Data 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 wish there were fewer 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, random projections, 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.

Bayes

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 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

A classic. Has a nice probabilistic interpretation “for free” via the Karhunen-Loève theorem. Matrix factorisations are the family that contains such methods. 🏗

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.

More interesting to me is Exponential Family PCA, which is a generalisation of PCA to non-Gaussian distributions (and I presume to non-additive relations). How does this even work? (Collins, Dasgupta, and Schapire 2001; Jun Li and Dacheng Tao 2013; Liu, Dobriban, and Singer 2017; Mohamed, Ghahramani, and Heller 2008).

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.

UMAP

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?)

As manifold learning

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

Multidimensional scaling

TDB.

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 colleague 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.

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

TBC.

For models rather than data

I do't want to reduce the order of data but rather a model? See model order reduction.

References

Azadkia, Mona, and Sourav Chatterjee. 2019. A Simple Measure of Conditional Dependence.” arXiv:1910.12327 [Cs, Math, Stat], December.
Bach, Francis R, and Michael I Jordan. 2002. “Kernel Independent Component Analysis.” Journal of Machine Learning Research 3 (July): 48.
Castro, Pablo de, and Tommaso Dorigo. 2019. INFERNO: Inference-Aware Neural Optimisation.” Computer Physics Communications 244 (November): 170–79.
Charpentier, Arthur, Stéphane Mussard, and Téa Ouraga. 2021. Principal Component Analysis: A Generalized Gini Approach.” European Journal of Operational Research, February.
Coifman, R. R., S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner, and S. W. Zucker. 2005. 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.
Coifman, Ronald R., and Stéphane Lafon. 2006. Diffusion Maps.” Applied and Computational Harmonic Analysis, Special Issue: Diffusion Maps and Wavelets, 21 (1): 5–30.
Collins, Michael, S. Dasgupta, and Robert E Schapire. 2001. A Generalization of Principal Components Analysis to the Exponential Family.” In Advances in Neural Information Processing Systems. Vol. 14. MIT Press.
Cook, R. Dennis. 2018. Principal Components, Sufficient Dimension Reduction, and Envelopes.” Annual Review of Statistics and Its Application 5 (1): 533–59.
Dwibedi, Debidatta, Yusuf Aytar, Jonathan Tompson, Pierre Sermanet, and Andrew Zisserman. 2019. Temporal Cycle-Consistency Learning,” April.
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.
Gordon, Geoffrey J. 2002. Generalized² Linear² Models.” In Proceedings of the 15th International Conference on Neural Information Processing Systems, 593–600. NIPS’02. Cambridge, MA, USA: MIT Press.
Goroshin, Ross, Joan Bruna, Jonathan Tompson, David Eigen, and Yann LeCun. 2014. Unsupervised Learning of Spatiotemporally Coherent Metrics.” arXiv:1412.6056 [Cs], December.
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.
Hinton, Geoffrey E., and Ruslan R. Salakhutdinov. 2006. Reducing the Dimensionality of Data with Neural Networks.” Science 313 (5786): 504–7.
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.
Hyvärinen, A, and E Oja. 2000. Independent Component Analysis: Algorithms and Applications.” Neural Networks 13 (4?5): 411–30.
Jun Li, and Dacheng Tao. 2013. Simple Exponential Family PCA.” IEEE Transactions on Neural Networks and Learning Systems 24 (3): 485–97.
Kandanaarachchi, Sevvandi, and Rob J. Hyndman. 2021. Dimension Reduction for Outlier Detection Using DOBIN.” Journal of Computational and Graphical Statistics 30 (1): 204–19.
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.
Lawrence, Neil. 2005. Probabilistic Non-Linear Principal Component Analysis with Gaussian Process Latent Variable Models.” Journal of Machine Learning Research 6 (Nov): 1783–1816.
Liu, Lydia T., Edgar Dobriban, and Amit Singer. 2017. \(e\)PCA: High Dimensional Exponential Family PCA.” arXiv.
Lopez-Paz, David, Suvrit Sra, Alex Smola, Zoubin Ghahramani, and Bernhard Schölkopf. 2014. Randomized Nonlinear Component Analysis.” arXiv:1402.0119 [Cs, Stat], February.
Maaten, Laurens van der, and Geoffrey Hinton. 2008. Visualizing Data Using t-SNE.” Journal of Machine Learning Research 9 (Nov): 2579–2605.
McInnes, Leland, John Healy, and James Melville. 2018. UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction.” arXiv:1802.03426 [Cs, Stat], December.
Mohamed, Shakir, Zoubin Ghahramani, and Katherine A Heller. 2008. Bayesian Exponential Family PCA.” In Advances in Neural Information Processing Systems. Vol. 21. Curran Associates, Inc.
Murdock, Calvin, and Fernando De la Torre. 2017. Additive Component Analysis.” In Conference on Computer Vision and Pattern Recognition (CVPR).
Oymak, Samet, and Joel A. Tropp. 2015. Universality Laws for Randomized Dimension Reduction, with Applications.” arXiv:1511.09433 [Cs, Math, Stat], November.
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.
Rohe, Karl, and Muzhe Zeng. 2020. Vintage Factor Analysis with Varimax Performs Statistical Inference.” arXiv:2004.05387 [Math, Stat], April.
Roy, Nicholas, Geoffrey Gordon, and Sebastian Thrun. 2005. Finding Approximate POMDP Solutions Through Belief Compression.” Journal of Artificial Intelligence Research 23 (1): 1–40.
Salakhutdinov, Ruslan, and Geoff Hinton. 2007. Learning a Nonlinear Embedding by Preserving Class Neighbourhood Structure.” In PMLR, 412–19.
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.
Sohn, Kihyuk, and Honglak Lee. 2012. Learning Invariant Representations with Local Transformations.” In Proceedings of the 29th International Conference on Machine Learning (ICML-12), 1311–18.
Sorzano, C. O. S., J. Vargas, and A. Pascual Montano. 2014. A Survey of Dimensionality Reduction Techniques.” arXiv:1403.2877 [Cs, q-Bio, Stat], March.
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.
Wasserman, Larry. 2018. Topological Data Analysis.” Annual Review of Statistics and Its Application 5 (1): 501–32.
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.

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