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1. Variational Optimization on Lie Groups, with Examples of Leading (Generalized) Eigenvalue Problems

   Tao, Molei ; Ohsawa, Tomoki ( August 2020 , Proceedings of Machine Learning Research)

   Chiappa, Silvia ; Calandra, Roberto (Ed.)

   The article considers smooth optimization of functions on Lie groups. By generalizing NAG variational principle in vector space (Wibisono et al., 2016) to general Lie groups, continuous Lie-NAG dynamics which are guaranteed to converge to local optimum are obtained. They correspond to momentum versions of gradient flow on Lie groups. A particular case of SO(𝑛) is then studied in details, with objective functions corresponding to leading Generalized EigenValue problems: the Lie-NAG dynamics are first made explicit in coordinates, and then discretized in structure preserving fashions, resulting in optimization algorithms with faithful energy behavior (due to conformal symplecticity) and exactly remaining on the Lie group. Stochastic gradient versions are also investigated. Numerical experiments on both synthetic data and practical problem (LDA for MNIST) demonstrate the effectiveness of the proposed methods as optimization algorithms (\emph{not} as a classification method).
2. An updated LLVM-based quantum research compiler with further OpenQASM support

   https://doi.org/10.1088/2058-9565/ab8c2c  

   Litteken, Andrew ; Fan, Yung-Ching ; Singh, Devina ; Martonosi, Margaret ; Chong, Frederic T. ( May 2020 , Quantum Science and Technology)

   Quantum computing is a rapidly growing field with the potential to change how we solve previously intractable problems. Emerging hardware is approaching a complexity that requires increasingly sophisticated programming and control. Scaffold is an older quantum programming language that was originally designed for resource estimation for far-future, large quantum machines, and ScaffCC is the corresponding LLVM-based compiler. For the first time, we provide a full and complete overview of the language itself, the compiler as well as its pass structure. While previous works Abhariet al(2015Parallel Comput.452–17), Abhariet al(2012 Scaffold: quantum programming languagehttps://cs.princeton.edu/research/techreps/TR-934-12), have piecemeal descriptions of different portions of this toolchain, we provide a more full and complete description in this paper. We also introduce updates to ScaffCC including conditional measurement and multidimensional qubit arrays designed to keep in step with modern quantum assembly languages, as well as an alternate toolchain targeted at maintaining correctness and low resource count for noisy-intermediate scale quantum (NISQ) machines, and compatibility with current versions of LLVM and Clang. Our goal is to provide the research community with a functional LLVM framework for quantum program analysis, optimization, and generation of executable code.
3. Algorithms to Approximate Column-Sparse Packing Problems

   Brubach, Brian ; Sankararaman, Karthik A ; Srinivasan, Aravind ; Xu, Pan ( January 2018 , ACM-SIAM Symposium on Discrete Algorithms (SODA))

   Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go “half the remaining distance” to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).
4. On the Distributed Complexity of Large-Scale Graph Computations

   https://doi.org/10.1145/3460900  

   Pandurangan, Gopal ; Robinson, Peter ; Scquizzato, Michele ( June 2021 , ACM Transactions on Parallel Computing)

   Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where k ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n >> k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. Our main contribution is the General Lower Bound Theorem , a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. This result is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic, and this theorem can be used in a “cookbook” fashion to show distributed lower bounds for several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds on the round complexity of two fundamental graph problems, namely, PageRank computation and triangle enumeration . These applications show that our approach can yield lower boundsmore »for problems where the application of communication complexity techniques seems not obvious or gives weak bounds, including and especially under a stochastic partition of the input. We then present distributed algorithms for PageRank and triangle enumeration with a round complexity that (almost) matches the respective lower bounds; these algorithms exhibit a round complexity that scales superlinearly in k , improving significantly over previous results [Klauck et al., SODA 2015]. Specifically, we show the following results: PageRank: We show a lower bound of Ὼ(n/k 2 ) rounds and present a distributed algorithm that computes an approximation of the PageRank of all the nodes of a graph in Õ(n/k 2 ) rounds. Triangle enumeration: We show that there exist graphs with m edges where any distributed algorithm requires Ὼ(m/k 5/3 ) rounds. This result also implies the first non-trivial lower bound of Ὼ(n 1/3 ) rounds for the congested clique model, which is tight up to logarithmic factors. We then present a distributed algorithm that enumerates all the triangles of a graph in Õ(m/k 5/3 + n/k 4/3 ) rounds.« less
5. QuadriFlow: A Scalable and Robust Method for Quadrangulation

   Huang, J. ; Zhou, Y. ; Niessner, M. ; Shewchuk, J. R. ; Guibas, L. J. ( July 2018 , Symposium on Geometry Processing)

   QuadriFlow is a scalable algorithm for generating quadrilateral surface meshes based on the Instant Field-Aligned Meshes of Jakob et al. (ACM Trans. Graph. 34(6):189, 2015). We modify the original algorithm such that it efficiently produces meshes with many fewer singularities. Singularities in quadrilateral meshes cause problems for many applications, including parametrization and rendering with Catmull-Clark subdivision surfaces. Singularities can rarely be entirely eliminated, but it is possible to keep their number small. Local optimization algorithms usually produce meshes with many singularities, whereas the best algorithms tend to require non-local optimization, and therefore are slow. We propose an efficient method to minimize singularities by combining the Instant Meshes objective with a system of linear and quadratic constraints. These constraints are enforced by solving a global minimum-cost network flow problem and local boolean satisfiability problems. We have verified the robustness and efficiency of our method on a subset of ShapeNet comprising 17,791 3D objects in the wild. Our evaluation shows that the quality of the quadrangulations generated by our method is as good as, if not better than, those from other methods, achieving about four times fewer singularities than Instant Meshes. Other algorithms that produce similarly few singularities are much slower; wemore »take less than ten seconds to process each model. Our source code is publicly available.« less