Spectral Regression: A Regression Framework for Regularized Subspace Learning
Introduction
Spectral methods have recently emerged as a powerful tool for
dimensionality reduction and manifold learning. These methods use
information contained in the eigenvectors of a data affinity (i.e.,
itemitem similarity) matrix to reveal low dimensional structure in
high dimensional data. The most popular manifold learning algorithms
include Locally Linear Embedding, Isomap, and Laplacian Eigenmap.
However, these algorithms only provide the embedding results of
training samples.
There are many extensions of these approaches
which try to solve the outofsample extension problem by seeking an
embedding function in reproducing kernel Hilbert space. A
disadvantage of all these approaches is that their computations
usually involve eigendecomposition of dense matrices which is
expensive in both time and memory.
Spectral Regression (SR) [1] [2] is a novel regression framework for efficient
regularized subspace learning. Some interesting aspects of SR include
 SR casts the problem of learning an embedding function into a
regression framework, which avoids eigendecomposition of dense
matrices. With different graph matrix W, SR provides the efficient solutions of Linear Discriminant Analysis (LDA)[3] [7],
Locality Preserving Projection (LPP)[7] [10], Neighborhood Preserving
Embedding (NPE)[7], Isometric Projection (IsoP)[4], Locality Sensitive Discriminant Analysis (LSDA) and much more.
 With regression as the building block, various
kinds of regularization techniques can be easily incorporated
in SR which makes it more flexible (e.g., $L_1$norm regularizer to produce sparse projections [9]).
 SR can be performed in supervised [3], unsupervised [10] and
semisupervised [1] [6] situations. It can make efficient use of both labeled
and unlabeled points to discover the intrinsic discriminant
structure in the data.
 SR may be conducted in the original space or in the reproducing kernel Hilbert space
(RKHS) into which data points are mapped. This gives rise to kernel SR [2] [5] [8]
(efficient solutions for many kernel subspace learning algorithms).
Codes
 SR_caller: SR caller (response generation).
 SR: Spectral Regression (Called by SR_caller)
 lsqr2: A iterative least squares solver, which will be called by SR
 lars: Least Angle Regression solver, which will be called by SR
 If you want to learn a sparse SR subspace by specifying the L1regularization parameter instead of providing the cardinality, you can set the 'LASSOway' as 'SLEP'.
Please see SLEP for more details on SLEP software. For your convenience, I extract the functions from SLEP
package which will be called by SR:
 LeastR, initFactor, sll_opts, ep1R.mexw64, ep1R.mexw32
 KSR_caller: SR caller (response generation).
 KSR: Kernel Spectral Regression (Called by KSR_caller)
NormalizeFea, constructW, constructKernel, EuDist2 and Eigenmap are also required.
The above codes are complicated and hard to follow. To successfully use (or, modify) these codes, one should be very familiar with the subspace learning area. Thus, I provide below some simple interfaces for different scenarios.
 Supervised and semisupervised learning : The following codes show how to use SR (linear) or KSR (nonlinear ) in supervised or semisupervised (utilize the unlabel data to learn the subspace) settings.
 Spectral Regression Discriminant Analisys: Supervised (semisupervised) linear SR.
 SRDAtrain: function to learn the SRDA model.
 SRDApredict uses SRDA as a classifier. We implement a nearest center rule in the SRDA subspace for classification.
 SRDAtest uses SRDA as a dimensionality reduction tool. It will embed the input data to a c1 dimensional SRDA subspace, where c is the number of classes.
 Example here
 Example on Sparse SRDA (Sparse LDA)

Deng Cai, Xiaofei He, and Jiawei Han, "SRDA: An Efficient Algorithm for LargeScale Discriminant Analysis", IEEE TKDE 2008. (pdf)
Bibtex source
 Spectral Regression Kernel Discriminant Analisys : Supervised (semisupervised) nonlinear SR.
 Unsupervised learning : The following codes show how to use SR (linear) or KSR (nonlinear ) in unsupervised settings.
 Unsupervised Spectral Regression : Unsupervised linear SR.
 Unsupervised Kernel Spectral Regression : Unsupervised nonlinear SR.
 UKSRtrain: function to learn the KSR model.
 UKSRtest: uses KSR as a dimensionality reduction tool. It will embed the input data to a d dimensional KSR subspace, where d is a specified parameter.
 Example here
If you find these algoirthms and data sets useful, we appreciate it very much if you can cite our related works:
 Deng Cai, "Spectral Regression: A Regression Framework for Efficient Regularized Subspace Learning", PhD Thesis, Department of Computer Science, UIUC, 2009. (pdf)
Bibtex source
 Deng Cai, Xiaofei He, Jiawei Han, "Speed Up Kernel Discriminant Analysis", The VLDB Journal, 2011. (pdf)
Bibtex source

Deng Cai, Xiaofei He, and Jiawei Han, "SRDA: An Efficient Algorithm for LargeScale Discriminant Analysis", IEEE TKDE 2008. (pdf)
Bibtex source
 Deng Cai, Xiaofei He and Jiawei Han, "Spectral Regression for Efficient Regularized Subspace Learning", ICCV'07. (pdf)
Bibtex source
 Deng Cai, Xiaofei He, Wei Vivian Zhang, and Jiawei Han, "Regularized Locality Preserving Indexing via Spectral Regression", CIKM'07. (pdf)
Bibtex source
Papers
 Deng Cai, Xiaofei He, and Jiawei Han. "SemiSupervised Regression using Spectral Techniques", Department of Computer Science Technical Report No. 2749, University of Illinois at UrbanaChampaign (UIUCDCSR20062749), July 2006. ( pdf )
 Deng Cai, Xiaofei He, and Jiawei Han. "Spectral Regression for Dimensionality Reduction", Department of Computer Science Technical Report No. 2856, University of Illinois at UrbanaChampaign (UIUCDCSR20072856), May 2007.
( pdf )
 Deng Cai, Xiaofei He, and Jiawei Han. "SRDA: An Efficient Algorithm for Large Scale Discriminant Analysis", Department of Computer Science Technical Report No. 2857, University of Illinois at UrbanaChampaign (UIUCDCSR20072857), May 2007.
( pdf )
 Deng Cai, Xiaofei He, and Jiawei Han. "Isometric Projection", Proc. 22nd Conference on Artifical Intelligence (AAAI'07), Vancouver, Canada, July 2007.
( pdf )
 Deng Cai, Xiaofei He, and Jiawei Han. "Efficient Kernel Discriminant Analysis via Spectral Regression", Department of Computer Science Technical Report No. 2888, University of Illinois at UrbanaChampaign (UIUCDCSR20072888), August 2007.
( pdf )
 Deng Cai, Xiaofei He, and Jiawei Han. "Spectral Regression: A Unified Subspace Learning Framework for ContentBased Image Retrieval", ACM Multimedia 2007, Augsburg, Germany, Sep. 2007.
( pdf )
 Deng Cai, Xiaofei He, and Jiawei Han. "Spectral Regression for Efficient Regularized Subspace Learning", IEEE International Conference on Computer Vision (ICCV'07), Rio de Janeiro, Brazil, Oct. 2007.
( pdf )
 Deng Cai, Xiaofei He, and Jiawei Han. "Efficient Kernel Discriminant Analysis via Spectral Regression", Proc. 2007 Int. Conf. on Data Mining (ICDM'07), Omaha, NE, Oct. 2007.
( pdf )
 Deng Cai, Xiaofei He, and Jiawei Han. "Spectral Regression: A Unified Approach for Sparse Subspace Learning", Proc. 2007 Int. Conf. on Data Mining (ICDM'07), Omaha, NE, Oct. 2007.
( pdf )
 Deng Cai, Xiaofei He, Wei Vivian Zhang, and Jiawei Han. "Regularized Locality Preserving Indexing via Spectral Regression", Proc. 2007 ACM Int. Conf. on Information and Knowledge Management (CIKM'07), Lisboa, Portugal, Nov. 2007.
( pdf )
 Deng Cai, Xiaofei He, and Jiawei Han. "SRDA: An Efficient Algorithm for Large Scale Discriminant Analysis", IEEE Transactions on Knowledge and Data Engineering, vol. 20, no. 1, pp. 112, January, 2008.
( pdf )
 Deng Cai, Xiaofei He, and Jiawei Han. "Speed Up Kernel Discriminant Analysis", The VLDB Journal, vol. 20, no. 1, pp. 2133, January, 2011.
( pdf )
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