Optimization on Lie manifolds and pattern recognition

Several pattern recognition problems can be reduced in a natural way to the problem of optimizing a nonlinear function over a Lie manifold. However, optimization on Lie manifolds involves, in general, a large number of nonlinear equality constraints and is hence one of the hardest optimization problems. We show that exploiting the special structure of Lie manifolds allows one to devise a method for optimizing on Lie manifolds in a computationally efficient manner. The new method relies on the differential geometry of Lie manifolds and the underlying connections between Lie groups and their associated Lie algebras. We describe an application of the new Lie group method to the problem of diagnosing malignancy in the cytological extracts of breast tumors. The diagnosis method that we present has a mean sensitivity of 98.086% and a predictive index of 0.0602, making it the most accurate and reliable diagnostic method reported thus far.

[1]  S. Shankar Sastry,et al.  Optimization Criteria, Sensitivity and Robustness of Motion and Structure Estimation , 1999, Workshop on Vision Algorithms.

[2]  O. Mangasarian,et al.  Robust linear programming discrimination of two linearly inseparable sets , 1992 .

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  D J Zahniser,et al.  High-resolution and contextual analysis for the diagnosis of fine needle aspirates of breast. , 1991, Analytical and quantitative cytology and histology.

[5]  D. Defays,et al.  An Efficient Algorithm for a Complete Link Method , 1977, Comput. J..

[6]  W. N. Street,et al.  Machine learning techniques to diagnose breast cancer from image-processed nuclear features of fine needle aspirates. , 1994, Cancer letters.

[7]  J. Frank Adams,et al.  Lectures on Lie groups , 1969 .

[8]  A. W. Knapp.  Lie groups beyond an introduction , 1988 .

[9]  J. H. Ward.  Hierarchical Grouping to Optimize an Objective Function , 1963 .

[10]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[11]  H. Trotter.  On the product of semi-groups of operators , 1959 .

[12]  Hans-Peter Kriegel,et al.  Knowledge Discovery in Large Spatial Databases: Focusing Techniques for Efficient Class Identification , 1995, SSD.

[13]  Hans-Peter Kriegel,et al.  A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise , 1996, KDD.

[14]  Jiawei Han,et al.  Efficient and Effective Clustering Methods for Spatial Data Mining , 1994, VLDB.

[15]  P. Bartels,et al.  Expert system support using Bayesian belief networks in the diagnosis of fine needle aspiration biopsy specimens of the breast. , 1994, Journal of clinical pathology.

[16]  Dimitrios Gunopulos,et al.  Automatic subspace clustering of high dimensional data for data mining applications , 1998, SIGMOD '98.

[17]  P. Cheung,et al.  The complementary role of fine needle aspiration cytology and Tru-cut needle biopsy in the management of breast masses. , 1987, The Australian and New Zealand journal of surgery.

[18]  S. Shankar Sastry,et al.  c ○ 2000 Kluwer Academic Publishers. Manufactured in The Netherlands. Linear Differential Algorithm for Motion Recovery: A Geometric Approach , 2022 .

[19]  W. N. Street,et al.  Breast cytology diagnosis with digital image analysis. , 1993, Analytical and quantitative cytology and histology.

[20]  R. Ward.  Numerical Computation of the Matrix Exponential with Accuracy Estimate , 1977 .

[21]  A. D. Gordon,et al.  Classification : Methods for the Exploratory Analysis of Multivariate Data , 1981 .

[22]  Kristin P. Bennett,et al.  Decision Tree Construction Via Linear Programming , 1992 .

[23]  R. Fang,et al.  Use of neural network analysis to diagnose breast cancer patients , 1993, Proceedings of TENCON '93. IEEE Region 10 International Conference on Computers, Communications and Automation.

[24]  M. Liou.  A novel method of evaluating transient response , 1966 .

[25]  W H Wolberg,et al.  Diagnostic schemes for fine needle aspirates of breast masses. , 1988, Analytical and quantitative cytology and histology.

[26]  Jana Kosecka,et al.  A lie theoretic approach to structure and motion in computer vision 1 , 1999 .

[27]  Teofilo F. GONZALEZ,et al.  Clustering to Minimize the Maximum Intercluster Distance , 1985, Theor. Comput. Sci..

[28]  M. Moskowitz.  The surjectivity of the exponential map for certain Lie groups , 1994 .

[29]  Tian Zhang,et al.  BIRCH: an efficient data clustering method for very large databases , 1996, SIGMOD '96.

[30]  J. Hermans,et al.  The value of aspiration cytologic examination of the breast a statistical review of the medical literature , 1992, Cancer.

[31]  P H Bartels,et al.  Reproducibility of Bayesian belief network assessment of breast fine needle aspirates. , 1996, Analytical and quantitative cytology and histology.

[32]  O. Mangasarian,et al.  Multisurface method of pattern separation for medical diagnosis applied to breast cytology. , 1990, Proceedings of the National Academy of Sciences of the United States of America.

[33]  D. Hazarika,et al.  View the PDF document Morphometric Studies of Cytological Specimens in Breast Carcinoma using Computerised Image Analysis System . , 1993 .

[34]  J. R. Quinlan.  DECISION TREES AS PROBABILISTIC CLASSIFIERS , 1987 .

[35]  S. Shankar Sastry,et al.  Euclidean Reconstruction and Reprojection Up to Subgroups , 2004, International Journal of Computer Vision.

[36]  Y. Oertel.  Fine needle aspiration of the breast , 1987 .

[37]  J. MacQueen.  Some methods for classification and analysis of multivariate observations , 1967 .

[38]  Brian Everitt,et al.  Cluster analysis , 1974 .

[39]  S. Shankar Sastry,et al.  Motion Recovery from Image Sequences: Discrete Viewpoint vs. Differential Viewpoint , 1998, ECCV.

[40]  D. Sattinger,et al.  Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics , 1986 .

[41]  Peter J. Rousseeuw,et al.  Finding Groups in Data: An Introduction to Cluster Analysis , 1990 .

[42]  Anil K. Jain,et al.  Algorithms for Clustering Data , 1988 .

[43]  I. G. MacDonald,et al.  Lectures on Lie groups and Lie algebras , 1995 .

[44]  C. Loan,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix , 1978 .

[45]  Ellen M. Voorhees,et al.  Implementing agglomerative hierarchic clustering algorithms for use in document retrieval , 1986, Inf. Process. Manag..

[46]  S Shapiro,et al.  Report of the International Workshop on Screening for Breast Cancer. , 1993, Journal of the National Cancer Institute.

[47]  Olvi L. Mangasarian,et al.  Nuclear feature extraction for breast tumor diagnosis , 1993, Electronic Imaging.