Foundation models for biology and physics optimize predictive accuracy, but their internal representations systematically fail to preserve the continuous geometry of the systems they model. We identify the root cause: the Geometric Alignment Tax, an intrinsic cost of forcing continuous manifolds through discrete categorical bottlenecks. Controlled ablations on synthetic dynamical systems demonstrate that replacing cross-entropy with a continuous head on an identical encoder reduces geometric distortion by up to 8.5x, while learned codebooks exhibit a non-monotonic double bind where finer quantization worsens geometry despite improving reconstruction. Under continuous objectives, three architectures differ by 1.3x; under discrete tokenization, they diverge by 3,000x. Evaluating 14 biological foundation models with rate-distortion theory and MINE, we identify three failure regimes: Local-Global Decoupling, Representational Compression, and Geometric Vacuity. A controlled experiment confirms that Evo 2's reverse-complement robustness on real DNA reflects conserved sequence composition, not learned symmetry. No model achieves simultaneously low distortion, high mutual information, and global coherence.

Decentralized Partially Observable Markov Decision Processes (Dec-POMDPs) are known to be NEXP-Complete and intractable to solve. However, for problems such as cooperative navigation, obstacle avoidance, and formation control, basic assumptions can be made about local visibility and local dependencies. The work DeWeese and Qu 2024 formalized these assumptions in the construction of the Locally Interdependent Multi-Agent MDP. In this setting, it establishes three closed-form policies that are tractable to compute in various situations and are exponentially close to optimal with respect to visibility. However, it is also shown that these solutions can have poor performance when the visibility is small and fixed, often getting stuck during simulations due to the so called "Penalty Jittering" phenomenon. In this work, we establish the Extended Cutoff Policy Class which is, to the best of our knowledge, the first non-trivial class of near optimal closed-form partially observable policies that are exponentially close to optimal with respect to the visibility for any Locally Interdependent Multi-Agent MDP. These policies are able to remember agents beyond their visibilities which allows them to perform significantly better in many small and fixed visibility settings, resolve Penalty Jittering occurrences, and under certain circumstances guarantee fully observable joint optimal behavior despite the partial observability. We also propose a generalized form of the Locally Interdependent Multi-Agent MDP that allows for transition dependence and extended reward dependence, then replicate our theoretical results in this setting.

Learning meaningful representations of data that can address challenges such as batch effect correction, data integration and counterfactual inference is a central problem in many domains including computational biology. Adopting a Conditional VAE framework, we identify the mathematical principle that unites these challenges: learning a representation that is marginally independent of a condition variable. We therefore propose the Contrastive Mixture of Posteriors (CoMP) method that uses a novel misalignment penalty to enforce this independence. This penalty is defined in terms of mixtures of the variational posteriors themselves, unlike prior work which uses external discrepancy measures such as MMD to ensure independence in latent space. We show that CoMP has attractive theoretical properties compared to previous approaches, especially when there is complex global structure in latent space. We further demonstrate state of the art performance on a number of real-world problems, including the challenging tasks of aligning human tumour samples with cancer cell-lines and performing counterfactual inference on single-cell RNA sequencing data. Incidentally, we find parallels with the fair representation learning literature, and demonstrate CoMP has competitive performance in learning fair yet expressive latent representations.

In many signal processing and machine learning applications, datasets containing private information are held at different locations, requiring the development of distributed privacy-preserving algorithms. Tensor and matrix factorizations are key components of many processing pipelines. In the distributed setting, differentially private algorithms suffer because they introduce noise to guarantee privacy. This paper designs new and improved distributed and differentially private algorithms for two popular matrix and tensor factorization methods: principal component analysis (PCA) and orthogonal tensor decomposition (OTD). The new algorithms employ a correlated noise design scheme to alleviate the effects of noise and can achieve the same noise level as the centralized scenario. Experiments on synthetic and real data illustrate the regimes in which the correlated noise allows performance matching with the centralized setting, outperforming previous methods and demonstrating that meaningful utility is possible while guaranteeing differential privacy.