We show the universality of depth-2 group convolutional neural networks (GCNNs) inaunified and constructivemanner based ontheridgelet theory.

Unsupervised domain adaptation (UDA) is a pivotal form in machine learning to extend the in-domain model to the distinctive target domains where the data distributions differ. Most prior works focus on capturing the inter-domain transferability but largely overlook rich intra-domain structures, which empirically results in even worse discriminability. In this work, we introduce a novel graph SPectral Alignment (SPA) framework to tackle the tradeoff. The core of our method is briefly condensed as follows: (i)-by casting the DA problem to graph primitives, SPA composes a coarse graph alignment mechanism with a novel spectral regularizer towards aligning the domain graphs in eigenspaces; (ii)-we further develop a fine-grained message propagation module --- upon a novel neighbor-aware self-training mechanism --- in order for enhanced discriminability in the target domain. On standardized benchmarks, the extensive experiments of SPA demonstrate that its performance has surpassed the existing cutting-edge DA methods. Coupled with dense model analysis, we conclude that our approach indeed possesses superior efficacy, robustness, discriminability, and transferability. Code and data are available at: https://github.com/CrownX/SPA.

In high-stakes scenarios-such as self-harm, legal, or medical queries-LLMs must be both trustworthy and helpful. However, these goals often conflict. We propose priority alignment, a new alignment paradigm that enforces a strict "trustworthy-before-helpful" ordering: optimization of helpfulness is conditioned on first meeting trustworthy thresholds (e.g., harmlessness or honesty). To realize this, we introduce Self-Priority Alignment (SPA)-a fully unsupervised framework that generates diverse responses, self-evaluates them and refines them by the model itself, and applies dual-criterion denoising to remove inconsistency and control variance. From this, SPA constructs lexicographically ordered preference pairs and fine-tunes the model using an uncertainty-weighted alignment loss that emphasizes high-confidence, high-gap decisions. Experiments across multiple benchmarks show that SPA improves helpfulness without compromising safety, outperforming strong baselines while preserving general capabilities. Our results demonstrate that SPA provides a scalable and interpretable alignment strategy for critical LLM applications.

Large Language Models (LLMs) as agents often struggle in out-of-distribution (OOD) scenarios. Real-world environments are complex and dynamic, governed by task-specific rules and stochasticity, which makes it difficult for LLMs to ground their internal knowledge in those dynamics. Under such OOD conditions, vanilla RL training often fails to scale; we observe Pass@k--the probability that at least one of (k) sampled trajectories succeeds--drops markedly across training steps, indicating brittle exploration and limited generalization. Inspired by model-based reinforcement learning, we hypothesize that equipping LLM agents with an internal world model can better align reasoning with environmental dynamics and improve decision-making. We show how to encode this world model by decomposing it into two components: state representation and transition modeling. Building on this, we introduce SPA, a simple reinforcement learning framework that cold-starts the policy via a Self-Play supervised finetuning (SFT) stage to learn the world model by interacting with the environment, then uses it to simulate future states prior to policy optimization. This simple initialization outperforms the online world-modeling baseline and greatly boosts the RL-based agent training performance. Experiments across diverse environments like Sokoban, FrozenLake, and Sudoku show that our approach significantly improves performance. For example, SPA boosts the Sokoban success rate from 25.6% to 59.8% and raises the FrozenLake score from 22.1% to 70.9% for the Qwen2.5-1.5B-Instruct model.

We show the universality of depth-2 group convolutional neural networks (GC-NNs) in a unified and constructive manner based on the ridgelet theory.

To build an automatic speech recognition (ASR) system that can serve everyone in the world, the ASR needs to be robust to a wide range of accents including unseen accents. We systematically study how three different variables in training data -- the number of speakers, the audio duration per each individual speaker, and the diversity of accents -- affect ASR robustness towards unseen accents in a low-resource training regime. We observe that for a fixed number of ASR training hours, it is more beneficial to increase the number of speakers (which means each speaker contributes less) than the number of hours contributed per speaker. We also observe that more speakers enables ASR performance gains from scaling number of hours. Surprisingly, we observe minimal benefits to prioritizing speakers with different accents when the number of speakers is controlled. Our work suggests that practitioners should prioritize increasing the speaker count in ASR training data composition for new languages.

Large Language Models (LLMs) are machine learning models that have seen widespread adoption due to their capability of handling previously difficult tasks. LLMs, due to their training, are sensitive to how exactly a question is presented, also known as prompting. However, prompting well is challenging, as it has been difficult to uncover principles behind prompting -- generally, trial-and-error is the most common way of improving prompts, despite its significant computational cost. In this context, we argue it would be useful to perform `predictive prompt analysis', in which an automated technique would perform a quick analysis of a prompt and predict how the LLM would react to it, relative to a goal provided by the user. As a demonstration of the concept, we present Syntactic Prevalence Analyzer (SPA), a predictive prompt analysis approach based on sparse autoencoders (SAEs). SPA accurately predicted how often an LLM would generate target syntactic structures during code synthesis, with up to 0.994 Pearson correlation between the predicted and actual prevalence of the target structure. At the same time, SPA requires only 0.4\% of the time it takes to run the LLM on a benchmark. As LLMs are increasingly used during and integrated into modern software development, our proposed predictive prompt analysis concept has the potential to significantly ease the use of LLMs for both practitioners and researchers.

Unsupervised domain adaptation (UDA) is a pivotal form in machine learning to extend the in-domain model to the distinctive target domains where the data distributions differ. Most prior works focus on capturing the inter-domain transferability but largely overlook rich intra-domain structures, which empirically results in even worse discriminability. In this work, we introduce a novel graph SPectral Alignment (SPA) framework to tackle the tradeoff. The core of our method is briefly condensed as follows: (i)-by casting the DA problem to graph primitives, SPA composes a coarse graph alignment mechanism with a novel spectral regularizer towards aligning the domain graphs in eigenspaces; (ii)-we further develop a fine-grained message propagation module --- upon a novel neighbor-aware self-training mechanism --- in order for enhanced discriminability in the target domain. On standardized benchmarks, the extensive experiments of SPA demonstrate that its performance has surpassed the existing cutting-edge DA methods.

The successive projection algorithm (SPA) is a workhorse algorithm to learn the $r$ vertices of the convex hull of a set of $(r-1)$-dimensional data points, a.k.a. a latent simplex, which has numerous applications in data science. In this paper, we revisit the robustness to noise of SPA and several of its variants. In particular, when $r \geq 3$, we prove the tightness of the existing error bounds for SPA and for two more robust preconditioned variants of SPA. We also provide significantly improved error bounds for SPA, by a factor proportional to the conditioning of the $r$ vertices, in two special cases: for the first extracted vertex, and when $r \leq 2$. We then provide further improvements for the error bounds of a translated version of SPA proposed by Arora et al. (''A practical algorithm for topic modeling with provable guarantees'', ICML, 2013) in two special cases: for the first two extracted vertices, and when $r \leq 3$. Finally, we propose a new more robust variant of SPA that first shifts and lifts the data points in order to minimize the conditioning of the problem. We illustrate our results on synthetic data.