Some computability-theoretic reductions between principles around ATR0\mathsf{ATR}_0

We study the computational content of various theorems with reverse mathematical strength around Arithmetical Transfinite Recursion ($\mathsf{ATR}_0$) from the point of view of computability-theoretic reducibilities, in particular Weihrauch reducibility. Our first main result states that it is equally hard to construct an embedding between two given well-orderings, as it is to construct a Turing jump hierarchy on a given well-ordering. This answers a question of Marcone. We obtain a similar result for Fra\"iss\'e's conjecture restricted to well-orderings. We then turn our attention to K\"onig's duality theorem, which generalizes K\"onig's theorem about matchings and covers to infinite bipartite graphs. Our second main result shows that the problem of constructing a K\"onig cover of a given bipartite graph is roughly as hard as the following "two-sided" version of the aforementioned jump hierarchy problem: given a linear ordering $L$, construct either a jump hierarchy on $L$ (which may be a pseudohierarchy), or an infinite $L$-descending sequence. We also obtain several results relating the above problems with choice on Baire space (choosing a path on a given ill-founded tree) and unique choice on Baire space (given a tree with a unique path, produce said path)

Stochastic Training of Graph Convolutional Networks with Variance Reduction

Graph convolutional networks (GCNs) are powerful deep neural networks for graph-structured data. However, GCN computes the representation of a node recursively from its neighbors, making the receptive field size grow exponentially with the number of layers. Previous attempts on reducing the receptive field size by subsampling neighbors do not have a convergence guarantee, and their receptive field size per node is still in the order of hundreds. In this paper, we develop control variate based algorithms which allow sampling an arbitrarily small neighbor size. Furthermore, we prove new theoretical guarantee for our algorithms to converge to a local optimum of GCN. Empirical results show that our algorithms enjoy a similar convergence with the exact algorithm using only two neighbors per node. The runtime of our algorithms on a large Reddit dataset is only one seventh of previous neighbor sampling algorithms

A new condition for the uniform convergence of certain trigonometric series

The present paper proposes a new condition to replace both the ($O$-regularly varying) quasimonotone condition and a certain type of bounded variation condition, and shows the same conclusion for the uniform convergence of certain trigonometric series still holds.Comment: 10 page

The universal Vassiliev-Kontsevich invariant for framed oriented links

We give a generalization of the Reshetikhin-Turaev functor for tangles to get a combinatorial formula for the universal Vassiliev-Kontsevich invariant of framed oriented links which is coincident with the Kontsevich integral. The universal Vassiliev-Kontsevich invariant is constructed using the Drinfeld associator. We prove the uniqueness of the Drinfeld associator. As a corollary one gets the rationality of the Kontsevich integral. Many properties of the universal Vassiliev-Kontsevich invariant are established. Connections to quantum group invariants and to multiple zeta values are discussed.Comment: 24 page

A Remark on "Two-Sided" Monotonicity Condition: An Application to LpL^{p} Convergence

To verify the universal validity of the "two-sided" monotonicity condition introduced in [8], we will apply it to include more classical examples. The present paper selects the $L^{p}$ convergence case for this purpose. Furthermore, Theorem 3 shows that our improvements are not trivial.Comment: 10 page

Some Remarks on the Best Approximation Rate of Certain Trigonometric Series

The main object of the present paper is to give a complete result regarding the best approximation rate of certain trigonometric series in general complex valued continuous function space under a new condition which gives an essential generalization to $O$-regularly varying quasimonotonicity. An application is present in Section 3.Comment: 14 page

On L1 Convergence of Fourier Series of Complex Valued Functions

In the present paper, we give a brief review of $L^{1}$-convergence of trigonometric series. Previous known results in this direction are improved and generalized by establishing a new condition.Comment: 13 pages, Accepted by Studia Sci. Math. Hunga

Deterministic consensus maximization with biconvex programming

Consensus maximization is one of the most widely used robust fitting paradigms in computer vision, and the development of algorithms for consensus maximization is an active research topic. In this paper, we propose an efficient deterministic optimization algorithm for consensus maximization. Given an initial solution, our method conducts a deterministic search that forcibly increases the consensus of the initial solution. We show how each iteration of the update can be formulated as an instance of biconvex programming, which we solve efficiently using a novel biconvex optimization algorithm. In contrast to our algorithm, previous consensus improvement techniques rely on random sampling or relaxations of the objective function, which reduce their ability to significantly improve the initial consensus. In fact, on challenging instances, the previous techniques may even return a worse off solution. Comprehensive experiments show that our algorithm can consistently and greatly improve the quality of the initial solution, without substantial cost.Comment: European Conference on Computer Vision (ECCV) 2018, oral presentatio

Investigation of antineutrino spectral anomaly with updated nuclear database

Recently, three successful antineutrino experiments (Daya Bay, Double Chooz, and RENO) measured the neutrino mixing angle theta_13; however, significant discrepancies were found, both in the absolute flux and spectral shape. In this study, the antineutrino spectra were calculated by using the updated nuclear database, and we found that the four isotopes antineutrino spectrum have all contribution to the 5--7 MeV bump with ENDF/B-VII.1 fission yield. The bump can be explained well using the updated library and more important isotopes contribution to the bump were also given. In the last, the fission yield correlation coefficient between the four isotopes were discussed, and found that the correlation coefficients are very large.Comment: 5 pages, 7 figures. arXiv admin note: text overlap with arXiv:1805.0997

Spectrum Cartography via Coupled Block-Term Tensor Decomposition

Spectrum cartography aims at estimating power propagation patterns over a geographical region across multiple frequency bands (i.e., a radio map)---from limited samples taken sparsely over the region. Classic cartography methods are mostly concerned with recovering the aggregate radio frequency (RF) information while ignoring the constituents of the radio map---but fine-grained emitter-level RF information is of great interest. In addition, many existing cartography methods work explicitly or implicitly assume random spatial sampling schemes that may be difficult to implement, due to legal/privacy/security issues. The theoretical aspects (e.g., identifiability of the radio map) of many existing methods are also unclear. In this work, we propose a joint radio map recovery and disaggregation method that is based on coupled block-term tensor decomposition. Our method guarantees identifiability of the individual radio map of \textit{each emitter} (thereby that of the aggregate radio map as well), under realistic conditions. The identifiability result holds under a large variety of geographical sampling patterns, including a number of pragmatic systematic sampling strategies. We also propose effective optimization algorithms to carry out the formulated radio map disaggregation problems. Extensive simulations are employed to showcase the effectiveness of the proposed approach.Comment: Accepted version; IEEE Transactions on Signal Processing (27-Apr-2020