Online learning has attracted great attention due to the increasing demand for systems that have the ability of learning and evolving. When the data to be processed is also high dimensional and dimension reduction is necessary for visualization or prediction enhancement, online dimension reduction will play an essential role. The purpose of this paper is to propose new online learning approaches for supervised dimension reduction. Our first algorithm is motivated by adapting the sliced inverse regression (SIR), a pioneer and effective algorithm for supervised dimension reduction, and making it implementable in an incremental manner. The new algorithm, called incremental sliced inverse regression (ISIR), is able to update the subspace of significant factors with intrinsic lower dimensionality fast and efficiently when new observations come in. We also refine the algorithm by using an overlapping technique and develop an incremental overlapping sliced inverse regression (IOSIR) algorithm. We verify the effectiveness and efficiency of both algorithms by simulations and real data applications.
This paper studies sufficient dimension reduction problem, and proposes an incremental sliced inverse regression algorithm. Numerical experiments are provided to demonstrate the effectiveness of the proposed algorithms.
The sliced inverse regression here is nothing but generalized eigenvalue decomposition:
Ax=lambda Bx.
Note that Multiclass Fisher Linear Discriminant Analysis, Canonical Correlation Analysis, Nonlinear Manifold Embedding and many subspace learning methods can also be formulated as generalized eigenvalue decomposition. All these methods need to compute covariance-like matrices in the additive form, which makes incremental update very convenient.
The incremental generalized eigenvalue decomposition has been extensively studied for over decades, especially between 1995 and 2005 in the face recognition community. I am just listing a few here:
Ye et al., IDR/QR: An Incremental Dimension Reduction Algorithm via QR Decomposition, 2005
Law and Jain, Incremental Nonlinear Dimensionality Reduction by Manifold Learning, 2006
Yan et al. Towards incremental and large scale face recognition, 2011
Ghassabeh et al. A New Incremental Face Recognition System, 2007
Song et al. A Novel Supervised Dimensionality Reduction Algorithm for Online Image Recognition, 2006.
Wang et al. Incremental two-dimensional linear discriminant analysis with applications to face recognition, 2010.
Salman et al. Efficient update of the covariance matrix inverse in iterated linear discriminant analysis, 2010
Park and Park, A comparison of generalized linear discriminant analysis algorithms, 2008
Wang, INCREMENTAL AND REGULARIZED LINEAR DISCRIMINANT ANALYSIS, 2012
These algorithms become less popular/known now, because (1) they are not scalable and efficient for large p, and (2) these classical dimensionality reduction methods perform poorly in many tasks, compared with the state of art results.
This paper only cites a few papers on incremental LDA,  but does not even mention that both LDA and SIR are essentially solving similar optimization problems. Moreover, it does not compare the results with any of the above references, either.
This paper even claims applying the Sherman–Morrison formula as the contribution. However, such an  update has been used in Salman et al. 2010, Park and Park 2008, Wang 2012.
In summary, this paper is far below the bar of ICLR.
Minor: There are numerous typos in this paper. The authors even misspell "Morrison" in the Sherman–Morrison formula as "Morison".