Nonnegative matrix factorization (NMF) approximates a nonnegative matrix, $X$, by the product of two nonnegative factors, $WH$, where $W$ has $r$ columns and $H$ has $r$ rows. In this paper, we consider NMF using the component-wise L1 norm as the error measure (L1-NMF), which is suited for data corrupted by heavy-tailed noise, such as Laplace noise or salt and pepper noise, or in the presence of outliers. Our first contribution is an NP-hardness proof for L1-NMF, even when $r=1$, in contrast to the standard NMF that uses least squares. Our second contribution is to show that L1-NMF strongly enforces sparsity in the factors for sparse input matrices, thereby favoring interpretability. However, if the data is affected by false zeros, too sparse solutions might degrade the model. Our third contribution is a new, more general, L1-NMF model for sparse data, dubbed weighted L1-NMF (wL1-NMF), where the sparsity of the factorization is controlled by adding a penalization parameter to the entries of $WH$ associated with zeros in the data. The fourth contribution is a new coordinate descent (CD) approach for wL1-NMF, denoted as sparse CD (sCD), where each subproblem is solved by a weighted median algorithm. To the best of our knowledge, sCD is the first algorithm for L1-NMF whose complexity scales with the number of nonzero entries in the data, making it efficient in handling large-scale, sparse data. We perform extensive numerical experiments on synthetic and real-world data to show the effectiveness of our new proposed model (wL1-NMF) and algorithm (sCD).

The poster child of the AI boom, valued at $850bn, needs to show strategic discipline after'casting its net too wide' If OpenAI is going to float this year, it has to get serious about its business model. The wow factor around the US company - the poster child of an AI industry boom that has stoked fears of a stock market bubble - has been long established, but when will the profits come? The developer of ChatGPT is one of the biggest startups in the world and is now valued at $850bn (£645bn). Meanwhile, it is reportedly spending $600bn on infrastructure (the amount it invests in datacentres and chips to power its AI models) by 2030. At least this is a reduction on an initial estimate of $1.4tn .

Online health communities (OHCs) are vital for fostering peer support and improving health outcomes. Support groups within these platforms can provide more personalized and cohesive peer support, yet traditional support group formation methods face challenges related to scalability, static categorization, and insufficient personalization. To overcome these limitations, we propose two novel machine learning models for automated support group formation: the Group specific Dirichlet Multinomial Regression (gDMR) and the Group specific Structured Topic Model (gSTM). These models integrate user generated textual content, demographic profiles, and interaction data represented through node embeddings derived from user networks to systematically automate personalized, semantically coherent support group formation. We evaluate the models on a large scale dataset from MedHelp, comprising over 2 million user posts. Both models substantially outperform baseline methods including LDA, DMR, and STM in predictive accuracy (held out log likelihood), semantic coherence (UMass metric), and internal group consistency. The gDMR model yields group covariates that facilitate practical implementation by leveraging relational patterns from network structures and demographic data. In contrast, gSTM emphasizes sparsity constraints to generate more distinct and thematically specific groups. Qualitative analysis further validates the alignment between model generated groups and manually coded themes, showing the practical relevance of the models in informing groups that address diverse health concerns such as chronic illness management, diagnostic uncertainty, and mental health. By reducing reliance on manual curation, these frameworks provide scalable solutions that enhance peer interactions within OHCs, with implications for patient engagement, community resilience, and health outcomes.

This study addresses the inverse problem of parameter estimation for Stochastic Differential Equations (SDEs) by minimizing a regularized discrepancy functional via Stochastic Gradient Descent (SGD). To achieve computational efficiency, we leverage the Wiener Chaos Expansion (WCE), a spectral decomposition technique that projects the stochastic solution onto an orthogonal basis of Hermite polynomials. This transformation effectively maps the stochastic dynamics into a hierarchical system of deterministic functions, termed the \textit{propagator}. By reducing the stochastic inference task to a deterministic optimization problem, our framework circumvents the heavy computational burden and sampling requirements of traditional simulation-based methods like MCMC or MLE. The robustness and scalability of the proposed approach are demonstrated through numerical experiments on various non-linear SDEs, including models for individual biological growth. Results show that the WCE-SGD framework provides accurate parameter recovery even from discrete, noisy observations, offering a significant paradigm shift in the efficient modeling of complex stochastic systems.

We study stochastic multi-armed bandits under simultaneous constraints on space and adaptivity: the learner interacts with the environment in $B$ batches and has only $W$ bits of persistent memory. Prior work shows that each constraint alone is surprisingly mild: near-minimax regret $\widetilde{O}(\sqrt{KT})$ is achievable with $O(\log T)$ bits of memory under fully adaptive interaction, and with a $K$-independent $O(\log\log T)$-type number of batches when memory is unrestricted. We show that this picture breaks down in the simultaneously constrained regime. We prove that any algorithm with a $W$-bit memory constraint must use at least $Ω(K/W)$ batches to achieve near-minimax regret $\widetilde{O}(\sqrt{KT})$, even under adaptive grids. In particular, logarithmic memory rules out $O(K^{1-\varepsilon})$ batch complexity. Our proof is based on an information bottleneck. We show that near-minimax regret forces the learner to acquire $Ω(K)$ bits of information about the hidden set of good arms under a suitable hard prior, whereas an algorithm with $B$ batches and $W$ bits of memory allows only $O(BW)$ bits of information. A key ingredient is a localized change-of-measure lemma that yields probability-level arm exploration guarantees, which is of independent interest. We also give an algorithm that, for any bit budget $W$ with $Ω(\log T) \le W \le O(K\log T)$, uses at most $W$ bits of memory and $\widetilde{O}(K/W)$ batches while achieving regret $\widetilde{O}(\sqrt{KT})$, nearly matching our lower bound up to polylogarithmic factors.

Electroencephalography (EEG)-based emotion recognition suffers from severe performance degradation when models are transferred across heterogeneous datasets due to physiological variability, experimental paradigm differences, and device inconsistencies. Existing domain adversarial methods primarily enforce global marginal alignment and often overlook class-conditional mismatch and decision boundary distortion, limiting cross-corpus generalization. In this work, we propose a unified Prototype-driven Adversarial Alignment (PAA) framework for cross-corpus EEG emotion recognition. The framework is progressively instantiated in three configurations: PAA-L, which performs prototype-guided local class-conditional alignment; PAA-C, which further incorporates contrastive semantic regularization to enhance intra-class compactness and inter-class separability; and PAA-M, the full boundary-aware configuration that integrates dual relation-aware classifiers within a three-stage adversarial optimization scheme to explicitly refine controversial samples near decision boundaries. By combining prototype-guided subdomain alignment, contrastive discriminative enhancement, and boundary-aware aggregation within a coherent adversarial architecture, the proposed framework reformulates emotion recognition as a relation-driven representation learning problem, reducing sensitivity to label noise and improving cross-domain stability. Extensive experiments on SEED, SEED-IV, and SEED-V demonstrate state-of-the-art performance under four cross-corpus evaluation protocols, with average improvements of 6.72\%, 5.59\%, 6.69\%, and 4.83\%, respectively. Furthermore, the proposed framework generalizes effectively to clinical depression identification scenarios, validating its robustness in real-world heterogeneous settings. The source code is available at \textit{https://github.com/WuCB-BCI/PAA}

This technical note studies the reliability limits of LLM-based multi-agent planning as a delegated decision problem. We model the LLM-based multi-agent architecture as a finite acyclic decision network in which multiple stages process shared model-context information, communicate through language interfaces with limited capacity, and may invoke human review. We show that, without new exogenous signals, any delegated network is decision-theoretically dominated by a centralized Bayes decision maker with access to the same information. In the common-evidence regime, this implies that optimizing over multi-agent directed acyclic graphs under a finite communication budget can be recast as choosing a budget-constrained stochastic experiment on the shared signal. We also characterize the loss induced by communication and information compression. Under proper scoring rules, the gap between the centralized Bayes value and the value after communication admits an expected posterior divergence representation, which reduces to conditional mutual information under logarithmic loss and to expected squared posterior error under the Brier score. These results characterize the fundamental reliability limits of delegated LLM planning. Experiments with LLMs on a controlled problem set further demonstrate these characterizations.

We study the effects of center initialization on the performance of a family of distributed gradient-based clustering algorithms introduced in [1], that work over connected networks of users. In the considered scenario, each user contains a local dataset and communicates only with its immediate neighbours, with the aim of finding a global clustering of the joint data. We perform extensive numerical experiments, evaluating the effects of center initialization on the performance of our family of methods, demonstrating that our methods are more resilient to the effects of initialization, compared to centralized gradient clustering [2]. Next, inspired by the $K$-means++ initialization [3], we propose a novel distributed center initialization scheme, which is shown to improve the performance of our methods, compared to the baseline random initialization.

We propose a control-theoretic framework for evolutionary clustering based on Mean Field Games (MFG). Moving beyond static or heuristic approaches, we formulate the problem as a population dynamics game governed by a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system. Driven by a variational cost functional rather than predefined statistical shapes, this continuous-time formulation provides a flexible basis for non-parametric cluster evolution. To validate the framework, we analyze the setting of time-dependent Gaussian mixtures, showing that the MFG dynamics recover the trajectories of the classical Expectation-Maximization (EM) algorithm while ensuring mass conservation. Furthermore, we introduce time-averaged log-likelihood functionals to regularize temporal fluctuations. Numerical experiments illustrate the stability of our approach and suggest a path toward more general non-parametric clustering applications where traditional EM methods may face limitations.

We revisit the standard perturbation-based approach of Abernethy et al. (2008) in the context of unconstrained Bandit Linear Optimization (uBLO). We show the surprising result that in the unconstrained setting, this approach effectively reduces Bandit Linear Optimization (BLO) to a standard Online Linear Optimization (OLO) problem. Our framework improves on prior work in several ways. First, we derive expected-regret guarantees when our perturbation scheme is combined with comparator-adaptive OLO algorithms, leading to new insights about the impact of different adversarial models on the resulting comparator-adaptive rates. We also extend our analysis to dynamic regret, obtaining the optimal $\sqrt{P_T}$ path-length dependencies without prior knowledge of $P_T$. We then develop the first high-probability guarantees for both static and dynamic regret in uBLO. Finally, we discuss lower bounds on the static regret, and prove the folklore $Ω(\sqrt{dT})$ rate for adversarial linear bandits on the unit Euclidean ball, which is of independent interest.