Non-stationary and Sparsely-correlated Multi-output Gaussian Process with Spike-and-Slab Prior

TL;DR

The paper tackles non-stationary, high-dimensional multi-output learning by introducing DMGP-SS, a convolution-process-based MGP with time-varying kernels and a dynamic spike-and-slab prior to selectively transfer information across sources. An EM algorithm fits the model, yielding a block-sparse covariance that captures dynamic, sparse cross-output correlations while mitigating negative transfer. The approach is validated on synthetic data and real gesture data, showing improved predictive accuracy and calibrated uncertainty, and is demonstrated in a Mountain Car reinforcement learning setting to highlight decision-making benefits. The work offers a scalable, data-driven framework for transfer learning in non-stationary time series with missing data and provides avenues for future extensions in online settings and uncertainty quantification.

Abstract

Multi-output Gaussian process (MGP) is commonly used as a transfer learning method to leverage information among multiple outputs. A key advantage of MGP is providing uncertainty quantification for prediction, which is highly important for subsequent decision-making tasks. However, traditional MGP may not be sufficiently flexible to handle multivariate data with dynamic characteristics, particularly when dealing with complex temporal correlations. Additionally, since some outputs may lack correlation, transferring information among them may lead to negative transfer. To address these issues, this study proposes a non-stationary MGP model that can capture both the dynamic and sparse correlation among outputs. Specifically, the covariance functions of MGP are constructed using convolutions of time-varying kernel functions. Then a dynamic spike-and-slab prior is placed on correlation parameters to automatically decide which sources are informative to the target output in the training process. An expectation-maximization (EM) algorithm is proposed for efficient model fitting. Both numerical studies and a real case demonstrate its efficacy in capturing dynamic and sparse correlation structure and mitigating negative transfer for high-dimensional time-series data. Finally, a mountain-car reinforcement learning case highlights its potential application in decision making problems.

Figures (9)

• Figure 1: The graphical structure of non-stationary MGP. Latent processes and kernel functions are with a gray background, while the parameters are with a white background. The parameters' priors are shown in rectangles.
• Figure 2: Dynamic correlation detection results with four sources: (a) Case 1, (b) Case 2. The first column is the true $a_{i,t}$, the second and fourth columns are the estimated ${\bm{\alpha}}_{m}$ for MGP-L1 and DMGP-SS respectively, and the third column shows the estimated $(\bm{A}_t \bm{A}_t^T)_{0:m-1, m}$ for DMGP-GP.
• Figure 3: Visualization of prediction results in Case 1 ($k=1$). The shaded region represents the $99\%$ confidence interval.
• Figure 4: The estimated $E_{\bm{\gamma}}{\bm{\gamma}}_{1:4}$ for DMGP-SS in Case 1.
• Figure 5: (a). The snapshots of "Stand", "Google", "Shoot" and "Throw". The twenty joints are represented by circles. We mark out the four selected joints for our study, where "L" and "R" represent "left-side" and "right-side" respectively, and "E" and "W" represent the elbow and wrist joints respectively. (b). The twelve movement signals of the selected joints in the three gestures' data, where the red signal is takes as the target signal and the others are the source signals. In the label of vertical axis, "x", "y" and "z" represent different coordinates. Each signal has a vertical range of $[-2,2]$, with the horizontal ticks representing time indexes.

Theorems & Definitions (1)

• Proposition 1