Symbolic Density Estimation: A Decompositional Approach

We introduce AI-Kolmogorov, a novel framework for Symbolic Density Estimation (SymDE). Symbolic regression (SR) has been effectively used to produce interpretable models in standard regression settings but its applicability to density estimation tasks has largely been unexplored. To address the SymDE task we introduce a multi-stage pipeline: (i) problem decomposition through clustering and/or probabilistic graphical model structure learning; (ii) nonparametric density estimation; (iii) support estimation; and finally (iv) SR on the density estimate. We demonstrate the efficacy of AI-Kolmogorov on synthetic mixture models, multivariate normal distributions, and three exotic distributions, two of which are motivated by applications in high-energy physics. We show that AI-Kolmogorov can discover underlying distributions or otherwise provide valuable insight into the mathematical expressions describing them.

FrontierCS: Evolving Challenges for Evolving Intelligence

We introduce FrontierCS, a benchmark of 156 open-ended problems across diverse areas of computer science, designed and reviewed by experts, including CS PhDs and top-tier competitive programming participants and problem setters. Unlike existing benchmarks that focus on tasks with known optimal solutions, FrontierCS targets problems where the optimal solution is unknown, but the quality of a solution can be objectively evaluated. Models solve these tasks by implementing executable programs rather than outputting a direct answer. FrontierCS includes algorithmic problems, which are often NP-hard variants of competitive programming problems with objective partial scoring, and research problems with the same property. For each problem we provide an expert reference solution and an automatic evaluator. Combining open-ended design, measurable progress, and expert curation, FrontierCS provides a benchmark at the frontier of computer-science difficulty. Empirically, we find that frontier reasoning models still lag far behind human experts on both the algorithmic and research tracks, that increasing reasoning budgets alone does not close this gap, and that models often over-optimize for generating merely workable code instead of discovering high-quality algorithms and system designs.

Neural Symbolic Regression using Control Variables

Symbolic regression (SR) is a powerful technique for discovering the analytical mathematical expression from data, finding various applications in natural sciences due to its good interpretability of results. However, existing methods face scalability issues when dealing with complex equations involving multiple variables. To address this challenge, we propose SRCV, a novel neural symbolic regression method that leverages control variables to enhance both accuracy and scalability. The core idea is to decompose multi-variable symbolic regression into a set of single-variable SR problems, which are then combined in a bottom-up manner. The proposed method involves a four-step process. First, we learn a data generator from observed data using deep neural networks (DNNs). Second, the data generator is used to generate samples for a certain variable by controlling the input variables. Thirdly, single-variable symbolic regression is applied to estimate the corresponding mathematical expression. Lastly, we repeat steps 2 and 3 by gradually adding variables one by one until completion. We evaluate the performance of our method on multiple benchmark datasets. Experimental results demonstrate that the proposed SRCV significantly outperforms state-of-the-art baselines in discovering mathematical expressions with multiple variables. Moreover, it can substantially reduce the search space for symbolic regression. The source code will be made publicly available upon publication.