Dataset condensation (DC) distills a large real-world dataset into a small synthetic dataset, with the goal of training a network from scratch on the latter that performs similarly to the former. State-of-the-art (SOTA) DC methods have achieved satisfactory results through techniques such as accuracy, gradient, training trajectory, or distribution matching. However, these works all perform matching in the high-dimension pixel space, ignoring that natural images are usually locally connected and have lower intrinsic dimensions, resulting in low condensation efficiency. In this work, we propose a simple-yet-efficient dataset condensation plugin that matches the raw and synthetic datasets in a low-dimensional manifold.

A.1 In-variance of Mutual Information Theorem 1 (In-variance of Mutual Information): Mutual information is invariant under reparametrization of the marginal variables.

Dataset distillation (DD) aims to synthesize a small dataset whose test performance is comparable to a full dataset using the same model. State-of-the-art (SoTA) methods optimize synthetic datasets primarily by matching heuristic indicators extracted from two networks: one from real data and one from synthetic data (see Figure 1, Left), such as gradients and training trajectories. DD is essentially a compression problem that emphasizes maximizing the preservation of information contained in the data. We argue that well-defined metrics which measure the amount of shared information between variables in information theory are necessary for success measurement but are never considered by previous works. Thus, we introduce mutual information (MI) as the metric to quantify the shared information between the synthetic and the real datasets, and devise MIM4DD numerically maximizing the MI via a newly designed optimizable objective within a contrastive learning framework to update the synthetic dataset. Specifically, we designate the samples in different datasets that share the same labels as positive pairs and vice versa negative pairs. Then we respectively pull and push those samples in positive and negative pairs into contrastive space via minimizing NCE loss. As a result, the targeted MI can be transformed into a lower bound represented by feature maps of samples, which is numerically feasible. Experiment results show that MIM4DD can be implemented as an add-on module to existing SoTADD methods.

Dataset distillation, a training-aware data compression technique, has recently attracted increasing attention as an effective tool for mitigating costs of optimization and data storage. However, progress remains largely empirical. Mechanisms underlying the extraction of task-relevant information from the training process and the efficient encoding of such information into synthetic data points remain elusive. In this paper, we theoretically analyze practical algorithms of dataset distillation applied to the gradient-based training of two-layer neural networks with width $L$. By focusing on a non-linear task structure called multi-index model, we prove that the low-dimensional structure of the problem is efficiently encoded into the resulting distilled data. This dataset reproduces a model with high generalization ability for a required memory complexity of $\tildeΘ$$(r^2d+L)$, where $d$ and $r$ are the input and intrinsic dimensions of the task. To the best of our knowledge, this is one of the first theoretical works that include a specific task structure, leverage its intrinsic dimensionality to quantify the compression rate and study dataset distillation implemented solely via gradient-based algorithms.

Understanding how and why data distillation methods work is vital not only for improving these methods but also for revealing fundamental characteristics of "good" training data.

To avoid redundancy in these synthetic datasets, it is crucial that each element contains unique features and remains diverse from others during the synthesis stage. In this paper, we provide a thorough theoretical and empirical analysis of diversity within synthesized datasets. We argue that enhancing diversity can improve the parallelizable yet isolated synthesizing approach.