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.
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.
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.
Stochastic neighbour embedding
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.
Autoencoder and word2vec
The “nonlinear PCA” interpretation of word2vec, I just heard from Junbin Gao.
\[L(x, x') = \|x-x\|^2=\|x-\sigma(U*\sigma*W^Tx+b)) + b')\|^2\]
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