The Retriever Should Remember: Experience-Amortized Reranking for Long-Term Agent Memory

Long-term language-model agents accumulate memories across interactions, but their retrievers typically do not accumulate retrieval experience. Semantic retrieval is efficient, but embedding similarity does not always reflect whether a memory contains evidence relevant to the current query. Large language model (LLM) rerankers provide stronger query-conditioned relevance scores, yet stateless reranking repeatedly scores a large candidate pool and discards these scores after each query. We introduce EARM, an experience-amortized reranking framework that treats previously acquired LLM relevance scores as reusable retrieval experience. EARM stores sparse query--memory relevance scores in an online matrix, learns their shared structure through causal matrix completion, and combines a small set of newly observed scores with estimated scores to rerank the remaining candidates. The scoring budget decreases as experience accumulates, changing LLM reranking from a repeated per-query expense into a retrieval capability learned over an agent's lifetime. Experiments on long-term conversational memory show that mixed observed-and-estimated reranking improves answer accuracy over semantic retrieval by up to 6.62% and remains effective when only 17.5% of candidates receive direct LLM relevance scores, thereby substantially reducing the inference overhead of LLM reranking. These results motivate a broader view of agent memory: a long-lived agent should remember not only past content, but also how that content has proved useful for retrieval.

Subzero matrix completion for sparse data analysis: large-scale learning of latent low-rank structure

We investigate when a sparse nonnegative matrix can be recovered from a real-valued matrix of much lower rank by zeroing out its negative elements. The potential for such decompositions suggests a mathematical connection between sparsity and rank; we analyze a number of sparse matrices with this latent low-rank structure and use them to illustrate the geometric origins of this connection. Previous algorithms have discovered these decompositions via an alternating minimization over the factors of a low-rank matrix, but to do so, they have also needed to compute and store another matrix, neither sparse nor low-rank, that is the size of their product. We develop a stochastic, alternating least-squares algorithm that operates on smaller blocks of this dense matrix and scales as a result to much larger problems. We also show how to further accelerate this algorithm with sparse optimizations and customized CUDA kernels. As one example, we use the algorithm to analyze the sparse matrix of synaptic weights for the recently published $\textit{Drosphilia}$ connectome. The nonzero elements of this matrix, with 139,255 rows and columns, record the number of synapses between cells in the nervous system of a female fruit fly. Despite a slowly decaying spectrum of singular values, this matrix exhibits a latent low-rank structure that is predictive of cell categories across multiple levels of specificity.

Graph-Based Correlation Matrix Generation: A Convex Optimization Approach

This work addresses the generation of theoretical correlation matrices with prescribed sparsity patterns associated to graph structures. We propose a novel convex optimization framework in which an initial matrix is projected onto an elliptope under a positive semidefiniteness constraint. Several numerical schemes are implemented and compared. The problem falls within the broader class of matrix completion, where off-diagonal entries corresponding to absent edges are fixed to zero and diagonal entries are fixed to one. Beyond this structural constraint, the approach offers greater flexibility than existing methods by allowing control over the mean of the off-diagonal entry distribution, enabling the generation of correlation matrices that better reflect realistic data. This procedure is not designed to yield a uniform distribution over the feasible set; rather, it provides a principled and tunable way to construct correlation matrices suitable for benchmarking statistical methods for graphical model inference. Theoretical guarantees on the existence of solutions are established, both in the general setting and under the additional mean constraint. Simulation studies illustrate the properties of the generated matrices with respect to graph structure. The methodology is applied to two real-world datasets from neuroscience and finance, and a comparison with GAN-based correlation matrix generation is provided.

Predicting Activities in Aqueous Electrolyte Solutions with Hybrid Machine Learning

Activities in aqueous electrolyte solutions, usually described by ionic activity and osmotic coefficients, are important properties for modeling many processes in industry and nature. Established activity models, such as those of Pitzer or Bromley, require fitting to experimental data for each electrolyte of interest and thus cannot predict properties for unstudied systems. While some predictive approaches exist, they are typically limited in scope and rely on additional ion-specific descriptors. In this work, we introduce a new hybrid model that combines the physics-based Bromley model with a matrix completion method (MCM) from machine learning. The MCM is employed to predict the electrolyte-specific parameters of the Bromley model, exploiting the fact that these parameters can be arranged in a matrix with cations and anions as rows and columns, respectively. Due to the lack of experimental data for many electrolytes, the initial parameter matrix is sparsely populated, making the prediction of the Bromley parameters for unstudied electrolytes a matrix completion problem. The hybrid model, Bromley-MCM, was trained end-to-end on experimental data for mean ionic activity coefficients and osmotic coefficients of aqueous solutions of 478 electrolytes at 298 K from the Dortmund Data Bank. As output, we obtain a completed matrix of Bromley parameters for 83 cations and 112 anions, enabling consistent prediction of concentration-dependent activities in aqueous solutions of 9,296 electrolytes at 298~K. This substantially extends the applicability of the Bromley model while maintaining high predictive accuracy, as demonstrated through evaluations on electrolytes excluded from model training.

Graph-Regularized Low-Rank Matrix Completion by Variable Projection

We address the low-rank matrix completion problem by incorporating graph regularization into the existing Riemannian Trust-Region Matrix Completion (RTRMC) framework. The latter uses the geometry of the low-rank constraint to remodel the problem as an unconstrained optimization problem on a single Grassmann manifold. Our approach, named Graph-Regularized RTRMC (GR-RTRMC), exploits the inherent relationships between rows and columns of the matrix. By using these relationships, we aim to improve the accuracy and robustness of matrix completion, particularly in scenarios where the underlying data exhibits strong correlations between rows or columns.

Latent Structural Categorical Matrix Completion with Application to Quasispecies Analysis

Matrix completion has been extensively studied for real-valued data, but existing methods are often limited in handling categorical variables. We propose LCMC, a double-loop optimization framework for categorical matrix completion via latent factorization based on a binary tensor representation. In this setting, each categorical entry is encoded as a one-hot vector along a third tensor mode, thereby preserving its discrete, non-ordinal nature. The outer loop adaptively estimates the latent dimension by iteratively updating it with feedback from the inner loop, while the inner loop reconstructs the categorical matrix through tensor factorization, supported by a corresponding theoretical analysis. To further improve scalability and robustness, we introduce enhancements including a split-merge-refine strategy and an adaptive data reduction technique. Experiments on synthetic and real-world datasets in viral quasispecies reconstruction, demonstrate that LCMC achieves superior accuracy and efficiency compared to existing methods.

Low-rank Distributional Matrix Completion

We study a distributional generalization of the matrix completion problem in which each entry of the target matrix is a probability distribution rather than a scalar. In this setting, only a subset of matrix entries is observed, and even for observed entries, the underlying distributions are not directly accessible; instead, we observe finitely many samples drawn from them. To represent distributional entries, we employ kernel mean embeddings and introduce a notion of Tucker rank for distribution-valued matrices to capture their low-rank structure. The infinite-dimensional nature of kernel embeddings poses significant methodological challenges. To address this, we introduce functional unfolding operators that link the proposed distributional low-rank structure to the classical Tucker rank for finite-dimensional tensors. Based on this framework, we propose a novel estimator for distributional matrix completion. We establish non-asymptotic error bounds that characterize the statistical performance of the estimator. Extensive experiments on synthetic data and a real-world application demonstrate the effectiveness of the proposed method.

Rank-Constrained Deep Matrix Completion for Group Recommendation

The growing popularity of group activities has increased the need for methods that provide recommendations to groups of users given their individual preferences. Many existing group recommender systems rely on aggregating individual user preferences, but they often struggle with high-dimensional and highly sparse rating data commonly found in real-world scenarios. We propose Group Rank-Constrained Deep Matrix Completion (Group RC-DMC), a novel framework that extends RC-DMC by integrating group-level representation learning via a Set-Transformer aggregator, jointly leveraging low-rank structure and attention-based nonlinear modeling. Unlike most existing group recommender systems, Group RC-DMC unifies explicit low-rank regularization, linear encoder-decoder architectures, and attention-based nonlinear group modeling within a single framework, yielding accurate predictions at both the individual and group levels. Group RC-DMC addresses data sparsity through low-rank matrix completion, computing per-user latent representations from observed ratings only, and enforcing a rank constraint on the latent space using a nuclear-norm proximal step based on periodic singular value thresholding. The decoder is parametrized as a low-rank factorization, enabling efficient inference. Experimental results on the MovieLens and Goodbooks datasets demonstrate that Group RC-DMC achieves superior reconstruction accuracy, measured by lower group RMSE, while remaining computationally efficient and competitive in group-level performance in terms of precision, recall, and F1 score compared with weighted-before-factorization (WBF) and after-factorization (AF) baselines. The results highlight the model's ability to recover the underlying low-rank structure of user-item interactions and provide robust group recommendations across small, medium, and large user groups.

Active multiple matrix completion with adaptive confidence sets

In this work, we formulate a new multi-task active learning setting in which the learner's goal is to solve multiple matrix completion problems simultaneously. At each round, the learner can choose from which matrix it receives a sample from an entry drawn uniformly at random. Our main practical motivation is market segmentation, where the matrices represent different regions with different preferences of the customers. The challenge in this setting is that each of the matrices can be of a different size and also of a different rank which is unknown. We provide and analyze a new algorithm, MAlocate that is able to adapt to the unknown ranks of the different matrices. We then give a lower-bound showing that our strategy is minimax-optimal and demonstrate its performance with synthetic experiments.