General intelligence requires systems that acquire new skills efficiently and generalize beyond their training distributions. Although program synthesis approaches have strong generalization power, they face scaling issues due to the large combinatorial spaces that quickly render them impractical, requiring human-generated DSLs or pre-trained priors to narrow this search space. On the other hand, deep learning methods have had high successes, but they lack structured test-time adaptation and rely on heavy stochastic sampling or expensive gradient updates for fine-tuning. In this work, we propose the Latent Program Network (LPN), a novel architecture that builds in test-time search directly into neural models. LPN learns a latent space of implicit programs--neurally mapping inputs to outputs--through which it can search using gradients at test time.

Breast cancer remains one of the leading causes of mortality among women worldwide, with early diagnosis being critical for effective treatment and improved survival rates. However, timely detection continues to be a challenge due to the complex nature of the disease and variability in patient risk factors . This study presents a fuzzy soft set theory - based expert system designed to assess the risk of breast cancer in patients using measurable clinical and physiological parameters. The proposed system integrates Body Mass Index (BMI), Insulin Level (IL), Lep tin Level (LL), Adiponectin Level (AL), and age as input variables to estimate breast cancer risk through a set of fuzzy inference rules and soft set computations. These parameters can be obtained from routine blood analyses, enabling a non - invasive and ac cessible method for preliminary assessment. The dataset used for model development and validation was obtained from the UCI Machine Learning Repository. The proposed expert system aims to support healthcare professionals in identifying high - risk patients a nd determining the necessity of further diagnostic procedures such as biopsies.

We study the problem of learning junta distributions on $\{0, 1\}^n$, where a distribution is a $k$-junta if its probability mass function depends on a subset of at most $k$ variables. We make two main contributions: - We show that learning $k$-junta distributions is \emph{computationally} equivalent to learning $k$-parity functions with noise (LPN), a landmark problem in computational learning theory. - We design an algorithm for learning junta distributions whose statistical complexity is optimal, up to polylogarithmic factors. Computationally, our algorithm matches the complexity of previous (non-sample-optimal) algorithms. Combined, our two contributions imply that our algorithm cannot be significantly improved, statistically or computationally, barring a breakthrough for LPN.

This paper introduces the Primender sequence, a novel integer sequence defined by a hybrid rule that combines classical primality with modular digit-based conditions. Specifically, a number n is included in the sequence if it is prime or ends with a prime number of unit digit or any length. In other words, numbers which are primes or have at least one prime suffix. The resulting sequence exhibits a deterministic yet non-trivial structure, blending number-theoretic properties with symbolic patterning. We propose the Primender sequence as a benchmark for evaluating the symbolic reasoning capabilities of Large Language Models (LLMs). The study is motivated by the need for interpretable, rule-based testbeds that can assess an LLM's ability to infer hidden rules, validate mathematical hypotheses, and generalize symbolic logic at scale. A key hypothesis explored is: Whenever a number in the Primender sequence is exactly one more than the largest prime less than or equal to it, the difference between it and the previous number in the sequence is also 1. We design a structured prompt and evaluation framework to test this hypothesis across multiple state-of-the-art LLMs, including ChatGPT, Copilot, DeepSeek, Gemini, Grok, and LLaMA. The models are tasked with identifying the underlying rule, validating the hypothesis, and generating the next 100,000 terms of the sequence. Comparative metrics such as rule inference accuracy, hypothesis evaluation, sequence validity, and symbolic explanation quality are used to assess model performance. This work contributes a novel mathematical construct and a reproducible methodology for benchmarking LLMs in symbolic reasoning, hypothesis testing, and scalable pattern generalization - bridging the domains of number theory, artificial intelligence, and software engineering.

Program synthesis methods aim to automatically generate programs restricted to a language that can explain a given specification of input-output pairs. While purely symbolic approaches suffer from a combinatorial search space, recent methods leverage neural networks to learn distributions over program structures to narrow this search space significantly, enabling more efficient search. However, for challenging problems, it remains difficult to train models to perform program synthesis in one shot, making test-time search essential. Most neural methods lack structured search mechanisms during inference, relying instead on stochastic sampling or gradient updates, which can be inefficient. In this work, we propose the Latent Program Network (LPN), a general algorithm for program induction that learns a distribution over latent programs in a continuous space, enabling efficient search and test-time adaptation. We explore how to train these networks to optimize for test-time computation and demonstrate the use of gradient-based search both during training and at test time. We evaluate LPN on ARC-AGI, a program synthesis benchmark that evaluates performance by generalizing programs to new inputs rather than explaining the underlying specification. We show that LPN can generalize beyond its training distribution and adapt to unseen tasks by utilizing test-time computation, outperforming algorithms without test-time adaptation mechanisms.

Proximal operators are ubiquitous in inverse problems, commonly appearing as part of algorithmic strategies to regularize problems that are otherwise ill-posed. Modern deep learning models have been brought to bear for these tasks too, as in the framework of plug-and-play or deep unrolling, where they loosely resemble proximal operators. Yet, something essential is lost in employing these purely data-driven approaches: there is no guarantee that a general deep network represents the proximal operator of any function, nor is there any characterization of the function for which the network might provide some approximate proximal. This not only makes guaranteeing convergence of iterative schemes challenging but, more fundamentally, complicates the analysis of what has been learned by these networks about their training data. Herein we provide a framework to develop learned proximal networks (LPN), prove that they provide exact proximal operators for a data-driven nonconvex regularizer, and show how a new training strategy, dubbed proximal matching, provably promotes the recovery of the log-prior of the true data distribution. Such LPN provide general, unsupervised, expressive proximal operators that can be used for general inverse problems with convergence guarantees. We illustrate our results in a series of cases of increasing complexity, demonstrating that these models not only result in state-of-the-art performance, but provide a window into the resulting priors learned from data.

The availability of real-time semantics greatly improves the core geometric functionality of SLAM systems, enabling numerous robotic and AR/VR applications. We present a new methodology for real-time semantic mapping from RGB-D sequences that combines a 2D neural network and a 3D network based on a SLAM system with 3D occupancy mapping. When segmenting a new frame we perform latent feature re-projection from previous frames based on differentiable rendering. Fusing re-projected feature maps from previous frames with current-frame features greatly improves image segmentation quality, compared to a baseline that processes images independently. For 3D map processing, we propose a novel geometric quasi-planar over-segmentation method that groups 3D map elements likely to belong to the same semantic classes, relying on surface normals. We also describe a novel neural network design for lightweight semantic map post-processing. Our system achieves state-of-the-art semantic mapping quality within 2D-3D networks-based systems and matches the performance of 3D convolutional networks on three real indoor datasets, while working in real-time. Moreover, it shows better cross-sensor generalization abilities compared to 3D CNNs, enabling training and inference with different depth sensors. Code and data will be released on project page: http://jingwenwang95.github.io/SeMLaPS

2D irregular shape packing is a necessary step to arrange UV patches of a 3D model within a texture atlas for memory-efficient appearance rendering in computer graphics. Being a joint, combinatorial decision-making problem involving all patch positions and orientations, this problem has well-known NP-hard complexity. Prior solutions either assume a heuristic packing order or modify the upstream mesh cut and UV mapping to simplify the problem, which either limits the packing ratio or incurs robustness or generality issues. Instead, we introduce a learning-assisted 2D irregular shape packing method that achieves a high packing quality with minimal requirements from the input. Our method iteratively selects and groups subsets of UV patches into near-rectangular super patches, essentially reducing the problem to bin-packing, based on which a joint optimization is employed to further improve the packing ratio. In order to efficiently deal with large problem instances with hundreds of patches, we train deep neural policies to predict nearly rectangular patch subsets and determine their relative poses, leading to linear time scaling with the number of patches. We demonstrate the effectiveness of our method on three datasets for UV packing, where our method achieves a higher packing ratio over several widely used baselines with competitive computational speed.