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notes about machine learning

On the equivalence of NMF and spectral clustering

derivation

Proof of $J_{\mathbf{K}}=\sum_{k=1}^{K} \sum_{i \in C_{k}}\left|\mathbf{x}{i}-\mathbf{m}{k}\right|^{2}=c_{2}-\sum_{k} \frac{1}{n_{k}} \sum_{i, j \in C_{k}} \mathbf{x}{i}^{T} \mathbf{x}{j}$. (Equati...

Posted by Kwan on August 19, 2020

Derivative of the Frobenius norm of a matrix involving the Hadamard products

derivation

Calculate $\frac{\partial\left|\mathbf{W} \circ \left(\mathbf{R}-\mathbf{U}^{\top} \mathbf{V}\right)\right|_ {F}^{2}}{\partial \mathbf{U}}$ and $\frac{\partial\left|\mathbf{W} \circ \left(\mathbf{R...

Posted by Kwan on August 6, 2020

服务器命令备忘录

备忘录

服务器命令备忘录 一键清除所有detached screens: screen -ls | grep Detached | cut -d. -f1 | awk '{print $1}' | xargs kill

Posted by Kwan on June 10, 2020

PRML $(2.61)$

derivation

We want to prove that \[\frac{1}{(2 \pi)^{D / 2}} \frac{1}{|\mathbf{\Sigma}|^{1 / 2}} \int \exp \left\lbrace-\frac{1}{2} \mathbf{z}^{\mathrm{T}} \mathbf{\Sigma}^{-1} \mathbf{z}\right\rbrace \mathb...

Posted by Kwan on June 5, 2020

Clustering and Projected Clustering with Adaptive Neighbors

derivation

脑子不太好使了，写一下记录一下怕忘。 Given a data set $\left\lbrace x_{1}, x_{2}, \ldots, x_{n}\right\rbrace$, we denote $X \in \mathbb{R}^{n \times d}$ as the data matrix. The probabilistic neighbors can be derive...

Posted by Kwan on May 8, 2020

The biggest eigenvalue of a normalized graph laplacian matrix

derivation

怕忘，写一个记录一下。 The (symmetric) normalized Laplacian is defined as \[L^{\mathrm{sym}}:=D^{-\frac{1}{2}} L D^{-\frac{1}{2}}=I-D^{-\frac{1}{2}} A D^{-\frac{1}{2}}.\] It is easy to verify that the norm...

Posted by Kwan on April 10, 2020

$\ell_{2,1}$-Norm Regularized Discriminative Feature Selection for Unsupervised Learning

derivation of some formulars

1. Notations According to the paper, the notations used are summarized as follows: denote $\mathcal{X}=\left \lbrace x_{1}, x_{2}, \ldots, x_{n}\right\rbrace$ as the training set, where $x_{i} \in ...

Posted by Kwan on March 24, 2020

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