\[y = \color{orange}{f}(x) + \epsilon\]
No restrictions on \(\color{orange}{f}\) but a prior distribution on a functional space: \(\color{orange}{f} \sim \mathcal{GP}(m(\cdot),C(\cdot,\cdot))\)
While \(m\) is often assumed to be \(0\), the covariance structure is critical and defined through tailored kernels. For instance, the Squared Exponential (or RBF) kernel is expressed as: \[C_{SE}(x, x^{\prime}) = s^2 \exp \Bigg(-\dfrac{(x - x^{\prime})^2}{2 \ell^2}\Bigg)\]
The Gaussian property induces that unobserved points have no influence on inference:
\[ \int \underbrace{p(f_{\color{grey}{obs}}, f_{\color{purple}{mis}})}_{\mathcal{GP}(m, C)} \ \mathrm{d}f_{\color{purple}{mis}} = \underbrace{p(f_{\color{grey}{obs}})}_{\mathcal{N}(m_{\color{grey}{obs}}, C_{\color{grey}{obs}})} \]
This crucial trick allows us to learn function properties from finite sets of observations. More generally, Gaussian processes are closed under conditioning and marginalisation.
\[\begin{bmatrix} f_{\color{grey}{o}} \\ f_{\color{purple}{m}} \\ \end{bmatrix} \sim \mathcal{N} \left( \begin{bmatrix} m_{\color{grey}{o}} \\ m_{\color{purple}{m}} \\ \end{bmatrix}, \begin{pmatrix} C_{\color{grey}{o, o}} & C_{\color{grey}{o}, \color{purple}{m}} \\ C_{\color{purple}{m}, \color{grey}{o}} & C_{\color{purple}{m, m}} \end{pmatrix} \right)\]
While marginalisation serves for training, conditioning leads
the key GP prediction formula:
\[f_{\color{purple}{m}} \mid f_{\color{grey}{o}} \sim \mathcal{N} \Big( m_{\color{purple}{m}} + C_{\color{purple}{m}, \color{grey}{o}} C_{\color{grey}{o, o}}^{-1} (f_{\color{grey}{o}} - m_{\color{grey}{o}}), \ \ C_{\color{purple}{m, m}} - C_{\color{purple}{m}, \color{grey}{o}} C_{\color{grey}{o, o}}^{-1} C_{\color{grey}{o}, \color{purple}{m}} \Big)\]
Instead of modelling multiple tasks (or individuals, or outputs) independently with a single GPs, we could leverage correlations existing across them.
A natural approach comes from defining multi-output kernels, explicitly modelling correlations between each task-input couple. A classical assumption is to consider separable kernels, such as:
\[k\left((i, t),\left(j, t^{\prime}\right)\right)=\operatorname{cov}\left(f_i(t), f_j\left(t^{\prime}\right)\right)=k_{\mathrm{o}}(i, j) k_{\mathrm{t}}\left(t, t^{\prime}\right)\]
By stacking outputs into one vector \(\textbf{y} = \{\textbf{y}_1, \dots, \textbf{y}_M\}^{\intercal}\), the matrices build from those kernels can be represented through a Kronecker product:
\[\textbf{K} = \textbf{K}_{\mathrm{o}} \otimes \textbf{K}_{\mathrm{t}}\]
There exist many ways to define multi-output kernels, though it is in practice costly and unstable to learn explicitly the inter-task correlations (i.e. the elements of \(\boldsymbol{K_o}\)) from data. Moreover, the time complexity for exact training is \(\mathcal{O}(\color{blue}{M}^3N^3)\).
Credits: Neil Lawrence
\[y_i = \mu_0 + f_i + \epsilon_i\]
with:
It follows that:
\[y_i \mid \mu_0 \sim \mathcal{GP}(\mu_0, \Sigma_{\theta_i} + \sigma_i^2 I), \ \perp \!\!\! \perp_i\]
\(\rightarrow\) Unified GP framework with a common mean process \(\mu_0\), and individual-specific process \(f_i\),
\(\rightarrow\) Naturaly handles irregular grids of input data.
Goal: Learn the
hyper-parameters, (and \(\mu_0\)’s
hyper-posterior).
Difficulty: The likelihood depends on \(\mu_0\), and individuals are not
independent.
E step: \[ \begin{align} p(\mu_0(\color{grey}{\mathbf{t}}) \mid \textbf{y}, \hat{\Theta}) &\propto \mathcal{N}(\mu_0(\color{grey}{\mathbf{t}}); m_0(\color{grey}{\textbf{t}}), \textbf{K}_{\hat{\theta}_0}^{\color{grey}{\textbf{t}}}) \times \prod\limits_{i =1}^M \mathcal{N}(\mathbf{y}_i; \mu_0( \color{purple}{\textbf{t}_i}), \boldsymbol{\Psi}_{\hat{\theta}_i, \hat{\sigma}_i^2}^{\color{purple}{\textbf{t}_i}}) \\ &= \mathcal{N}(\mu_0(\color{grey}{\mathbf{t}}); \hat{m}_0(\color{grey}{\textbf{t}}), \hat{\textbf{K}}^{\color{grey}{\textbf{t}}}), \end{align} \] M step:
\[ \begin{align*} \hat{\Theta} &= \underset{\Theta}{\arg\max} \ \ \log \mathcal{N} \left( \hat{m}_0(\color{grey}{\textbf{t}}); m_0(\color{grey}{\textbf{t}}), \mathbf{K}_{\theta_0}^{\color{grey}{\textbf{t}}} \right) - \dfrac{1}{2} Tr \left( \hat{\mathbf{K}}^{\color{grey}{\textbf{t}}} {\mathbf{K}_{\theta_0}^{\color{grey}{\textbf{t}}}}^{-1} \right) \\ & \ \ \ + \sum\limits_{i = 1}^{M}\left\{ \log \mathcal{N} \left( \mathbf{y}_i; \hat{m}_0(\color{purple}{\mathbf{t}_i}), \boldsymbol{\Psi}_{\theta_i, \sigma^2}^{\color{purple}{\mathbf{t}_i}} \right) - \dfrac{1}{2} Tr \left( \hat{\mathbf{K}}^{\color{purple}{\mathbf{t}_i}} {\boldsymbol{\Psi}_{\theta_i, \sigma^2}^{\color{purple}{\mathbf{t}_i}}}^{-1} \right) \right\}. \end{align*} \]
Defining a multi-task prior distribution by:
\[\begin{align} p(y_* (\textbf{t}_*^{p}) \mid \textbf{y}) &= \int p\left(y_* (\textbf{t}_*^{p}) \mid \textbf{y}, \mu_0(\textbf{t}_*^{p})\right) p(\mu_0 (\textbf{t}_*^{p}) \mid \textbf{y}) \ d \mu_0(\mathbf{t}^{p}_{*}) \\ &= \int \underbrace{ p \left(y_* (\textbf{t}_*^{p}) \mid \mu_0 (\textbf{t}_*^{p}) \right)}_{\mathcal{N}(y_*; \mu_0, \Psi_*)} \ \underbrace{p(\mu_0 (\textbf{t}_*^{p}) \mid \textbf{y})}_{\mathcal{N}(\mu_0; \hat{m}_0, \hat{K})} \ d \mu_0(\mathbf{t}^{p}_{*}) \\ &= \mathcal{N}( \hat{m}_0 (\mathbf{t}^{p}_{*}), \underbrace{\Psi_* + \hat{K}}_{\Gamma}) \end{align}\]
\[p \left( \begin{bmatrix} y_*(\color{grey}{\mathbf{t}_{*}}) \\ y_*(\color{purple}{\mathbf{t}^{p}}) \\ \end{bmatrix} \mid \textbf{y} \right) = \mathcal{N} \left( \begin{bmatrix} y_*(\color{grey}{\mathbf{t}_{*}}) \\ y_*(\color{purple}{\mathbf{t}^{p}}) \\ \end{bmatrix}; \ \begin{bmatrix} \hat{m}_0(\color{grey}{\mathbf{t}_{*}}) \\ \hat{m}_0(\color{purple}{\mathbf{t}^{p}}) \\ \end{bmatrix}, \begin{pmatrix} \Gamma_{\color{grey}{**}} & \Gamma_{\color{grey}{*}\color{purple}{p}} \\ \Gamma_{\color{purple}{p}\color{grey}{*}} & \Gamma_{\color{purple}{pp}} \end{pmatrix} \right)\]
\[p(y_*(\color{purple}{\mathbf{t}^{p}}) \mid y_*(\color{grey}{\mathbf{t}_{*}}), \textbf{y}) = \mathcal{N} \Big( y_*(\color{purple}{\mathbf{t}^{p}}); \ \hat{\mu}_{*}(\color{purple}{\mathbf{t}^{p}}) , \hat{\Gamma}_{\color{purple}{pp}} \Big)\]
with:
Leroy et al. - MAGMA: Inference and Prediction using Multi-Task Gaussian Processes with Common Mean - Machine Learning - 2022
Leroy et al. - Cluster-Specific Predictions with Multi-Task Gaussian Processes - Journal of Machine Learning Research - 2023
A unique underlying mean process might be too restrictive.
\(\rightarrow\) Mixture of multi-task GPs:
\[y_i = \mu_0 + f_i + \epsilon_i\]
with:
It follows that:
\[y_i \mid \mu_0 \sim \mathcal{GP}(\mu_0, \Psi_i), \ \perp \!\!\! \perp_i\]
Leroy et al. - Cluster-Specific Predictions with Multi-Task Gaussian Processes - Journal of Machine Learning Research - 2023
A unique underlying mean process might be too restrictive.
\(\rightarrow\) Mixture of multi-task GPs:
\[y_i \mid \{\color{green}{Z_{ik}} = 1 \} = \mu_{\color{green}{k}} + f_i + \epsilon_i\]
with:
It follows that:
\[y_i \mid \mu_0 \sim \mathcal{GP}(\mu_0, \Psi_i), \ \perp \!\!\! \perp_i\]
A unique underlying mean process might be too restrictive.
\(\rightarrow\) Mixture of multi-task GPs:
\[y_i \mid \{\color{green}{Z_{ik}} = 1 \} = \mu_{\color{green}{k}} + f_i + \epsilon_i\]
with:
It follows that:
\[y_i \mid \{ \boldsymbol{\mu} , \color{green}{\boldsymbol{\pi}} \} \sim \sum\limits_{k=1}^K{ \color{green}{\pi_k} \ \mathcal{GP}\Big(\mu_{\color{green}{k}}, \Psi_i^\color{green}{k} \Big)}, \ \perp \!\!\! \perp_i\]
The integrated likelihood is not tractable anymore due to posterior dependencies between \( \boldsymbol{\mu} = \{\mu_\color{green}{k}\}_\color{green}{k}\) and \(\mathbf{Z}= \{Z_i\}_i\).
Variational inference still allows us to maintain closed-form approximations. For any distribution \(q\):
\[\log p(\textbf{y} \mid \Theta) = \mathcal{L}(q; \Theta) + KL \big( q \mid \mid p(\boldsymbol{\mu}, \boldsymbol{Z} \mid \textbf{y}, \Theta)\big)\]
The posterior independence is forced by an approximation assumption:
\[q(\boldsymbol{\mu}, \boldsymbol{Z}) = q_{\boldsymbol{\mu}}(\boldsymbol{\mu})q_{\boldsymbol{Z}}(\boldsymbol{Z}).\]
Maximising the lower bound \(\mathcal{L}(q; \Theta)\) induces natural factorisations over clusters and individuals for the variational distributions.
E step: \[ \begin{align} \hat{q}_{\boldsymbol{\mu}}(\boldsymbol{\mu}) &= \color{green}{\prod\limits_{k = 1}^K} \mathcal{N}(\mu_\color{green}{k};\hat{m}_\color{green}{k}, \hat{\textbf{C}}_\color{green}{k}) , \hspace{2cm} \hat{q}_{\boldsymbol{Z}}(\boldsymbol{Z}) = \prod\limits_{i = 1}^M \mathcal{M}(Z_i;1, \color{green}{\boldsymbol{\tau}_i}) \end{align} \] M step:
\[ \begin{align*} \hat{\Theta} &= \underset{\Theta}{\arg\max} \sum\limits_{k = 1}^{K}\ \mathcal{N} \left( \hat{m}_k; \ m_k, \boldsymbol{C}_{\color{green}{\gamma_k}} \right) - \dfrac{1}{2} \textrm{tr}\left( \mathbf{\hat{C}}_k\boldsymbol{C}_{\color{green}{\gamma_k}}^{-1}\right) \\ & \hspace{1cm} + \sum\limits_{k = 1}^{K}\sum\limits_{i = 1}^{M}\tau_{ik}\ \mathcal{N} \left( \mathbf{y}_i; \ \hat{m}_k, \boldsymbol{\Psi}_{\color{blue}{\theta_i}, \color{blue}{\sigma_i^2}} \right) - \dfrac{1}{2} \textrm{tr}\left( \mathbf{\hat{C}}_k\boldsymbol{\Psi}_{\color{blue}{\theta_i}, \color{blue}{\sigma_i^2}}^{-1}\right) \\ & \hspace{1cm} + \sum\limits_{k = 1}^{K}\sum\limits_{i = 1}^{M}\tau_{ik}\log \color{green}{\pi_{k}} \end{align*} \]
Sharing the covariance structures offers a compromise between flexibility and parsimony:
\[ p(y_*(\mathbf{t}^{p}) \mid \color{green}{Z_{*k}} = 1, y_*(\mathbf{t}_{*}), \textbf{y}) = \mathcal{N} \Big( y_*(\mathbf{t}^{p}); \ \hat{\mu}_{*}^\color{green}{k}(\mathbf{t}^{p}) , \hat{\Gamma}_{pp}^\color{green}{k} \Big), \forall \color{green}{k}, \]
\(\hat{\mu}_{*}^\color{green}{k}(\mathbf{t}^{p}) =
\hat{m}_\color{green}{k}(\mathbf{t}^{p}) + \Gamma^\color{green}{k}_{p*}
{\Gamma^\color{green}{k}_{**}}^{-1} (y_*(\mathbf{t}_{*}) -
\hat{m}_\color{green}{k} (\mathbf{t}_{*}))\)
\(\hat{\Gamma}_{pp}^\color{green}{k} =
\Gamma_{pp}^\color{green}{k} - \Gamma_{p*}^\color{green}{k}
{\Gamma^{\color{green}{k}}_{**}}^{-1}
\Gamma^{\color{green}{k}}_{*p}\)
\[p(y_*(\textbf{t}^p) \mid y_*(\textbf{t}_*), \textbf{y}) = \color{green}{\sum\limits_{k = 1}^{K} \tau_{*k}} \ \mathcal{N} \big( y_*(\mathbf{t}^{p}); \ \hat{\mu}_{*}^\color{green}{k}(\textbf{t}^p) , \hat{\Gamma}_{pp}^\color{green}{k}(\textbf{t}^p) \big).\]
By identifying the underlying clustering structure, MagmaClust discards unnecessary information and provides enhanced predictions as well as a lower uncertainty.
Each cluster-specific prediction is weighted by its membership probability \(\color{green}{\tau_{*k}}\).
Implemented as an R package MagmaClustR: https://arthurleroy.github.io/MagmaClustR
Different sources of correlation might exist in the data (e.g. multiple genes and individuals)
\[y_{\color{blue}{i}\color{red}{j}} = \mu_{0} + f_\color{blue}{i} + g_\color{red}{j} + \epsilon_{\color{blue}{i}\color{red}{j}}\]
with:
Key idea for training: define \(\color{blue}{M}+\color{red}{P} + 1\) different hyper-posterior distributions for \(\mu_0\) by conditioning over the adequate sub-sample of data.
\[p(\mu_0 \mid \{y_{\color{blue}{i}\color{red}{j}} \}_{\color{red}{j} = 1,\dots, \color{red}{P}}^{\color{blue}{i} = 1,\dots, \color{blue}{M}}) = \mathcal{N}\Big(\mu_{0}; \ \hat{m}_{0}, \hat{K}_0 \Big).\]
\[p(\mu_0 \mid \{y_{\color{blue}{i}\color{red}{j}} \}_{\color{blue}{i} = 1,\dots, \color{blue}{M}}) = \mathcal{N}\Big(\mu_{0}; \ \hat{m}_{\color{red}{j}}, \hat{K}_\color{red}{j} \Big), \ \forall \color{red}{j} \in 1, \dots, \color{red}{P}\]
\[p(\mu_0 \mid \{y_{\color{blue}{i}\color{red}{j}} \}_{\color{red}{j} = 1,\dots, \color{red}{P}}) = \mathcal{N}\Big(\mu_{0}; \ \hat{m}_{\color{blue}{i}}, \hat{K}_\color{blue}{i} \Big), \forall \color{blue}{i} \in 1, \dots, \color{blue}{M} \]
Each sub-sample of data leads to a specific hyper-posterior distribution of the mean process \(\mu_0\).
Although sharing the same mean process, different tasks still lead to different predictions.
Multi-mean GP provides adaptive predictions according to the relevant context.
And also:
\(\rightarrow\) All methods
scale linearly with the number of tasks
and clusters.
\(\rightarrow\) Parallel computing
can be used to speed up training.
Overall, the computational
complexity is: