Random embeddings and hashing

December 6, 2016 — December 1, 2020

feature construction
functional analysis
linear algebra
probabilistic algorithms
probability
sparser than thou
statistics
Figure 1: Separation of inputs by random projection

See also matrix factorisations, for some extra ideas on why random projections have a role in motivating compressed sensing, randomised regressions etc.

1 Random projections

I am especially interested in random projections for kernel approximation.

Occasionally we might use non-linear projections to increase the dimensionality of our data in the hope of making a non-linear regression approximately linear, which dates back to (Cover 1965).

1.1 Low-dimensional embeddings

Over at compressed sensing we mention some useful tools for these problems, such as the Johnson-Lindenstrauss lemma.

See low-d projections.

2 Locality-sensitive hashing

TBD

3 Nonlinear embeddings

Cover’s Theorem (Cover 1965):

It was shown that, for a random set of linear inequalities in \(d\) unknowns, the expected number of extreme inequalities, which are necessary and sufficient to imply the entire set, tends to \(2d\) as the number of consistent inequalities tends to infinity, thus bounding the expected necessary storage capacity for linear decision algorithms in separable problems. The results, even those dealing with randomly positioned points, have been combinatorial in nature, and have been essentially independent of the configuration of the set of points in the space.

TBC

4 References

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