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1. H. K. Ho, L. Zhang, K. Ramamohanarao, and S. Martin (2013), "A Survey of Machine Learning Methods for Secondary and Supersecondary Protein Structure Prediction," Protein Supersecondary Structure, A. Kister, Ed., Humana Press (Publisher)
   
   In this chapter we provide a survey of protein secondary and supersecondary structure prediction using methods from machine learning. Our focus is on machine learning methods applicable to β-hairpin and β-sheet prediction, but we also discuss methods for more general supersecondary structure prediction. We provide background on the secondary and supersecondary structures that we discuss, the features used to describe them, and the basic theory behind the machine learning methods used. We survey the machine learning methods available for secondary and supersecondary structure prediction and compare them where possible.




2. M. Misra, S. Martin, and J.-L. Faulon (2011), "Graphs: Flexible Representations of Molecular Structures and Biological Networks," Computational Approaches in Cheminformatics and Bioinformatics , R. Guha and A. Bender, Eds., Wiley. & Sons. (Publisher)
   
   The past two decades have seen a large accumulation of biological sequences and chemical compounds in many publicly available databases. For a long time, the two communities of bioinformatics and cheminformatics have developed in parallel working largely with sequence data alone or mainly in the chemical space, respectively. As the need to study biological networks has increased, however, a concurrent need to develop tools and algorithms capable of handling the combined sequence and chemical space has arisen. We present here a graph-based technique, named molecular signature, which is sufficiently adaptable to permit combined description, for high-throughput analyses, of both sequences and chemicals.



3. S. Martin (2010), "Machine Learning based Bioinformatics Algorithms: Application to Chemicals," Handbook of Cheminformatics Algorithms, J.-L. Faulon and A. Bender, Eds., CRC Press. (publisher)
   
   In this chapter we present a targeted overview of clustering, classification and regression algorithms. The target of our overview is algorithms, which have been used in either bioinformatics or chemoinformatics applications. In particular, we compare and contrast the efforts in both fields.




4. S. Martin, W. M. Brown, and J.-L. Faulon (2008), "Predicting Protein Interactions using Product Kernels," Advances in Biochemical Engineering/Biotechnology: Protein-Protein Interactions, M. Werther and H. Seitz, Eds., vol. 110, Springer-Verlag. (publisher, presentation)
   
   In this chapter, we provide a brief discussion of the relative merits of different experimental and computational methods available for identifying protein interactions. We then focus on the application of our particular (computational) method using Support Vector Machine product kernels. We describe our method in detail and discuss the application of the method for predicting protein–protein interactions, β-strand interactions, and protein–chemical interactions.




5. G. S. Davidson, S. Martin, K. Boyack, B. N. Wylie, J. Martinez, A. Aragon, M. Werner-Washburne, M. Mosquera-Caro, and C. L. Willman (2007), "Robust Methods in Microarray Analysis," Genomics and Proteomics Engineering in Medicine and Biology M. Akay, Ed., Wiley/IEEE. (publisher)
   
   High throughput analysis techniques are required in order to make good use of the genomic sequences that have recently become available for many species, including humans.  Unfortunately, microarray data are also notoriously inaccurate, and it is possible to spend far too much time contemplating the results of a given microarray analysis method, only to arrive at a dead end.  In this chapter, we discuss several methods for microarray analysis we have developed, which are meant to provide more accurate results and/or quality assessments of the results obtained.

Journal Articles

1. S. Martin (2012), "Lattice Enumeration for Inverse Molecular Design Using the Signature Descriptor," Journal of Chemical Information and Modeling, 52(7):1787-1797. (journal, software)
   
   We describe an inverse quantitative structure–activity relationship (QSAR) framework developed for the design of molecular structures with desired properties. This framework uses chemical fragments encoded with a molecular descriptor known as a signature. It solves a system of linear constrained Diophantine equations to reorganize the fragments into novel molecular structures. The method has been previously applied to problems in drug and materials design but has inherent computational limitations due to the necessity of solving the Diophantine constraints. We propose a new approach to overcome these limitations using the Fincke–Pohst algorithm for lattice enumeration. We benchmark the new approach against previous results on LFA-1/ICAM-1 inhibitory peptides, linear homopolymers, and hydrofluoroether foam blowing agents.



2. S. Martin and J.-P. Watson (2011), "Non-Manifold Surface Reconstruction from High Dimensional Point Cloud Data," Computational Geometry: Theory and Applications, 44(8):427-441. (journal, software)
   
   We describe an algorithm capable of reconstructing a non-manifold surface embedded as a point cloud in a high dimensional space. Our algorithm will work for non-orientable surfaces, and for surfaces with certain types of self-intersection. The self-intersections must be ordinary double curves and are fitted locally by intersecting planes using a degenerate quadratic surface.




3. S. Martin, A. Thompson, E. A. Coutsias, and J.-P. Watson (2010), "Topology of Cyclo-Octane Energy Landscape," Journal of Chemical Physics 132:234115. (journal, software, presentation)
   
   Understanding energy landscapes is a major challenge in chemistry and biology. Although a wide variety of methods have been invented and applied to this problem, very little is understood about the actual mathematical structures underlying such landscapes. We have discovered an example of an energy landscape which is nonmanifold, demonstrating previously unknown mathematical complexity. The example occurs in the energy landscape of cyclo-octane, which was found to have the structure of a reducible algebraic variety, composed of the union of a sphere and a Klein bottle, intersecting in two rings.



4. S. Martin, G. Chandler, and M. S. Derzon (2008), "Simulation of High Pressure Micro-Capillary 3He Counters," Journal of Physics G: Nuclear and Particle Physics 35:115103. (journal)
   
   Low pressure (1-4 atm) cylindrical 3He counters are widely used as neutron detectors.  These detectors are relatively large (1-2.5 cm diameter) and can be subject to noise induced by microphonics.  Meanwhile, new advancements in micro-fabrication are enabling the manufacture of high pressure (over 3000 atm) micro-capillaries (~100 micron diameter).  Can these micro-capillaries be used as accurate and high-efficiency 3He counters?  To investigate these questions, we have developed a mathematical model/computer simulation.




5. W. M. Brown, S. Martin, S. N. Pollock, E. A. Coutsias, and J.-P. Watson (2008), "Algorithmic Dimensionality Reduction for Molecular Structure Analysis," Journal of Chemical Physics 129(6):064118.  (journal)
   
   Linear dimensionality reduction approaches have been used to exploit the redundancy in a Cartesian coordinate representation of molecular motion by producing low-dimensional representations of molecular motion.  Here, we investigate the effectiveness of several automated algorithms for nonlinear dimensionality reduction for representation of trans,trans-1,2,4-trifluorocyclooctane conformation - a molecule whose structure can be described on a 2-manifold in a Cartesian coordinate phase space




6. W. M. Brown, A. Sasson, D. R. Bellew, L. A. Hunsaker, S. Martin, A. Leitao, L. M. Deck, D. L. Vander Jagt, and T. Oprea (2008), "Efficient Calculation of Molecular Properties from Simulation using Kernel Molecular Dynamics," Journal of Chemical Information and Modeling 48(8):1626-1637. (journal)
   
   Understanding the relationship between chemical structure and function is a ubiquitous problem in chemistry and biology.  Here, we present a novel approach that uses aspects of simulation and informatics in order to formulate structure−property relationships.  We show how supervised learning can be utilized to overcome the sampling problem in simulation approaches.  Likewise, we show how learning can be achieved based on molecular descriptions that are rooted in the physics of dynamic intermolecular forces.



7. J.-L. Faulon, M. Misra, S. Martin, K. Sale, and R. Sapra (2008), "Genome Scale Enzyme-Metabolite and Drug-Target Interaction Predictions using the Signature Molecular Descriptor," Bioinformatics 24(2):225-233. (journal, pdf)
   
   Identifying protein enzymatic or pharmacological activities are important areas of research in biology and chemistry. Biological and chemical databases are increasingly being populated with linkages between protein sequences and chemical structures. There is now sufficient information to apply machine-learning techniques to predict interactions between chemicals and proteins at a genome scale. Current machine-learning techniques use as input either protein sequences and structures or chemical information. We propose here a method to infer protein–chemical interactions using heterogeneous input consisting of both protein sequence and chemical information.



8. S. Martin, Z. Zhang, A. Martino, and J.-L. Faulon (2007), "Boolean Dynamics of Genetic Regulatory Networks Inferred from Microarray Time Series Data," Bioinformatics 23(7):866-874.  (journal, pdf, supplement)
   
   Methods available for the inference of genetic regulatory networks strive to produce a single network, usually by optimizing some quantity to fit the experimental observations.  In this paper we investigate the possibility that multiple networks can be inferred, all resulting in similar dynamics.  This idea is motivated by theoretical work which suggests that biological networks are robust and adaptable to change, and that the overall behavior of a genetic regu-latory network might be captured in terms of dynamical basins of attraction.



9. S. Martin, Z. Mao, L. S. Chan, and S. Rasheed (2007), "Inferring Protein-Protein Interaction Networks using Protein Complex Data," International Journal of Bioinformatics Research and Applications 3(4):480-492.   Expanded version of BIOT 2006 conference paper with same authors. (journal)
   
   Present day approaches for the determination of protein-protein interaction networks are usually based on two hybrid experimental measurements.  Here we consider a computational method that uses another type of experimental data: instead of direct information about protein-protein interactions, we consider data in the form of protein complexes.  We propose a method for using these complexes to provide predictions of protein-protein interactions.  When applied to a dataset obtained from a cat melanoma cell line we find that we are able to predict when a protein pair belongs to a complex with ∼96% accuracy. 



10. S. Martin, R. D. Carr, and J.-L. Faulon (2006), "Random Removal of Edges from Scale Free Graphs," Physica A 371(2):870-876. (journal)
    
    It has been discovered that many naturally occurring networks (the internet, the power grid of the western US, various biological networks, etc.) satisfy a power-law degree distribution.  Such scale-free networks have many interesting properties, one of which is robustness to random damage.  This problem has been analyzed from the point of view of node deletion and connectedness.  Recently, it has also been considered from the point of view of node deletion and scale preservation.  In this paper we consider the problem from the point of view of edge deletion and scale preservation.  In agreement with the work on node deletion and scale preservation, we show that a scale-free graph should not be expected to remain scale free when edges are removed at random.



11. C. Wilson, G. S. Davidson, S. Martin, E. Andries, J. Potter, R. Harvey, K. Ar, Y. Xu, K. J. Kopecky, D. P. Ankerst, H. Gundacker, M. L. Slovak, M. Mosquera-Caro, I-M. Chen, D. L. Stirewalt, M. Murphy, F. A. Shultz, H. Kang, X. Wang, J. P. Radich, F. R. Appelbaum, S. R. Atlas, J. Godwin, and C. L. Willman (2006), “Gene Expression Profiling of Adult Acute Myeloid Leukemia Identifies Novel Biologic Clusters for Risk Classification and Outcome Prediction,” Blood  108(2): 685-696. (journal, pdf)
    
    To determine whether gene expression profiling could improve risk classification and outcome prediction in older acute myeloid leukemia (AML) patients, expression profiles were obtained in pretreatment leukemic samples from 170 patients whose median age was 65 years.  These expression profiles were analyzed using unsupervised clustering methods were used to classify patients into 6 cluster groups that varied significantly in rates of resistant disease.  These gene expression signatures provide insights into novel groups of AML not predicted by traditional studies that impact prognosis and potential therapy.



12. W. M. Brown, S. Martin, Mark D. Rintoul, and J.-L. Faulon (2006), "Designing Novel Polymers with Targeted Properties using the Signature Molecular Descriptor," Journal of Chemical Information and Modeling 46(2): 826-835. (journal)
    
    A method for solving the inverse quantitative structure−property relationship (QSPR) problem is presented which facilitates the design of novel polymers with targeted properties.  Here, we demonstrate the efficacy of the approach using the targeted design of polymers exhibiting a desired glass transition temperature, heat capacity, and density.  We show how the inverse problem can be solved to design poly(N-methyl hexamethylene sebacamide) despite the fact that the polymer was used not used in the training of this model.



13. W. M. Brown, S. Martin, J. Chabarek, C. Strauss, and J.-L. Faulon (2006), "Prediction of Beta-Strand Packing Interactions using the Signature Product," Journal of Molecular Modeling 12(3):355-361. (journal, poster)
    
    The prediction of β-sheet topology requires the consideration of long-range interactions between β-strands that are not necessarily consecutive in sequence. Since these interactions are difficult to simulate using ab initio methods, we propose a supplementary method able to assign β-sheet topology using only sequence information.  Our method is based on the signature molecular descriptor, which has been used previously to predict protein–protein interactions successfully, and to develop quantitative structure–activity relationships for small organic drugs and peptide inhibitors.



14. J.-L. Faulon, W. M. Brown, and S. Martin (2005), "Reverse Engineering Chemical Structures from Molecular Descriptors: How Many Solutions?," Journal of Computer Aided Molecular Design 19(9-10):637-650. (journal)
    
    Physical, chemical and biological properties are the ultimate information of interest for chemical compounds.  Molecular descriptors that map structural information to activities and properties are obvious candidates for information sharing.  In this paper, we consider the feasibility of using molecular descriptors to safely exchange chemical information in such a way that the original chemical structures cannot be reverse engineered.



15. S. Martin, D. Roe, and J.-L. Faulon (2005), "Predicting Protein-Protein Interactions using Signature Products," Bioinformatics 21(2):218-226. (journal, pdf, software)
    
    Proteome-wide prediction of protein–protein interaction is a difficult and important problem in biology.  Although there have been recent advances in both experimental and computational methods for predicting protein–protein interactions, we are only beginning to see a confluence of these techniques.  In this paper, we describe a very general, high-throughput method for predicting protein–protein interactions.  Our method combines a sequence-based description of proteins with experimental information that can be gathered from any type of protein–protein interaction screen.



16. C. Churchwell, M. D. Rintoul, S. Martin, D. P. Visco Jr., A. Kotu, R. S. Larson, L. O. Sillerud, D. C. Brown, and J.-L. Faulon (2004), "The Signature Molecular Descriptor 3. Inverse-Quantitative Structure-Activity Relationship of ICAM-1 Inhibitory Peptides," Journal of Molecular Graphics and Modeling 43(3):721-734. (journal)
    
    We present a methodology for solving the inverse-quantitative structure–activity relationship (QSAR) problem using the molecular descriptor called signature.  First, we create a QSAR equation that correlates the occurrence of a signature to the activity values using a stepwise multilinear regression technique.  Second, we construct constraint equations, specifically the graphicality and consistency equations, which facilitate the reconstruction of the solution compounds directly from the signatures.  Third, we solve the set of constraint equations, which are both linear and Diophantine in nature.  Last, we reconstruct and enumerate the solution molecules and calculate their activity values from the QSAR equation.



17. S. Martin, M. Kirby, and R. Miranda (2000), "Symmetric Veronese Classifiers with Application to Materials Design," Engineering Applications of Artificial Intelligence 13(5):513-520. (journal)
    
    To solve the materials classification problem, we propose a fast, exhaustive approach.  We propose to test every feature (chemical property), every pair of features, every three features, etc., against every classifier architecture from a certain group of classifiers known as Support Vector Machines.  This approach generalizes Pierre Villars’ work to higher dimensions and more operations.  We have duplicated his result in identifying the Mendeleev Number as the single best feature, and we have produced a new result for the case of two features: namely, we have identified the Mendeleev number with the valence electron number as the best combination of two features.

Letter to the Editor

1. S. Martin, M. P. Mosquera-Caro, J. W. Potter, G. S. Davidson, E. Andries, H. Kang, P. Helman, R. L. Veroff, S. R. Atlas, M. Murphy, X. Wang, K. Ar, Y. Xu, I-M. Chen, F. A. Schultz, C. S. Wilson, R. Harvey, E. Bedrick, J. Shuster, A. J. Carroll, B. Camitta, and C. L. Willman (2007), "Gene Expression Overlap affects Karyotype Prediction in Pediatric ALL," Leukemia 21:1341-1344. (journal)
   
   Treatment of acute lymphoblastic leukemia (ALL) involves the assignment of patients to risk groups based on cytogentic abnormalities.  Here we report the results of a gene expression experiment in which we have discovered that the predictions of karyotype are insensitive, in that there are a large number of false positive classifications among patients with poorly defined cytogenetic abnormalities.

Conference Proceedings

1. S. Martin, V. Subramanya, and S. Mills (2012), "Using Graph Layout to Generalise Focus+Context Image Magnification and Distortion," Image and Vision Computing New Zealand (IVCNZ): 97-102. (proceedings, presentation)
   
   We present a novel framework for performing distortion-oriented focus+context image magnification. Our framework uses algorithms from graph drawing to manipulate the mesh underlying an image. Specifically, we apply a spectral graph layout algorithm to a weighted graph, where vertices in the graph correspond to pixels in the image, and edges connect directly adjacent vertices/pixels. By assigning appropriate weights to the edges, we can replicate the results of previous distortion-oriented approaches. In addition, we can perform image-aware distortion by using pixel values to influence the edge weights of our graph. We compare our approach to previous methods and demonstrate new results using image-based edge weighting schemes.



2. S. Martin, W. M. Brown, R. Klavans, and K. Boyack (2011), "OpenOrd: An Open-Source Toolbox for Large Graph Layout," Visualization and Data Analysis (VDA): 7868-06. (proceedings, software)
   
   We document an open-source toolbox for drawing large-scale undirected graphs.  This toolbox is based on a previously implemented closed-source algorithm known as VxOrd.  Our toolbox, which we call OpenOrd, extends the capabilities of VxOrd to large graph layout by incorporating edge-cutting, a multi-level approach, average-link clustering, and a parallel implementation.  At each level, vertices are grouped using force-directed layout and average-link clustering.  The clustered vertices are then re-drawn and the process is repeated.  When a suitable drawing of the coarsened graph is obtained, the algorithm is reversed to obtain a drawing of the original graph.  This approach results in layouts of large graphs which incorporate both local and global structure.



3. S. Martin and S. McKenna, (2007), "Predicting Building Contamination using Machine Learning," International Conference on Machine Learning and Applications (ICMLA): 192-197. (proceedings, presentation)
   
   Potential events involving biological or chemical contamination of buildings are of major concern in the area of homeland security. Tools are needed to provide rapid, onsite predictions of contaminant levels given only approximate measurements in limited locations throughout a building. In principal, such tools could use calculations based on physical process models to provide accurate predictions. In practice, however, physical process models are too complex and computationally costly to be used in a real-time scenario. We investigate the feasibility of using machine learning to provide easily computed but approximate models that would be applicable in the field.
   



4. J. Joo, S. Plimpton, S. Martin, L. Swiler, and J.-L. Faulon (2007), "Sensitivity Analysis of a Computational Model of the IKK-NF-kB-IkBa-A20 Signal Transduction Network," Annals of the New York Academy of Sciences 1115:221-239. (proceedings)
   
   The NF-kB signaling network plays an important role in many different compartments of the immune system during immune activation.  Using a computational model of the NF-kB signaling network involving two negative regulators, IkBa and A20, we performed sensitivity analyses with three different sampling methods and present a ranking of the kinetic rate variables by the strength of their influence on the NF-kB signaling response.  We also present a classification of temporal-response profiles of nuclear NF-kB concentration into six clusters, which can be regrouped to three biologically relevant clusters.



5. S. Martin (2006), "An Approximate Version of Kernel PCA," Proceedings of the 5th International Conference on Machine Learning and Applications (ICMLA):239-244. (proceedings, presentation, poster)
   
   We propose an analog of kernel principal component analysis (kernel PCA). Our algorithm is based on an approximation of PCA which uses Gram-Schmidt orthonormalization. We combine this approximation with support vector machine kernels to obtain a nonlinear generalization of PCA. By using our approximation to PCA we are able to provide a more easily computed (in the case of many data points) and readily interpretable version of kernel PCA.



6. S. Martin, Z. Mao, L. S. Chan, S. Rasheed (2006), "Protein Interactions Extrapolated from Feline Protein Complexes," Proceedings of the 3rd Biotechnology and Bioinformatics Symposium (BIOT):45-52. (pdf, presentation)
   
   The determination of protein-protein interaction networks is a difficult problem in biology.  Present day approaches to this problem are usually based on two hybrid experimental measurements coupled with refinement and extrapolation using computational techniques.  Here we consider a  computational method for similar refinement and extrapolation using experimental data from which protein interactions can not be directly inferred.



7. S. Martin (2006), "The Numerical Stability of Kernel Methods," Proceedings of the 9th International Symposium on Artificial Intelligence and Mathematics (AIMATH):P01. (pdf, presentation)
   
   Kernel methods use kernel functions to provide nonlinear versions of different methods in machine learning and data mining, such as Principal Component Analysis and Support Vector Machines.  These kernel functions require the calculation of some or all of the entries of a matrix of the form X^TX .  The formation of this type of matrix is known to result in potential numerical instability in the case of least squares problems. How does the computation of the kernel matrix impact the stability of kernel methods?  We investigate this question in detail in the case of kernel PCA and also provide some analysis of kernel use in Support Vector Machines.



8. S. Martin (2005), "Training Support Vector Machines using Gilbert's Algorithm," Proceedings of the 5th IEEE International Conference on Data Mining (ICDM):306-313.  (proceedings, presentation, software) Support vector machines are classifiers designed around the computation of an optimal separating hyperplane.  This hyperplane is typically obtained by solving a constrained quadratic programming problem, but may also be located by solving a nearest point problem.  Gilbert's algorithm can be used to solve this nearest point problem but is unreasonably slow.  In this paper we present a modified version of Gilbert's algorithm for the fast computation of the support vector machine hyperplane.



9. S. Martin and A. Backer (2005), "Estimating Manifold Dimension by Inversion Error," Proceedings of the 20th annual ACM Symposium on Applied Computing (SAC):22-26. (proceedings, presentation)
   
   There has been recent interest in the application of a class of nonlinear dimensionality reduction algorithms which assume that a dataset has been sampled from a manifold.  From this assumption, it follows that estimating the dimension of the manifold is the first step in analyzing an image dataset.  Once an estimate of the dimension is obtained, it is used as a parameter for the nonlinear dimensionality reduction algorithm.  In this paper, we consider reversing this approach.  Instead of estimating the dimension of the manifold in order to obtain a low dimensional representation, we consider producing low dimensional representations in order to estimate of the dimensionality of the manifold.



10. S. Martin, M. Kirby, and R. Miranda (2000), "Kernel/Feature Selection for Support Vector Machines Applied to Materials Design," Proceedings of 9th IFAC Symposium on Artificial Intelligence in Real Time Control (AIRTC):29-34. (pdf)
    
    Support Vector Machines are classifiers with architectures determined by kernel functions.  In these proceedings we propose a method for selecting the best SVM kernel for a given classification problem.  Our method searches for the best kernel by remapping the data via a kernel variant of the classical Gram-Schmidt orthonormalization procedure then using Fisher’s linear discriminant on the remapped data.



Extended Abstracts

1. S. Martin, W. M. Brown, J.-L. Faulon, D. Weis, D. Visco, and J. Kenneke (2005), "Inverse Design of Large Molecules using Linear Diophantine Equations," Proceedings of the 4th IEEE Computational Systems Bioinformatics Workshops (CSBW):11-16. (proceedings, poster)
   
   We have previously developed a method for the inverse design of small ligands.  A key step in our method involves computing the Hilbert basis of a system of linear Diophantine equations.  In our previous application, the ligands considered were small peptide rings, so that the resulting system of Diophantine equations was relatively small and easy to solve.  When considering larger molecules, however, the Diophantine system is larger and more difficult to solve.  In this work we present a method for reducing the system of Diophantine equations before they are solved, allowing the inverse design of larger compounds.



2. S. Martin, G. S. Davidson, E. E. May, J.-L. Faulon, and M. Werner-Washburne (2004), "Inferring Genetic Networks from Microarray Data," Proceedings of the 3rd IEEE Computational Systems Bioinformatics (CSB):566-569. (proceedings, poster)
   
   In theory, it should be possible to infer realistic genetic networks from time series microarray data.  In practice, however, network discovery has proved problematic.  The three major challenges are 1) inferring the network; 2) estimating the stability of the inferred network; and 3) making the network visually accessible to the user.  Here we describe a method, tested on publicly available time series microarray data, which addresses these concerns.



3. J.-L. Faulon, S. Martin, and R. D. Carr (2004), "Dynamical Robustness in Gene Regulatory Networks," Proceedings of the 3rd IEEE Computational Systems Bioinformatics (CSB):626-627. (proceedings, pdf, poster)
   
   We investigate the robustness of biological networks, emphasizing gene regulatory networks.  We define the robustness of a dynamical network as the magnitude of perturbation in terms of rates and concentrations that will not change the steady state dynamics of the network.  We find the number of dynamical networks versus their dynamical robustness follows a power law.



Dissertation and M.Sc. Paper

1. S. Martin (2001), Techniques in Support Vector Classification, Ph. D. Dissertation, Colorado State University. (pdf)
   
   Here we consider three problems in Support Vector Classification: feature selection, kernel selection, and training.  Feature selection is done using Fisher's discriminant adapted to SVMs.  Kernel selection is done using a kernel version of Gram-Schmidt orthonormalization, and training is done using a geometrical interpretation of the quadratic optimization program normally used to solve for the SVM.





2. S. Martin (1997), "Concerning the Quadratic Relations which define the Grassman Manifold," M.S. Paper, Colorado State University. (pdf)
   
   The Plucker embedding gives a bijective correspondence between the d-planes of a projective space Pⁿ and the points of the Grassman Manifold in a higher dimensional space P^N.  The Grassman Manifold can be defined as the set of points in P^N whose homogeneous coordinates satisfy certain quadratic relations, those relations being generated by sequences in {0,...,n}.  Here we present a minimal set of generating sequences for the quadratic relations and subsequently investigate the linear independence of said relations.



Last Updated Jan. 24, 2013.