**Discrete Morse Theory**

Discrete Morse theory is a tool for determining equivalences between topological spaces arising from discrete mathematical structures. This theory was developed by Robin Forman in the 1990s as a combinatorial analog to Morse theory, developed by Marston Morse in the 1920s. The original theory deals with analyzing such equivalences for general topological spaces, while discrete Morse theory provides similar methods of analysis for topological spaces endowed with additional, discrete structure. For these structures, applications of the discrete theory are often more natural, as well as simpler and more straightforward to apply. Discrete Morse theory has applications throughout many fields of pure and applied mathematics. Within pure mathematics, for example, the theory has been widely applied to problems in geometry, topology, and knot theory; and within computer science, the theory has been used to evaluate data compression algorithms and to bound the complexity of algorithms that determine whether graphs have certain properties – for example, whether all components of a graph are connected. If we wish to know whether a given property holds for a certain topological space, our question can often be reduced to the question of whether the space is equivalent to another space for which the property holds. For example, whether a simple algorithm exists for determining if a graph is connected depends on whether the structure that represents the space of not-connected graphs can be shrunken to a point. Alas, it cannot, so any algorithm for testing graph connectedness must, at least in some cases, conduct an exhaustive search. This result has real-world implications: for example, it means that if we want to test a communications system – say, immediately after a disaster – to determine whether it is still connected, there is no guaranteed way of finding the answer without testing every component individually.

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http://…/s48forman.pdf … **Dynamic Sampling Convolutional Neural Network (DSCNN)**

We present Dynamic Sampling Convolutional Neural Networks (DSCNN), where the position-specific kernels learn from not only the current position but also multiple sampled neighbour regions. During sampling, residual learning is introduced to ease training and an attention mechanism is applied to fuse features from different samples. And the kernels are further factorized to reduce parameters. The multiple sampling strategy enlarges the effective receptive fields significantly without requiring more parameters. While DSCNNs inherit the advantages of DFN, namely avoiding feature map blurring by position-specific kernels while keeping translation invariance, it also efficiently alleviates the overfitting issue caused by much more parameters than normal CNNs. Our model is efficient and can be trained end-to-end via standard back-propagation. We demonstrate the merits of our DSCNNs on both sparse and dense prediction tasks involving object detection and flow estimation. Our results show that DSCNNs enjoy stronger recognition abilities and achieve 81.7% in VOC2012 detection dataset. Also, DSCNNs obtain much sharper responses in flow estimation on FlyingChairs dataset compared to multiple FlowNet models’ baselines. … **MOPLS-N**

Multi-Objective Optimization (MOO) is very difficult for expensive functions because most current MOO methods rely on a large number of function evaluations to get an accurate solution. We address this problem with surrogate approximation and parallel computation. We develop an MOO algorithm MOPLS-N for expensive functions that combines iteratively updated surrogate approximations of the objective functions with a structure for efficiently selecting a population of $N$ points so that the expensive objectives for all points are simultaneously evaluated on $N$ processors in each iteration. MOPLS incorporates Radial Basis Function (RBF) approximation, Tabu Search and local candidate search around multiple points to strike a balance between exploration, exploitation and diversification during each algorithm iteration. Eleven test problems (with 8 to 24 decision variables and two real-world watershed problems are used to compare performance of MOPLS to ParEGO, GOMORS, Borg, MOEA/D, and NSGA-III on a limited budget of evaluations with between 1 (serial) and 64 processors. MOPLS in serial is better than all non-RBF serial methods tested. Parallel speedup of MOPLS is higher than all other parallel algorithms with 16 and 64 processors. With both algorithms on 64 processors MOPLS is at least 2 times faster than NSGA-III on the watershed problems. … **Sourcegraph**

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