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In this work, we focus on the Bipartite Stochastic Block Model (BiSBM), a popular model for bipartite graphs with a community structure. We consider the high dimensional setting where the number $n_1$ of type I nodes is far smaller than the number $n_2$ of type II nodes. The recent work of Braun and Tyagi (2022) established a sufficient and necessary condition on the sparsity level $p_{max}$ of the bipartite graph to be able to recover the latent partition of type I nodes. They proposed an iterative method that extends the one proposed by Ndaoud et al. (2022) to achieve this goal. Their method requires a good enough initialization, usually obtained by a spectral method, but empirical results showed that the refinement algorithm doesn't improve much the performance of the spectral method. This suggests that the spectral achieves exact recovery in the same regime as the refinement method. We show that it is indeed the case by providing new entrywise bounds on the eigenvectors of the similarity matrix used by the spectral method. Our analysis extend the framework of Lei (2019) that only applies to symmetric matrices with limited dependencies. As an important technical step, we also derive an improved concentration inequality for similarity matrices.

The graph traversal edit distance (GTED) is an elegant distance measure defined as the minimum edit distance between strings reconstructed from Eulerian trails in two edge-labeled graphs. GTED can be used to infer evolutionary relationships between species by comparing de Bruijn graphs directly without the computationally costly and error-prone process of genome assembly. Ebrahimpour Boroojeny et al.~(2018) suggest two ILP formulations for GTED and claim that GTED is polynomially solvable because the linear programming relaxation of one of the ILP always yields optimal integer solutions. The result that GTED is polynomially solvable is contradictory to the complexity results of existing string-to-graph matching problems. We resolve this conflict in complexity results by proving that GTED is NP-complete and showing that the ILPs proposed by Ebrahimpour Boroojeny et al. do not solve GTED but instead solve for a lower bound of GTED and are not solvable in polynomial time. In addition, we provide the first two, correct ILP formulations of GTED and evaluate their empirical efficiency. These results provide solid algorithmic foundations for comparing genome graphs and point to the direction of approximation heuristics. The source code to reproduce experimental results is available at //github.com/Kingsford-Group/gtednewilp/.

Matrix recovery from sparse observations is an extensively studied topic emerging in various applications, such as recommendation system and signal processing, which includes the matrix completion and compressed sensing models as special cases. In this work we propose a general framework for dynamic matrix recovery of low-rank matrices that evolve smoothly over time. We start from the setting that the observations are independent across time, then extend to the setting that both the design matrix and noise possess certain temporal correlation via modified concentration inequalities. By pooling neighboring observations, we obtain sharp estimation error bounds of both settings, showing the influence of the underlying smoothness, the dependence and effective samples. We propose a dynamic fast iterative shrinkage thresholding algorithm that is computationally efficient, and characterize the interplay between algorithmic and statistical convergence. Simulated and real data examples are provided to support such findings.

The Erd\"os Renyi graph is a popular choice to model network data as it is parsimoniously parametrized, straightforward to interprete and easy to estimate. However, it has limited suitability in practice, since it often fails to capture crucial characteristics of real-world networks. To check the adequacy of this model, we propose a novel class of goodness-of-fit tests for homogeneous Erd\"os Renyi models against heterogeneous alternatives that allow for nonconstant edge probabilities. We allow for asymptotically dense and sparse networks. The tests are based on graph functionals that cover a broad class of network statistics for which we derive limiting distributions in a unified manner. The resulting class of asymptotic tests includes several existing tests as special cases. Further, we propose a parametric bootstrap and prove its consistency, which allows for performance improvements particularly for small network sizes and avoids the often tedious variance estimation for asymptotic tests. Moreover, we analyse the sensitivity of different goodness-of-fit test statistics that rely on popular choices of subgraphs. We evaluate the proposed class of tests and illustrate our theoretical findings by extensive simulations.

A common method of generalizing binary to multi-class classification is the error correcting code (ECC). ECCs may be optimized in a number of ways, for instance by making them orthogonal. Here we test two types of orthogonal ECCs on seven different datasets using three types of binary classifier and compare them with three other multi-class methods: 1 vs. 1, one-versus-the-rest and random ECCs. The first type of orthogonal ECC, in which the codes contain no zeros, admits a fast and simple method of solving for the probabilities. Orthogonal ECCs are always more accurate than random ECCs as predicted by recent literature. Improvments in uncertainty coefficient (U.C.) range between 0.4--17.5% (0.004--0.139, absolute), while improvements in Brier score between 0.7--10.7%. Unfortunately, orthogonal ECCs are rarely more accurate than 1 vs. 1. Disparities are worst when the methods are paired with logistic regression, with orthogonal ECCs never beating 1 vs. 1. When the methods are paired with SVM, the losses are less significant, peaking at 1.5%, relative, 0.011 absolute in uncertainty coefficient and 6.5% in Brier scores. Orthogonal ECCs are always the fastest of the five multi-class methods when paired with linear classifiers. When paired with a piecewise linear classifier, whose classification speed does not depend on the number of training samples, classifications using orthogonal ECCs were always more accurate than the other methods and also faster than 1 vs. 1. Losses against 1 vs. 1 here were higher, peaking at 1.9% (0.017, absolute), in U.C. and 39% in Brier score. Gains in speed ranged between 1.1% and over 100%. Whether the speed increase is worth the penalty in accuracy will depend on the application.

This paper is concerned with a class of DC composite optimization problems which, as an extension of convex composite optimization problems and DC programs with nonsmooth components, often arises in robust factorization models of low-rank matrix recovery. For this class of nonconvex and nonsmooth problems, we propose an inexact linearized proximal algorithm (iLPA) by computing in each step an inexact minimizer of a strongly convex majorization constructed with a partial linearization of their objective functions, and establish the global convergence of the generated iterate sequence under the Kurdyka-\L\"ojasiewicz (KL) property of a potential function. In particular, by leveraging the composite structure, we provide a verifiable condition for the potential function to have the KL property of exponent $1/2$ at the limit point, so for the iterate sequence to have a local R-linear convergence rate, and clarify its relationship with the regularity used in the convergence analysis of algorithms for convex composite optimization. Finally, our iLPA is applied to a robust factorization model for matrix completions with outliers, and numerical comparison with the Polyak subgradient method confirms its superiority in computing time and quality of solutions.

The Fisher information matrix is a quantity of fundamental importance for information geometry and asymptotic statistics. In practice, it is widely used to quickly estimate the expected information available in a data set and guide experimental design choices. In many modern applications, it is intractable to analytically compute the Fisher information and Monte Carlo methods are used instead. The standard Monte Carlo method produces estimates of the Fisher information that can be biased when the Monte-Carlo noise is non-negligible. Most problematic is noise in the derivatives as this leads to an overestimation of the available constraining power, given by the inverse Fisher information. In this work we find another simple estimate that is oppositely biased and produces an underestimate of the constraining power. This estimator can either be used to give approximate bounds on the parameter constraints or can be combined with the standard estimator to give improved, approximately unbiased estimates. Both the alternative and the combined estimators are asymptotically unbiased so can be also used as a convergence check of the standard approach. We discuss potential limitations of these estimators and provide methods to assess their reliability. These methods accelerate the convergence of Fisher forecasts, as unbiased estimates can be achieved with fewer Monte Carlo samples, and so can be used to reduce the simulated data set size by several orders of magnitude.

In 1954, Alston S. Householder published Principles of Numerical Analysis, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower and upper triangular matrices. And now, matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in numerical linear algebra and matrix analysis in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of the Euclidean space, Hermitian space, Hilbert space, and things in the complex domain. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields.

Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.