Mutual Information Maximization

Bayesian optimization often involves multiple objectives, constraints, and fidelity levels. We address the challenge of jointly selecting where and at which fidelity to evaluate to identify the highest-fidelity feasible Pareto frontier in this combined setting. From a unified information-theoretic perspective, we measure query utility by the information gain about this frontier, provided by an observation. Since this mutual information is intractable, we derive a variational lower bound using a mixture of under- and over-truncated approximations to the Pareto-consistent region. Multi-fidelity surrogate models propagate the information to arbitrary fidelities, yielding a cost-aware acquisition function without separate heuristics for fidelity selection or constraint handling. Experiments on synthetic, benchmark, and real-world problems demonstrate effectiveness across diverse objective, constraint, and fidelity settings.

Mutual information (MI) provides an objective for suppressing or encouraging statistical dependence in implicit generative models. However, direct MI evaluation is challenging in implicit models due to typically intractable densities. A remedy is estimating the generator gradient from the difference between conditional and marginal scores. This score difference can, in turn, be estimated by differentiating a log density ratio learned through classification. This construction nevertheless faces two difficulties: (i) singular distributions need not admit the required score functions, and (ii) poor overlap can hinder density-ratio estimation. We therefore introduce Spread Mutual Information (SMI), a weighted integral of MI across noise levels obtained by applying a common spreading kernel to the generated variable. Gaussian spreading yields smooth, strictly positive conditional and marginal densities, extending the gradient construction to distributions that may originally be singular. Across a variaty of experiments, SMI consistently achieves effective dependence control among MI-based methods and remains competitive with established task-specific approaches.

Submodular Information Measures (SIMs) have recently emerged as a powerful framework for representation learning and multimodal learning. In particular, the SCORE framework~\cite{majee2024score} demonstrated that SIMs can serve as effective objectives for supervised contrastive learning. Despite their empirical success, however, the geometric and statistical properties induced by different submodular information measures remain poorly understood. In this work, we develop a unified theoretical framework connecting SIMs to classical concepts in representation learning and statistical pattern recognition. We show that Total Information (TI) objectives characterize intra-class structure: Graph Cut TI recovers within-class variance, LogDet TI recovers generalized variance and covariance volume, and Facility Location TI induces imbalance-aware separation that emphasizes rare and confusable classes. We further show that Mutual Information (MI) objectives capture complementary notions of inter-class structure: Graph Cut MI is closely related to centroid separation and Fisher-style discrimination, LogDet MI captures covariance-aware separation through Mahalanobis distance, and Facility Location MI measures nearest-mode representational overlap. We validate these theoretical characterizations using controlled synthetic experiments that independently vary variance, covariance, class imbalance, class separation, and multimodal overlap. Across all settings, the empirical behavior closely matches the proposed theory. Our results provide the first unified geometric and statistical understanding of submodular information measures and offer principled guidance for selecting and designing SIM-based objectives for representation learning.

Aligning large language models with diverse and heterogeneous human values requires multi-objective alignment methods to effectively trade off conflicting preference dimensions. Current methods achieve this trade-off by training policies conditioned on preference vectors and leveraging online direct preference optimization. However, exploration uncertainty can cause the reward distributions of responses generated under different preference vectors to overlap, and the generated responses may fail to effectively align with the corresponding preference vectors. In this paper, we propose Multi-Objective Exploration and Preference Optimization via Mutual Information (MI-EPO), an information-theoretic framework. It unifies multi-objective exploration and alignment by maximizing the joint conditional mutual information among generated responses, preference feedback, and preference vectors. By incorporating a probabilistic routing mechanism, MI-EPO naturally decomposes objective alignment and preference-aware exploration, encouraging the model to generate responses that are distinguishable and aligned with different preference conditions. Experiments on safe alignment and helpful assistant tasks show that MI-EPO significantly improves the alignment between generated responses and preference vectors, makes the outputs more controllable, and achieves stable trade-offs across multiple objectives.

Alzheimer's disease (AD) is a progressive neurodegenerative disorder and the leading cause of dementia, affecting memory, reasoning, communication, and daily functioning. Early diagnosis is particularly important, as timely intervention may help slow cognitive decline and improve patient care. Recent studies have demonstrated that spontaneous speech contains valuable linguistic and acoustic biomarkers associated with dementia. However, existing approaches often rely on independently trained modality-specific models, feature concatenation strategies, ensemble methods, or attention-based fusion mechanisms that do not explicitly maximize the dependency between speech and transcript representations. In this work, we propose a multimodal deep learning framework for automatic dementia detection that jointly exploits speech and transcript information in an end-to-end trainable manner. Specifically, speech recordings are divided into 10-second segments and passed through a pre-trained HuBERT model to extract contextualized acoustic representations. To better capture informative temporal speech characteristics, attentive statistics pooling is employed to aggregate frame-level acoustic embeddings. For the textual modality, transcripts are encoded using a pre-trained BERT model, where the [CLS] token representation is used as the linguistic embedding. The acoustic and textual representations are subsequently combined using an attention-based Audio-Text Fusion (AT-Fusion) mechanism. In addition, we introduce a MINE objective to maximize the mutual information between modalities and improve multimodal representation alignment. The fused multimodal representation is finally used for dementia classification. Experiments conducted on the publicly available ADReSS Challenge and PROCESS-2 dataset demonstrate the effectiveness and robustness of the proposed approach for speech-based dementia assessment.

The quality of graph-structured data is fundamental to the success of modern graph analysis techniques such as Graph Neural Networks (GNNs). However, real-world graph data is often suboptimal, suffering from issues such as noise and incomplete connections. Graph Structure Learning (GSL) has emerged as a promising technique that adaptively optimizes node connections. However, we observe that the effectiveness of GSL often comes at the cost of a dramatic expansion in edge count, resulting in significant storage and computational overhead. In this work, we reveal that this limitation stems from the prevalent use of similarity-based edge construction, which predominantly connects highly similar neighbors based on their embeddings, introducing substantial structure redundancy. To address this, we propose a novel Informative Graph Structure Learning method (InGSL), which jointly considers both similarity and diversity in edge construction by incorporating a mutual-information-guided learning strategy. Notably, InGSL serves as a plug-in module that can be seamlessly integrated into existing GSL frameworks. Through extensive experiments on six representative GSL methods, we demonstrate that InGSL achieves significant performance improvements at a reduced number of edges.

Reinforcement learning with verifiable rewards (RLVR) recently thrives in large language model (LLM) reasoning tasks. However, the reward sparsity and the long reasoning horizon make effective exploration challenging. In practice, this challenge manifests as the \emph{entropy collapse} phenomenon, where RLVR improves single-rollout accuracy but fails to expand coverage on successful reasoning trajectories. Passive exploration techniques like entropy regularization tend to dismiss generation quality, resulting in noisy rollouts. In response to this issue, we propose an Information-Maximizing Augmented eXploration (IMAX) framework to train a pool of soft prefixes that reshapes the base model's prior over reasoning trajectories. Rather than relying on RL to incentivize exploration on top of the base model, each prefix acts as a trainable control knob that induces a distinct rollout distribution from the same backbone model. To encourage discovery of diverse and task-relevant reasoning behaviors, we derive an Information Maximization (InfoMax) reward to complement the verifiable rewards for RL training. IMAX is in general algorithm-agnostic and can be seamlessly integrated into existing RLVR pipelines. Experiment results have shown that across three backbone scales, IMAX consistently improves reasoning performance over standard RLVR, with gains up to 11.60% in Pass@4 and 10.57% in Avg@4.

Modern self-supervised representation learning methods often relies on empirical heuristics that are not theoretically grounded. In this study we propose HyDeS, a theoretically grounded method based on multi-view mutual information maximization within an hyperspherical space using Shannon differential entropy with a non-parametric von Mises-Fisher density estimator. We show that HyDeS bias the trained model towards focusing on foreground features of the images and perform well on segmentation tasks such as VOC PASCAL, while it lags in fine-grained classification. We provide a detailed analysis of the induced latent space geometry and learning dynamics, that can be used for designing other theoretically grounded self-supervised learning methods.

Data selection for language models is often framed as balancing example quality and diversity. We argue that both are consequences of a more fundamental principle: selected examples should preserve the downstream query space induced by instructions, task signals, or retrieval needs. We present Greedy Information Projection (GIP), a query-aligned method that requires only candidate embeddings and a score signal from LLM judgments, metadata, or intrinsic geometry. GIP greedily selects examples whose embedding span explains the largest residual component of task/query scores, yielding a fast matching-pursuit selector. A Gaussian projection view connects this update to maximizing mutual information, equivalently minimizing the residual volume of the query subspace left unexplained by selected data; this explains how quality and diversity emerge from one objective. Empirically, GIP matches or surpasses full-data fine-tuning using small instruction and reasoning subsets, while pretraining and RAG passage-selection experiments show that the same residual principle transfers beyond supervised fine-tuning.

While post-training has successfully improved large language models (LLMs) across a variety of domains, these gains heavily rely on human-labeled data or external verifiers. Existing data has already been exploited, and new data is expensive to collect. Moreover, true intelligence goes far beyond verifiable tasks. Therefore, we need self-improvement frameworks that are less dependent on external signals and more broadly applicable to both verifiable and non-verifiable domains. We propose Mutual Information Preference Optimization (MIPO), a contrastive data augmentation method that constructs preference pairs by generating a positive response conditioning on the correct prompt, and a negative response by conditioning on a random, unrelated prompt. We show that using Direct Preference Optimization to learn from this paired data maximizes pointwise mutual information under the base LLM between prompts and model responses. Experiments with with 1-7B parameter Llama and Qwen instruct models show that MIPO achieves 3-16% gains (and 51% increase for Qwen2.5-1.5B-Instruct) on personalization compared to prompting baselines. Surprisingly, MIPO can also be useful in verifiable domains, such as math and multiple-choice question answering, yielding 1-20% gains without any additional data or external supervision. These results suggest a promising direction for self-improvement using intrinsic signals derived from contrastive data pairs.

Mutual information (MI) is a principled and widely used objective for robotic information gathering (RIG), providing strong theoretical guarantees for sensor placement (SP) and informative path planning (IPP). However, its high computational cost - dominated by repeated log-determinant evaluations - has limited its use in real-time planning. This paper presents Schur-MI, a Gaussian process (GP) MI formulation that (i) leverages the iterative structure of RIG to precompute and reuse expensive intermediate quantities across planning steps, and (ii) uses a Schur-complement factorization to avoid large determinant computations. Together, these methods reduce the per-evaluation cost of MI from $\mathcal{O}(|\mathcal{V}|^3)$ to $\mathcal{O}(|\mathcal{A}|^3)$, where $\mathcal{V}$ and $\mathcal{A}$ denote the candidate and selected sensing locations, respectively. Experiments on real-world bathymetry datasets show that Schur-MI achieves up to a $12.7\times$ speedup over the standard MI formulation. Field trials with an autonomous surface vehicle (ASV) performing adaptive IPP further demonstrate the method's practicality. By making MI computation tractable for online planning, Schur-MI helps bridge the gap between information-theoretic objectives and real-time robotic exploration. Our code is available at: www.sgp-tools.com

We propose FDRMFL, a task-driven multimodal feature extraction framework for federated regression under non-IID data distributions. Extracting predictive features from high-dimensional multimodal inputs is particularly challenging in this setting: data cannot leave each client, local samples are scarce and heterogeneously distributed, and unsupervised dimensionality reduction discards task-relevant information while federated training introduces representation drift across communication rounds. FDRMFL addresses these challenges through a unified four-term local objective: MSE prediction loss, a correlation-based mutual information surrogate that preserves dependence between the fused representation and the continuous target, a symmetric KL penalty that aligns cross-modal latent distributions before fusion, and an InfoNCE-style contrastive loss that anchors local representations to the global consensus. Experiments on three synthetic and two real-world near-infrared spectroscopy datasets under non-IID federated partitions, with comprehensive ablation and sensitivity analyses, demonstrate that each component contributes to the framework's effectiveness. FDRMFL reduces mean MSE by 33.8% relative to the best traditional baseline (PCA) and by 43.0% relative to VAE in simulation, and attains the lowest overall mean MSE among six federated algorithms including FedAvg, FedProx, MOON, SCAFFOLD, and FedBN.

The natural language generation domain has witnessed great success thanks to Transformer models. Although they have achieved state-of-the-art generative quality, they often neglect generative diversity. Prior attempts to tackle this issue suffer from either low model capacity or over-complicated architectures. Some recent methods employ the VAE framework to enhance diversity, but their latent variables fully depend on the input context, restricting exploration of the latent space. In this paper, we introduce VOLTA, a framework that elevates generative diversity by bridging Transformer with VAE via a more effective cross-attention-based connection, departing from conventional embedding concatenation or summation. Additionally, we propose integrating InfoGAN-style latent codes to enable input-independent variability, further diversifying the generation. Moreover, our framework accommodates discrete inputs alongside its existing support for continuous inputs. We perform comprehensive experiments with two types of Transformers on six datasets from three different NLG tasks to show that our approach can significantly improve generative diversity while maintaining generative quality.