Infinity Search: Approximate Vector Search with Projections on q-Metric Spaces

Despite the ubiquity of vector search applications, prevailing search algorithms overlook the metric structure of vector embeddings, treating it as a constraint rather than exploiting its underlying properties. In this paper, we demonstrate that in $q$-metric spaces, metric trees can leverage a stronger version of the triangle inequality to reduce comparisons for exact search. Notably, as $q$ approaches infinity, the search complexity becomes logarithmic. Therefore, we propose a novel projection method that embeds vector datasets with arbitrary dissimilarity measures into $q$-metric spaces while preserving the nearest neighbor. We propose to learn an approximation of this projection to efficiently transform query points to a space where euclidean distances satisfy the desired properties. Our experimental results with text and image vector embeddings show that learning $q$-metric approximations enables classic metric tree algorithms -- which typically underperform with high-dimensional data -- to achieve competitive performance against state-of-the-art search methods.

Alignment of large language models with constrained learning

We study the problem of computing an optimal large language model (LLM) policy for a constrained alignment problem, where the goal is to maximize a primary reward objective while satisfying constraints on secondary utilities. Despite the popularity of Lagrangian-based LLM policy search in constrained alignment, iterative primal-dual methods often fail to converge, and non-iterative dual-based methods do not achieve optimality in the LLM parameter space. To address these challenges, we employ Lagrangian duality to develop an iterative dual-based alignment method that alternates between updating the LLM policy via Lagrangian maximization and updating the dual variable via dual descent. In theory, we characterize the primal-dual gap between the primal value in the distribution space and the dual value in the LLM parameter space. We further quantify the optimality gap of the learned LLM policies at near-optimal dual variables with respect to both the objective and the constraint functions. These results prove that dual-based alignment methods can find an optimal constrained LLM policy, up to an LLM parametrization gap. We demonstrate the effectiveness and merits of our approach through extensive experiments conducted on the PKU-SafeRLHF dataset.

Self-GIVE: Associative Thinking from Limited Structured Knowledge for Enhanced Large Language Model Reasoning

When addressing complex questions that require new information, people often associate the question with existing knowledge to derive a sensible answer. For instance, when evaluating whether melatonin aids insomnia, one might associate "hormones helping mental disorders" with "melatonin being a hormone and insomnia a mental disorder" to complete the reasoning. Large Language Models (LLMs) also require such associative thinking, particularly in resolving scientific inquiries when retrieved knowledge is insufficient and does not directly answer the question. Graph Inspired Veracity Extrapolation (GIVE) addresses this by using a knowledge graph (KG) to extrapolate structured knowledge. However, it involves the construction and pruning of many hypothetical triplets, which limits efficiency and generalizability. We propose Self-GIVE, a retrieve-RL framework that enhances LLMs with automatic associative thinking through reinforcement learning. Self-GIVE extracts structured information and entity sets to assist the model in linking to the queried concepts. We address GIVE's key limitations: (1) extensive LLM calls and token overhead for knowledge extrapolation, (2) difficulty in deploying on smaller LLMs (3B or 7B) due to complex instructions, and (3) inaccurate knowledge from LLM pruning. Specifically, after fine-tuning using self-GIVE with a 135 node UMLS KG, it improves the performance of the Qwen2.5 3B and 7B models by up to $\textbf{28.5%$\rightarrow$71.4%}$ and $\textbf{78.6$\rightarrow$90.5%}$ in samples $\textbf{unseen}$ in challenging biomedical QA tasks. In particular, Self-GIVE allows the 7B model to match or outperform GPT3.5 turbo with GIVE, while cutting token usage by over 90\%. Self-GIVE enhances the scalable integration of structured retrieval and reasoning with associative thinking.

Wireless Link Scheduling with State-Augmented Graph Neural Networks

We consider the problem of optimal link scheduling in large-scale wireless ad hoc networks. We specifically aim for the maximum long-term average performance, subject to a minimum transmission requirement for each link to ensure fairness. With a graph structure utilized to represent the conflicts of links, we formulate a constrained optimization problem to learn the scheduling policy, which is parameterized with a graph neural network (GNN). To address the challenge of long-term performance, we use the state-augmentation technique. In particular, by augmenting the Lagrangian dual variables as dynamic inputs to the scheduling policy, the GNN can be trained to gradually adapt the scheduling decisions to achieve the minimum transmission requirements. We verify the efficacy of our proposed policy through numerical simulations and compare its performance with several baselines in various network settings.

Generative Diffusion Models for Resource Allocation in Wireless Networks

This paper proposes a supervised training algorithm for learning stochastic resource allocation policies with generative diffusion models (GDMs). We formulate the allocation problem as the maximization of an ergodic utility function subject to ergodic Quality of Service (QoS) constraints. Given samples from a stochastic expert policy that yields a near-optimal solution to the problem, we train a GDM policy to imitate the expert and generate new samples from the optimal distribution. We achieve near-optimal performance through sequential execution of the generated samples. To enable generalization to a family of network configurations, we parameterize the backward diffusion process with a graph neural network (GNN) architecture. We present numerical results in a case study of power control in multi-user interference networks.

Opportunistic Routing in Wireless Communications via Learnable State-Augmented Policies

This paper addresses the challenge of packet-based information routing in large-scale wireless communication networks. The problem is framed as a constrained statistical learning task, where each network node operates using only local information. Opportunistic routing exploits the broadcast nature of wireless communication to dynamically select optimal forwarding nodes, enabling the information to reach the destination through multiple relay nodes simultaneously. To solve this, we propose a State-Augmentation (SA) based distributed optimization approach aimed at maximizing the total information handled by the source nodes in the network. The problem formulation leverages Graph Neural Networks (GNNs), which perform graph convolutions based on the topological connections between network nodes. Using an unsupervised learning paradigm, we extract routing policies from the GNN architecture, enabling optimal decisions for source nodes across various flows. Numerical experiments demonstrate that the proposed method achieves superior performance when training a GNN-parameterized model, particularly when compared to baseline algorithms. Additionally, applying the method to real-world network topologies and wireless ad-hoc network test beds validates its effectiveness, highlighting the robustness and transferability of GNNs.

Learning Efficient Positional Encodings with Graph Neural Networks

Positional encodings (PEs) are essential for effective graph representation learning because they provide position awareness in inherently position-agnostic transformer architectures and increase the expressive capacity of Graph Neural Networks (GNNs). However, designing powerful and efficient PEs for graphs poses significant challenges due to the absence of canonical node ordering and the scale of the graph. {In this work, we identify four key properties that graph PEs should satisfy}: stability, expressive power, scalability, and genericness. We find that existing eigenvector-based PE methods often fall short of jointly satisfying these criteria. To address this gap, we introduce PEARL, a novel framework of learnable PEs for graphs. Our primary insight is that message-passing GNNs function as nonlinear mappings of eigenvectors, enabling the design of GNN architectures for generating powerful and efficient PEs. A crucial challenge lies in initializing node attributes in a manner that is both expressive and permutation equivariant. We tackle this by initializing GNNs with random node inputs or standard basis vectors, thereby unlocking the expressive power of message-passing operations, while employing statistical pooling functions to maintain permutation equivariance. Our analysis demonstrates that PEARL approximates equivariant functions of eigenvectors with linear complexity, while rigorously establishing its stability and high expressive power. Experimental evaluations show that PEARL outperforms lightweight versions of eigenvector-based PEs and achieves comparable performance to full eigenvector-based PEs, but with one or two orders of magnitude lower complexity. Our code is available at https://github.com/ehejin/Pearl-PE.

Feasible Learning

We introduce Feasible Learning (FL), a sample-centric learning paradigm where models are trained by solving a feasibility problem that bounds the loss for each training sample. In contrast to the ubiquitous Empirical Risk Minimization (ERM) framework, which optimizes for average performance, FL demands satisfactory performance on every individual data point. Since any model that meets the prescribed performance threshold is a valid FL solution, the choice of optimization algorithm and its dynamics play a crucial role in shaping the properties of the resulting solutions. In particular, we study a primal-dual approach which dynamically re-weights the importance of each sample during training. To address the challenge of setting a meaningful threshold in practice, we introduce a relaxation of FL that incorporates slack variables of minimal norm. Our empirical analysis, spanning image classification, age regression, and preference optimization in large language models, demonstrates that models trained via FL can learn from data while displaying improved tail behavior compared to ERM, with only a marginal impact on average performance.

Explainable Brain Age Gap Prediction in Neurodegenerative Conditions using coVariance Neural Networks

Brain age is the estimate of biological age derived from neuroimaging datasets using machine learning algorithms. Increasing \textit{brain age gap} characterized by an elevated brain age relative to the chronological age can reflect increased vulnerability to neurodegeneration and cognitive decline. Hence, brain age gap is a promising biomarker for monitoring brain health. However, black-box machine learning approaches to brain age gap prediction have limited practical utility. Recent studies on coVariance neural networks (VNN) have proposed a relatively transparent deep learning pipeline for neuroimaging data analyses, which possesses two key features: (i) inherent \textit{anatomically interpretablity} of derived biomarkers; and (ii) a methodologically interpretable perspective based on \textit{linkage with eigenvectors of anatomic covariance matrix}. In this paper, we apply the VNN-based approach to study brain age gap using cortical thickness features for various prevalent neurodegenerative conditions. Our results reveal distinct anatomic patterns for brain age gap in Alzheimer's disease, frontotemporal dementia, and atypical Parkinsonian disorders. Furthermore, we demonstrate that the distinct anatomic patterns of brain age gap are linked with the differences in how VNN leverages the eigenspectrum of the anatomic covariance matrix, thus lending explainability to the reported results.

Convolutional Filtering with RKHS Algebras

In this paper, we develop a generalized theory of convolutional signal processing and neural networks for Reproducing Kernel Hilbert Spaces (RKHS). Leveraging the theory of algebraic signal processing (ASP), we show that any RKHS allows the formal definition of multiple algebraic convolutional models. We show that any RKHS induces algebras whose elements determine convolutional operators acting on RKHS elements. This approach allows us to achieve scalable filtering and learning as a byproduct of the convolutional model, and simultaneously take advantage of the well-known benefits of processing information in an RKHS. To emphasize the generality and usefulness of our approach, we show how algebraic RKHS can be used to define convolutional signal models on groups, graphons, and traditional Euclidean signal spaces. Furthermore, using algebraic RKHS models, we build convolutional networks, formally defining the notion of pointwise nonlinearities and deriving explicit expressions for the training. Such derivations are obtained in terms of the algebraic representation of the RKHS. We present a set of numerical experiments on real data in which wireless coverage is predicted from measurements captured by unmaned aerial vehicles. This particular real-life scenario emphasizes the benefits of the convolutional RKHS models in neural networks compared to fully connected and standard convolutional operators.