Importance sampling, which involves sampling from a probability density function (PDF) proportional to the product of an importance weight function and a base PDF, is a powerful technique with applications in variance reduction, biased or customized sampling, data augmentation, and beyond. Inspired by the growing availability of score-based generative models (SGMs), we propose an entirely training-free Importance sampling framework that relies solely on an SGM for the base PDF. Our key innovation is realizing the importance sampling process as a backward diffusion process, expressed in terms of the score function of the base PDF and the specified importance weight function--both readily available--eliminating the need for any additional training. We conduct a thorough analysis demonstrating the method's scalability and effectiveness across diverse datasets and tasks, including importance sampling for industrial and natural images with neural importance weight functions. The training-free aspect of our method is particularly compelling in real-world scenarios where a single base distribution underlies multiple biased sampling tasks, each requiring a different importance weight function. To the best of our knowledge our approach is the first importance sampling framework to achieve this.

While neural lossy compression techniques have markedly advanced the efficiency of Channel State Information (CSI) compression and reconstruction for feedback in MIMO communications, efficient algorithms for more challenging and practical tasks-such as CSI compression for future channel prediction and reconstruction with relevant side information-remain underexplored, often resulting in suboptimal performance when existing methods are extended to these scenarios. To that end, we propose a novel framework for compression with side information, featuring an encoding process with fixed-rate compression using a trainable codebook for codeword quantization, and a decoding procedure modeled as a backward diffusion process conditioned on both the codeword and the side information. Experimental results show that our method significantly outperforms existing CSI compression algorithms, often yielding over twofold performance improvement by achieving comparable distortion at less than half the data rate of competing methods in certain scenarios. These findings underscore the potential of diffusion-based compression for practical deployment in communication systems.

In steganography, selecting an optimal cover image--referred to as cover selection--is pivotal for effective message concealment. Traditional methods have typically employed exhaustive searches to identify images that conform to specific perceptual or complexity metrics. However, the relationship between these metrics and the actual message hiding efficacy of an image is unclear, often yielding less-than-ideal steganographic outcomes. Inspired by recent advancements in generative models, we introduce a novel cover selection framework, which involves optimizing within the latent space of pretrained generative models to identify the most suitable cover images, distinguishing itself from traditional exhaustive search methods. Our method shows significant advantages in message recovery and image quality. We also conduct an information-theoretic analysis of the generated cover images, revealing that message hiding predominantly occurs in low-variance pixels, reflecting the waterfilling algorithm's principles in parallel Gaussian channels.

In recent years, transformer-based models have revolutionized deep learning, particularly in sequence modeling. To better understand this phenomenon, there is a growing interest in using Markov input processes to study transformers. However, our current understanding in this regard remains limited with many fundamental questions about how transformers learn Markov chains still unanswered. In this paper, we address this by focusing on first-order Markov chains and single-layer transformers, providing a comprehensive characterization of the learning dynamics in this context. Specifically, we prove that transformer parameters trained on next-token prediction loss can either converge to global or local minima, contingent on the initialization and the Markovian data properties, and we characterize the precise conditions under which this occurs. To the best of our knowledge, this is the first result of its kind highlighting the role of initialization. We further demonstrate that our theoretical findings are corroborated by empirical evidence. Based on these insights, we provide guidelines for the initialization of transformer parameters and demonstrate their effectiveness. Finally, we outline several open problems in this arena. Code is available at: https://github.com/Bond1995/Markov.

The design of reliable and efficient codes for channels with feedback remains a longstanding challenge in communication theory. While significant improvements have been achieved by leveraging deep learning techniques, neural codes often suffer from high computational costs, a lack of interpretability, and limited practicality in resource-constrained settings. We focus on designing low-complexity coding schemes that are interpretable and more suitable for communication systems. We advance both analytical and neural codes. First, we demonstrate that POWERBLAST, an analytical coding scheme inspired by Schalkwijk-Kailath (SK) and Gallager-Nakiboglu (GN) schemes, achieves notable reliability improvements over both SK and GN schemes, outperforming neural codes in high signal-to-noise ratio (SNR) regions. Next, to enhance reliability in low-SNR regions, we propose LIGHTCODE, a lightweight neural code that achieves state-of-the-art reliability while using a fraction of memory and compute compared to existing deep-learning-based codes. Finally, we systematically analyze the learned codes, establishing connections between LIGHTCODE and POWERBLAST, identifying components crucial for performance, and providing interpretation aided by linear regression analysis.

Polar codes, developed on the foundation of Arikan's polarization kernel, represent a breakthrough in coding theory and have emerged as the state-of-the-art error-correction-code in short-to-medium block length regimes. Importantly, recent research has indicated that the reliability of polar codes can be further enhanced by substituting Arikan's kernel with a larger one, leading to a faster polarization. However, for short-to-medium block length regimes, the development of polar codes that effectively employ large kernel sizes has not yet been realized. In this paper, we explore a novel, non-linear generalization of polar codes with an expanded kernel size, which we call DeepPolar codes. Our results show that DeepPolar codes effectively utilize the benefits of larger kernel size, resulting in enhanced reliability compared to both the existing neural codes and conventional polar codes.

In recent years, attention-based transformers have achieved tremendous success across a variety of disciplines including natural languages. A key ingredient behind their success is the generative pretraining procedure, during which these models are trained on a large text corpus in an auto-regressive manner. To shed light on this phenomenon, we propose a new framework that allows both theory and systematic experiments to study the sequential modeling capabilities of transformers through the lens of Markov chains. Inspired by the Markovianity of natural languages, we model the data as a Markovian source and utilize this framework to systematically study the interplay between the data-distributional properties, the transformer architecture, the learnt distribution, and the final model performance. In particular, we theoretically characterize the loss landscape of single-layer transformers and show the existence of global minima and bad local minima contingent upon the specific data characteristics and the transformer architecture. Backed by experiments, we demonstrate that our theoretical findings are in congruence with the empirical results. We further investigate these findings in the broader context of higher order Markov chains and deeper architectures, and outline open problems in this arena. Code is available at \url{https://github.com/Bond1995/Markov}.

Data-parallel SGD is the de facto algorithm for distributed optimization, especially for large scale machine learning. Despite its merits, communication bottleneck is one of its persistent issues. Most compression schemes to alleviate this either assume noiseless communication links, or fail to achieve good performance on practical tasks. In this paper, we close this gap and introduce LASER: LineAr CompreSsion in WirEless DistRibuted Optimization. LASER capitalizes on the inherent low-rank structure of gradients and transmits them efficiently over the noisy channels. Whilst enjoying theoretical guarantees similar to those of the classical SGD, LASER shows consistent gains over baselines on a variety of practical benchmarks. In particular, it outperforms the state-of-the-art compression schemes on challenging computer vision and GPT language modeling tasks. On the latter, we obtain $50$-$64 \%$ improvement in perplexity over our baselines for noisy channels.

Distributed source coding (DSC) is the task of encoding an input in the absence of correlated side information that is only available to the decoder. Remarkably, Slepian and Wolf showed in 1973 that an encoder without access to the side information can asymptotically achieve the same compression rate as when the side information is available to it. While there is vast prior work on this topic, practical DSC has been limited to synthetic datasets and specific correlation structures. Here we present a framework for lossy DSC that is agnostic to the correlation structure and can scale to high dimensions. Rather than relying on hand-crafted source modeling, our method utilizes a conditional Vector-Quantized Variational Autoencoder (VQ-VAE) to learn the distributed encoder and decoder. We evaluate our method on multiple datasets and show that our method can handle complex correlations and achieves state-of-the-art PSNR.

In decoding linear block codes, it was shown that noticeable reliability gains can be achieved by introducing learnable parameters to the Belief Propagation (BP) decoder. Despite the success of these methods, there are two key open problems. The first is the lack of interpretation of the learned weights, and the other is the lack of analysis for non-AWGN channels. In this work, we aim to bridge this gap by providing insights into the weights learned and their connection to the structure of the underlying code. We show that the weights are heavily influenced by the distribution of short cycles in the code. We next look at the performance of these decoders in non-AWGN channels, both synthetic and over-the-air channels, and study the complexity vs. performance trade-offs, demonstrating that increasing the number of parameters helps significantly in complex channels. Finally, we show that the decoders with learned weights achieve higher reliability than those with weights optimized analytically under the Gaussian approximation.