A Strategy for Manifold Clustering with Sample Algorithms

Sheng YuUM Statistics

Outline

Motivation

Strategy

Sample Algorithms

Motivation (pattern)

Most current clustering methods are only able to detect agglomerated patterns.

New generation methods, such as normalized cut, have more flexibility, but are still not able to detect twisted, perhaps also entangled manifolds.

Such manifold patterns are not rare.

This is a manifold...

Example:

Try to cluster a pair of symmetric double spiral.

Example:

result fromk-means

Example:

result from normalized cut

Motivation (noise)

Theoretically, hierarchical clustering method using “single linkage” as the merging criterion is able to cluster twisted patterns. However, since “single linkage” is extremely sensitive to noisy, it is not actually a usable method.

MotivationTo design a new method that is not only able

to accomplish traditional “easy” tasks, but also handles twisted, entangled patterns as well.

Also, this new method should not be ruined by noise (moderate level, in terms of signal-noise ratio).

Outline

Motivation

Strategy

Sample Algorithms

Strategy (rationale)

Strategy (design)Engine: Searches

paths between each pair of points. More powerful engine provides faster speed.

Filter: Tells the engine which neighboring points can be connected from a specific start point. Controls the quality.

Engine

Filter

Example (easy one)

Example (not so easy one)

Example (not so easy one)

Example (hard one)

Outline

Motivation

Strategy

Sample Algorithms

Algorithms (filter)The filter I currently use is still primitive.

But it does a lot of jobs, such as the above examples.

The strategy is an open framework. We can build better filters to detect even more difficult patterns and have more resistance to noise.

Algorithmsthe importance of the engine

Sample Size 320

Sample Size 640

Brute force 97” Death touch

Fission 1.5” 65’’

Algebraic fission

Never minded 0.5”

Strategy (rationale)

AlgorithmsThe true benefit of a super fast engine is that

it allows us to do iteration.

We need to set up a range of acceptable number of clusters.

We do not need our initial parameters to be precise. The algorithm will do heuristic search for us.

a demo of visual aids for choosing parameters