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Improvements in language models are often driven by improving the quality of the data we train them on, which can be limiting when strong supervision is scarce. In this work, we show that paired preference data consisting of individually weak data points can enable gains beyond the strength of each individual data point. We formulate the delta learning hypothesis to explain this phenomenon, positing that the relative quality delta between points suffices to drive learning via preference tuning--even when supervised finetuning on the weak data hurts. We validate our hypothesis in controlled experiments and at scale, where we post-train 8B models on preference data generated by pairing a small 3B model's responses with outputs from an even smaller 1.5B model to create a meaningful delta. Strikingly, on a standard 11-benchmark evaluation suite (MATH, MMLU, etc.), our simple recipe matches the performance of Tulu 3, a state-of-the-art open model tuned from the same base model while relying on much stronger supervisors (e.g., GPT-4o). Thus, delta learning enables simpler and cheaper open recipes for state-of-the-art post-training. To better understand delta learning, we prove in logistic regression that the performance gap between two weak teacher models provides useful signal for improving a stronger student. Overall, our work shows that models can learn surprisingly well from paired data that might typically be considered weak. 

This article introduces a novel concatenated coding scheme called sparse regression LDPC (SR-LDPC) codes. An SR-LDPC code consists of an outer non-binary LDPC code and an inner sparse regression code (SPARC), whose respective field size and section sizes are equal. For such codes, an efficient decoding algorithm is proposed based on approximate message passing (AMP) that dynamically shares soft information between inner and outer decoders. This dynamic exchange of information is facilitated by a denoiser that runs belief propagation (BP) on the factor graph of the outer LDPC code within each AMP iteration. It is shown that this BP denoiser falls within the framework of non-separable denoising functions and subsequently, that state evolution holds for the proposed AMP-BP algorithm. Leveraging the rich structure of SR-LDPC codes, this article proposes an efficient low-dimensional approximate state evolution recursion that can be used for efficient hyperparameter tuning, thus paving the way for future work on optimal code design. Finally, numerical simulations demonstrate that SR-LDPC codes outperform contemporary codes over the AWGN channel for parameters of practical interest. SR-LDPC codes are shown to be viable means for obtaining shaping gains over the AWGN channel. 

Novel sparse regression LDPC (SR-LDPC) codes exhibit excellent performance over additive white Gaussian noise (AWGN) channels in part due to their natural provision of shaping gains. Though SR-LDPC-like codes have been considered within the context of single-user error correction and massive random access, they are yet to be examined as candidates for coordinated multi-user communication scenarios. This article explores this gap in the literature and demonstrates that SR-LDPC codes, when combined with coded demixing techniques, offer a new framework for efficient non-orthogonal multiple access (NOMA) in the context of coordinated multi-user communication channels. The ensuing communication scheme is referred to as MU-SR-LDPC coding. Empirical evidence suggests that MU-SR-LDPC coding can increase the sum-rate for a fixed Eb/N0 when compared to orthogonal multiple access (OMA) techniques such as time division multiple access (TDMA) or frequency division multiple access (FDMA). Importantly, MU-SR-LDPC coding enables a pragmatic solution path for user-centric cell-free communication systems with (local) joint decoding. Results are supported by numerical simulations. 

This study critically appraises employing chitosan as a composite with bentonite, biochar, or both materials as an alternative to conventional barrier materials. A comprehensive literature review was conducted to identify the studies reporting chitosan-bentonite composite (CBC), chitosan amended biochar (CAB), and chitosan-bentonite-biochar composite (CBBC) for effective removal of various contaminants. The study aims to review the synthesis of these composites, identify fundamental properties affecting their adsorption capacities, and examine how these properties affect or enhance the removal abilities of other materials within the composite. Notably, CBC composites have the advantage of adsorbing both cationic and anionic species, such as heavy metals and dyes, due to the cationic nature of chitosan and the anionic nature of montmorillonite, along with the increased accessible surface area due to the clay. CAB composites have the unique advantage of being low-cost sorbents with high specific surface area, affinity for a wide range of contaminants owing to the high surface area and microporosity of biochar, and abundant available functional groups from the chitosan. Limited studies have reported the utilization of CBBC composites to remove various contaminants. These composites can be prepared by combining the steps employed in preparing CBC and CAB composites. They can benefit from the favorable adsorption properties of all three materials while also satisfying the mechanical requirements of a barrier material. This study serves as a knowledge base for future research to develop novel composite barrier materials by incorporating chitosan and biochar as amendments to bentonite. 

Containment barrier systems, such as vertical slurry walls and low-permeable liners in waste containment systems, are commonly used to prevent groundwater contamination. However, traditional low-permeable clays used in these barriers have limitations in effectively removing various contaminants, including phosphate, which is a contaminant of global concern. The overarching goal of this work is to create a novel chitosan-bentonite composite barrier for improving the performance of containment systems. Chitosan, a material derived by deacetylating chitin, is a promising barrier material due to its ability to adsorb various contaminants. The purpose of this study is to investigate incorporating chitosan into these barriers to enhance their contaminant adsorption capacity. Previous studies were performed on three chitosans with varying degree of deacetylation (DOD) and molecular weights (MW) and one type of bentonite. The current study presents results from batch tests on four additional chitosan materials and a different source of bentonite. These tests assessed their individual phosphate removal capabilities and were compared with earlier findings. The chitosans exhibited varying phosphate removal efficiencies based on DOD, MW, surface area, and source. The highest removal efficiency ranging from 20.9% to 85.6%, at different initial phosphate concentrations, was achieved by one of the chitosan variants. In contrast, bentonite achieved 15.3% to 41.6% removal at different phosphate concentrations. Results suggest a composite material of chitosan and bentonite in engineered barriers could significantly enhance phosphate removal, especially at lower concentrations (0.5 mg/l), compared to a simple bentonite-based barrier. 

Excessive levels of phosphate in stormwater runoff can negatively impact receiving surface water bodies, such as retention ponds, and may also seep into groundwater. Liner systems composed of materials with greater phosphate selectivity have the potential to mitigate infiltration and eliminate phosphate. One potential material is chitosan, an abundant naturally occurring biopolymer. This study evaluated five materials for their ability to remove phosphate from synthetic stormwater using batch tests with different initial phosphate concentrations ranging from 0.5 to 12 mg/L and a fixed 24-h exposure time. The materials included two types of clayey soils (kaolin and bentonite) and three different varieties of chitosan with varying molecular weights (low, medium, and high). The phosphate removal efficiency of kaolin was found to be the highest, with efficiencies ranging from 100% to 56% at different concentrations, while bentonite was found to be the least effective, with removal efficiencies ranging from 40% to 12%. The removal efficiencies of all three types of chitosans analyzed were higher than those of bentonite but lower than those of kaolin. The removal efficiencies ranged from 77% to 19% for low-molecular-weight chitosan, 84% to 31% for medium-molecular-weight chitosan, and 55% to 18% for high-molecular-weight chitosan. The removal mechanism of phosphate by kaolin and bentonite was attributed to surface adsorption and precipitation. In chitosan, the likely mechanism is electrostatic attraction. The maximum adsorption capacity for kaolin was not reached under the tested phosphate concentration range, indicating potential adsorption sites remained available on the particle surfaces. The results for bentonite, low-molecular-weight chitosan, and high-molecular-weight chitosan showed that these materials nearly reached their maximum adsorption capacities, indicating that fewer adsorption sites were remaining. The Langmuir adsorption isotherm was found to be the best-fit model for phosphate adsorption in all the materials tested compared to the Freundlich isotherm. According to the Langmuir model, the maximum adsorption capacities for kaolin, bentonite, low-molecular-weight chitosan, medium-molecular-weight chitosan, and high-molecular-weight chitosan were found to be 140.85, 33, 48.78, 82.64, and 51.28 mg/kg, respectively.