Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for a broad class of encoder-decoder architectures. In this study, we focus on approximating continuous operators in $\mathcal{C}(\mathcal{X}, \mathcal{Y})$, where $\mathcal{X}$ and $\mathcal{Y}$ are infinite-dimensional normed or metric spaces, and we consider uniform convergence on compact subsets of $\mathcal{X}$. Unlike standard results in the operator learning literature, we investigate the case where the approximating operator sequence can be chosen independently of the compact sets. Taking a topological perspective, we analyze different types of operator approximation and show that compact-set-independent approximation is a strictly stronger property in most relevant operator learning frameworks. To establish our results, we introduce a new approximation property tailored to encoder-decoder architectures, which enables us to prove a universal operator approximation theorem ensuring uniform convergence on every compact subset. This result unifies and extends existing universal operator approximation theorems for various encoder-decoder architectures, including classical DeepONets, BasisONets, special cases of MIONets, architectures based on frames and other related approaches.

Graph layouts and node embeddings are two distinct paradigms for non-parametric graph representation learning. In the former, nodes are embedded into 2D space for visualization purposes. In the latter, nodes are embedded into a high-dimensional vector space for downstream processing. State-of-the-art algorithms for these two paradigms, force-directed layouts and random-walk-based contrastive learning (such as DeepWalk and node2vec), have little in common. In this work, we show that both paradigms can be approached with a single coherent framework based on established neighbor embedding methods. Specifically, we introduce graph t-SNE, a neighbor embedding method for two-dimensional graph layouts, and graph CNE, a contrastive neighbor embedding method that produces high-dimensional node representations by optimizing the InfoNCE objective. We show that both graph t-SNE and graph CNE strongly outperform state-of-the-art algorithms in terms of local structure preservation, while being conceptually simpler.

--We propose Lib2V ec, a novel self-supervised framework to efficiently learn meaningful vector representations of library cells, enabling ML models to capture essential cell semantics. The framework comprises three key components: (1) an automated method for generating regularity tests to quantitatively evaluate how well cell representations reflect inter-cell relationships; (2) a self-supervised learning scheme that systematically extracts training data from Liberty files, removing the need for costly labeling; and (3) an attention-based model architecture that accommodates various pin counts and enables the creation of property-specific cell and arc embeddings. Experimental results demonstrate that Lib2V ec effectively captures functional and electrical similarities. Moreover, linear algebraic operations on cell vectors reveal meaningful relationships, such as vector(BUF) - vector(INV) + vector(NAND) approximating the vector of AND, showcasing the framework's nuanced representation capabilities. Lib2V ec also enhances downstream circuit learning applications, especially when labeled data is scarce. Library cell representations are vital for effective machine learning (ML)-based circuit analysis and optimization, as library cells are the fundamental building blocks of circuit netlists. Traditional methods often rely on manually defined features [1]-[4], requiring extensive expertise and feature engineering. Alternatively, one-hot encoding [5] demands large amounts of domain-specific training data, which may not always be available.

Many real-world and artificial systems and processes can be represented as graphs. Some examples of such systems include social networks, financial transactions, supply chains, and molecular structures. In many of these cases, one needs to consider a collection of graphs, rather than a single network. This could be a collection of distinct but related graphs, such as different protein structures or graphs resulting from dynamic processes on the same network. Examples of the latter include the evolution of social networks, community-induced graphs, or ego-nets around various nodes. A significant challenge commonly encountered is the absence of ground-truth labels for graphs or nodes, necessitating the use of unsupervised techniques to analyze such systems. Moreover, even when ground-truth labels are available, many existing graph machine learning methods depend on complex deep learning models, complicating model explainability and interpretability. To address some of these challenges, we have introduced NEExT (Network Embedding Exploration Tool) for embedding collections of graphs via user-defined node features. The advantages of the framework are twofold: (i) the ability to easily define your own interpretable node-based features in view of the task at hand, and (ii) fast embedding of graphs provided by the Vectorizers library. In this paper, we demonstrate the usefulness of NEExT on collections of synthetic and real-world graphs. For supervised tasks, we demonstrate that performance in graph classification tasks could be achieved similarly to other state-of-the-art techniques while maintaining model interpretability. Furthermore, our framework can also be used to generate high-quality embeddings in an unsupervised way, where target variables are not available.

Laplacian-based methods are popular for dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance approximates the geodesic distance on the underlying submanifold which the data are assumed to lie on. However, for some applications, other metrics, such as the Wasserstein distance, may provide a more appropriate notion of distance than the Euclidean distance. We provide a framework that generalizes the problem of manifold learning to metric spaces and study when a metric satisfies sufficient conditions for the pointwise convergence of the graph Laplacian.

We propose a new algorithm for fine-tuning large language models using reinforcement learning. Tapered Off-Policy REINFORCE (TOPR) uses an asymmetric, tapered variant of importance sampling to speed up learning while maintaining stable learning dynamics, even without the use of KL regularization. TOPR can be applied in a fully offline fashion, allows the handling of positive and negative examples in a unified framework, and benefits from the implementational simplicity that is typical of Monte Carlo algorithms. We demonstrate the effectiveness of our approach with a series of experiments on the GSM8K and MATH reasoning benchmarks, finding performance gains for training both a model for solution generation and as a generative verifier. We show that properly leveraging positive and negative examples alike in the off-policy regime simultaneously increases test-time accuracy and training data efficiency, all the while avoiding the ``wasted inference'' that comes with discarding negative examples. We find that this advantage persists over multiple iterations of training and can be amplified by dataset curation techniques, enabling us to match 70B-parameter model performance with 8B language models. As a corollary to this work, we find that REINFORCE's baseline parameter plays an important and unexpected role in defining dataset composition in the presence of negative examples, and is consequently critical in driving off-policy performance.

Large language models (LLMs) undergo a three-phase training process: unsupervised pre-training, supervised fine-tuning (SFT), and learning from human feedback (RLHF/DPO). Notably, it is during the final phase that these models are exposed to negative examples -- incorrect, rejected, or suboptimal responses to queries. This paper delves into the role of negative examples in the training of LLMs, using a likelihood-ratio (Likra) model on multiple-choice question answering benchmarks to precisely manage the influence and the volume of negative examples. Our findings reveal three key insights: (1) During a critical phase in training, Likra with negative examples demonstrates a significantly larger improvement per training example compared to SFT using only positive examples. This leads to a sharp jump in the learning curve for Likra unlike the smooth and gradual improvement of SFT; (2) negative examples that are plausible but incorrect (near-misses) exert a greater influence; and (3) while training with positive examples fails to significantly decrease the likelihood of plausible but incorrect answers, training with negative examples more accurately identifies them. These results indicate a potentially significant role for negative examples in improving accuracy and reducing hallucinations for LLMs.

Hierarchy-aware representations ensure that the semantically closer classes are mapped closer in the feature space, thereby reducing the severity of mistakes while enabling consistent coarse-level class predictions. Towards this end, we propose a novel framework, Hierarchical Composition of Orthogonal Subspaces (Hier-COS), which learns to map deep feature embeddings into a vector space that is, by design, consistent with the structure of a given taxonomy tree. Our approach augments neural network backbones with a simple transformation module that maps learned discriminative features to subspaces defined using a fixed orthogonal frame. This construction naturally improves the severity of mistakes and promotes hierarchical consistency. Furthermore, we highlight the fundamental limitations of existing hierarchical evaluation metrics popularly used by the vision community and introduce a preference-based metric, Hierarchically Ordered Preference Score (HOPS), to overcome these limitations. We benchmark our method on multiple large and challenging datasets having deep label hierarchies (ranging from 3 - 12 levels) and compare with several baselines and SOTA. Through extensive experiments, we demonstrate that Hier-COS achieves state-of-the-art hierarchical performance across all the datasets while simultaneously beating top-1 accuracy in all but one case. We also demonstrate the performance of a Vision Transformer (ViT) backbone and show that learning a transformation module alone can map the learned features from a pre-trained ViT to Hier-COS and yield substantial performance benefits.

In recent years the information asymmetric Lipschitz bandits In this paper we studied the Lipschitz bandit problem applied to the multiplayer information asymmetric problem studied in \cite{chang2022online, chang2023optimal}. More specifically we consider information asymmetry in rewards, actions, or both. We adopt the CAB algorithm given in \cite{kleinberg2004nearly} which uses a fixed discretization to give regret bounds of the same order (in the dimension of the action) space in all 3 problem settings. We also adopt their zooming algorithm \cite{ kleinberg2008multi}which uses an adaptive discretization and apply it to information asymmetry in rewards and information asymmetry in actions.

Visitors from far and wide have been traveling to Japan, with the country breaking a tourism record in 2024. Between Jan. 1 and Nov. 30, projections indicated that nearly 33.4 million travelers visited Japan, according to the country's government site. Nearly three million Americans visited the country in 2024. Hokuto Asano, first secretary at the Embassy of Japan, told Fox News Digital that the number of visitors last year ended up reaching 36 million. Yukiyoshi Noguchi, who is the counselor at the embassy, said 2024 was declared the "U.S.-Japan Tourism Year" by both governments.