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Matlab-Toolbox-for-Dimensionality-Reduction

Matlab Toolbox for Dimensionality Reduction

Matlab Toolbox for Dimensionality Reduction (v0.8.1b)

Information

Author: Laurens van der Maaten Affiliation: Delft University of Technology Contact: lvdmaaten@gmail.com Release date: March 21, 2013 Version: 0.8.1b

Installation

Copy the drtoolbox/ folder into the $MATLAB_DIR/toolbox directory (where $MATLAB_DIR indicates your Matlab installation directory). Start Matlab and select 'Set path...' from the File menu. Click the 'Add with subfolders...' button, select the folder $MATLAB_DIR/toolbox/drtoolbox in the file dialog, and press Open. Subsequently, press the Save button in order to save your changes to the Matlab search path. The toolbox is now installed. Some of the functions in the toolbox use MEX-files. Precompiled versions of these MEX-files are distributed with this release, but the compiled version for your platform might be missing. In order to compile all MEX-files, type cd([matlabroot '/toolbox/drtoolbox']) in your Matlab prompt, and execute the function MEXALL.

Features

This Matlab toolbox implements 34 techniques for dimensionality reduction and metric learning. These techniques are all available through the COMPUTE_MAPPING function or through the GUI. The following techniques are available:

• Principal Component Analysis ('PCA')
• Linear Discriminant Analysis ('LDA')
• Multidimensional scaling ('MDS')
• Probabilistic PCA ('ProbPCA')
• Factor analysis ('FactorAnalysis')
• Sammon mapping ('Sammon')
• Isomap ('Isomap')
• Landmark Isomap ('LandmarkIsomap')
• Locally Linear Embedding ('LLE')
• Laplacian Eigenmaps ('Laplacian')
• Hessian LLE ('HessianLLE')
• Local Tangent Space Alignment ('LTSA')
• Diffusion maps ('DiffusionMaps')
• Kernel PCA ('KernelPCA')
• Generalized Discriminant Analysis ('KernelLDA')
• Stochastic Neighbor Embedding ('SNE')
• Symmetric Stochastic Neighbor Embedding ('SymSNE')
• t-Distributed Stochastic Neighbor Embedding ('tSNE')
• Neighborhood Preserving Embedding ('NPE')
• Locality Preserving Projection ('LPP')
• Stochastic Proximity Embedding ('SPE')
• Linear Local Tangent Space Alignment ('LLTSA')
• Conformal Eigenmaps ('CCA', implemented as an extension of LLE)
• Maximum Variance Unfolding ('MVU', implemented as an extension of LLE)
• Landmark Maximum Variance Unfolding ('LandmarkMVU')
• Fast Maximum Variance Unfolding ('FastMVU')
• Locally Linear Coordination ('LLC')
• Manifold charting ('ManifoldChart')
• Coordinated Factor Analysis ('CFA')
• Gaussian Process Latent Variable Model ('GPLVM')
• Deep autoencoders ('Autoencoder')
• Neighborhood Components Analysis ('NCA')
• Maximally Collapsing Metric Learning ('MCML')
• Large Margin Nearest Neighhbor metric learning ('LMNN')

Furthermore, the toolbox contains 6 techniques for intrinsic dimensionality estimation. These techniques are available through the function INTRINSIC_DIM. The following techniques are available:

• Eigenvalue-based estimation ('EigValue')
• Maximum Likelihood Estimator ('MLE')
• Estimator based on correlation dimension ('CorrDim')
• Estimator based on nearest neighbor evaluation ('NearNb')
• Estimator based on packing numbers ('PackingNumbers')
• Estimator based on geodesic minimum spanning tree ('GMST')

In addition to these techniques, the toolbox contains functions for prewhitening of data (the function PREWHITEN), exact and estimate out-of-sample extension (the functions OUT_OF_SAMPLE and OUT_OF_SAMPLE_EST), and a function that generates toy datasets (the function GENERATE_DATA). The graphical user interface of the toolbox is accessible through the DRGUI function.

Usage

All the functions that you should call as a user of the toolbox are located in the same folder as this Readme-file. The folder contains the following files:

• compute_mapping.m This function performs the specified dimension reduction technique on the specified data set. Type HELP COMPUTE_MAPPING to get details on supported techniques and on the parameters of the techniques.
• drgui.m This function allows you to use some of the toolbox functionality via a graphical user interface.
• generate_data.m This function generates some artificial data sets such as the Swiss roll data set.
• intrinsic_dim.m This function performs intrinsic dimensionality estimation using the specified estimator on the specified data set.
• mexall.m This function compiles all the MEX-files that are required to use the toolbox. Please run immediately after installation.
• out_of_sample.m This function takes as input a dimension reduction mapping and a set of new test points, and outputs the locations of the test points in the reduced space. This function is only supported by parametric and spectral techniques.
• out_of_sample_est.m This function takes as input a training set, a reduced version of that training set, and a set of new test points, and finds an approximate locations of the test points in the reduced space. Only use this function for techniques that do not support out-of-sample-extensions.
• prewhiten.m This function whitens data, i.e., it makes it zero-mean, identity-covariance.
• reconstruct_data.m This function computes reconstructions of reduced data for linear techniques and autoencoders.
• test_toolbox.m This function runs a full test of all functionalities of the toolbox.

Here is an example on how to use the toolbox:

[X, labels] = generate_data('helix', 2000); figure, scatter3(X(:,1), X(:,2), X(:,3), 5, labels); title('Original dataset'), drawnow no_dims = round(intrinsic_dim(X, 'MLE')); disp(['MLE estimate of intrinsic dimensionality: ' num2str(no_dims)]); [mappedX, mapping] = compute_mapping(X, 'PCA', no_dims);
figure, scatter(mappedX(:,1), mappedX(:,2), 5, labels); title('Result of PCA'); [mappedX, mapping] = compute_mapping(X, 'Laplacian', no_dims, 7); figure, scatter(mappedX(:,1), mappedX(:,2), 5, labels(mapping.conn_comp)); title('Result of Laplacian Eigenmaps'); drawnow

It will create a helix dataset, estimate the intrinsic dimensionality of the dataset, run Laplacian Eigenmaps on the dataset, and plot the results. All functions in the toolbox can work both on data matrices as on PRTools datasets (http://prtools.org). For more information on the options for dimensionality reduction, type HELP COMPUTE_MAPPING in your Matlab prompt. Information on the intrinsic dimensionality estimators can be obtained by typing the HELP INTRINSIC_DIM.

Pitfalls

When you run certain code, you might receive an error that a certain file is missing. This is because in some parts of the code, MEX-functions are used. I provide a number of precompiled versions of these MEX-functions in the toolbox. However, the MEX-file for your platform might be missing. To fix this, type in your Matlab:

mexall

Now you have compiled versions of the MEX-files as well. This fix also solves slow execution of the shortest path computations in Isomap.

If you encounter an error concerning CSDP while running the FastMVU-algorithm, the binary of CSDP for your platform is missing. If so, please obtain a binary distribution of CSDP from https://projects.coin-or.org/Csdp/ and place it in the drtoolbox/techniques directory. Make sure it has the right name for your platform (csdp.exe for Windows, csdpmac for Mac OS X (PowerPC), csdpmaci for Mac OS X (Intel), and csdplinux for Linux).

Many methods for dimensionality reduction perform spectral analyses of sparse matrices. You might think that eigenanalysis is a well-studied problem that can easily be solved. However, eigenanalysis of large matrices turns out to be tedious. The toolbox allows you to use two different methods for eigenanalysis:

• The original Matlab functions (based on Arnoldi methods)
• The JDQR functions (based on Jacobi-Davidson methods)

For problems up to 10,000 data points, we recommend using the 'Matlab' setting. For larger problems, switching to 'JDQR' is often worth trying.

Some other common questions are answered in the FAQ on http://homepage.tudelft.nl/19j49/Matlab_Toolbox_for_Dimensionality_Reduction.html

Papers

For more information on the implemented techniques and for theoretical and empirical comparisons, please have a look at the following papers:

• L.J.P. van der Maaten, E.O. Postma, and H.J. van den Herik. Dimensionality Reduction: A Comparative Review. Tilburg University Technical Report, TiCC-TR 2009-005, 2009.
• L.J.P. van der Maaten and G.E. Hinton. Visualizing High-Dimensional Data Using t-SNE. Journal of Machine Learning Research 9(Nov):2579-2605, 2008.

Version history

Version 0.8.1b:

• Several bugfixes.

Version 0.8b:

• Several bugfixes.
• Added function RECONSTRUCT_DATA to reconstruct data from low-dimensional embedding.
• Added implementation of large-margin nearest neighbor (LMNN) metric learning.
• Updated implementation of deep autoencoders to use pretraining based on denoising autoencoders.

Version 0.7.1b:

• Small bugfixes.

Version 0.7b:

• Many small bugfixes and speed improvements.
• Added out-of-sample extension for manifold charting.
• Added first version of graphical user interface for the toolbox. The GUI was developed by Maxim Vedenev with the help of Susanth Vemulapalli and Maarten Huybrecht. I made some changes in the initial version of the GUI code.
• Added implementation of Gaussian Process Latent Variable Model (GPLVM).
• Removed Simple PCA as probabilistic PCA is more appropriate.

Version 0.6b:

• Resolved bug in LLE that was introduced with v0.6b.
• Added implementation of t-SNE.
• Resolved small bug in data generation function.
• Improved RBM implementation in autoencoders (note that successful training of an RBM still depends on parameter settings such as weight_cost and learning rate that can only be set in the train_rbm.m code).
• Added implementation of Sammon mapping.
• Removed dependency on the Statistics toolbox in Laplacian Eigenmaps.
• Resolved bug in implementation of SPE.
• Various speed and memory improvements by exploiting Matlab's new BSXFUN functionality.

Version 0.5b:

• Resolved issues with unconnected neighborhood graph for LLE and Laplacian Eigenmaps (now works like Isomap).
• Resolved bug in prewhitening of data.
• Improved implementations of SNE and symmetric SNE.
• Resolved two bugs in nearest neighbor intrinsic dimensionality estimator.
• Replaced MDS implementation by implementation for classical MDS.

Version 0.4b:

• Added Symmetric SNE ('SymSNE') implementation.
• Added Landmark MVU ('LandmarkMVU') implementation.
• Added completely new implementation of autoencoders using RBM training.
• Added out-of-sample extensions for (Landmark) Isomap, LLE, Laplacian Eigenmaps, Landmark MVU, and FastMVU.
• Added new 'difficult' dataset to data generation function.
• Improved implementations of NPE, LPP, and LLTSA.
• Resolved issue with parameter parsing in manifold charting.
• Resolved issue with adaptive neighborhood selection combined target dimensionalities higher than 40.
• The number of timesteps t can now be specified in diffusion maps.
• Speed up the implementations of Kernel PCA and Kernel LDA for datasets with more than 3,000 instances (with factor ~5).
• Resolved efficiency issue eigendecomposition performed by diffusion maps.
• Speed improvement in nearest neighbor search for datasets with more than 2,000 datapoints (with assistance from James Monaco).
• Speed improvement of Hessian LLE implementation.
• The toolbox now works without using the Statistics Toolbox.
• Data generation function now also returns the true underlying manifold.
• Resolved issue that might occur when Isomap or FastMVU are employed on a PRTools dataset.

Version 0.3b:

• Improved PCA implementation for cases in which D > N.
• Added implementation of probabilistic PCA (using EM algorithm).
• Added implementation of manifold charting.
• Added function for adaptive neighborhood selection (with assistance from Nathan Mekuz).
• Various speed improvements (with assistance from Nathan Mekuz).
• Added welcome message.
• Added contents information for VER command.
• Fixed issue with divisions by zero in intrinsic dimensionality estimators.
• Removed implementation of ICA from the toolbox.

Version 0.2b:

• Resolved issues in LPP, NPE, LTSA, and Kernel PCA implementations.
• Added implementations of LLTSA and Simple PCA.
• Added Conformal Eigenmaps (CCA) as a postprocessing step for LLE.
• Added MVU as a postprocessing step for LLE.
• Added function for prewhitening of data.
• Added function for precise out-of-sample extensions for PCA, LDA, NPE, LPP, LLTSA, Simple PCA, autoencoders, and Kernel PCA.
• Added six techniques for intrinsic dimension estimation.

Version 0.1b:

• The initial release of the toolbox.

Disclaimer

(C) Laurens van der Maaten, 2013 You are free to use, modify, or redistribute this code in any way you want for non-commercial purposes. If you do so, I would appreciate it if you refer to the original author or refer to one of the papers mentioned above.

Parts of the code were taken from other authors, but often I made numerous improvements and modifications. A list of files in which I use source code from other authors is given below:

• minimize.m: C.E. Rasmussen
• hlle.m, mgs.m: C. Grimes and D. Donoho
• dijk.m: M.G. Kay
• dijkstra.cpp: J. Boyer
• L2_distance.m: R. Bunschoten
• jdqr.m, jdqz.m: G. Sleijpen
• components.m: J. Gilbert
• hillclimber2c.m, lmvu.m, fastmvu.m, computegr.c, csdp.m, mexCCACollectData2.c, writesdpa.m, sparse_nn.m, readsol.m, sdecca2.m, hill_obj.m: K. Weinberger
• llc.m, infermfa.m, mppca.m: Y. Teh
• cca.m, mexCCACollectData.c: F. Sha
• combn.m: J. van der Geest
• sammon.m: G.C. Cawley
• GUI: M. Vedenev

Contact

If you encounter any problems, please consult the online FAQ first! If you have any bugs, questions, suggestions, or modifications, please contact me:

lvdmaaten@gmail.com