Conditional Distribution Compression via the Kernel Conditional Mean Embedding
This work addresses a gap in distribution compression for machine learning applications involving conditional data, though it appears incremental as it builds on existing kernel methods.
The paper tackles the problem of compressing conditional distributions for labelled data, which was not addressed by existing methods like Kernel Herding, by introducing new metrics and algorithms such as ACKH and ACKIP, with experiments showing that ACKIP outperforms joint distribution compression methods and greedy selection.
Existing distribution compression methods, like Kernel Herding (KH), were originally developed for unlabelled data. However, no existing approach directly compresses the conditional distribution of labelled data. To address this gap, we first introduce the Average Maximum Conditional Mean Discrepancy (AMCMD), a natural metric for comparing conditional distributions. We then derive a consistent estimator for the AMCMD and establish its rate of convergence. Next, we make a key observation: in the context of distribution compression, the cost of constructing a compressed set targeting the AMCMD can be reduced from $\mathcal{O}(n^3)$ to $\mathcal{O}(n)$. Building on this, we extend the idea of KH to develop Average Conditional Kernel Herding (ACKH), a linear-time greedy algorithm that constructs a compressed set targeting the AMCMD. To better understand the advantages of directly compressing the conditional distribution rather than doing so via the joint distribution, we introduce Joint Kernel Herding (JKH), a straightforward adaptation of KH designed to compress the joint distribution of labelled data. While herding methods provide a simple and interpretable selection process, they rely on a greedy heuristic. To explore alternative optimisation strategies, we propose Joint Kernel Inducing Points (JKIP) and Average Conditional Kernel Inducing Points (ACKIP), which jointly optimise the compressed set while maintaining linear complexity. Experiments show that directly preserving conditional distributions with ACKIP outperforms both joint distribution compression (via JKH and JKIP) and the greedy selection used in ACKH. Moreover, we see that JKIP consistently outperforms JKH.