arXiv subjects

Jie Wu

Publications and source records attributed to Jie Wu.

At least 235 records · Page 13Linked to original sources

arXiv2021

Cohomology ring of manifold arrangements

We study the cohomology ring of the complement $\mathcal{M}(\mathcal{A})$ of a manifold arrangement $\mathcal{A}$ in a smooth manifold $M$ without boundary. We first give the concept of monoidal cosheaf on a locally geometric poset $\mathfrak{L}$, and then define the generalized Orlik--Solomon algebra $A^*(\mathfrak{L}, \mathcal{C})$ over a commutative ring with unit, which is built by the classical Orlik--Solomon algebra and a monoidal cosheaf $\mathcal{C}$ as coefficients. Furthermore, we construct a monoidal cosheaf $\hat{\mathcal{C}}(\mathcal{A})$ associated with $\mathcal{A}$, so that the generalized Orlik--Solomon algebra $A^*(\mathfrak{L}, \hat{\mathcal{C}}(\mathcal{A}))$ becomes a double complex with suitable multiplication structure and the associated total complex $Tot(A^*(\mathfrak{L}, \hat{\mathcal{C}}(\mathcal{A})))$ is a differential algebra. Our main result is that $H^*(Tot(A^*(\mathfrak{L}, \hat{\mathcal{C}}(\mathcal{A}))))$ is isomorphic to $H^*(\mathcal{M}(\mathcal{A}))$ as algebras. Our argument is of topological with the use of a spectral sequence induced by a geometric filtration associated with $\mathcal{A}$. In particular, we also discuss the mixed Hodge complex structure on our model if $M$ and all elements in $\mathcal{A}$ are complex smooth varieties, and show that it induces the canonical mixed Hodge structure of $\mathcal{M}(\mathcal{A})$. As an application, we calculate the cohomology of chromatic configuration spaces, which agrees with many known results in some special cases. In addition, some explicit formulas with respect to Poincaré polynomial and chromatic polynomial are also given.

math.AT↗

arXiv2021

On some sums involving the integral part function

Denote by $τ$ k (n), $ω$(n) and $μ$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = $ω$, 2 $ω$ , $μ$ 2 , $τ$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $ε$ (x $θ$ f +$ε$) for x $\rightarrow$ $\infty$, where $θ$ $ω$ = 53 110 , $θ$ 2 $ω$ = 9 19 , $θ$ $μ$2 = 2 5 , $θ$ $τ$ k = 5k--1 10k--1 and $ε$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{è}s.

math.NT↗

arXiv2021

Online Multi-Granularity Distillation for GAN Compression

Generative Adversarial Networks (GANs) have witnessed prevailing success in yielding outstanding images, however, they are burdensome to deploy on resource-constrained devices due to ponderous computational costs and hulking memory usage. Although recent efforts on compressing GANs have acquired remarkable results, they still exist potential model redundancies and can be further compressed. To solve this issue, we propose a novel online multi-granularity distillation (OMGD) scheme to obtain lightweight GANs, which contributes to generating high-fidelity images with low computational demands. We offer the first attempt to popularize single-stage online distillation for GAN-oriented compression, where the progressively promoted teacher generator helps to refine the discriminator-free based student generator. Complementary teacher generators and network layers provide comprehensive and multi-granularity concepts to enhance visual fidelity from diverse dimensions. Experimental results on four benchmark datasets demonstrate that OMGD successes to compress 40x MACs and 82.5X parameters on Pix2Pix and CycleGAN, without loss of image quality. It reveals that OMGD provides a feasible solution for the deployment of real-time image translation on resource-constrained devices. Our code and models are made public at: https://github.com/bytedance/OMGD.

cs.CV↗

arXiv2021

Classification of solutions of the 2D steady Navier-Stokes equations with separated variables in cone-like domains

We investigate the problem of classification of solutions for the steady Navier-Stokes equations in any cone-like domains. In the form of separated variables, $$u(x,y)=\left( \begin{array}{c} φ_1(r)v_1(θ) φ_2(r)v_2(θ) \end{array} \right) ,$$ where $x=r\cosθ$ and $y=r\sinθ$ in polar coordinates, we obtain the expressions of all smooth solutions with $C^0$ Dirichlet boundary condition. In particular, it shows that (i) some solutions are found, which are Hölder continuous on the boundary, but their gradients blow up at the corner; (ii) all solutions in the entire plane of $\mathbb{R}^2$ like harmonic functions or Stokes equations, are polynomial expressions.

math.AP↗

arXiv2021

Weakly-Supervised Spatio-Temporal Anomaly Detection in Surveillance Video

In this paper, we introduce a novel task, referred to as Weakly-Supervised Spatio-Temporal Anomaly Detection (WSSTAD) in surveillance video. Specifically, given an untrimmed video, WSSTAD aims to localize a spatio-temporal tube (i.e., a sequence of bounding boxes at consecutive times) that encloses the abnormal event, with only coarse video-level annotations as supervision during training. To address this challenging task, we propose a dual-branch network which takes as input the proposals with multi-granularities in both spatial-temporal domains. Each branch employs a relationship reasoning module to capture the correlation between tubes/videolets, which can provide rich contextual information and complex entity relationships for the concept learning of abnormal behaviors. Mutually-guided Progressive Refinement framework is set up to employ dual-path mutual guidance in a recurrent manner, iteratively sharing auxiliary supervision information across branches. It impels the learned concepts of each branch to serve as a guide for its counterpart, which progressively refines the corresponding branch and the whole framework. Furthermore, we contribute two datasets, i.e., ST-UCF-Crime and STRA, consisting of videos containing spatio-temporal abnormal annotations to serve as the benchmarks for WSSTAD. We conduct extensive qualitative and quantitative evaluations to demonstrate the effectiveness of the proposed approach and analyze the key factors that contribute more to handle this task.

cs.CV↗

arXiv2021

On The Discrete Morse Functions for Hypergraphs

A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. In this paper, we study the embedded homology as well as the homology of the (lower-)associated simplicial complexes for hypergraphs. We generalize the discrete Morse functions on simplicial complexes. We study the discrete Morse functions on hypergraphs as well as the discrete Morse functions on the (lower-)associated simplicial complexes of the hypergraphs.

math.CO↗

arXiv2021

A variant of the prime number theorem

Let $Λ(n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. In this note, we prove that for any $\varepsilon>0$ the asymptotic formula $$ \sum_{n\le x} Λ\Big(\Big[\frac{x}{n}\Big]\Big) = x\sum_{d\ge 1} \frac{Λ(d)}{d(d+1)} + O_{\varepsilon}\big(x^{9/19+\varepsilon}\big) \qquad (x\to\infty)$$ holds. This improves a recent result of Bordellès, which requires $\frac{97}{203}$ in place of $\frac{9}{19}$.

math.NT↗

arXiv2021

Arithmetic exponent pairs for algebraic trace functions and applications

We study short sums of algebraic trace functions via the $q$-analogue of van der Corput method, and develop methods of arithmetic exponent pairs that coincide with the classical case while the moduli has sufficiently good factorizations. As an application, we prove a quadratic analogue of Brun-Titchmarsh theorem on average, bounding the number of primes $p\leqslant X$ with $p^2+1\equiv0\pmod q$. The other two applications include a larger level of distribution of divisor functions in arithmetic progressions and a sub-Weyl subconvex bound of Dirichlet $L$-functions studied previously by Irving.

math.NT↗

arXiv2021

Addendum to "Arithmetic exponent pairs of algebraic trace functions and applications"

This addendum devotes to a detailed proof for the inequality (9.14) in our joint work: Arithmetic exponent pairs for algebraic trace functions and applications, with an appendix by Will Sawin, arXiv:1603.07060 [math.NT], which will appear in Algebra and Number Theory. We do not intend to publish this addendum in any journals; arXiv should be a good place for those reader who want to find such details. The proof involves various averages of arithmetic functions.

math.NT↗

arXiv2021

Optimal Data Placement for Data-Sharing Scientific Workflows in Heterogeneous Edge-Cloud Computing Environments

The heterogeneous edge-cloud computing paradigm can provide a more optimal direction to deploy scientific workflows than traditional distributed computing or cloud computing environments. Due to the different sizes of scientific datasets and some of these datasets must keep private, it is still a difficult problem to finding an data placement strategy that can minimize data transmission as well as placement cost. To address this issue, this paper combines advantages of both edge and cloud computing to construct a data placement model, which can balance data transfer time and data placement cost using intelligent computation. The most difficult research challenge the model solved is to consider many constrain in this hybrid computing environments, which including shared datasets within individual and among multiple workflows across various geographical regions. According to the constructed model, the study propose a new data placement strategy named DE-DPSO-DPS, which using a discrete particle swarm optimization algorithm with differential evolution (DE-DPSO-DPA) to distribute these scientific datasets. The strategy also not only consider the characteristics such as the number and storage capacity of edge micro-datacenters, the bandwidth between different datacenters and the proportion of private datasets, but also analysis the performance of algorithm during the workflows execution. Comprehensive experiments are designed in simulated heterogeneous edge-cloud computing environments demonstrate that the data placement strategy can effectively reduce the data transmission time and placement cost as compared to traditional strategies for data-sharing scientific workflows.

cs.DC↗

arXiv2021

A Unified Topological Approach to Data Science

We establish a new theory which gives a unified topological approach to data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and hypergraphs to super-hypergraphs and establish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology.

math.AT↗

arXiv2021

On homotopy braids

Homotopy braid group is the subject of the paper. First, linearity of homotopy braid group over the integers is proved. Then we prove that the group homotopy braid group on three strands is torsion free.

math.GR↗

arXiv2020

On topological representation theory from quivers

In this work, we introduce {\em topological representations of a quiver} as a system consisting of topological spaces and its relationships determined by the quiver. Such a setting gives a natural connection between topological representations of a quiver and diagrams of topological spaces. First, we investigate the relation between the category of topological representations and that of linear representations of a quiver via $P(Γ)$-$\mathcal{TOP}^o$ and $kΓ$-Mod, concerning (positively) graded or vertex (positively) graded modules. Second, we discuss the homological theory of topological representations of quivers via $Γ$-limit $Lim^Γ$ and using it, define the homology groups of topological representations of quivers via $H_n$. It is found that some properties of a quiver can be read from homology groups. Third, we investigate the homotopy theory of topological representations of quivers. We define the homotopy equivalence between two morphisms in $\textbf{Top}\mathrm{-}\textbf{Rep}Γ$ and show that the parallel Homotopy Axiom also holds for top-representations based on the homotopy equivalence. Last, we mainly obtain the functor $At^Γ$ from $\textbf{Top}\mathrm{-}\textbf{Rep}Γ$ to $\textbf{Top}$ and show that $At^Γ$ preserves homotopy equivalence between morphisms. The relationship is established between the homotopy groups of a top-representation $(T,f)$ and the homotopy groups of $At^Γ(T,f)$.

math.RT↗

arXiv2020

Sequential scaled sparse factor regression

Large-scale association analysis between multivariate responses and predictors is of great practical importance, as exemplified by modern business applications including social media marketing and crisis management. Despite the rapid methodological advances, how to obtain scalable estimators with free tuning of the regularization parameters remains unclear under general noise covariance structures. In this paper, we develop a new methodology called sequential scaled sparse factor regression (SESS) based on a new viewpoint that the problem of recovering a jointly low-rank and sparse regression coefficient matrix can be decomposed into several univariate response sparse regressions through regular eigenvalue decomposition. It combines the strengths of sequential estimation and scaled sparse regression, thus sharing the scalability and the tuning free property for sparsity parameters inherited from the two approaches. The stepwise convex formulation, sequential factor regression framework, and tuning insensitiveness make SESS highly scalable for big data applications. Comprehensive theoretical justifications with new insights into high-dimensional multi-response regressions are also provided. We demonstrate the scalability and effectiveness of the proposed method by simulation studies and stock short interest data analysis.

stat.ME↗

arXiv2020

Robust Sequence Submodular Maximization

Submodularity is an important property of set functions and has been extensively studied in the literature. It models set functions that exhibit a diminishing returns property, where the marginal value of adding an element to a set decreases as the set expands. This notion has been generalized to considering sequence functions, where the order of adding elements plays a crucial role and determines the function value; the generalized notion is called sequence (or string) submodularity. In this paper, we study a new problem of robust sequence submodular maximization with cardinality constraints. The robustness is against the removal of a subset of elements in the selected sequence (e.g., due to malfunctions or adversarial attacks). Compared to robust submodular maximization for set function, new challenges arise when sequence functions are concerned. Specifically, there are multiple definitions of submodularity for sequence functions, which exhibit subtle yet critical differences. Another challenge comes from two directions of monotonicity: forward monotonicity and backward monotonicity, both of which are important to proving performance guarantees. To address these unique challenges, we design two robust greedy algorithms: while one algorithm achieves a constant approximation ratio but is robust only against the removal of a subset of contiguous elements, the other is robust against the removal of an arbitrary subset of the selected elements but requires a stronger assumption and achieves an approximation ratio that depends on the number of the removed elements. Finally, we generalize the analyses to considering sequence functions under weaker assumptions based on approximate versions of sequence submodularity and backward monotonicity

cs.DS↗

arXiv2020

Reinforcement Learning for Weakly Supervised Temporal Grounding of Natural Language in Untrimmed Videos

Temporal grounding of natural language in untrimmed videos is a fundamental yet challenging multimedia task facilitating cross-media visual content retrieval. We focus on the weakly supervised setting of this task that merely accesses to coarse video-level language description annotation without temporal boundary, which is more consistent with reality as such weak labels are more readily available in practice. In this paper, we propose a \emph{Boundary Adaptive Refinement} (BAR) framework that resorts to reinforcement learning (RL) to guide the process of progressively refining the temporal boundary. To the best of our knowledge, we offer the first attempt to extend RL to temporal localization task with weak supervision. As it is non-trivial to obtain a straightforward reward function in the absence of pairwise granular boundary-query annotations, a cross-modal alignment evaluator is crafted to measure the alignment degree of segment-query pair to provide tailor-designed rewards. This refinement scheme completely abandons traditional sliding window based solution pattern and contributes to acquiring more efficient, boundary-flexible and content-aware grounding results. Extensive experiments on two public benchmarks Charades-STA and ActivityNet demonstrate that BAR outperforms the state-of-the-art weakly-supervised method and even beats some competitive fully-supervised ones.

cs.CV↗

arXiv2020

Fine-Grained Image Captioning with Global-Local Discriminative Objective

Significant progress has been made in recent years in image captioning, an active topic in the fields of vision and language. However, existing methods tend to yield overly general captions and consist of some of the most frequent words/phrases, resulting in inaccurate and indistinguishable descriptions (see Figure 1). This is primarily due to (i) the conservative characteristic of traditional training objectives that drives the model to generate correct but hardly discriminative captions for similar images and (ii) the uneven word distribution of the ground-truth captions, which encourages generating highly frequent words/phrases while suppressing the less frequent but more concrete ones. In this work, we propose a novel global-local discriminative objective that is formulated on top of a reference model to facilitate generating fine-grained descriptive captions. Specifically, from a global perspective, we design a novel global discriminative constraint that pulls the generated sentence to better discern the corresponding image from all others in the entire dataset. From the local perspective, a local discriminative constraint is proposed to increase attention such that it emphasizes the less frequent but more concrete words/phrases, thus facilitating the generation of captions that better describe the visual details of the given images. We evaluate the proposed method on the widely used MS-COCO dataset, where it outperforms the baseline methods by a sizable margin and achieves competitive performance over existing leading approaches. We also conduct self-retrieval experiments to demonstrate the discriminability of the proposed method.

cs.CV↗

arXiv2020

Adversarially Trained Multi-Singer Sequence-To-Sequence Singing Synthesizer

This paper presents a high quality singing synthesizer that is able to model a voice with limited available recordings. Based on the sequence-to-sequence singing model, we design a multi-singer framework to leverage all the existing singing data of different singers. To attenuate the issue of musical score unbalance among singers, we incorporate an adversarial task of singer classification to make encoder output less singer dependent. Furthermore, we apply multiple random window discriminators (MRWDs) on the generated acoustic features to make the network be a GAN. Both objective and subjective evaluations indicate that the proposed synthesizer can generate higher quality singing voice than baseline (4.12 vs 3.53 in MOS). Especially, the articulation of high-pitched vowels is significantly enhanced.

eess.AS↗