Teacher: 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.
Teacher: Now, understand the problem? If you are still confused, see the following example:
We propose a local coherence model based on a convolutional neural network that operates over the entity grid representation of a text. The model captures long range entity transitions along with entity-specific features without loosing generalization, thanks to the power of distributed representation. We present a pairwise ranking method to train the model in an end-to-end fashion on a task and learn task-specific high level features. Our evaluation on three different coherence assessment tasks demonstrates that our model achieves state of the art results outperforming existing models by a good margin.
Solution: A Neural Local Coherence Model
Reason: This statement "A Neural Local Coherence Model" is taken from the subtext "convolutional neural network" and its explanation in the passage. This is a positive example because the title belongs to the mentions in the passage

Now, solve this instance: 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 BF (1 + ) OPT, where OPT = infrank-k A A AF . 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/ )q1) tensor B for which A BF (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 BF < , where is any positive, arbitrarily small function of n. 2. We give an algorithm for any > 0 which outputs a rank k tensor B for which ABF (1+ ) OPT and runs in (nnz(A)+n poly(k/ )+exp(k/ )) n time in the unit cost RAM model, whenever OPT > 2O(n ) and there is a rank-k tensor B = k i=1 ui vi wi for which A BF (1 + /2) OPT and ui2, vi2, wi2 2 ). If OPT 2(n), then B instead satisfies ABF 2(n ). 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 1o(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
Student:
Relative Error Tensor Low Rank Approximation