cSVD

Clustered Singular Value Decomposition (cSVD) is a matrix decomposition that factorizes a given dataset using its Singular Value Decomposition (SVD) and reconstructs the features by projection matrices of clustered subsets. In this method, k-means clustering is used to label the whole data into several subsets in a low-dimensional space.

1. SVD

$\rho=Vv$, $v \in D$ (a given dataset)

$v \approx \tilde v=V^\top\rho$

2. cSVD

$\rho=Vv$

$v \approx \tilde v=W_l\top(W_lV^\top\rho)$

where $\rho=W_lw$, $w \in D_l$ satisfying $D=\cup_{l=1}^k D_l$, $\cap_{l=1}^k D_l \neq \emptyset$, $l=1,2,\cdots , k.$ and $k$ is the number of clusters.

3. Getting started

import csvd

  ...

cs = csvd.ClusteredSVD(data, n_svd_components=n_svd_components, num_clusters=num_clusters)
V = cs.V
W = cs.wlist
rec_svd = (V.T @ cs.rho).squeeze(-1)
rec_csvd = cs.rec

  ...

Parameters

---

• data: B x N dataset matrix (B: number of instances, N: number of features)
• n_svd_components: reduced parameter dimension
• num_clusters: number of clusters

Returns

---

• V: a projection matrix V obtained by SVD with the whole data
• rho: reduced dimensional data
• labels: clustering labels
• wlist: a list of k projection matrices Ws obtained by SVD with each clustered dataset

4. Test

4.1. Sklearn Datasets

(#instances, #features), n_components, num_clusters

• D1. wine dataset: (178, 13), n_components=5, num_clusters=4
• D2. breast cancer dataset: (569, 30), n_components=5, num_clusters=3
• D3. mnist dataset: (1797, 64), n_components=20, num_clusters=10
• D4. covertype dataset: (581012, 54), n_components=5, num_clusters=10

4.2. Reconstruction Error (MSE)

type	D1	D2	D3	D4
SVD	0.1745	0.2760	1.9888	78.5639
cSVD	0.1705	0.1863	1.6580	73.0463

5. Reference

Yongho Kim and Jan Heiland (2023), Convolutional Autoencoders, Clustering, and POD for Low-dimensional Parametrization of Navier-Stokes Equations