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.

Input: Consider Input: 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

Output: Relative Error Tensor Low Rank Approximation


Input: Consider Input: The ability to monitor the progress of students’ academic performance is a critical issue to the academic community of higher learning. A system for analyzing students’ results based on cluster analysis and uses standard statistical algorithms to arrange their scores data according to the level of their performance is described. In this paper, we also implemented k-mean clustering algorithm for analyzing students’ result data. The model was combined with the deterministic model to analyze the students’ results of a private Institution in %igeria which is a good benchmark to monitor the progression of academic performance of students in higher Institution for the purpose of making an effective decision by the academic planners. Keywordsk – mean, clustering, academic performance, algorithm.

Output: Application of k-Means Clustering algorithm for prediction of Students’ Academic Performance


Input: Consider Input: In this paper, we propose a novel neural approach for paraphrase generation. Conventional paraphrase generation methods either leverage hand-written rules and thesauri-based alignments, or use statistical machine learning principles. To the best of our knowledge, this work is the first to explore deep learning models for paraphrase generation. Our primary contribution is a stacked residual LSTM network, where we add residual connections between LSTM layers. This allows for efficient training of deep LSTMs. We experiment with our model and other state-of-the-art deep learning models on three different datasets: PPDB, WikiAnswers and MSCOCO. Evaluation results demonstrate that our model outperforms sequence to sequence, attention-based and bi-directional LSTM models on BLEU, METEOR, TER and an embedding-based sentence similarity metric.


Output: Neural Paraphrase Generation with Stacked Residual LSTM Networks