# Dimensionality reduction

### Wherein I teach myself, amongst other things, how a *sparse* PCA works, and decide where to file multidimensional scaling

Usefulness: 🔧
Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

Related: “clustering”? Also with the notion of similarity as seen in kernel tricks. Inducing a differential metric. Matrix factorisations and random features, high-dimensional statistics and discuss random projections and their role in compressed sensing etc.

## PCA and cousins

Kernel PCA, linear algebra and probabilistic formulations.

Linear algebra version of PCA puts us in the world of matrix factorisationa.

### Non-linear versions

• Principal component analysis (PCA) is one of the most versatile tools for unsupervised learning with applications ranging from dimensionality reduction to exploratory data analysis and visualization. While much effort has been devoted to encouraging meaningful representations through regularization (e.g. non-negativity or sparsity), underlying linearity assumptions can limit their effectiveness. To address this issue, we propose Additive Component Analysis (ACA), a novel nonlinear extension of PCA. Inspired by multivariate nonparametric regression with additive models, ACA fits a smooth manifold to data by learning an explicit mapping from a low-dimensional latent space to the input space, which trivially enables applications like denoising. Furthermore, ACA can be used as a drop-in replacement in many algorithms that use linear component analysis methods as a subroutine via the local tangent space of the learned manifold. Unlike many other nonlinear dimensionality reduction techniques, ACA can be efficiently applied to large datasets since it does not require computing pairwise similarities or storing training data during testing. Multiple ACA layers can also be composed and learned jointly with essentially the same procedure for improved representational power, demonstrating the encouraging potential of nonparametric deep learning. We evaluate ACA on a variety of datasets, showing improved robustness, reconstruction performance, and interpretability.

## Autoencoder and word2vec

“The nonlinear PCA” interpretation, I just heard from Junbin Gao.

$L(x, x') = \|x-x\|^2=\|x-\sigma(U*sigma*W^Tx+b)) + b')\|^2$

## Locality Preserving projections

Try to preserve the nearness of points if they are connected on some (weight) graph.

$\sum_{i,j}(y_i-y_j)^2 w_{i,j}$

So we seen an optimal projection vector.

(requirement for sparse similarity matrix?)

TDB.

## Topological data analysis

Start with the distances between points and try to find a lower dimensional manifold which preserves their distances. Local MDS? TDB.

## Stochastic neighbour embedding

Probabilisitically preserving closeness.

t-SNE

# Refs

Cook, R. Dennis. 2018. “Principal Components, Sufficient Dimension Reduction, and Envelopes.” Annual Review of Statistics and Its Application 5 (1): 533–59. https://doi.org/10.1146/annurev-statistics-031017-100257.

Globerson, Amir, and Sam T. Roweis. 2006. “Metric Learning by Collapsing Classes.” In Advances in Neural Information Processing Systems, 451–58. NIPS’05. Cambridge, MA, USA: MIT Press. http://papers.nips.cc/paper/2947-metric-learning-by-collapsing-classes.pdf.

Goroshin, Ross, Joan Bruna, Jonathan Tompson, David Eigen, and Yann LeCun. 2014. “Unsupervised Learning of Spatiotemporally Coherent Metrics,” December. http://arxiv.org/abs/1412.6056.

Hadsell, R., S. Chopra, and Y. LeCun. 2006. “Dimensionality Reduction by Learning an Invariant Mapping.” In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2:1735–42. https://doi.org/10.1109/CVPR.2006.100.

Hinton, Geoffrey E., and Ruslan R. Salakhutdinov. 2006. “Reducing the Dimensionality of Data with Neural Networks.” Science 313 (5786): 504–7. https://doi.org/10.1126/science.1127647.

Hinton, Geoffrey, and Sam Roweis. 2002. “Stochastic Neighbor Embedding.” In Proceedings of the 15th International Conference on Neural Information Processing Systems, 857–64. NIPS’02. Cambridge, MA, USA: MIT Press. http://papers.nips.cc/paper/2276-stochastic-neighbor-embedding.pdf.

Lawrence, Neil. 2005. “Probabilistic Non-Linear Principal Component Analysis with Gaussian Process Latent Variable Models.” Journal of Machine Learning Research 6 (Nov): 1783–1816. http://www.jmlr.org/papers/v6/lawrence05a.html.

Lopez-Paz, David, Suvrit Sra, Alex Smola, Zoubin Ghahramani, and Bernhard Schölkopf. 2014. “Randomized Nonlinear Component Analysis,” February. http://arxiv.org/abs/1402.0119.

Maaten, Laurens van der, and Geoffrey Hinton. 2008. “Visualizing Data Using T-SNE.” Journal of Machine Learning Research 9 (Nov): 2579–2605. http://www.jmlr.org/papers/v9/vandermaaten08a.html.

Murdock, Calvin, and Fernando De la Torre. 2017. “Additive Component Analysis.” In Conference on Computer Vision and Pattern Recognition (CVPR). http://www.calvinmurdock.com/content/uploads/publications/cvpr2017aca.pdf.

Oymak, Samet, and Joel A. Tropp. 2015. “Universality Laws for Randomized Dimension Reduction, with Applications,” November. http://arxiv.org/abs/1511.09433.

Peluffo-Ordónez, Diego H., John A. Lee, and Michel Verleysen. 2014. “Short Review of Dimensionality Reduction Methods Based on Stochastic Neighbour Embedding.” In Advances in Self-Organizing Maps and Learning Vector Quantization, 65–74. Springer. http://link.springer.com/chapter/10.1007/978-3-319-07695-9_6.

Salakhutdinov, Ruslan, and Geoff Hinton. 2007. “Learning a Nonlinear Embedding by Preserving Class Neighbourhood Structure.” In PMLR, 412–19. http://proceedings.mlr.press/v2/salakhutdinov07a.html.

Smola, Alex J., Robert C. Williamson, Sebastian Mika, and Bernhard Schölkopf. 1999. “Regularized Principal Manifolds.” In Computational Learning Theory, edited by Paul Fischer and Hans Ulrich Simon, 214–29. Lecture Notes in Computer Science 1572. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/3-540-49097-3_17.

Sohn, Kihyuk, and Honglak Lee. 2012. “Learning Invariant Representations with Local Transformations.” In Proceedings of the 29th International Conference on Machine Learning (ICML-12), 1311–8. http://machinelearning.wustl.edu/mlpapers/paper_files/ICML2012Sohn_659.pdf.

Sorzano, C. O. S., J. Vargas, and A. Pascual Montano. 2014. “A Survey of Dimensionality Reduction Techniques,” March. http://arxiv.org/abs/1403.2877.

Wang, Boyue, Yongli Hu, Junbin Gao, Yanfeng Sun, Haoran Chen, and Baocai Yin. 2017. “Locality Preserving Projections for Grassmann Manifold.” In PRoceedings of IJCAI, 2017. http://arxiv.org/abs/1704.08458.

Wasserman, Larry. 2018. “Topological Data Analysis.” Annual Review of Statistics and Its Application 5 (1): 501–32. https://doi.org/10.1146/annurev-statistics-031017-100045.

Weinberger, Kilian, Anirban Dasgupta, John Langford, Alex Smola, and Josh Attenberg. 2009. “Feature Hashing for Large Scale Multitask Learning.” In Proceedings of the 26th Annual International Conference on Machine Learning, 1113–20. ICML ’09. New York, NY, USA: ACM. https://doi.org/10.1145/1553374.1553516.