can_ppt

Clustering and Projected Clustering with Adaptive Neighbors
Probabilistic Neighbors Clustering with Adaptive Neighbors Projected Clustering with Adaptive Neighbors (PCAN) Experimental R
n sT i 1=1,0≤si ≤1
min
xi − xj
j =1
2 2 sij
(1.2)
Feiping Nie, Xiaoqian Wang, Heng Huang
Clustering and Projected Clustering with Adaptive Neighbors
Probabilistic Neighbors Clustering with Adaptive Neighbors Projected Clustering with Adaptive Neighbors (PCAN) Experimental R
min
xi − xj
j =1
2 2 sij
+ γs2 ij
(1.3)
where γ is a parameter to balance the two terms.
2 x n×1 as a vector Denote dx ij = xi − xj 2 , and denote di ∈ R with the j -th element as dx ij , then the problem (1.3) can be written in vector form as
Feiping Nie, Xiaoqian Wang, Heng Huang
Department of Computer Science and Engineering, University of Texas, Arlington, USA
August 25, 2014
Feiping Nie, Xiaoqian Wang, Heng Huang
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Feiping Nie, Xiaoqian Wang, Heng Huang
Clustering and Projected Clustering with Adaptive Neighbors
Probabilistic Neighbors Clustering with Adaptive Neighbors Projected Clustering with Adaptive Neighbors (PCAN) Experimental R
Computing the Probabilistic Neighbors
Combining the two rules, the probabilistic neighbors of xi can be computed by solving the following problem:
n sT i 1=1,0≤si ≤1
n sT i 1=1,0≤siБайду номын сангаас≤1
min
s2 ij
j =1
(1.1)
A smaller distance xi − xj 2 2 should be assigned a larger probability sij . It can be achieved by solving the following problem:
Probabilistic Neighbors Clustering with Adaptive Neighbors Projected Clustering with Adaptive Neighbors (PCAN) Experimental R
Clustering and Projected Clustering with Adaptive Neighbors
Probabilistic Neighbors Clustering with Adaptive Neighbors Projected Clustering with Adaptive Neighbors (PCAN) Experimental R
Probabilistic Neighbors Assignment for All the Data
n S ∈Rn×n i,j =1 s.t. ∀i, sT i 1
min
xi − xj
2 2 sij
+ γs2 ij
(2.2)
= 1, 0 ≤ si ≤ 1, S ∈ Ω
Feiping Nie, Xiaoqian Wang, Heng Huang
Clustering and Projected Clustering with Adaptive Neighbors
The problem (2.2) is very difficult to solve since the constraint S ∈ Ω is involved. We propose a novel algorithm as follows to solve this challenge. More details can be found in the paper. Optimization Algorithm for the Problem (2.2) Input: Data matrix X ∈ Rn×d , cluster number c, parameter γ , a large enough λ. Output: S ∈ Rn×n with exact c connected components. 1. Initialize S by the optimal solution to the problem (2.1). 2. Update F , which is formed by the c eigenvectors of T +S LS = D S − S 2 corresponding to the c smallest eigenvalues. 3. For each i, update the i-th row of S by solving the problem similar to (1.4). 4. Iteratively perform 2-3 until converges.
n ∀i,sT i 1=1,0≤si ≤1
min
xi − xj
i,j =1
2 2 sij
+ γs2 ij
(2.1)
The obtained matrix S can be seen as a similarity matrix. However, the matrix S can not directly used for clustering, since in most cases the data are all connected with the S .
Deterministic Neighbors and Probabilistic Neighbors
Exploring the local connectivity of data is a successful strategy for clustering task, e.g., spectral clustering. For the i-th data point xi , all the data points {x1 , x2 , ..., xn } can be connected to xi as a neighbor with a weight sij . Deterministic Neighbors. The weights sij have only two values: 0 or 1. Probabilistic Neighbors. The weights sij have probabilistic n×1 is a vector with values: sT i 1 = 1, 0 ≤ si ≤ 1, where si ∈ R the j -th element as sij .
sT i 1=1,0≤si ≤1
min
1 x si + d 2γ i
2
(1.4)
2
This problem can be solved with a closed form solution.
Feiping Nie, Xiaoqian Wang, Heng Huang Clustering and Projected Clustering with Adaptive Neighbors
Feiping Nie, Xiaoqian Wang, Heng Huang
Clustering and Projected Clustering with Adaptive Neighbors
Probabilistic Neighbors Clustering with Adaptive Neighbors Projected Clustering with Adaptive Neighbors (PCAN) Experimental R
Probabilistic Neighbors Clustering with Adaptive Neighbors Projected Clustering with Adaptive Neighbors (PCAN) Experimental R
The CAN Clustering Algorithm
Rules for Computing the Probabilistic Neighbors
If there is no any distance information in the data, as a prior, all the data points can be the neighbors of xi with the same 1 probability n . It can be achieved by solving the following problem: