Diffusion maps is a dimensionality reduction or feature extraction algorithm introduced by R. R. Coifman and S. Lafon. It computes a family of embeddings of a data set into Euclidean space (often low-dimensional) whose coordinates can be computed from the eigenvectors and eigenvalues of a diffusion operator on the data. The Euclidean distance between points in the embedded space is equal to the "diffusion distance" between probability distributions centered at those points. Different from linear dimensionality reduction methods such as principal component analysis (PCA) and multi-dimensional scaling (MDS), diffusion maps is part of the family of nonlinear dimensionality reduction methods which focus on discovering the underlying manifold that the data has been sampled from. By integrating local similarities at different scales, diffusion maps gives a global description of the data-set. Compared with other methods, the diffusion maps algorithm is robust to noise perturbation and is computationally inexpensive.
Contents
Definition of diffusion maps
Following and, diffusion maps can be defined in four steps.
Connectivity
Diffusion maps exploit the relationship between heat diffusion and random walk Markov chain. The basic observation is that if we take a random walk on the data, walking to a nearby data-point is more likely than walking to another that is far away. Let
Based on this, the connectivity
More generally, the kernel function has the following properties
(
(
The kernel constitutes the prior definition of the local geometry of data-set. Since a given kernel will capture a specific feature of the data set, its choice should be guided by the application that one has in mind. This is a major difference with methods such as principal component analysis, where correlations between all data points are taken into account at once.
Given
and define:
Although the new normalized kernel does not inherit the symmetric property, it does inherit the positivity-preserving property and gains a conservation property:
Diffusion process
From
We define the diffusion matrix
We then define the new kernel
or equivalently,
where D is a diagonal matrix and
We apply the graph Laplacian normalization to this new kernel:
where
One of the main ideas of diffusion framework is that running the chain forward in time (taking larger and larger powers of
The eigendecomposition of the matrix
where
Parameter α {\displaystyle \alpha } and the Diffusion Operator
The reason to introduce the normalization step involving
Diffusion distance
The diffusion distance at time
where
Intuitively,
- Points are closer at a given scale (as specified by
D t ( x i , x j ) ) if they are highly connected in the graph, therefore emphasizing the concept of a cluster. - This distance is robust to noise, since the distance between two points depends on all possible paths of length
t between the points. - From a machine learning point of view, the distance takes into account all evidences linking
x i x j
Diffusion process and low-dimensional embedding
The diffusion distance can be calculated using the eigenvectors by
So the eigenvectors can be used as a new set of coordinates for the data. The diffusion map is defined as:
Because of the spectrum decay, it is sufficient to use only the first k eigenvectors and eigenvalues. Thus we get the diffusion map from the original data to a k-dimensional space which is embedded in the original space.
In, it is proved that
so the Euclidean distance in the diffusion coordinates approximates the diffusion distance.
Algorithm
The basic algorithm framework of diffusion map is as:
Step 1. Given the similarity matrix L
Step 2. Normalize the matrix according to parameter
Step 3. Form the normalized matrix
Step 4. Compute the k largest eigenvalues of
Step 5. Use diffusion map to get the embedding
Application
In the paper, they showed how to design a kernel that reproduces the diffusion induced by a Fokker-Planck equation. Also, they explained that when the data approximate a manifold, then one can recover the geometry of this manifold by computing an approximation of the Laplace-Beltrami operator. This computation is completely insensitive to the distribution of the points and therefore provides a separation of the statistics and the geometry of the data. Since Diffusion map gives a global description of the data-set, it can measure the distances between pair of sample points in the manifold the data is embedded. Based on diffusion map, there are many applications, such as face recognition, spectral clustering, low dimensional representation of images, image segmentation, 3D model segmentation, speaker verification and identification, sampling on manifolds,anomaly detection, image inpainting, and so on. Through their use of the Fokker–Planck equation in the context of the Meila-Shi algorithm, it is claimed that the method of Coifman and Lafon is related to computational quantum mechanics.