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Recent efforts in neural compression have focused on the rate-distortion-perception (RDP) tradeoff, where the perception constraint ensures the source and reconstruction distributions are close in terms of a statistical divergence. Theoretical work on RDP describes properties of RDP-optimal compressors without providing constructive and low complexity solutions. While classical rate-distortion theory shows that optimal compressors should efficiently pack space, RDP theory additionally shows that infinite randomness shared between the encoder and decoder may be necessary for RDP optimality. In this paper, we propose neural compressors that are low complexity and benefit from high packing efficiency through lattice coding and shared randomness through shared dithering over the lattice cells. For two important settings, namely infinite shared and zero shared randomness, we analyze the RDP tradeoff achieved by our proposed neural compressors and show optimality in both cases. Experimentally, we investigate the roles that these two components of our design, lattice coding and randomness, play in the performance of neural compressors on synthetic and real-world data. We observe that performance improves with more shared randomness and better lattice packing.

With the emergence of new storage and communication methods, the insertion, deletion, and substitution (IDS) channel has attracted considerable attention. However, many topics on the IDS channel and the associated Levenshtein distance remain open, making the invention of a novel IDS-correcting code a hard task.

Dependence among marginally constrained observations can break a finite-sample barrier. To formalize this phenomenon, we introduce the \emph{minimum list entropy coupling} $H(P\|Q_1,\dots,Q_m)$, the minimum conditional entropy $H(X|Y_1,\dots,Y_m)$ over all joint distributions with prescribed discrete marginals $X\sim P$ and $Y_i\sim Q_i$. Unlike classical formulations based on independent observations, our model allows $Y_1,\dots,Y_m$ to be arbitrarily dependent while keeping each marginal fixed. This enlarged coupling space reveals a sharp dichotomy: independent observations reduce residual uncertainty exponentially, whereas dependent observations can eliminate it exactly after finitely many samples. We characterize this zero-entropy regime through necessary and sufficient conditions and give concrete structural criteria under which it occurs. In particular, under mild support assumptions, zero entropy is achieved with $O(\log(1/P_{\min}))$ observations, where $P_{\min}$ is the minimum nonzero mass of $P$. We also develop a greedy algorithm with monotone approximation guarantees for computing $H(P\|Q_1,\dots,Q_m)$. Finally, we show that the same framework formalizes finite-sample limits in distribution-matching representation learning and randomness extraction, where zero entropy corresponds to exact recovery and exact extraction.

We study high-dimensional LASSO under differential privacy via objective perturbation with heterogeneous covariate scales. In practical scenarios, covariates often exhibit diverse scales; however, standard preprocessing is problematic under privacy constraints, as it consumes additional privacy budget. This heterogeneity induces effective anisotropy in the objective perturbation via the inverse Gram matrix of covariates, which can degrade the stability and accuracy of algorithms. To address this, we propose a Gram-based anisotropic objective perturbation, a ``pre-distortion" strategy that counteracts the distortion from the covariate structure to restore isotropy in the estimation process. Using an Approximate Message Passing (AMP) framework and state evolution analysis, we demonstrate that our proposed perturbation significantly stabilizes convergence and improves both statistical efficiency and privacy performance compared to standard uniform noise injection. Our results provide theoretical insights into designing stable and efficient private estimators without relying on data-dependent preprocessing.

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.