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Recently, neural network architectures have been developed to accommodate when the data has the structure of a graph or, more generally, a hypergraph. While useful, graph structures can be potentially limiting. Hypergraph structures in general do not account for higher order relations between their hyperedges. Simplicial complexes offer a middle ground, with a rich theory to draw on. We develop a convolutional neural network layer on simplicial 2-complexes.
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Deep Neural Networks (DNNs) have achieved im- pressive accuracy in many application domains including im- age classification. Training of DNNs is an extremely compute- intensive process and is solved using variants of the stochastic gradient descent (SGD) algorithm. A lot of recent research has focussed on improving the performance of DNN training. In this paper, we present optimization techniques to improve the performance of the data parallel synchronous SGD algorithm using the Torch framework: (i) we maintain data in-memory to avoid file I/O overheads, (ii) we present a multi-color based MPI Allreduce algorithm to minimize communication overheads, and (iii) we propose optimizations to the Torch data parallel table framework that handles multi-threading. We evaluate the performance of our optimizations on a Power 8 Minsky cluster with 32 nodes and 128 NVidia Pascal P100 GPUs. With our optimizations, we are able to train 90 epochs of the ResNet-50 model on the Imagenet-1k dataset using 256 GPUs in just 48 minutes. This significantly improves on the previously best known performance of training 90 epochs of the ResNet-50 model on the same dataset using 256 GPUs in 65 minutes. To the best of our knowledge, this is the best known training performance demonstrated for the Imagenet- 1k dataset.
We show that the Connes-Consani semi-norm on singular homology with real coefficients, defined via s-modules, coincides with the ordinary $ell^1$-semi-norm on singular homology in all dimensions.
In this paper we develop further the multi-parameter model of random simplicial complexes. Firstly, we give an intrinsic characterisation of the multi-parameter probability measure. Secondly, we show that in multi-parameter random simplicial complexes the links of simplexes and their intersections are also multi-parameter random simplicial complexes. Thirdly, we find conditions under which a multi-parameter random simplicial complex is connected and simply connected.
There are many ways to present model categories, each with a different point of view. Here wed like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras, it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) Were going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well.
Given a simplicial complex K with weights on its simplices and a chain on it, the Optimal Homologous Chain Problem (OHCP) is to find a chain with minimal weight that is homologous (over the integers) to the given chain. The OHCP is NP-complete, but if the boundary matrix of K is totally unimodular (TU), it becomes solvable in polynomial time when modeled as a linear program (LP). We define a condition on the simplicial complex called non total-unimodularity neutralized, or NTU neutralized, which ensures that even when the boundary matrix is not TU, the OHCP LP must contain an integral optimal vertex for every input chain. This condition is a property of K, and is independent of the input chain and the weights on the simplices. This condition is strictly weaker than the boundary matrix being TU. More interestingly, the polytope of the OHCP LP may not be integral under this condition. Still, an integral optimal vertex exists for every right-hand side, i.e., for every input chain. Hence a much larger class of OHCP instances can be solved in polynomial time than previously considered possible. As a special case, we show that 2-complexes with trivial first homology group are guaranteed to be NTU neutralized.
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