MIC: Maximizing Informational Capacity in Adaptive Representations via Isotropic Subspace Alignment

Representation learning is fundamental to NLP, but building embeddings that work well at different computational budgets is challenging. Matryoshka Representation Learning (MRL) offers a flexible inference paradigm through nested embeddings; however, learning such structures requires explicit coordination of how information is arranged across embedding dimensionality and model depth. In this work, we propose MIPIC (Matryoshka Representation Learning via Self-Distilled Intra-Relational Alignment and Progressive Information Chaining), a unified training framework designed to produce structurally coherent and semantically compact Matryoshka representations. MIPIC promotes cross-dimensional structural consistency through Self-Distilled Intra-Relational Alignment (SIA), which aligns token-level geometric and attention-driven relations between full and truncated representations using top-k CKA self-distillation. Complementarily, it enables depth-wise semantic consolidation via Progressive Information Chaining (PIC), a scaffolded alignment strategy that incrementally transfers mature task semantics from deeper layers into earlier layers. Extensive experiments on STS, NLI, and classification benchmarks (spanning models from TinyBERT to BGEM3, Qwen3) demonstrate that MIPIC yields Matryoshka representations that are highly competitive across all capacities, with significant performance advantages observed under extreme low-dimensional.

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.

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.