In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

[Q]: We consider relative error low rank approximation of tensors with respect to the Frobenius norm. Namely, given an order-q tensor A ∈ R ∏q i=1 ni , output a rank-k tensor B for which ‖A − B‖F ≤ (1 + ) OPT, where OPT = infrank-k A′ ‖A − A‖F . Despite much success on obtaining relative error low rank approximations for matrices, no such results were known for tensors. One structural issue is that there may be no rank-k tensor Ak achieving the above infinum. Another, computational issue, is that an efficient relative error low rank approximation algorithm for tensors would allow one to compute the rank of a tensor, which is NP-hard. We bypass these two issues via (1) bicriteria and (2) parameterized complexity solutions: 1. We give an algorithm which outputs a rank k′ = O((k/ )q−1) tensor B for which ‖A − B‖F ≤ (1+ ) OPT in nnz(A)+n ·poly(k/ ) time in the real RAM model, whenever either Ak exists or OPT > 0. Here nnz(A) denotes the number of non-zero entries in A. If both Ak does not exist and OPT = 0, then B instead satisfies ‖A − B‖F < γ, where γ is any positive, arbitrarily small function of n. 2. We give an algorithm for any δ > 0 which outputs a rank k tensor B for which ‖A−B‖F ≤ (1+ ) OPT and runs in (nnz(A)+n poly(k/ )+exp(k/ )) ·nδ time in the unit cost RAM model, whenever OPT > 2−O(n ) and there is a rank-k tensor B = ∑k i=1 ui ⊗ vi ⊗ wi for which ‖A − B‖F ≤ (1 + /2) OPT and ‖ui‖2, ‖vi‖2, ‖wi‖2 ≤ 2 δ). If OPT ≤ 2−Ω(nδ), then B instead satisfies ‖A−B‖F ≤ 2−Ω(n δ). Our first result is polynomial time, and in fact input sparsity time, in n, k, and 1/ , for any k ≥ 1 and any 0 < < 1, while our second result is fixed parameter tractable in k and 1/ . For outputting a rank-k tensor, or even a bicriteria solution with rank-Ck for a certain constant C > 1, we show a 2 1−o(1)) time lower bound under the Exponential Time Hypothesis. Our results are based on an “iterative existential argument”, and give the first relative error low rank approximations for tensors for a large number of error measures for which nothing was known. In particular, we give the first relative error approximation algorithms on tensors for: column row and tube subset selection, entrywise `p-low rank approximation for 1 ≤ p < 2, low rank approximation with respect to sum of Euclidean norms of faces or tubes, weighted low rank approximation, and low rank approximation in distributed and streaming models. We also obtain several new results for matrices, such as nnz(A)-time CUR decompositions, improving the previous nnz(A) log n-time CUR decompositions, which may be of independent interest. ∗Work done while visiting IBM Almaden, and supported in part by UTCS TAship (CS361 Spring 17 Introduction to Computer Security). †Supported in part by Simons Foundation, and NSF CCF-1617955. ar X iv :1 70 4. 08 24 6v 1 [ cs .D S] 2 6 A pr 2 01 7
[A]: Relative Error Tensor Low Rank Approximation


[Q]: The use of microblogging platforms such as Twitter during crises has become widespread. More importantly, information disseminated by affected people contains useful information like reports of missing and found people, requests for urgent needs etc. For rapid crisis response, humanitarian organizations look for situational awareness information to understand and assess the severity of the crisis. In this paper, we present a novel framework (i) to generate abstractive summaries useful for situational awareness, and (ii) to capture sub-topics and present a short informative summary for each of these topics. A summary is generated using a two stage framework that first extracts a set of important tweets from the whole set of information through an Integer-linear programming (ILP) based optimization technique and then follows a word graph and concept event based abstractive summarization technique to produce the final summary. High accuracies obtained for all the tasks show the effectiveness of the proposed framework.
[A]: Summarizing Situational and Topical Information During Crises


[Q]: The stable model (SM) semantics lacks the properties of existence, relevance and cumulativity. If we prospectively consider the class of conservative extensions of the SM semantics (i.e., semantics that for each normal logic program P retrieve a superset of the set of stable models of P), one may wander how do the semantics of this class behave in what concerns the aforementioned properties. That is the type of issue dealt with in this paper. We define a large class of conservative extensions of the SM semantics, dubbed affix stable model semantics (ASM), and study the above referred properties into two non-disjoint subfamilies of the class ASM, here dubbed ASMh and ASMm. From this study a number of results stem which facilitate the assessment of semantics in the class ASMh ∪ASMm with respect to the properties of existence, relevance and cumulativity, whilst unveiling relations among these properties. As a result of the approach taken in our work, light is shed on the characterization of the SM semantics, as we show that the properties of (lack of) existence and (lack of) cautious monotony are equivalent, which opposes statements on this issue that may be found in the literature. We also characterize the relevance failure of SM semantics in a more clear way than usually stated in the literature.
[A]:
Properties of Stable Model Semantics Extensions