README( Aug 18, 2006)--------------------------------------------------------------------I.   IntroductionII.  InstallationIII. Usage & ExamplesIV.  Contacts & CopyrightsV.   References--------------------------------------------------------------------I. IntroductionFunction [Z, CCAEIGEN, CCADETAILS] = CCA(X, Y, EDGES, OPTS) computes a lowdimensional embedding Z in R^d that maximally preserves angles among  input data X that lives in R^D, with the algorithm Conformal Component Analysis (CCA).The key idea is to constrain Z = L*Y where Y is a partial basis that spans the space of R^d. Such Y can be computed from graph Laplacian (such as the outputs of Laplacian eigenmap [3] and Locally Linear Embedding, ie, LLE [4]). The parameterization matrix L is optimized to maximallypreserve angles between edges coded in the sparse matrix EDGES.Ref.[1-2] shows that the optimization problem can be formulated as a semidefinite programming problem (SDP) in P = L'*L.  SDPs are convex optimization problems; they can be solved efficiently.Recently, the linear parameterization idea (Z=L*Y) has also been used in findingmaximum variance unfolding (MVU, a.k.a, Semidefinite Embedding) [5] and hasbeen applied to sensor localization problems [6]. MVU aims to compute (local) distance mapping between high dimensional data points and low dimensional representation, while CCA computes angle preserving mappings.     This distribution also implements a variant of MVU, which is described in [6].--------------------------------------------------------------------II. InstallationThe distribution of this package includes following files1. README: this file2. cca.m:  main Matlab function3. mexCCACollectData.c:  a C source file 4. demoCCA.m:  demo function    To install, make sure cca.m (optionally, demoCCA.m) is in your Matlab path.Also, in Matlab's command window, issue command "mex mexCCACollectData.c" to the C source file compile and make sure the compiled object is in your Matlabpath.A SDP solver is needed. This implementation uses CSDP solver [7]. Make sure themain Matlab function csdp.m is in your Matlab path.This distribution has been tested on MAC OS X 10.3.x or above, with Matlab 7.0 or above, as well as Intel-based Linux (kernel 2.4 or above) with Matlab 7.0 orabove. The CSDP solver that has been tested is version 4.9 and 5.0.--------------------------------------------------------------------III. Usage & Examples.1. To run examples, type demoCCANote: You need an implementaiton of LLE to run demoCCA().This distribution provides one based on Sam Roweis's implementation at http://www.cs.toronto.edu/~roweis/lle/ .2. To call the function cca(), use following syntax:<=== Inputs:X: input data stored in matrix  (D x N) where D is the dimensionalityY: partial basis stored in matrix (d x N)   Note that if graph Laplacians/LLE are used to compute Y, then make sure Y(1,:)   is not a constant vector.EDGES: a sparse matrix of (N x N). In each column i, the row indices j to  nonzero entrices define data points that are in the nearest neighbors of  data point i.OPTS:   OPTS.method        == 'CCA':   implements CCA algorithm [1,2]          == 'MVU':   implements MVU [6]                default is 'CCA'   OPTS.regularizer: the tradeoff factor when maximizing trace (as in MVU [5])    while preserving distances (it is okay to use a nonzero regularizer in CCA    algorithm, where we will get an "unfolded" CCA to enforce low rank in    solutions)        default is 0.           OPTS.relative: In MVU, either absolute distance can be preserved     (OPTS.relative ==0) or relative distance can be preserved (OPTS.relative == 1)        default is 0        ====> Outputs:Z: low dimensional embedding (d X N)CCAEIGN: eigenspectra of the matrix P = L'*L. If P is low-rank (say d' < d),  then Z can be cutoff at d' dimension as dimensionality reduced further.CCADetails:   CCADetails.cost: final objective function value   CCADetails.c: if OPTS.method == 'CCA', this is the local scaling    coefficient for each data point   CCADetails.P: matrix P = L'*L    CCADetails.opts: options used to run the algorithm, same as OPTS with    checked parameters   CCADetails.sdpflag: SDP solver exit flag. (check CSDP solver manual. In   general, an exit flag of 0 means normal exit.)--------------------------------------------------------------------IV. Misc.1. Please send bug report to feisha@cs.berkeley.edu2. Feel free to use it for educational and research purpose.--------------------------------------------------------------------V. References   [1]    Fei Sha and Lawrence K. Saul (2005).   Analysis and Extension of Spectral Methods for Nonlinear Dimensionality Reduction.  Proceedings of 22nd  International Conference on Machine Learning (ICML 2005), Bonn, Germany[2] Fei Sha, Kilian Weinberge and Lawrence K. Saul (200x). Manifold learning by  semidefinite programming. Manuscript in preparation.[3].  M. Belkin and P. Niyogi, Laplacian Eigenmaps for Dimensionality Reduction and  Data Representation, Neural Comp., 15(6):1373-1396, 2003.    [4]. L. K. Saul and S. T. Roweis (2003). Think globally, fit locally: unsupervised  learning of low dimensional manifolds. Journal of Machine Learning Research 4:119-155[5].  K. Q. Weinberger and L. K. Saul (2006). Unsupervised learning of image  manifolds by semidefinite programming. International Journal of Computer  Vision 70(1): 77-90 [6]. K. Q. Weinberger, Fei Sha, Qihui Zhu and L. K. Saul (2006). Localization in  Large Scale Sensor Networks via Semidefinite Programming and Graph  Regularization. Submitted manuscript. [7]. Borchers, B., CSDP, A C Library for Semidefinite Programming. Optimization  Methods and Software 11(1):613-623, 1999.  http://infohost.nmt.edu/~borchers/csdp.html --------------------------------------------------------------------