. These methods are however still far from scaling to realistic neural networks. To facilitate progress on this crucial area, we make two key contributions. First, we present a unified framework that encompasses previous methods. This analysis results in the identification of new methods that combine the strengths of multiple existing approaches, accomplishing a speedup of two orders of magnitude compared to the previous state of the art. Second, we propose a new data set of benchmarks which includes a collection of previously released testcases. We use the benchmark to provide the first experimental comparison of existing algorithms and identify the factors impacting the hardness of verification problems.Comment: Updated version of "Piecewise Linear Neural Network verification: A comparative study

A Test Vector Minimization Algorithm Based On Delta Debugging For Post-Silicon Validation Of Pcie Rootport

In silicon hardware design, such as designing PCIe devices, design verification is an essential part of the design process, whereby the devices are subjected to a series of tests that verify the functionality. However, manual debugging is still widely used in post-silicon validation and is a major bottleneck in the validation process. The reason is a large number of tests vectors have to be analyzed, and this slows process down. To solve the problem, a test vector minimizer algorithm is proposed to eliminate redundant test vectors that do not contribute to reproduction of a test failure, hence, improving the debug throughput. The proposed methodology is inspired by the Delta Debugging algorithm which is has been used in automated software debugging but not in post-silicon hardware debugging. The minimizer operates on the principle of binary partitioning of the test vectors, and iteratively testing each subset (or complement of set) on a post-silicon System-Under-Test (SUT), to identify and eliminate redundant test vectors. Test results using test vector sets containing deliberately introduced erroneous test vectors show that the minimizer is able to isolate the erroneous test vectors. In test cases containing up to 10,000 test vectors, the minimizer requires about 16ns per test vector in the test case when only one erroneous test vector is present. In a test case with 1000 vectors including erroneous vectors, the same minimizer requires about 140μs per erroneous test vector that is injected. Thus, the minimizer’s CPU consumption is significantly smaller than the typical amount of time of a test running on SUT. The factors that significantly impact the performance of the algorithm are number of erroneous test vectors and distribution (spacing) of the erroneous vectors. The effect of total number of test vectors and position of the erroneous vectors are relatively minor compared to the other two. The minimization algorithm therefore was most effective for cases where there are only a few erroneous test vectors, with large number of test vectors in the set

Double Bubbles Minimize

The classical isoperimetric inequality in R^3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 120 degrees.Comment: 57 pages, 32 figures. Includes the complete code for a C++ program as described in the article. You can obtain this code by viewing the source of this articl