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Linear representation learning is widely studied due to its conceptual simplicity and empirical utility in tasks such as compression, classification, and feature extraction. Given a set of points $$[\x_1, \x_2, \ldots, \x_n] = \X \in \R^{d \times n}$$ and a vector $$\y \in \R^d$$, the goal is to find coefficients $$\w \in \R^n$$ so that $$\X \w \approx \y$$, subject to some desired structure on $$\w$$. In this work we seek $$\w$$ that forms a local reconstruction of $$\y$$ by solving a regularized least squares regression problem. We obtain local solutions through a locality function that promotes the use of columns of $$\X$$ that are close to $$\y$$ when used as a regularization term. We prove that, for all levels of regularization and under a mild condition that the columns of $$\X$$ have a unique Delaunay triangulation, the optimal coefficients' number of non-zero entries is upper bounded by $d+1$, thereby providing local sparse solutions when $$d \ll n$$. Under the same condition we also show that for any $$\y$$ contained in the convex hull of $$\X$$ there exists a regime of regularization parameter such that the optimal coefficients are supported on the vertices of the Delaunay simplex containing $$\y$$. This provides an interpretation of the sparsity as having structure obtained implicitly from the Delaunay triangulation of $$\X$$. We demonstrate that our locality regularized problem can be solved in comparable time to other methods that identify the containing Delaunay simplex. 

Background Team leadership during medical emergencies like cardiac arrest resuscitation is cognitively demanding, especially for trainees. These cognitive processes remain poorly characterized due to measurement challenges. Using virtual reality simulation, this study aimed to elucidate and compare communication and cognitive processes-such as decision-making, cognitive load, perceived pitfalls, and strategies-between expert and novice code team leaders to inform strategies for accelerating proficiency development. Methods A simulation-based mixed methods approach was utilized within a single large academic medical center, involving twelve standardized virtual reality cardiac arrest simulations. These 10- to 15-minutes simulation sessions were performed by seven experts and five novices. Following the simulations, a cognitive task analysis was conducted using a cued-recall protocol to identify the challenges, decision-making processes, and cognitive load experienced across the seven stages of each simulation. Results The analysis revealed 250 unique cognitive processes. In terms of reasoning patterns, experts used inductive reasoning, while novices tended to use deductive reasoning, considering treatments before assessments. Experts also demonstrated earlier consideration of potential reversible causes of cardiac arrest. Regarding team communication, experts reported more critical communications, with no shared subthemes between groups. Experts identified more teamwork pitfalls, and suggested more strategies compared to novices. For cognitive load, experts reported lower median cognitive load (53) compared to novices (80) across all stages, with the exception of the initial presentation phase. Conclusions The identified patterns of expert performance — superior teamwork skills, inductive clinical reasoning, and distributed cognitive strategiesn — can inform training programs aimed at accelerating expertise development.