Gaussian Process

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Gaussian Process

Author: Changsheng Lu (卢长胜)

Definition

Gaussian process (GP) is a set of random variables following Gaussian distribution.

Gaussian process for machine learning

Essentially, it uses the correlation between training data and test data to model the prediction of test data, and each prediction is subject to a Gaussian distribution.

  • Tutorial can be found here

Algorithm

Before introducing the algorithm, we first define the squared exponential kernel $k$:

\[k(A, B) = e^{-\frac{1}{2p}(a \ominus b)}= \left[ \begin{matrix} e^{-\frac{1}{2p}\|a_1 - b_1\|} & e^{-\frac{1}{2p}\|a_1 - b_2\|} & \cdots & e^{-\frac{1}{2p}\|a_1 - b_n\|} \\ e^{-\frac{1}{2p}\|a_2 - b_1\|} & e^{-\frac{1}{2p}\|a_2 - b_2\|} & \cdots & e^{-\frac{1}{2p}\|a_2 - b_n\|} \\ \cdots & \cdots & \cdots & \cdots \\ e^{-\frac{1}{2p}\|a_m - b_1\|} & e^{-\frac{1}{2p}\|a_m - b_2\|} & \cdots & e^{-\frac{1}{2p}\|a_m - b_n\|} \end{matrix} \right]\]

where $A$ has $m$ data points and $B$ has $n$ data points, and each element $k_{ij}$ ranges from 0~1.

Assuming the training set is ${X_1, Y_1}$ and testing set is $X_2$, now our task is to predict the label $\hat{Y_2}$ of $X_2$. The prediction value of each data can be modeled to satisfy the Gaussian distribution according to Gaussian process, which can be formulated as:

\[\left(\begin{matrix} f \\ f_s \end{matrix}\right) \sim \mathcal{N}\left( \left(\begin{matrix} \mu \\ \mu_s\end{matrix}\right), \left(\begin{matrix} K & K_s \\ K^\text{T}_s & K_{ss}\end{matrix}\right) \right)\]

The overall algorithm is as follows:
Step 1: compute variance $K_{ss}$ for testing data, $K_{ss} = e^{-\frac{1}{2p}(X_2 \ominus X_2)}$, with the size of $N \times N$.
Step 2: (optional) compute prior

\(L_{ss}L^{\text{T}}_{ss} = K_{ss}\)
\(f_{prior} = \mu_s + L_{ss}\mathcal{N}(0, I)\)

where $L_{ss}$ can be obtained using Cholesky decomposition and $\mu_s$ is unknown so far (however we can assume to be zero here to visualize the $f_{prior}$). $f_{prior}$ has the size of $N \times 1$.
Step 3: compute variance $K$ for training data, $K = e^{-\frac{1}{2p}(X_1 \ominus X_1)}$, with the size of $M \times M$.

\[LL^{\text{T}} = K\]

So that $L = CholeskyDecom(K)$, with the size of $M \times M$.
Step 4: compute covariance between training data and testing data $K_s$, $K_s = e^{-\frac{1}{2p}(X_1 \ominus X_2)}$, with the size of $M \times N$.

\[LL_{s} = K_s\]

We can obtain $L_{s}=linearSolve(L, K_s)$, with the size of $M \times N$.
Step 5: compute the mean at test points

\[LL'=y\] \[\mu_s = L^{\text{T}}_sL'\]

where $\mu_s$ has the size of $N \times 1$.
Step 6: compute the posterior

\[L'_{ss}L'^{\text{T}}_{ss} = K_{ss}-L^{\text{T}}_sL_{s}\]

So that $L{‘}{ss} = CholeskyDecom(K{ss}-L^{\text{T}}{s}L{s})$, with the size of $N \times N$. The posterior function is

\[f_{post} = \mu_s + L'_{ss}\mathcal{N}(0, I)\]

The standard deviation can be computed by

\[s2=diag(K_{ss}) - diag(L^{\text{T}}_sL_s)=diag(K_{ss}-L^{\text{T}}_sL_{s})\] \[stdv = sqrt(s2)\]

$diag(.)$ means utilizing the diagnal elements to form a vector so that $s2$ has the size of $N \times 1$. $stdv$ is the computed standard diviation. The oscillation range of posterior function can be written as

\[\mu_s \pm j*stdv\]

For example, $\mu_{s} \pm 1stdv$, $\mu_{s} \pm 2stdv$, $\mu_{s} \pm 3*stdv$, etc.

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