Piecewise Sparse Recovery in Unions of Bases

Sparse recovery is widely applied in many fields, since many signals or vectors can be sparsely represented under some frames or dictionaries. Most of fast algorithms at present are based on solving $l^0$ or $l^1$ minimization problems and they are efficient in sparse recovery. However, compared with the practical results, the theoretical sufficient conditions on the sparsity of the signal for $l^0$ or $l^1$ minimization problems and algorithms are too strict. \par In many applications, there are signals with certain structures as piecewise sparsity. Piecewise sparsity means that the sparse signal $\mathbf{x}$ is a union of several sparse sub-signals, i.e., $\mathbf{x}=(\mathbf{x}_1^T,\ldots,\mathbf{x}_N^T)^T$, corresponding to the matrix $A$ which is composed of union of bases $A=[A_1,\ldots,A_N]$. In this paper, we consider the uniqueness and feasible conditions for piecewise sparse recovery. We introduce the mutual coherence for the sub-matrices $A_i\ (i=1,\ldots,N)$to study the new upper bounds of$\|\mathbf{x}\|_0$(number of nonzero entries of signal) recovered by$l^0$or$l^1$optimizations. The structured information of measurement matrix$A$is used to improve the sufficient conditions for successful piecewise sparse recovery and also improve the reliability of$l_0$and$l_1$ optimization models on recovering global sparse vectors

Genus-One Stable Maps, Local Equations, and Vakil-Zinger's desingularization

We describe an algebro-geometric approach to Vakil-Zinger's desingularization of the main component of the moduli of genus one stable maps to projective space. The new approach provides complete local structural results for this moduli space as well as for the desingularization of the entire moduli space and should fully extend to higher genera.Comment: 31 pages, 2 figure

Derived Resolution Property for Stacks, Euler Classes and Applications

By resolving an arbitrary perfect derived object over a Deligne-Mumford stack, we define its Euler class. We then apply it to define the Euler numbers for a smooth Calabi-Yau threefold in the 4-dimensional projective space. These numbers are conjectured to be the reduced Gromov-Witten invariants and to determine the usual Gromov-Witten numbers of the smooth quintic as speculated by J. Li and A. Zinger.Comment: 16 page

Genus Two Stable Maps, Local Equations and Modular Resolutions

We geometrically describe a canonical sequence of modular blowups of the relative Picard stack of the Artin stack of pre-stable genus two curves. The final blowup stack locally diagonalizes certain tautological derived objects. This implies a resolution of the primary component of the moduli space of genus two stable maps to projective space and meanwhile makes the whole moduli space admit only normal crossing singularities. Our approach should extend to higher genera.Comment: 81 pages, 7 figure

Fast Asynchronous Parallel Stochastic Gradient Decent

Stochastic gradient descent~(SGD) and its variants have become more and more popular in machine learning due to their efficiency and effectiveness. To handle large-scale problems, researchers have recently proposed several parallel SGD methods for multicore systems. However, existing parallel SGD methods cannot achieve satisfactory performance in real applications. In this paper, we propose a fast asynchronous parallel SGD method, called AsySVRG, by designing an asynchronous strategy to parallelize the recently proposed SGD variant called stochastic variance reduced gradient~(SVRG). Both theoretical and empirical results show that AsySVRG can outperform existing state-of-the-art parallel SGD methods like Hogwild! in terms of convergence rate and computation cost

Bayesian Analysis of Rank Data with Covariates and Heterogeneous Rankers

Data in the form of ranking lists are frequently encountered, and combining ranking results from different sources can potentially generate a better ranking list and help understand behaviors of the rankers. Of interest here are the rank data under the following settings: (i) covariate information available for the ranked entities; (ii) rankers of varying qualities or having different opinions; and (iii) incomplete ranking lists for non-overlapping subgroups. We review some key ideas built around the Thurstone model family by researchers in the past few decades and provide a unifying approach for Bayesian Analysis of Rank data with Covariates (BARC) and its extensions in handling heterogeneous rankers. With this Bayesian framework, we can study rankers' varying quality, cluster rankers' heterogeneous opinions, and measure the corresponding uncertainties. To enable an efficient Bayesian inference, we advocate a parameter-expanded Gibbs sampler to sample from the target posterior distribution. The posterior samples also result in a Bayesian aggregated ranking list, with credible intervals quantifying its uncertainty. We investigate and compare performances of the proposed methods and other rank aggregation methods in both simulation studies and two real-data examples

Dynamics of quantum correlations for central two-qubit coupled to an isotropic Lipkin-Meshkov-Glick bath

We investigate the dynamics of quantum discord and entanglement for two central spin qubits coupled to an isotropic Lipkin-Meshkov-Glick bath. It is found that both quantum discord and entanglement have quite distinct behaviors with respect to the two different phases of the bath. In the case of the symmetry broken phase bath, quantum discord and entanglement can remain as constant. In the case of the symmetric phase bath, quantum discord and entanglement always periodically oscillate with time. The critical point of quantum phase transition of the bath can be revealed clearly by the distinct behaviors of quantum correlations. Furthermore, it is observed that quantum discord is significantly enhanced during the evolution while entanglement periodically vanishes

An Optimal Distributed (Δ+1)(\Delta+1)-Coloring Algorithm?

Vertex coloring is one of the classic symmetry breaking problems studied in distributed computing. In this paper we present a new algorithm for $(\Delta+1)$-list coloring in the randomized ${\sf LOCAL}$ model running in $O(\mathsf{Det}_{\scriptscriptstyle d}(\text{poly} \log n))$ time, where $\mathsf{Det}_{\scriptscriptstyle d}(n')$ is the deterministic complexity of $(\text{deg}+1)$-list coloring on $n'$-vertex graphs. (In this problem, each $v$ has a palette of size $\text{deg}(v)+1$.) This improves upon a previous randomized algorithm of Harris, Schneider, and Su [STOC'16, JACM'18] with complexity $O(\sqrt{\log \Delta} + \log\log n + \mathsf{Det}_{\scriptscriptstyle d}(\text{poly} \log n))$, and, for some range of $\Delta$, is much faster than the best known deterministic algorithm of Fraigniaud, Heinrich, and Kosowski [FOCS'16] and Barenboim, Elkin, and Goldenberg [PODC'18], with complexity $O(\sqrt{\Delta\log \Delta}\log^\ast \Delta + \log^* n)$. Our algorithm "appears to be" optimal, in view of the $\Omega(\mathsf{Det}(\text{poly} \log n))$ randomized lower bound due to Chang, Kopelowitz, and Pettie [FOCS'16], where $\mathsf{Det}$ is the deterministic complexity of $(\Delta+1)$-list coloring. At present, the best upper bounds on $\mathsf{Det}_{\scriptscriptstyle d}(n')$ and $\mathsf{Det}(n')$ are both $2^{O(\sqrt{\log n'})}$ and use a black box application of network decompositions (Panconesi and Srinivasan [Journal of Algorithms'96]). It is quite possible that the true complexities of both problems are the same, asymptotically, which would imply the randomized optimality of our $(\Delta+1)$-list coloring algorithm

Dynamics of quantum correlations for two-qubit coupled to a spin chain with Dzyaloshinskii-Moriya interaction

We study the dynamics of quantum discord and entanglement for two spin qubits coupled to a spin chain with Dzyaloshinsky-Moriya (DM) interaction. We numerically and analytically investigate the time evolution of quantum discord and entanglement for two-qubit initially prepared in a class of $X-$structure state. In the case of evolution from a pure state, quantum correlations decay to zero in a very short time at the critical point of the environment. In the case of evolution from a mixed state, It is found that quantum discord may get maximized due to the quantum critical behavior of the environment while entanglement vanishes under the same condition. Moreover, we observed sudden transition between classical and quantum decoherence when single qubit interacts with the environment. The effects of DM interaction on quantum correlations are also considered and revealed in the two cases. It can enhance the decay of quantum correlations and its effect on quantum correlations can be strengthened by anisotropy parameter

Autocomplete 3D Sculpting

Digital sculpting is a popular means to create 3D models but remains a challenging task for many users. This can be alleviated by recent advances in data-driven and procedural modeling, albeit bounded by the underlying data and procedures. We propose a 3D sculpting system that assists users in freely creating models without predefined scope. With a brushing interface similar to common sculpting tools, our system silently records and analyzes users' workflows, and predicts what they might or should do in the future to reduce input labor or enhance output quality. Users can accept, ignore, or modify the suggestions and thus maintain full control and individual style. They can also explicitly select and clone past workflows over output model regions. Our key idea is to consider how a model is authored via dynamic workflows in addition to what it is shaped in static geometry, for more accurate analysis of user intentions and more general synthesis of shape structures. The workflows contain potential repetitions for analysis and synthesis, including user inputs (e.g. pen strokes on a pressure sensing tablet), model outputs (e.g. extrusions on an object surface), and camera viewpoints. We evaluate our method via user feedbacks and authored models.Comment: 10 page