Dominic Broadbent | Searcharxiv

arXiv2025

Bilateral Distribution Compression: Reducing Both Data Size and Dimensionality

Existing distribution compression methods reduce the number of observations in a dataset by minimising the Maximum Mean Discrepancy (MMD) between original and compressed sets, but modern datasets are often large in both sample size and dimensionality. We propose Bilateral Distribution Compression (BDC), a two-stage framework that compresses along both axes while preserving the underlying distribution, with overall linear time and memory complexity in dataset size and dimension. Central to BDC is the Decoded MMD (DMMD), which we introduce to quantify the discrepancy between the original data and a compressed set decoded from a low-dimensional latent space. BDC proceeds by (i) learning a low-dimensional projection using the Reconstruction MMD (RMMD), and (ii) optimising a latent compressed set with the Encoded MMD (EMMD). We show that this procedure minimises the DMMD, guaranteeing that the compressed set faithfully represents the original distribution. Experiments show that BDC can achieve comparable or superior downstream task performance to ambient-space compression at substantially lower cost and with significantly higher rates of compression.

stat.ML↗

arXiv2025

Conditional Distribution Compression via the Kernel Conditional Mean Embedding

Existing distribution compression methods, like Kernel Herding (KH), were originally developed for unlabelled data. However, no existing approach directly compresses the conditional distribution of \textit{labelled} data. To address this gap, we first introduce the Average Maximum Conditional Mean Discrepancy (AMCMD), a metric for comparing conditional distributions, and derive a closed form estimator. 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 cubic to linear. Leveraging this, we extend KH to propose Average Conditional Kernel Herding (ACKH), a linear-time greedy algorithm for constructing compressed sets that target 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), an 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 also 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 and the greedy selection used in ACKH. Moreover, we see that JKIP consistently outperforms JKH.

stat.ML↗