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Kernel based data fusion

Kernel based data fusion. Discussion of a Paper by G. Lanckriet. Paper. Overview. Problem : Aggregation of heterogeneous data Idea : Different data are represented by different kernels Question : How to combine different kernels in an elegant/efficient way?

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Kernel based data fusion

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  1. Kernel based data fusion Discussion of a Paper by G. Lanckriet

  2. Paper

  3. Overview Problem: Aggregation of heterogeneous data Idea: Different data are represented by different kernels Question: How to combine different kernels in an elegant/efficient way? Solution: Linear combination and SDP Application: Recognition of ribosomal and membrane proteins

  4. Linear combination of kernels • Resulting kernel K is positive definite (xTKx > 0 for x, provided i > 0 and xTKi x > 0 ) • Elegant aggregation of heterogeneous data • More efficient than training of individual SVMs • KCCA uses unweighted sum over individual kernels xTKx = x2K x2K weight kernel 0 x

  5. Support Vector Machine square norm vector penalty term Hyperplane slack variables

  6. Dual form quadratic, convex scalar  0 positive definite Lagrange multipliers • Maximization instead of minimization • Equality constraints • Lagrange multipliers instead of w,b, • Quadratic program (QP)

  7. Inserting linear combination ugly Fixed trace,avoids trivial solution Combined kernel must be within the cone of positive semidefinite matrices

  8. Cone and other stuff Positive semidefinite cone: A Positive semidefinite: xTAx ≥ 0, x The set of all symmetric positive semidefinite matrices of particular dimension is called the positive semidefinite cone. http://www.convexoptimization.com/dattorro/positive_semidefinate_cone.html

  9. Semidefinite program (SDP) Fixed trace,avoids trivial solution positive semidefinite constraints

  10. Dual form • Quadratically constraint quadratic program (QCQP) • QCQPs can be solved more efficiently than SDPs(O(n3) <-> O(n4.5)) • Interior point methods quadratic constraint

  11. Interior point algorithm • Classical Simplex method follows edges of polyhedron • Interior point methods walk through the interior of the feasible region Linear program: maximize cTx subject to Ax < b x ≥ 0

  12. Application • Recognition of ribosomal and membrane proteins in yeast • 3 Types of data • Amino acid sequences • Protein protein interactions • mRNA expression profiles • 7 Kernels • Empirical kernel map -> sequence homology • BLAST(B), Smith-Waterman(SW), Pfam • FFT -> sequence hydropathy • KD hydropathy profiles, padding, low-pass filter, FFT, RBF • Interaction kernel(LI) -> PPI • Diffusion(D) -> PPI • RBF(E) -> gene expression

  13. Results • Combination of kernels performs better than individual kernels • Gene expression (E) most important for ribosomal protein recognition • PPI (D) most important for membrane protein recognition

  14. Results • Small improvement compared to weights = 1 • SDP robust in the presence of noise • How performs SDP versus kernel weights derived from accuracy of individual SVMs? • Membrane protein recognition • Other methods use sequence information only • TMHMM designed for topology prediction • TMHMM not trained on yeast only

  15. Why is this cool? Everything you ever dreamed of: • Optimization of C included(2-norm soft margin SVM =1/C) • Hyperkernels (optimize the kernel itself) • Transduction (learn from labeled & unlabeled samples in polynomial time) • SDP has many applications(Graph theory, combinatorial optimization, …)

  16. Literature • Learning the kernel matrix with semidefinite programming G.R.G.Lanckrit et. al, 2004 • Kernel-based data fusion and its application to protein function prediction in yeastG.R.G.Lanckrit et. al, 2004 • Machine learning using HyperkernelsC.S.Ong, A.J.Smola, 2003 • Semidefinite optimizationM.J.Todd, 2001 • http://www-user.tu-chemnitz.de/~helmberg/semidef.html

  17. Software • SeDuMi (SDP) • Mosek (QCQP, Java,C++, commercial) • YALMIP (Matlab) … http://www-user.tu-chemnitz.de/~helmberg/semidef.html

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