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A Bluffer’s Guide to Dimension Reduction Leland McInnes

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Bluffer’s Guides are lighthearted and humorous surveys providing a condensed overview of a potentially complicated subject.

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Focus on the intuition and core ideas

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* = I’m lying, but in a good way

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There are only two Dimension reduction techniques*

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Matrix Factorization Neighbour Graphs

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Matrix Factorization Principal Component Analysis Non-negative Matrix Factorization Latent Dirichlet Allocation Word2Vec GloVe Generalised Low Rank Models Linear Autoencoder Probablistic PCA Sparse PCA

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Neighbour Graphs Locally Linear Embedding Laplacian Eigenmaps Hessian Eigenmaps Local Tangent Space Alignment t-SNE UMAP Isomap JSE Spectral Embedding LargeVis NerV

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Autoencoders?

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Matrix Factorization

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X ≈ UV Where X is an NxD matrix U is an Nxd matrix V is an dxD matrix X U V N × D N × d d × D

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N ∑ i=1 D ∑ j=1 Loss (Xij , (UV)ij) Subject to constraints… Minimize

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Generalized Low Rank Models Udell, Horn, Zadeh, Boyd 2016

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Principal Component Analysis

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We can do an awful lot with mean squared error

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Classic PCA N ∑ i=1 D ∑ j=1 (Xij − (UV)ij) 2 with no constraints Minimize

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We can make PCA more interpretable by constraining how many archetypes can be combined

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Sparse PCA N ∑ i=1 D ∑ j=1 (Xij − (UV)ij) 2 Subject to Minimize ∥U∥2 = 1 and ∥U∥0 ≤ k

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What if we turn the dial to 11?

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K-Means* N ∑ i=1 D ∑ j=1 (Xij − (UV)ij) 2 Subject to Minimize ∥U∥2 = 1 and ∥U∥0 = 1

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Non-Negative Matrix Factorization

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Only allowing additive combinations of archetypes might be more interpretable…

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NMF N ∑ i=1 D ∑ j=1 (Xij − (UV)ij) 2 Subject to Minimize Uij ≥ 0 and Vij ≥ 0

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NMF N ∑ i=1 D ∑ j=1 (UV)ij − Xij log ((UV)ij) Subject to Minimize Uij ≥ 0 and Vij ≥ 0

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Exponential Family PCA

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X ∼ Pr( ⋅ ∣ Θ) where Suppose Θ = UV

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Let the loss be the negative log likelihood of observing X given O X Θ

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How to parameterize Pr(.|O) ? Use the exponential family of distributions! Pr( ⋅ ∣ Θ)

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−log(P(Xi ∣ Θi )) ∝ G(Θi ) − Xi ⋅ Θi In general for an exponential family distribution

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Normal Matrix Factorization N ∑ i=1 D ∑ j=1 1 2 ((UV)ij )2 − Xij ⋅ (UV)ij With no constraints Minimize

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Normal Matrix Factorization N ∑ i=1 D ∑ j=1 1 2 ((UV)ij )2 − Xij ⋅ (UV)ij + 1 2 (Xij )2 With no constraints Minimize

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Normal Matrix Factorization N ∑ i=1 D ∑ j=1 1 2 (Xij − (UV)ij) 2 With no constraints Minimize

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Poisson Matrix Factorization N ∑ i=1 D ∑ j=1 exp(UV)ij − Xij ⋅ (UV)ij With no constraints Minimize

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Binomial Matrix Factorization Bernoulli Matrix Factorization Gamma Matrix Factorization Beta Matrix Factorization Exponential Matrix Factorization …

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Latent Dirichlet Allocation

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What if Oi were parameters for a multinomial distribution? Θi

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Multinomial Matrix Factorization N ∑ i=1 D ∑ j=1 − (UV)ij ⋅ log (Xij) Subject to Minimize (UV)1 = 1 and (UV)ij ≥ 0

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We can add a latent variable k

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Let Uik = P(i|k), Vkj = P(k|j) Then Θij = ∑ k Uik ⋅ Vkj = ∑ k P(i|k) ⋅ P(k|j) = P(i|j)

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Probabilistic Latent Semantic Indexing N ∑ i=1 D ∑ j=1 − (UV)ij ⋅ log (Xij) Subject to Minimize U1 = 1, V1 = 1 and Uij ≥ 0,Vij ≥ 0

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Let’s be Bayesian!

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We can apply a Dirichlet prior over the multinomial distributions for U and V U V

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And that’s LDA* (modulo all the technical details involved in the Bayesian inference used for optimization)

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Neighbour Graphs

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*

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*

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How is the graph constructed?

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How is the graph laid out in a low dimensional space?

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Isomap

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Graph Construction K-Nearest Neighbours weighted by ambient distance

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Complete graph weighted by shortest path length

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Consider the weighted adjacency matrix Aij = { w(i, j) if (i, j) ∈ E 0 otherwise

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Factor the matrix! (largest eigenvectors)

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Spectral Embedding

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Graph Construction Kernel weighted edges*

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Compute the graph Laplacian* Lij = −w(i, j) di × dj if i ≠ j 1 − w(i, i) di if i = j Where di is the total weight of row i di i

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We have a matrix again…

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Factor the matrix! (smallest eigenvectors*)

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t-SNE (t-distributed Stochastic Neighbour Embedding)

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Graph Construction K-Nearest Neighbours* weighted by a kernel with bandwidth adapted to the K neighbours

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Graph Construction Normalize outgoing edge weights to sum to one

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Graph Construction Symmetrize by averaging edge weights between each pair of vertices

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Graph Construction Renormalize so the total edge weight is one

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Use a force directed graph layout!*

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UMAP (Uniform Manifold Approximation and Projection)

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Graph Construction K-Nearest Neighbours weighted according to fancy math* I have fun mathematics to explain this which this margin is too small to contain

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Use a force directed graph layout!*

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Summary

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Dimension reduction is built on only a couple of primitives

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Framing the problem as a matrix factorization or neighbour graph algorithm captures most of the core intuitions

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This provides a general framework for understanding almost all dimension reduction techniques

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Questions? [email protected] @leland_mcinnes