Motivated by recent work on stochastic gradient descent methods, we develop two stochastic variants of greedy algorithms for possibly non-convex optimization problems with sparsity constraints. We prove linear convergence in expectation to the solution within a specified tolerance. This generalized framework applies to problems such as sparse signal recovery in compressed sensing, low-rank matrix recovery, and covariance matrix estimation, giving methods with provable convergence guarantees that often outperform their deterministic counterparts. We also analyze the settings where gradients and projections can only be computed approximately, and prove the methods are robust to these approximations. We include many numerical experiments which align with the theoretical analysis and demonstrate these improvements in several different settings.
There are several methods proposed to solve special instances of ). @cite_6 and @cite_35 propose the forward selection method for sparse vector and low-rank matrix recovery. The method selects each nonzero entry or each rank-one matrix in an iterative fashion. @cite_36 generalizes this algorithm to the more general dictionary @math . @cite_27 proposes the forward-backward method in which an atom can be added or removed from the set, depending on how much it contributes to decrease the loss function. @cite_44 extends this algorithms beyond the quadratic loss studied in @cite_27 . @cite_42 extends the CoSaMP algorithm for a more general loss function. Very recently, @cite_38 further generalizes CoSaMP and proposes the Gradient Matching Pursuit (GradMP) algorithm to solve ). This is perhaps the first greedy algorithm for ) - the very general form of sparse recovery. They show that under a restricted convexity assumption of the objective function, the algorithm linearly converges to the optimal solution. This desirable property is also possessed by CoSaMP. We note that there are other algorithms having also been extended to the setting of sparsity in arbitrary @math but only limited to the quadratic loss setting, see e.g. @cite_5 @cite_7 @cite_10 @cite_14 .
@cite_35 @cite_38 @cite_14 @cite_7 @cite_36 @cite_42 @cite_6 @cite_44 @cite_27 @cite_5 @cite_10
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A citing sentence is one that appears in a scientific article and cites previous work. Citing sentences have been studied and used in many applications. For example, they have been used in scientific paper summarization, automatic survey generation, paraphrase identification, and citation function classification. Citing sentences that cite multiple papers are common in scientific writing. This observation should be taken into consideration when using citing sentences in applications. For instance, when a citing sentence is used in a summary of a scientific paper, only the fragments of the sentence that are relevant to the summarized paper should be included in the summary. In this paper, we present and compare three different approaches for identifying the fragments of a citing sentence that are related to a given target reference. Our methods are: word classification, sequence labeling, and segment classification. Our experiments show that segment classification achieves the best results. Abstract Documents in computer-readable form can be used to provide information about other documents, i.e. those they cite. To do this efficiently requires procedures for computer recognition of citing statements. This is not easy, especially for multi-sentence citing statements. Computer recognition procedures have been developed which are accurate to the following extent: 73 of the words in statements selected by computer procedures as being citing statements are words which are correctly attributable to the corresponding documents. The retrieval effectiveness of computer-recognized citing statements was tested in the following way. First, for eight retrieval requests in inorganiic chemistry, average recall by search of Chemical Abstracts Service indexing and Chemical Abstracts abstract text words was found to be 50 . Words from citing statements referring to the papers to be retrieved were then added to the index terms and abstract words as additional access points, and searching was repeated. Average recall increased to 70 . Only words from citing statements published within a year of the cited papers were used. The retrieval effect of citing statement words alone (published within a year) without index or abstract terms was the following: average recall was 40 . When just the words of the titles of the cited papers were added to those citing statement words, average recall increased to 50 . Purpose – The purpose of this paper is to suggest an alternative to the widely used Times Cited criterion for analysing citation networks. The approach involves taking account of the natures of the papers that cite a given paper, so as to differentiate between papers that attract the same number of citations.Design methodology approach – ArticleRank is an algorithm that has been derived from Google's PageRank algorithm to measure the influence of journal articles. ArticleRank is applied to two datasets – a citation network based on an early paper on webometrics, and a self‐citation network based on the 19 most cited papers in the Journal of Documentation – using citation data taken from the Web of Knowledge database.Findings – ArticleRank values provide a different ranking of a set of papers from that provided by the corresponding Times Cited values, and overcomes the inability of the latter to differentiate between papers with the same numbers of citations. The difference in rankings between Times Cited ... We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math .