## Abstract

It is expensive to obtain labeled real-world visual data for use in training of supervised algorithms. Therefore, it is valuable to leverage existing databases of labeled data. However, the data in the source databases is often obtained under conditions that differ from those in the new task. Transfer learning provides techniques for transferring learned knowledge from a *source* domain to a *target* domain by finding a mapping between them. In this paper, we discuss a method for projecting both source and target data to a generalized subspace where each target sample can be represented by some combination of source samples. By employing a low-rank constraint during this transfer, the structure of source and target domains are preserved. This approach has three benefits. First, good alignment between the domains is ensured through the use of only relevant data in some subspace of the source domain in reconstructing the data in the target domain. Second, the discriminative power of the source domain is naturally passed on to the target domain. Third, noisy information will be filtered out during knowledge transfer. Extensive experiments on synthetic data, and important computer vision problems such as face recognition application and visual domain adaptation for object recognition demonstrate the superiority of the proposed approach over the existing, well-established methods.

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So far, we still consider larger energies as being better for the subspace learning method. However, this will change later once we start minimizing rather than maximizing the objective function.

We use minimal of PCA instead of maximal to fit the LTSL.

Note that we use ULPP and UNPE to denote unsupervised LPP and NPE, and SLPP and SNPE to denote supervised LPP and NPE.

## References

Argyriou, A., Evgeniou, T., & Pontil, M. (2007). Multi-task feature learning. In

*Advances in neural information processing systems*, Cambridge, MA: The MIT Press.Arnold, A., Nallapati, R., & Cohen, W. (2007). A comparative study of methods for transductive transfer learning.

*IEEE International Conference on Data Mining (Workshops),*77–82.Aytar, Y., & Zisserman, A. (2011). Tabula rasa: Model transfer for object category detection.

*IEEE International Conference on Computer Vision,*2252–2259.Bartels, R. H., & Stewart, G. (1972). Solution of the matrix equation ax+ xb= c [f4].

*Communications of the ACM*,*15*(9), 820–826.Belhumeur, P., Hespanha, J., & Kriegman, D. (2002). Eigenfaces vs. Fisherfaces: Recognition using class specific linear projection.

*IEEE Transactions on Pattern Analysis and Machine Intelligence*,*19*(7), 711–720.Belkin, M., & Niyogi, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation.

*Neural Computation*,*15*(6), 1373–1396.Blitzer, J., McDonald, R., & Pereira, F. (2006). Domain adaptation with structural correspondence learning. In

*Conference on Empirical Methods in Natural Language Processing, Association for Computational Linguistics*, (pp. 120–128).Blitzer, J., Foster, D., & Kakade, S. (2011). Domain adaptation with coupled subspaces.

*Journal of Machine Learning Research-Proceedings Track*,*15*, 173–181.Cai, J. F., Candès, E. J., & Shen, Z. (2010). A singular value thresholding algorithm for matrix completion.

*SIAM Journal on Optimization*,*20*, 1956–1982.Candès, E., & Recht, B. (2009). Exact matrix completion via convex optimization.

*Foundations of Computational Mathematics*,*9*(6), 717–772.Candes, E., Li, X., Ma, Y., & Wright, J. (2011). Robust principal component analysis?

*Journal of the ACM*.Chen, M., Weinberger, K., & Blitzer, J. (2011). Co-training for domain adaptation.

*Advances in Neural Information Processing Systems*.Coppersmith, D., & Winograd, S. (1990). Matrix multiplication via arithmetic progressions.

*Journal of Symbolic Computation*,*9*(3), 251–280.Dai, W., Xue, G., Yang, Q., & Yu, Y. (2007). Co-clustering based classification for out-of-domain documents. In

*ACM SIGKDD International Conference on Knowledge Discovery And Data Mining (ACM)*, (pp. 210–219).Dai, W., Xue, G.R., Yang, Q., & Yu, Y. (2007b). Transferring naive bayes classifiers for text classification. In

*AAAI Conference on Artificial Intelligence*(pp. 540–545).Dai, W., Yang, Q., Xue, G., & Yu, Y. (2007c). Boosting for transfer learning. In

*International Conference on Machine learning, ACM*(pp. 193–200).Daumé, H. (2007). Frustratingly easy domain adaptation.

*Annual Meeting-Association for Computational Linguistics*,*45*, 256–263.Daumé, H, I. I. I., & Marcu, D. (2006). Domain adaptation for statistical classifiers.

*Journal of Artificial Intelligence Research*,*26*(1), 101–126.Duan, L., Tsang, I.W., Xu, D., & Chua, T.S. (2009). Domain adaptation from multiple sources via auxiliary classifiers. In

*International Conference on Machine Learning, ACM*(pp. 289–296).Duan, L., Xu, D., & Chang, S.F. (2012a). Exploiting web images for event recognition in consumer videos: A multiple source domain adaptation approach.

*IEEE Conference on Computer Vision and Pattern Recognition,*1338–1345.Duan, L., Xu, D., & Tsang, I. (2012b). Domain adaptation from multiple sources: A domain-dependent regularization approach.

*IEEE Transactions on Neural Networks and Learning Systems*,*23*(3), 504–518.Duan, L., Xu, D., Tsang, I. W. H., & Luo, J. (2012c). Visual event recognition in videos by learning from web data.

*IEEE Transactions on Pattern Analysis and Machine Intelligence*,*34*(9), 1667–1680.Eckstein, J., & Bertsekas, D. (1992). On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators.

*Mathematical Programming*,*55*(1), 293–318.Gao, J., Fan, W., Jiang, J., & Han, J. (2008). Knowledge transfer via multiple model local structure mapping. In

*ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM*(pp. 283–291).Glorot, X., Bordes, A., & Bengio, Y. (2011). Domain adaptation for large-scale sentiment classification: A deep learning approach. In

*International Conference on Machine Learning, ACM*(pp. 513–520).Gong, B., Shi, Y., Sha, F., & Grauman, K. (2012). Geodesic flow kernel for unsupervised domain adaptation. In

*IEEE Conference on Computer Vision and Pattern Recognition*(pp. 2066–2073).Gopalan, R., Li, R., & Chellappa, R. (2011). Domain adaptation for object recognition: An unsupervised approach. In

*IEEE International Conference on Computer Vision*(pp. 999–1006).Griffin, G., Holub, A., & Perona, P. (2007).

*Caltech-256 object category dataset*. California Institute of Technology: Tech. rep.He, X., & Niyogi, P. (2004). Locality preserving projections. In

*Advances in neural information processing systems, vol 16*. Cambridge, MA: The MIT Press.He, X., Cai, D., Yan, S., & Zhang, H. (2005). Neighborhood preserving embedding.

*IEEE International Conference on Computer Vision*,*2*, 1208–1213.Ho, J., Yang, M., Lim, J., Lee, K., & Kriegman, D. (2003). Clustering appearances of objects under varying illumination conditions, vol 1. In

*IEEE Conference on Computer Vision and Pattern Recognition*(pp. 1–11).Hoffman, J., Rodner, E., Donahue, J., Saenko, K., & Darrell, T. (2013). Efficient learning of domain-invariant image representations. arXiv, preprint arXiv:13013224.

Huang, D., Sun, J., & Wang, Y. (2012). The BUAA-VISNIR face database instructions. http://irip.buaa.edu.cn/research/The_BUAA-VisNir_Face_Database_Instructions.pdf.

Jhuo, I. H., Liu, D., Lee, D., & Chang. S. F. (2012) Robust visual domain adaptation with low-rank reconstruction. In

*IEEE Conference on Computer Vision and Pattern Recognition*(pp. 2168–2175).Jiang, J., & Zhai, C. (2007). Instance weighting for domain adaptation in NLP.

*Annual Meeting-Association for Computational Linguistics*,*45*, 264–271.Jiang, W., Zavesky, E., Chang, S. F., & Loui, A. (2008). Cross-domain learning methods for high-level visual concept classification. In

*IEEE International Conference on Image Processing*(pp. 161–164).Keshavan, R., Montanari, A., & Oh, S. (2010). Matrix completion from noisy entries.

*The Journal of Machine Learning Research*,*99*, 2057–2078.Kulis, B., Jain, P., & Grauman, K. (2009). Fast similarity search for learned metrics.

*IEEE Transactions on Pattern Analysis and Machine Intelligence*,*31*(12), 2143–2157.Kulis, B., Saenko, K., & Darrell, T. (2011). What you saw is not what you get: Domain adaptation using asymmetric kernel transforms. In

*IEEE Conference on Computer Vision and Pattern Recognition*(pp. 1785–1792).Lawrence, N., Platt, J. (2004). Learning to learn with the informative vector machine. In

*International Conference on Machine learning, ACM*(pp. 65–72).Lim, J., Salakhutdinov, R., & Torralba, A. (2011). Transfer learning by borrowing examples for multiclass object detection. In

*Advances in neural information processing systems*. Cambridge, MA: The MIT Press.Lin, Z., Chen, M., Wu, L., & Ma, Y. (2009). The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. Technical Report, UILU-ENG-09-2215.

Liu, G., Lin, Z., & Yu, Y. (2010). Robust subspace segmentation by low-rank representation. In

*International Conference on Machine Learning*(pp. 663–670).Liu, G., Lin, Z., Yan, S., Sun, J., Yu, Y., & Ma, Y. (2013). Robust recovery of subspace structures by low-rank representation.

*IEEE Transactions on Pattern Analysis and Machine Intelligence*,*35*(1), 171–184.Lopez-Paz, D., Hernndez-Lobato, J., & Schölkopf, B. (2012). Semi-supervised domain adaptation with non-parametric copulas. In:

*Advances in neural information processing systems*. Cambridge, MA: The MIT Press.Lu, L., & Vidal, R. (2006). Combined central and subspace clustering for computer vision applications. In

*International Conference on Machine Learning, ACM*(pp. 593–600).Mihalkova, L., Huynh, T., & Mooney, R. (2007). Mapping and revising markov logic networks for transfer learning. In

*AAAI Conference on Artificial Intelligence*(pp. 608–614).Pan, S. J., & Yang, Q. (2010). A survey on transfer learning.

*IEEE Transactions on Knowledge and Data Engineering*,*22*(10), 1345–1359.Qi, G.J., Aggarwal, C., Rui, Y., Tian, Q., Chang, S., & Huang, T. (2011). Towards cross-category knowledge propagation for learning visual concepts. In

*IEEE Conference on Computer Vision and Pattern Recognition*(pp. 897–904).Raina, R., Battle, A., Lee, H., Packer, B., & Ng, A. (2007). Self-taught learning: Transfer learning from unlabeled data. In

*International Conference on Machine Learning*(pp. 759–766).Roweis, S. T., & Saul, L. K. (2000). Nonlinear dimensionality reduction by locally linear embedding.

*Science*,*290*(5500), 2323–2326.Saenko, K., Kulis, B., Fritz, M., & Darrell, T. (2010). Adapting visual category models to new domains. In

*European Computer Vision Conference*(pp. 213–226).Shao, M., Xia, S., & Fu, Y., (2011). Genealogical face recognition based on ub kinface database. In

*IEEE Conference on Computer Vision and Pattern Recognition (Workshop on Biometrics)*(pp. 65–70).Shao, M., Castillo, C., Gu, Z., & Fu, Y. (2012). Low-rank transfer subspace learning. In

*IEEE International Conference on Data Mining*(pp. 1104–1109).Si, S., Tao, D., & Geng, B. (2010). Bregman divergence-based regularization for transfer subspace learning.

*IEEE Transactions on Knowledge and Data Engineering*,*22*(7), 929–942.Sun, Q., Chattopadhyay, R., Panchanathan, S., & Ye, J. (2011). A two-stage weighting framework for multi-source domain adaptation. In

*Advances in neural information processing systems*. Cambridge, MA: The MIT Press.Turk, M., & Pentland, A. (1991). Eigenfaces for recognition.

*Journal of Cognitive Neuroscience*,*3*(1), 71–86.Wang, Z., Song, Y., & Zhang, C. (2008). Transferred dimensionality reduction. In

*Machine learning and knowledge discovery in databases*(pp. 550–565). New York: Springer.Wright, J., Ganesh, A., Rao, S., Peng, Y., & Ma, Y. (2009). Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization.

*Advances in Neural Information Processing Systems*,*22*, 2080–2088.Yan, S., Xu, D., Zhang, B., Zhang, H., Yang, Q., & Lin, S. (2007). Graph embedding and extensions: A general framework for dimensionality reduction.

*IEEE Transactions on Pattern Analysis and Machine Intelligence*,*29*(1), 40–51.Yang, J., Yan, R., & Hauptmann, A.G. (2007). Cross-domain video concept detection using adaptive svms. In

*International Conference on Multimedia, ACM*(pp. 188–197).Yang, J., Yin, W., Zhang, Y., & Wang, Y. (2009). A fast algorithm for edge-preserving variational multichannel image restoration.

*SIAM Journal on Imaging Sciences*,*2*(2), 569–592.Zhang, C., Ye, J., & Zhang, L. (2012). Generalization bounds for domain adaptation. In

*Advances in neural information processing systems*, Cambridge, MA: The MIT Press.Zhang, T., Tao, D., & Yang, J. (2008). Discriminative locality alignment. In

*European conference on computer vision*(pp. 725–738). New York: Springer.

## Acknowledgments

This research is supported in part by the NSF CNS award 1314484, Office of Naval Research award N00014-12-1-1028, Air Force Office of Scientific Research award FA9550-12-1-0201, U.S. Army Research Office grant W911NF-13-1-0160, and IC Postdoc Program Grant 2011-11071400006.

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## Appendix

### Appendix

### 1.1 Proof of Lemma 1

###
*Proof*

\(\square \)

Next, due to space limitations, we only show a proof of Theorem 2 based on a specific learning method, Principal Component Analysis (PCA). Proof of others follows a similar scheme.

### 1.2 Proof of Theorem 2

###
*Proof*

Suppose \(X_\mathrm{s}\) and \(X_\mathrm{t}\) are strictly drawn from \(\mathcal {S}_i\) and \(\mathcal {T}_i\). We use \(Y_\mathrm{s} = P^\mathrm{T} X_\mathrm{s}\) and \(Y_\mathrm{t} = P^\mathrm{T} X_\mathrm{t}\) to denote their low-dimensional representations in the subspace \(P\). Both \(Y_\mathrm{s}\) and \(Y_\mathrm{t}\) are of size \(m \times n\). Therefore, the energy of PCA in the source domain \(\mathcal {S}_i\) is:

where \(\mathbf {e}\) is \(\{\mathbf {e}_i = 1 | i = 1,2, \ldots , n\}\). For simplicity, we remove the constant term \(\frac{1}{n-1}\) and replace \(\frac{1}{n}\mathbf {e}\mathbf {e^\mathrm{T}}\) with the matrix \(C\). Then the energy of PCA in \(\mathcal {S}_i\) and \(\mathcal {T}_i\) can be rewritten as:

Since \(\widehat{Z}_i\) is non-singular, we have \(\widehat{Z}_i\widehat{Z}_i^{-1} = \mathbf {I}\) and the above function can be rewritten as:

Note that \(\Vert Y_\mathrm{s} \widehat{Z}_i - Y_\mathrm{s}*C \widehat{Z}_i \Vert _F^2 \) is the PCA energy of the data from the target domain that has been perturbed by \(\widehat{Z}_i\). Therefore, \(F(P, \widehat{\mathcal {T}}_i) = \Vert Y_\mathrm{s} \widehat{Z}_i - Y_\mathrm{s}*C \widehat{Z}_i \Vert _F^2\). Combining that with the inequality in (20) results in:

If we add \(F(P, \mathcal {{T}}_i)\) to both sides of Inequality (21), then we derive Inequality (8). \(\square \)

There are three points to be made. First, the difference between \(F(P, \mathcal {T}_i)\) and \(F(P, \widehat{\mathcal {T}}_i)\) is that the first one is the PCA energy of \(Y_\mathrm{s} Z_i\) while the second one is the PCA energy of \(Y_\mathrm{s} \widehat{Z}_i\) where \(\widehat{Z}_i = Z_i + \gamma \mathbf {I}\). Compared with \(Y_\mathrm{s} Z_i\), \(Y_\mathrm{s} \widehat{Z}_i\) adds a small term to each vector. However, this will not cause a significant change in \(F(P, \mathcal {T}_i)\). Therefore, term \(\xi \) in Theorem 2 will not be very large. Second, although we only compare the PCA energy of \(\mathcal {S}_i\) and \(\mathcal {T}_i\) that are drawn from one subspace, this theorem is easily extended to any other subspace included in the subspace union. Finally, other subspace learning methods can be similarly proven, since they are all unified in the linear graph embedding framework (Yan et al. 2007). For example, the proof for LDA holds the value of \(\mathbf {Tr}(P^\mathrm{T}SP)\) fixed, where \(S=S_\mathrm{b} + S_\mathrm{w}\), and then maximizes \(\mathbf {Tr}(P^\mathrm{T}S_\mathrm{b}P)\) in the same way as was done for PCA.

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Shao, M., Kit, D. & Fu, Y. Generalized Transfer Subspace Learning Through Low-Rank Constraint.
*Int J Comput Vis* **109**, 74–93 (2014). https://doi.org/10.1007/s11263-014-0696-6

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DOI: https://doi.org/10.1007/s11263-014-0696-6