How Much Information Can a Vision Token Hold? A Scaling Law for Recognition Limits in VLMs

Recent vision-centric approaches have made significant strides in long-context modeling. Represented by DeepSeek-OCR, these models encode rendered text into continuous vision tokens, achieving high compression rates without sacrificing recognition precision. However, viewing the vision encoder as a lossy channel with finite representational capacity raises a fundamental question: what is the information upper bound of visual tokens? To investigate this limit, we conduct controlled stress tests by progressively increasing the information quantity (character count) within an image. We observe a distinct phase-transition phenomenon characterized by three regimes: a near-perfect Stable Phase, an Instability Phase marked by increased error variance, and a total Collapse Phase. We analyze the mechanical origins of these transitions and identify key factors. Furthermore, we formulate a probabilistic scaling law that unifies average vision token load and visual density into a latent difficulty metric. Extensive experiments across various Vision-Language Models demonstrate the universality of this scaling law, providing critical empirical guidance for optimizing the efficiency-accuracy trade-off in visual context compression.

InfoDPCCA: Information-Theoretic Dynamic Probabilistic Canonical Correlation Analysis

Extracting meaningful latent representations from high-dimensional sequential data is a crucial challenge in machine learning, with applications spanning natural science and engineering. We introduce InfoDPCCA, a dynamic probabilistic Canonical Correlation Analysis (CCA) framework designed to model two interdependent sequences of observations. InfoDPCCA leverages a novel information-theoretic objective to extract a shared latent representation that captures the mutual structure between the data streams and balances representation compression and predictive sufficiency while also learning separate latent components that encode information specific to each sequence. Unlike prior dynamic CCA models, such as DPCCA, our approach explicitly enforces the shared latent space to encode only the mutual information between the sequences, improving interpretability and robustness. We further introduce a two-step training scheme to bridge the gap between information-theoretic representation learning and generative modeling, along with a residual connection mechanism to enhance training stability. Through experiments on synthetic and medical fMRI data, we demonstrate that InfoDPCCA excels as a tool for representation learning. Code of InfoDPCCA is available at https://github.com/marcusstang/InfoDPCCA.

Deep Dynamic Probabilistic Canonical Correlation Analysis

This paper presents Deep Dynamic Probabilistic Canonical Correlation Analysis (D2PCCA), a model that integrates deep learning with probabilistic modeling to analyze nonlinear dynamical systems. Building on the probabilistic extensions of Canonical Correlation Analysis (CCA), D2PCCA captures nonlinear latent dynamics and supports enhancements such as KL annealing for improved convergence and normalizing flows for a more flexible posterior approximation. D2PCCA naturally extends to multiple observed variables, making it a versatile tool for encoding prior knowledge about sequential datasets and providing a probabilistic understanding of the system's dynamics. Experimental validation on real financial datasets demonstrates the effectiveness of D2PCCA and its extensions in capturing latent dynamics.