K-Tensors

## Project description

# K-Tensors: Clustering Positive Semi-Definite Matrices

## Install the Package from PyPI

- Go to the terminal (mac) or command line (windows), copy and paste:
`pip3 install KTensors`

or`pip3 install KTensors==0.1.7`

for a specific version

## install the Package from GitHub

- Go to the terminal (mac) or command line (windows), copy and paste:
`pip3 install git+https://github.com/Hanchao-Zhang/KTensors.git`

- Open a python console, copy and paste:
`from KTensors import KTensors`

## About the Package

`KTensors`

`KTensors(Psis, K, max_iter=1000).clustering()`

input:

- Psis: a 3D array of size (n, p, p) where n is the number of matrices and p is the dimension of the positive semi-definite matrices.
- K: number of clusters
- max_iter: maximum number of iterations, default is 1000, usually finish within 10 iterations

return:

- group: a vector of length n, each element is an index of group membership
- CPCs: K orthonormal basis matrices of size p by p for each cluster
- $\mathbf F$: $\mathbf F = \mathbf B^\top \mathbf\Psi \mathbf B$
- $\text{diag}(\mathbf F)$: $\text{diag}(\mathbf F) = (\mathbf B^\top \mathbf\Psi \mathbf B) \circ \mathbf I$ the diagonal of matrix $\mathbf F$
- centers: Mean of each cluster
- loss: loss function for each iteration

`KMetrics`

`KMetrics(Psis, K, metrics)`

input:

- Psis: a 3D array of size (n, p, p) where n is the number of matrices and p is the dimension of the positive semi-definite matrices.
- K: number of clusters
- metrics: a string of metrics, including 'euclidean', 'affine_invariant', 'log_euclidean', 'log_det', and 'symmetric_stein'.

output:

- a vector of length n, each element is an index of group membership
- loss function for each iteration

## Some Tecnical Details

### Loss Function

$$\begin{aligned}\mathcal L(\mathbf \Psi, \mathcal P_{\mathbf B} (\mathbf \Psi) ) = \Vert \mathbf \Psi - \mathbf B((\mathbf B^\top \mathbf \Psi \mathbf B) \circ \mathbb I )\mathbf B^\top \Vert_F^2 = \Vert\mathbf B^\top \mathbf \Psi \mathbf B - (\mathbf B^\top \mathbf \Psi \mathbf B)\circ \mathbb I \Vert_F^2 \end{aligned}$$

where $\mathbb I$ is a identity matrix, and $\circ$ is the Hadamard product.

For more technical details, please refer to the paper.

## Project details

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