UMAP
Uniform Manifold Approximation and Projection
for dimension reduction
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Who am I?
I am a research mathematician at the Tutte
Institute for Mathematics and Computing
My Ph.D. was in Profinite Lie Rings
(no, you don’t care)
I now work on applying topological techniques
to unsupervised learning problems
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What is Dimension
Reduction?
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Find the “latent”
features in your data
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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
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Neighbour
Graphs
Locally Linear Embedding
Laplacian Eigenmaps
Hessian Eigenmaps
Local Tangent Space Alignment
t-SNE
UMAP
Isomap
JSE
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PCA is the prototypical
matrix factorization
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PCA on MNIST digits
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PCA on Fashion MNIST
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t-SNE is the current
state-of-the art for
neighbour graphs
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t-SNE on MNIST digits
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t-SNE on Fashion MNIST
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Uniform Manifold
Approximation and
Projection
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UMAP builds
mathematical theory to
justify the graph based
approach
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First, a little bit of
topological data
analysis…
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Simplices
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Theorem 1 (Nerve theorem). Let U = {Ui
}i2I be
a cover of a topological space X. If, for all ⇢ I
T
i2
Ui is either contractible or empty, then N(U)
is homtopically equivalent to X.
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If the data is uniformly
distributed on the
manifold then the
cover will be “good”
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When is data that
nicely behaved?
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Assumption:
Data is uniformly
distributed on the
manifold
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Define a Riemannian
metric on the manifold
to make this
assumption true
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Why choose a fixed
radius? Why not have a
fuzzy cover?
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Theorem 2 (UMAP Adjunction). The
functors FinReal : sFuzz ! FinEPMet
and FinSing : FinEPMet ! sFuzz
form an adjunction FinReal a FinSing.
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Assumption:
The manifold is locally
connected
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But our local metrics
are all incompatible!
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Theorem 2 (UMAP Adjunction). The
functors FinReal : sFuzz ! FinEPMet
and FinSing : FinEPMet ! sFuzz
form an adjunction FinReal a FinSing.
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f(↵, ) = ↵ + ↵ ·
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Under a probabilistic fuzzy
union the combination of
weights on edges is given by
Slide 45
Slide 45 text
No content
Slide 46
Slide 46 text
Suppose we were given
a low dimensional
representation
Slide 47
Slide 47 text
We can apply the same
process to get a fuzzy
graph!
Slide 48
Slide 48 text
Except we know the
manifold, and don’t
know the “correct”
nearest neighbour
distance
Slide 49
Slide 49 text
Now measure the
distance between the
graphs using cross-
entropy and optimize
X
a2A
µ(a) log
✓
µ(a)
⌫(a)
◆
+ (1 µ(a)) log
✓
1 µ(a)
1 ⌫(a)
◆
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Get the clumps right
Get the gaps right
Slide 53
Slide 53 text
No content
Slide 54
Slide 54 text
No content
Slide 55
Slide 55 text
No content
Slide 56
Slide 56 text
On real data?
Slide 57
Slide 57 text
UMAP on MNIST digits
Slide 58
Slide 58 text
UMAP on Fashion MNIST
Slide 59
Slide 59 text
Implementation
Slide 60
Slide 60 text
Need to find (approximate)
nearest neighbours very
efficiently
Even in high dimensional space