Understanding Submodular Information Measure Based Objectives for Representation Learning: A Variance and Separation Perspective

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.

SMA: Submodular Modality Aligner For Data Efficient Multimodal Learning

Despite the recent success of Multimodal Foundation Models (FMs), their reliance on massive paired datasets limits their applicability in low-data and rare-scenario settings where aligned data is scarce and expensive. A key bottleneck is the adoption of an instance-level formulation, which learns alignment by maximizing correlation between individual image-text pairs while neglecting the underlying geometric structure across modalities resulting in a modality gap across input modalities. In this paper, we propose a combinatorial paradigm for multimodal alignment that moves beyond pairwise learning and introduce the \emph{Submodular Modality Aligner (SMA)}, which treats multiple augmentations and descriptions of an entity as a set, leveraging multiple descriptions of the data to capture richer cross-modal structure. We instantiate SMA using a principled objective based on Submodular Mutual Information (SMI), which jointly maximizes inter-modality mutual information while reducing cross-modal divergence. This formulation enables the model to effectively utilize multiple positive associations and extract significantly more information from limited data. We evaluate SMA on 14 zero-shot classification and retrieval tasks from the CLIP benchmark and demonstrate consistent gains in the low-data regime. Notably, SMA achieves strong multimodal generalization using only tens of thousands of samples. This is orders of magnitude fewer than standard approaches. Our results highlight the importance of set-based formulations and submodular objectives for data-efficient multimodal learning.

Theoretical Analysis of Submodular Information Measures for Targeted Data Subset Selection

With increasing volume of data being used across machine learning tasks, the capability to target specific subsets of data becomes more important. To aid in this capability, the recently proposed Submodular Mutual Information (SMI) has been effectively applied across numerous tasks in literature to perform targeted subset selection with the aid of a exemplar query set. However, all such works are deficient in providing theoretical guarantees for SMI in terms of its sensitivity to a subset's relevance and coverage of the targeted data. For the first time, we provide such guarantees by deriving similarity-based bounds on quantities related to relevance and coverage of the targeted data. With these bounds, we show that the SMI functions, which have empirically shown success in multiple applications, are theoretically sound in achieving good query relevance and query coverage.