Local Guarantees in Graph Cuts and Clustering

Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Min $s-t$ Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization. Here, we are given a graph with edges labeled $+$ or $-$ and the goal is to produce a clustering that agrees with the labels as much as possible: $+$ edges within clusters and $-$ edges across clusters. The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize a global objective. We depart from this and study local objectives: minimizing the maximum number of disagreements for edges incident on a single node, and the analogous max min agreements objective. This naturally gives rise to a family of basic min-max graph cut problems. A prototypical representative is Min Max $s-t$ Cut: find an $s-t$ cut minimizing the largest number of cut edges incident on any node. We present the following results: $(1)$ an $O(\sqrt{n})$-approximation for the problem of minimizing the maximum total weight of disagreement edges incident on any node (thus providing the first known approximation for the above family of min-max graph cut problems), $(2)$ a remarkably simple $7$-approximation for minimizing local disagreements in complete graphs (improving upon the previous best known approximation of $48$), and $(3)$ a $1/(2+\varepsilon)$-approximation for maximizing the minimum total weight of agreement edges incident on any node, hence improving upon the $1/(4+\varepsilon)$-approximation that follows from the study of approximate pure Nash equilibria in cut and party affiliation games

Improving A Multi-objective Multipopulation Artificial Immune Network For Biclustering

The biclustering technique was developed to avoid some of the drawbacks presented by standard clustering techniques. Given that biclustering requires the optimization of at least two conflicting objectives and that multiple independent solutions are desirable as the outcome, a few multi-objective evolutionary algorithms for biclustering were proposed in the literature. However, apart from the individual characteristics of the biclusters that should be optimized during their construction, several other global aspects should also be considered, such as the coverage of the dataset and the overlap among biclusters. These requirements will be addressed in this work with the MOM-aiNet+ algorithm, which is an improvement of the original multi-objective multipopulation artificial immune network denoted MOM-aiNet. Here, the MOM-aiNet+ algorithm will be described in detail, its main differences from the original MOM-aiNet will be highlighted, and both algorithms will be compared, together with three other proposals from the literature. © 2009 IEEE.27482755De França, F.O., Bezerra, G., Von Zuben, F.J., New Perspectives for the Biclustering Problem (2006) Proceedings of the IEEE Congress on Evolutionary Computation (CEC), pp. 753-760. , Vancouver, Canada(2003) The Analysis of Gene Expression Data, , G. Parmigiani, E. S. Garett, R. A. Irizarry, and S. L. Zeger, Eds., SpringerHerlocker, J., Konstan, J., Borchers, A., Riedl, J., An algorithmic framework for performing collaborative filtering (1999) Proceedings of the 1999 Conference on Research and Development in Information Retrieval, pp. 230-237Hartigan, J.A., Direct clustering of a data matrix (1972) Journal of the American Statistical Association (JASA), 67 (337), pp. 123-129Mirkin, B., (1996) Mathematical Classification and Clustering, , ser. Nonconvex Optimization and Its Applications. SpringerCheng, Y., Church, G.M., Biclustering of expression data (2000) Proc. of the 8th Int. Conf. on Intelligent Systems for Molecular Biology, pp. 93-103Mitra, S., Banka, H., Multi-objective evolutionary biclustering of gene expression data (2006) Pattern Recognition, 39 (12), pp. 2464-2477. , DOI 10.1016/j.patcog.2006.03.003, PII S0031320306000872, BioinformaticsDivina, F., Aguilar-Ruiz, J.S., A multi-objective approach to discover biclusters in microarray data (2007) Proceedings of the Genetic and Evolutionary Computation Conference (GECCO'07), pp. 385-392. , London, UKMaulik, U., Mukhopadhyay, A., Bandyopadhyay, S., Zhang, M.Q., Zhang, X., Multiobjective fuzzy biclustering in microarray data: Method and a new performance measure (2008) Proceedings of the 2008 IEEE Congress on Evolutionary Computation (CEC 2008), pp. 1536-1543. , Hong Kong, ChinaCoelho, G.P., De França, F.O., Von Zuben, F.J., A multi- objective multipopulation approach for biclustering (2008) Artificial Immune Systems, Proc. of the 7th International Conference on Artificial Immune Systems (ICARIS), 5132, pp. 71-82. , ser. Lecture Notes in Computer Science, P. J. Bentley, D. Lee, and S. Jung, Eds., Phuket, ThailandDe Castro, P.A.D., De França, F.O., Ferreira, H.M., Von Zuben, F.J., Applying biclustering to text mining: An immune-inspired approach (2007) Artificial Immune Systems, Proc. of the 6th International Conference on Artificial Immune Systems (ICARIS), 4628, pp. 83-94. , ser. Lecture Notes in Computer Science, L. N. de Castro, F. J. Von Zuben, and H. Knidel, Eds., Santos, BrazilAgrawal, R., Gehrke, J., Gunopulus, D., Raghavan, P., Automatic subspace clustering of high dimensional data for data mining applications (1998) Proc. of the ACM/SIGMOD Int. Conference on Management of Data, pp. 94-105Feldman, R., Sanger, J., (2006) Text Mining Handbook, , Cambridge University PressDe Castro, P.A.D., De França, F.O., Ferreira, H.M., Von Zuben, F.J., Evaluating the performance of a biclustering algorithm applied to collaborative filtering: A comparative analysis (2007) Proc. of the 7th International Conference on Hybrid Intelligent Systems, pp. 65-70. , Kaiserslautern, GermanySymeonidis, P., Nanopoulos, A., Papadopoulos, A., Manolopoulos, Y., Nearest-biclusters collaborative filtering with constant values (2007) Advances in Web Mining and Web Usage Analysis, 4811, pp. 36-55. , Philadelphia, USA: Springer-Verlag, ser. Lecture Notes in Computer ScienceMadeira, S.C., Oliveira, A.L., Biclustering algorithms for biological data analysis: A survey (2004) IEEE Transactions on Computational Biology and Bioinformatics, 1 (1), pp. 24-45Deb, K., (2001) Multi-Objective Optimization using Evolutionary Algorithms, , Chichester, UK: John-Wiley & SonsDe Castro, L.N., Timmis, J., (2002) Artificial Immune Systems: A New Computational Intelligence Approach, , SpringerBurnet, F.M., Clonal selection and after (1978) Theoretical Immunology, pp. 63-85. , G. I. Bell, A. S. Perelson, and G. H. Pimgley Jr, Eds. Marcel Dekker IncJerne, N.K., Towards a network theory of the immune system (1974) Ann. Immunol., Inst. Pasteur, 125 C, pp. 373-389De Castro, L.N., Von Zuben, F.J., Learning and optimization using the clonal selection principle (2002) IEEE Transactions on Evolutionary Computation, 6 (3), pp. 239-251AiNet: An artificial immune network for data analysis (2001) Data Mining: A Heuristic Approach, pp. 231-259. , H. A. Abbass, R. A. Sarker, and C. S. Newton, Eds. Idea Group PublishingDeb, K., Pratap, A., Agarwal, S., Meyarivan, T., A fast and elitist multiobjective genetic algorithm: NSGA-II (2002) IEEE Transactions on Evolutionary Computation, 6 (2), pp. 182-197Cho, R., Campbell, M., Winzeler, E., Steinmetz, L., Conway, A., Wodicka, L., Wolfsberg, T., Davis, R., A genome-wide transcriptional analysis of the mitotic cell cycle (1998) Molecular Cell, 2, pp. 65-73Alizadeh, A.A., Eisen, M.B., Davis, R.E., Ma, C., Lossos, I.S., Rosenwald, A., Boldrick, J.C., Staudt, L.M., Distinct types of diffuse large B-cell lymphoma identified by gene expression profiling (2000) Nature, 403, pp. 503-51

Multi-objective Biclustering: When Non-dominated Solutions Are Not Enough

The biclustering technique was developed to avoid some of the drawbacks presented by standard clustering techniques, such as their impossibility of finding correlating data under a subset of features, and, consequently, to allow the extraction of more accurate information from datasets. Given that biclustering requires the optimization of at least two conflicting objectives (residue and volume) and that multiple independent solutions are desirable as the outcome, a few multi-objective evolutionary algorithms for biclustering were proposed in the literature. However, these algorithms only focus their search in the generation of a global set of non-dominated biclusters, which may be insufficient for most of the problems as the coverage of the dataset can be compromised. In order to overcome such problem, a multi-objective artificial immune system capable of performing a multipopulation search, named MOM-aiNet, was proposed. In this work, the MOM-aiNet algorithm will be described in detail, and an extensive set of experimental comparisons will be performed, with the obtained results of MOM-aiNet being confronted with those produced by the popular CC algorithm, by another immune-inspired approach for biclustering (BIC-aiNet), and by the multi-objective approach for biclustering proposed by Mitra & Banka. © 2009 Springer Science+Business Media B.V.82175202Han, J., Kamber, M., (2006) Data Mining: Concepts and Techniques, , Morgan Kaufmann San FranciscoDe França, F.O., Bezerra, G., Von Zuben, F.J., New perspectives for the biclustering problem (2006) Proceedings of the IEEE Congress on Evolutionary Computation (CEC), pp. 753-760. , IEEE, VancouverParmigiani, G., Garett, E.S., Irizarry, R.A., Zeger, S.L., (2003) The Analysis of Gene Expression Data, , Springer New YorkHerlocker, J., Konstan, J., Borchers, A., Riedl, J., An algorithmic framework for performing collaborative filtering (1999) Proceedings of the 1999 Conference on Research and Development in Information Retrieval, pp. 230-237. , Berkeley, 15-19 AugustFeldman, R., Sanger, J., (2006) The Text Mining Handbook, , Cambridge University Press CambridgeHartigan, J.A., Direct clustering of a data matrix (1972) J. Am. Stat. Assoc. (JASA), 67 (337), pp. 123-129Mirkin, B., (1996) Mathematical Classification and Clustering. Nonconvex Optimization and Its Applications., , Springer New YorkCheng, Y., Church, G.M., Biclustering of expression data (2000) Proc. of the 8th Int. Conf. on Intelligent Systems for Molecular Biology, pp. 93-103. , La Jolla, 19-23 AugustDeb, K., (2001) Multi-Objective Optimization Using Evolutionary Algorithms, , Wiley ChichesterMitra, S., Banka, H., Multi-objective evolutionary biclustering of gene expression data (2006) Pattern Recognition, 39 (12), pp. 2464-2477. , DOI 10.1016/j.patcog.2006.03.003, PII S0031320306000872, BioinformaticsDeb, K., Pratap, A., Agarwal, S., Meyarivan, T., A fast and elitist multiobjective genetic algorithm: NSGA-II (2002) IEEE Transactions on Evolutionary Computation, 6 (2), pp. 182-197. , DOI 10.1109/4235.996017, PII S1089778X02041012Mitra, S., Banka, H., Pal, S.K., A MOE framework for biclustering of microarray data (2006) Proceedings of the 18th International Conference on Pattern Recognition (ICPR'06), pp. 1154-1157. , Hong Kong, 20-24 AugustDivina, F., Aguilar-Ruiz, J.S., A multi-objective approach to discover biclusters in microarray data (2007) Proceedings of the Genetic and Evolutionary Computation Conference (GECCO'07), pp. 385-392. , London, 7-11 JulyGiráldez, R., Evolutionary search of biclusters by minimal intrafluctuation (2007) Proceedings of the IEEE International Fuzzy Systems Conference (FUZZ-IEEE 2007), pp. 1-6. , IEEE, LondonMaulik, U., Mukhopadhyay, A., Bandyopadhyay, S., Zhang, M.Q., Zhang, X., Multiobjective fuzzy biclustering in microarray data: Method and a new performance measure (2008) Proceedings of the 2008 IEEE Congress on Evolutionary Computation (CEC 2008), pp. 1536-1543. , IEEE, Hong KongKrishnapuram, R., Joshi, A., Yi, L., A fuzzy relative of the k-medoids algorithm with application to document and snippet clustering (1999) Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE'99), pp. 1281-1286. , IEEE, SeoulDe França, F.O., Von Zuben, F.J., Coelho, G.P., A multi-objective multipopulation approach for biclustering (2008) Lecture Notes in Computer Science, 5132, pp. 71-82. , Bentley, P.J., Lee, D., Jung, S. (eds.) Artificial Immune Systems, Proc. of the 7th International Conference on Artificial Immune Systems (ICARIS) Phuket, 10-13 AugustDe Castro, L.N., Timmis, J., (2002) Artificial Immune Systems: A New Computational Intelligence Approach, , Springer New YorkDe França, F.O., Ferreira, H.M., Von Zuben, F.J., Castro, P.A.D., Applying biclustering to text mining: An immune-inspired approach (2007) Lecture Notes in Computer Science, 4628, pp. 83-94. , de Castro, L.N., Von Zuben, F.J., Knidel, H. (eds.) Artificial Immune Systems, Proc. of the 6th International Conference on Artificial Immune Systems (ICARIS) Santos, 26-29 AugustAgrawal, R., Gehrke, J., Gunopulus, D., Raghavan, P., Automatic subspace clustering of high dimensional data for data mining applications (1998) Proc. of the ACM/SIGMOD Int. Conference on Management of Data, pp. 94-105. , Seattle, 2-4 JuneDhillon, I.S., Co-clustering documents and words using bipartite spectral graph partitioning (2001) Proc. of the 7th Int. Con. on Knowledge Discovery and Data Mining, pp. 269-274. , San Francisco, 26-29 AugustDe França, F.O., Ferreira, H.M., Von Zuben, F.J., Castro, P.A.D., Applying biclustering to perform collaborative filtering (2007) Proc. of the 7th International Conference on Intelligent Systems Design and Applications, pp. 421-426. , Rio de Janeiro, 22-24 OctoberDe França, F.O., Ferreira, H.M., Von Zuben, F.J., Castro, P.A.D., Evaluating the performance of a biclustering algorithm applied to collaborative filtering: A comparative analysis (2007) Proc. of the 7th International Conference on Hybrid Intelligent Systems, pp. 65-70. , Kaiserslautern, 17-19 SeptemberSymeonidis, P., Nanopoulos, A., Papadopoulos, A., Manolopoulos, Y., Nearest-biclusters collaborative filtering with constant values (2007) Lecture Notes in Computer Science, 4811, pp. 36-55. , Advances in Web Mining and Web Usage Analysis Springer PhiladelphiaTang, C., Zhang, L., Zhang, I., Ramanathan, M., Interrelated two-way clustering: An unsupervised approach for gene expression data analysis (2001) Proc. of the 2nd IEEE Int. Symposium on Bioinformatics and Bioengineering, pp. 41-48. , IEEE PiscatawayMadeira, S.C., Oliveira, A.L., Biclustering algorithms for biological data analysis: A survey (2004) IEEE Trans. Comput. Biol. Bioinformatics, 1 (1), pp. 24-45Edgeworth, F.Y., (1881) Mathematical Physics, , P. Keagan LondonPareto, V., (1896) Cours d'Economie Politique, , F. Rouge LausanneBurnet, F.M., Bell, G.I., Perelson, A.S., Pimgley Jr., G.H., Clonal selection and after (1978) Theoretical Immunology, pp. 63-85. , Marcel Dekker New YorkJerne, N.K., Towards a network theory of the immune system (1974) Ann. Immunol. Inst. Pasteur, 125 C, pp. 373-389De Castro, L.N., Von Zuben, F.J., Learning and optimization using the Clonal selection principle (2002) IEEE Trans. Evol. Comput., 6 (3), pp. 239-251De Castro, L.N., Von Zuben, F.J., AiNet: An artificial immune network for data analysis (2001) Data Mining: A Heuristic Approach, pp. 231-259. , Abbass, H.A., Sarker, R.A., Newton, C.S. (eds.) Idea Group, HarrisburgCoelho, G.P., Von Zuben, F.J., Omni-aiNet: An immune-inspired approach for omni optimization (2006) Lecture Notes in Computer Science, 4163, pp. 294-308. , Bersini, H., Carneiro, J. (eds.) Artificial Immune Systems, Proc. of the 5th International Conference on Artificial Immune Systems (ICARIS) Oeiras, PortugalCho, R.J., Campbell, M.J., Winzeler, E.A., Steinmetz, L., Conway, A., Wodicka, L., Wolfsberg, T.G., Davis, R.W., A genome-wide transcriptional analysis of the mitotic cell cycle (1998) Molecular Cell, 2 (1), pp. 65-73Alizadeh, A.A., Elsen, M.B., Davis, R.E., Ma, C.L., Lossos, I.S., Rosenwald, A., Boldrick, J.C., Staudt, L.M., Distinct types of diffuse large B-cell lymphoma identified by gene expression profiling (2000) Nature, 403 (6769), pp. 503-511. , DOI 10.1038/35000501Snedecor, G.S., Cochran, W.G., (1989) Statistical Methods, , Iowa University Press Iow