Fig 1.
Where g is the current generation number initialized with 0; gmax is the maximum number of generations which is defined by the user; |P| is the number of solutions in the population. After step-8, g is incremented by 1 and the process continues until maximum number of generations is reached.
Fig 2.
Figure showing dominance and non-dominance relationships in two objective space.
Here, both the functions have to be maximized.
Table 1.
Brief descriptions of datasets used.
Here, #DocSentences is the total number of sentences in the document.
Table 2.
ROUGE scores attained by different methods for DUC2001 and DUC2002 data sets.
Here our proposed methods are executed using Normalized Google Distance (NGD), Cosine Similarity (CS) and Word Mover Distance (WMD), and, SMaxRouge strategy is used for selecting a single best solution from the final Pareto front. Here, † denotes the best results; it also indicates that results are statistically significant at 5% significance level; xx indicates results are not available in reference paper. For LeastMedSq and Linear Regression methods, results in the reference paper are presented up to 4 decimal points, therefore, to make a fair comparison up to 5 decimal points, we have added 0 as the last decimal digit such that their results remain unchanged. Similar case also applicable to NN-SE and SummaRuNNer methods.
Table 3.
ROUGE scores attained by proposed Approach-1 and Approach-2 utilizing word mover distance (WMD) on CNN dataset.
Here, SMaxRouge strategy is used for selecting a single best solution from the final Pareto front.
Table 4.
BLUE scores attained by proposed Approach-1 and Approach-2 utilizing word mover distance (WMD) on three datasets.
Here, SMaxRouge strategy is used for selecting a single best solution (based on maximum BLEU score) from the final Pareto front.
Fig 3.
Pareto optimal fronts obtained after application of the proposed approach.
Here, Proposed approach refers to Approach-1 (WMD) with SOM-based operators. Sub-figures (a), (b), (c) and (d) are the Pareto optimal fronts obtained after first, fourteen, nineteen and twenty-fifth generation, respectively. Red color dots represent Pareto optimal solutions; three axes represent three objective functional values, namely, sentence position, readability, coverage.
Table 5.
Average readability and coverage scores of the summaries obtained by our proposed approaches utilizing WMD on three datasets.
Here, the used summaries are obtained using SMaxRouge strategy.
Table 6.
ROUGE scores obtained using Approach-1 (WMD) when the best solution is selected using any of the strategies under UMaxRouge strategy.
All the strategies explored here for selecting a single best solution from the final Pareto front are unsupervised in nature. Bold entries indicate they are able to beat the state-of-the-art algorithms.
Table 7.
Population size and number of fitness evaluation (NFE) used by different optimization approaches.
‘-’ indicates value not mentioned in the reference paper.
Fig 4.
Sub-figures (a), (b), (c) and (d) show the convergence plots for four random documents. At each generation/iteration, maximum Rouge-1 and Rouge-2 scores are plotted.
Table 8.
Improvements attained by the proposed approach, Approach-1 (WMD) with SOM based operators over other methods considering ROUGE scores.
Here, xx indicates non-availability of results on the DUC2001 dataset.
Fig 5.
An example of reference summary and predicted summary for document AP881109—0149 of topic d21d under DUC2001 dataset.
Fig 6.
An example of reference summary and predicted summary for document SJMN91—06106024 of topic d60k under DUC2001 dataset.
Table 9.
The p-values obtained by Approach-1 (WMD) with SOM based operators (under SMaxRouge scheme) considering ROUGE-1 and ROUGE-2 score values.
Fig 7.
Sub-figures (a) and (b) for DUC2001 and DUC2002 dataset, respectively, show the variations of average Rouge-1/Rouge-2 values of highest ranked (rank-1) solutions in each document. In each colored box, the horizontal colored line indicates the median value of rank-1 solutions.
Fig 8.
Sub-figures (a), (b) and (c) show the Rouge-1/Rouge-2 score variations per document over DUC2001 dataset. In each colored box, the horizontal colored line indicates the median value of Rouge-1/Rouge-2 score using rank-1 solutions of a document.
Fig 9.
Sub-figures (a), (b) and (c) show the Rouge-1/Rouge-2 score variations per document over DUC2002 dataset. In each colored box, the horizontal colored line indicates the median value of Rouge-1/Rouge-2 score using rank-1 solutions of a document.
Table 10.
The p-values obtained by Approach-1 (WMD) with SOM and without SOM based operators (under SMaxRouge scheme) considering ROUGE-1 and ROUGE-2 score values.
Table 11.
Ranking of different methods.