Many practical data analysis tasks reduce to learning, from observed samples, how a collection of variables depend on each other. A widely used approach is to fit a Gaussian graphical model, which represents the dependence structure as a graph connecting the variables. In a number of important applications, such as financial returns, gene co-expression, and climate or network analysis, the dependencies tend to be positive: variables move together rather than offset each other. Encoding this positivity through the constraint of multivariate total positivity of order two (MTP2) yields an attractive estimator that produces accurate fits with no tuning required. The resulting graphs are, however, typically much denser than the underlying ground-truth model, which makes them hard to interpret and slow to use in any downstream task that operates on the graph. In this work, we propose a novel highly-scalable approach for learning Gaussian graphical models from data using spectral sparsification; we call it Spectral-MTP2. Spectral graph sparsification is a fundamental method which aims to preserve meaningful properties of a dense graph with a sparser subgraph. We theoretically and empirically investigate and validate our method, and show that learning Gaussian Graphical Models under MTP2 using spectral sparsification preserves MTP2 and approximates well the original model in terms of Kullback-Leibler divergence and Gaussian log-likelihood. In simulations and applications to equity returns and gene expression, we observe that Spectral-MTP2 retains most of the fit quality of the denser MTP2 baseline, while producing substantially sparser and more interpretable graphs.

We use the expectation operator in different contexts in the proof. Here, we show the full derivation of the lower bound for negative mutual-information. We derive the lower bound for the general case where there are both observed and unobserved confounders. The VDE optimization involves the expectations of distributions with parameters with respect to a distribution that also has parameters. In our experiments, we let the control function be a categorical variable.

Weconsidertwoscenarios where causal effects are estimable.

In consequential domains, it is often impossible to compel individuals to take treatment, so that optimal policy rules are merely suggestions in the presence of human non-adherence to treatment recommendations. In these same domains, there may be heterogeneity both in who responds in taking-up treatment, and heterogeneity in treatment efficacy. For example, in social services, a persistent puzzle is the gap in take-up of beneficial services among those who may benefit from them the most. When in addition the decision-maker has distributional preferences over both access and average outcomes, the optimal decision rule changes. We study identification, doubly-robust estimation, and robust estimation under potential violations of positivity. We consider fairness constraints such as demographic parity in treatment take-up, and other constraints, via constrained optimization. Our framework can be extended to handle algorithmic recommendations under an often-reasonable covariate-conditional exclusion restriction, using our robustness checks for lack of positivity in the recommendation. We develop a two-stage, online learning-based algorithm for solving over parametrized policy classes under general constraints to obtain variance-sensitive regret bounds. We assess improved recommendation rules in a stylized case study of optimizing recommendation of supervised release in the PSA-DMF pretrial risk-assessment tool while reducing surveillance disparities.

LLM-based formal proof assistants (e.g., in Lean) hold great promise for automating mathematical discovery. But beyond syntactic correctness, do these systems truly understand mathematical structure as humans do? We investigate this question in context of mathematical inequalities -- specifically the prover's ability to recognize that the given problem simplifies by applying a known inequality such as AM/GM. Specifically, we are interested in their ability to do this in a compositional setting where multiple inequalities must be applied as part of a solution. We introduce Ineq-Comp, a benchmark built from elementary inequalities through systematic transformations, including variable duplication, algebraic rewriting, and multi-step composition. Although these problems remain easy for humans, we find that most provers -- including Goedel, STP, and Kimina-7B -- struggle significantly. DeepSeek-Prover-V2-7B shows relative robustness, but still suffers a 20% performance drop (pass@32). Even for DeepSeek-Prover-V2-671B model, the gap between compositional variants and seed problems exists, implying that simply scaling up the model size alone does not fully solve the compositional weakness. Strikingly, performance remains poor for all models even when formal proofs of the constituent parts are provided in context, revealing that the source of weakness is indeed in compositional reasoning. Our results expose a persisting gap between the generalization behavior of current AI provers and human mathematical intuition. All data and evaluation code can be found at https://github.com/haoyuzhao123/LeanIneqComp.

We present a risk-aware safety certification method for autonomous, learning enabled control systems. Focusing on two realistic risks, state/input delays and interval matrix uncertainty, we model the neural network (NN) controller with local sector bounds and exploit positivity structure to derive linear, delay-independent certificates that guarantee local exponential stability across admissible uncertainties. To benchmark performance, we adopt and implement a state-of-the-art IQC NN verification pipeline. On representative cases, our positivity-based tests run orders of magnitude faster than SDP-based IQC while certifying regimes the latter cannot-providing scalable safety guarantees that complement risk-aware control.

Causal inference relies on two fundamental assumptions: ignorability and positivity .

We thank all reviewers for their comments and acknowledgement of our contribution. Theorem 3 and Corollary 4, as Reviewer 3 suggested. How to choose the proper Bregman divergence? It is yet unclear whether there exist ways to systematically design the "best Bregman divergence in a (k 1) This is also commonly adopted in the literature. Is continuity of the intensity function restrictive?