Data dimensionality reduction

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

March 23, 2015 — September 11, 2020

feature construction
functional analysis
linear algebra
machine learning
neural nets
sparser than thou
Figure 1


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.

1 Bayes

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

2 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.

3 Feature selection

Figure 2

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.

4 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).

5 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.

5.1 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.

5.2 For indexing my database

See learnable indexes.

5.3 Locality Preserving projections

Figure 3

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?)

5.4 Diffusion maps

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

Mikhail Belkin connects this to the graph laplacian literature.

5.5 As manifold learning

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

5.6 Multidimensional scaling

Figure 5


5.7 Random projection

See random embeddings

5.8 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.

6 Autoencoder and word2vec

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

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


7 For models rather than data

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

8 Incoming

9 References

Azadkia, and Chatterjee. 2019. A Simple Measure of Conditional Dependence.” arXiv:1910.12327 [Cs, Math, Stat].
Bach, and Jordan. 2002. “Kernel Independent Component Analysis.” Journal of Machine Learning Research.
Charpentier, Mussard, and Ouraga. 2021. Principal Component Analysis: A Generalized Gini Approach.” European Journal of Operational Research.
Coifman, Ronald R., and Lafon. 2006. Diffusion Maps.” Applied and Computational Harmonic Analysis, Special Issue: Diffusion Maps and Wavelets,.
Coifman, R. R., Lafon, Lee, et al. 2005. Geometric Diffusions as a Tool for Harmonic Analysis and Structure Definition of Data: Diffusion Maps.” Proceedings of the National Academy of Sciences.
Collins, Dasgupta, and Schapire. 2001. A Generalization of Principal Components Analysis to the Exponential Family.” In Advances in Neural Information Processing Systems.
Cook. 2018. Principal Components, Sufficient Dimension Reduction, and Envelopes.” Annual Review of Statistics and Its Application.
de Castro, and Dorigo. 2019. INFERNO: Inference-Aware Neural Optimisation.” Computer Physics Communications.
Dwibedi, Aytar, Tompson, et al. 2019. Temporal Cycle-Consistency Learning.”
Gerber, Pospisil, Navandar, et al. 2020. Low-Cost Scalable Discretization, Prediction, and Feature Selection for Complex Systems.” Science Advances.
Globerson, and Roweis. 2006. Metric Learning by Collapsing Classes.” In Advances in Neural Information Processing Systems. NIPS’05.
Gordon. 2002. Generalized² Linear² Models.” In Proceedings of the 15th International Conference on Neural Information Processing Systems. NIPS’02.
Goroshin, Bruna, Tompson, et al. 2014. Unsupervised Learning of Spatiotemporally Coherent Metrics.” arXiv:1412.6056 [Cs].
Hadsell, Chopra, and LeCun. 2006. Dimensionality Reduction by Learning an Invariant Mapping.” In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.
Hinton, Geoffrey, and Roweis. 2002. Stochastic Neighbor Embedding.” In Proceedings of the 15th International Conference on Neural Information Processing Systems. NIPS’02.
Hinton, Geoffrey E., and Salakhutdinov. 2006. Reducing the Dimensionality of Data with Neural Networks.” Science.
Hyvärinen, and Oja. 2000. Independent Component Analysis: Algorithms and Applications.” Neural Networks.
Jun Li, and Dacheng Tao. 2013. Simple Exponential Family PCA.” IEEE Transactions on Neural Networks and Learning Systems.
Kandanaarachchi, and Hyndman. 2021. Dimension Reduction for Outlier Detection Using DOBIN.” Journal of Computational and Graphical Statistics.
Kim, and Klabjan. 2020. A Simple and Fast Algorithm for L1-Norm Kernel PCA.” IEEE Transactions on Pattern Analysis and Machine Intelligence.
Lawrence. 2005. Probabilistic Non-Linear Principal Component Analysis with Gaussian Process Latent Variable Models.” Journal of Machine Learning Research.
Liu, Dobriban, and Singer. 2017. \(e\)PCA: High Dimensional Exponential Family PCA.”
Lopez-Paz, Sra, Smola, et al. 2014. Randomized Nonlinear Component Analysis.” arXiv:1402.0119 [Cs, Stat].
McInnes, Healy, and Melville. 2018. UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction.” arXiv:1802.03426 [Cs, Stat].
Mohamed, Ghahramani, and Heller. 2008. Bayesian Exponential Family PCA.” In Advances in Neural Information Processing Systems.
Murdock, and Torre. 2017. Additive Component Analysis.” In Conference on Computer Vision and Pattern Recognition (CVPR).
Oymak, and Tropp. 2015. Universality Laws for Randomized Dimension Reduction, with Applications.” arXiv:1511.09433 [Cs, Math, Stat].
Peluffo-Ordónez, Lee, and Verleysen. 2014. Short Review of Dimensionality Reduction Methods Based on Stochastic Neighbour Embedding.” In Advances in Self-Organizing Maps and Learning Vector Quantization.
Rohe, and Zeng. 2020. Vintage Factor Analysis with Varimax Performs Statistical Inference.” arXiv:2004.05387 [Math, Stat].
Roy, Gordon, and Thrun. 2005. Finding Approximate POMDP Solutions Through Belief Compression.” Journal of Artificial Intelligence Research.
Salakhutdinov, and Hinton. 2007. Learning a Nonlinear Embedding by Preserving Class Neighbourhood Structure.” In PMLR.
Shalizi. 2021. A Note on Simulation-Based Inference by Matching Random Features.”
Smola, Williamson, Mika, et al. 1999. Regularized Principal Manifolds.” In Computational Learning Theory. Lecture Notes in Computer Science 1572.
Sohn, and Lee. 2012. Learning Invariant Representations with Local Transformations.” In Proceedings of the 29th International Conference on Machine Learning (ICML-12).
Sorzano, Vargas, and Montano. 2014. A Survey of Dimensionality Reduction Techniques.” arXiv:1403.2877 [Cs, q-Bio, Stat].
van der Maaten, and Hinton. 2008. Visualizing Data Using t-SNE.” Journal of Machine Learning Research.
Wang, Hu, Gao, et al. 2017. Locality Preserving Projections for Grassmann Manifold.” In PRoceedings of IJCAI, 2017.
Wasserman. 2018. Topological Data Analysis.” Annual Review of Statistics and Its Application.
Weinberger, Dasgupta, Langford, et al. 2009. Feature Hashing for Large Scale Multitask Learning.” In Proceedings of the 26th Annual International Conference on Machine Learning. ICML ’09.