Dimensionality reduction

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


I will restructure learning on manifolds and dimensionality reduction into a more useful distinction.

You have lots of predictors in your regression model! Too many predictors You want less predictors! Maybe then it would be faster, or at least more compact. Can you throw some out, or summarise them in some sens? Also with the notion of similarity as seen in kernel tricks. What you might do to learn an index. Inducing a differential metric. Matrix factorisations and random features, high-dimensional statistics. Ultimately, this is always (at least implicitly) learning a manifold. A good dimension reduction can produce a nearly sufficient statistic for indirect inference.


Throwing out data in a classical Bayes context is a subtle matter, but it can be done. See Bayesian model selection.

Learning a summary statistic

See learning summary statistics. As seen in approximate Bayes. Note this is not at all the same thing as discarding predictors; rather it is about learning a useful statistic to make inferences over some more intractable ones.

Feature selection

Deciding whether to include or discard predictors. This one is very old and has been included in regression models for a long time. Model selection is a classic one, and regularised sparse model selection is the surprisingly effective recent evolution. But it continues! FOCI is an application of an interesting new independence test (Azadkia and Chatterjee 2019) that is very much en vogue despite being in an area that we all thought was thoroughly mined out.

PCA and cousins

The classic. Kernel PCA, linear algebra and probabilistic formulations. Has a nice probabilistic interpretation “for free” via the Karhunen—Loève theorem.

Matrix factorisations are a generalisation here, from rank 1 operators to higher rank operators. 🏗

There are various extensions such as additive component analysis:

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.

Learning a distance metric

A related notion is to learn a simpler way of quantifying, in some sense, how similar are two datapoints. This usually involves learning an embedding in some low dimensional ambient space as a by-product.


Uniform Manifold approximation and projection for dimension reduction (McInnes, Healy, and Melville 2018). Apparently super hot right now. (HT James Nichols). Nikolay Oskolkov’s introduction is neat. John Baez discusses the category theoretic underpinning.

For indexing my database

See learnable indexes.

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?)

Diffusion maps

This manifold-learning technique seemed fashionable for a while. (Coifman and Lafon 2006; R. R. Coifman et al. 2005a, 2005b)

Mikhail Belkin connects this to the graph laplacian literature.

As manifold learning

Same thing, with some different emphases and history, over at manifold learning.

Multidimensional scaling


Random projection

See random embeddings

Stochastic neighbour embedding and other visualisation-oriented methods

These methods are designed to make high-dimensional data sets look comprehensible in low-dimensional representation.

Probabilistically preserving closeness. The height of this technique is the famous t-SNE, although as far as I understand it has been superseded by UMAP.

My colleage Ben Harwood advises:

Instead of reducing and visualising higher dimensional data with t-SNE or PCA, here are three relatively recent non-linear dimension reduction techniques that are designed for visualising high dimensional data in 2D or 3D:

Trimap and LargeVis are learned mappings that I would expect to be more representative of the original data than what t-SNE provides. UMAP assumes connectedness of the manifold so it’s probably less suitable for data that contains distinct clusters but otherwise still a great option.

Autoencoder and word2vec

The “nonlinear PCA” interpretation of word2vec, I just heard from Junbin Gao.

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



Azadkia, Mona, and Sourav Chatterjee. 2019. “A Simple Measure of Conditional Dependence.” December 20, 2019. http://arxiv.org/abs/1910.12327.

Castro, Pablo de, and Tommaso Dorigo. 2019. “INFERNO: Inference-Aware Neural Optimisation.” Computer Physics Communications 244 (November): 170–79. https://doi.org/10.1016/j.cpc.2019.06.007.

Coifman, Ronald R., and Stéphane Lafon. 2006. “Diffusion Maps.” Applied and Computational Harmonic Analysis, Special Issue: Diffusion Maps and Wavelets, 21 (1): 5–30. https://doi.org/10.1016/j.acha.2006.04.006.

Coifman, R. R., S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner, and S. W. Zucker. 2005a. “Geometric Diffusions as a Tool for Harmonic Analysis and Structure Definition of Data: Diffusion Maps.” Proceedings of the National Academy of Sciences 102 (21): 7426–31. https://doi.org/10.1073/pnas.0500334102.

———. 2005b. “Geometric Diffusions as a Tool for Harmonic Analysis and Structure Definition of Data: Multiscale Methods.” Proceedings of the National Academy of Sciences 102 (21): 7432–7. https://doi.org/10.1073/pnas.0500896102.

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.

Dwibedi, Debidatta, Yusuf Aytar, Jonathan Tompson, Pierre Sermanet, and Andrew Zisserman. 2019. “Temporal Cycle-Consistency Learning,” April. https://arxiv.org/abs/1904.07846v1.

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 18, 2014. 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.

Kim, Cheolmin, and Diego Klabjan. 2020. “A Simple and Fast Algorithm for L1-Norm Kernel PCA.” IEEE Transactions on Pattern Analysis and Machine Intelligence 42 (8): 1842–55. https://doi.org/10.1109/TPAMI.2019.2903505.

Lawrence, Neil. 2005. “Probabilistic Non-Linear Principal Component Analysis with Gaussian Process Latent Variable Models.” Journal of Machine Learning Research 6 (Nov, 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 1, 2014. 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.

McInnes, Leland, John Healy, and James Melville. 2018. “UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction.” December 6, 2018. http://arxiv.org/abs/1802.03426.

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 30, 2015. 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.

Rohe, Karl, and Muzhe Zeng. 2020. “Vintage Factor Analysis with Varimax Performs Statistical Inference.” April 11, 2020. http://arxiv.org/abs/2004.05387.

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 12, 2014. 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.