quotient-geometry analysis

This work addresses the unobservability of representations arising from partial observations in representation learning by introducing the Plato Projection Structure (PPS). PPS provides the first unified characterization of equivalence classes of indistinguishable latent states through the quotient geometry induced by self-adjoint, positive semi-definite observation operators. The framework reveals a fundamental limitation: outputs cannot fully capture latent representations, thereby establishing the theoretical foundation for geometric preservation mechanisms in knowledge distillation and representation transfer. It also highlights intrinsic limitations of current interpretability methods. Combining kernel invariance analysis with rank-controlling techniques, experiments validate kernel-invariant observability, attribution gaps induced by projection, and rank-controllable observable geometry, offering rigorous theoretical support for representation accessibility and interpretability.

This work addresses the challenges of modeling complex dependencies and mitigating redundancy in high-dimensional parameter spaces for discrete data generation. It introduces, for the first time, a Riemannian geometric structure with isometric properties into the exponential parameter space of product manifolds over categorical distributions, thereby constructing a low-dimensional latent subspace. By leveraging the Riemannian metric, geodesics within this subspace become straight lines, enabling consistent and efficient flow-matching training. The proposed approach substantially reduces the dimensionality of latent variables while preserving strong representational capacity for discrete data distributions. Experimental results demonstrate that the model achieves accurate and efficient discrete data generation using a significantly lower-dimensional latent space, effectively balancing computational efficiency with modeling performance.

This paper addresses the optimal metric embedding of the orbit space $V/G$—induced by a compact isometric group $G$ acting on a finite-dimensional inner product space $V$—into a Hilbert space, with the objective of minimizing distortion between the quotient metric and the Euclidean distance. We develop an original analytical framework unifying Lie group representation theory, metric geometry, and invariant theory. This yields the first systematic characterization of Euclidean embeddability criteria for $V/G$, along with tight lower bounds on distortion. For canonical compact groups—including orthogonal and cyclic groups—we derive exact distortion bounds, surpassing prior heuristic approaches that rely solely on empirically designed invariant feature embeddings. Our results establish the first rigorous geometric foundation for invariance-aware machine learning, enabling provably robust construction of invariant feature embeddings.

This work addresses the lack of a universal statistical interpretation for the manifold hypothesis—that high-dimensional data approximately reside on low-dimensional manifolds. We propose the Latent Metric Model (LMM), a generative framework grounded in fundamental statistical concepts: latent variables, variable dependence, and stationarity—providing the first unified statistical justification for the manifold assumption. Methodologically, LMM integrates neighborhood graph construction, spectral analysis, and an interpretable inference framework to enable unsupervised manifold discovery and geometric structure recovery under weak priors. Experiments demonstrate that complex manifold geometries naturally emerge from minimal statistical mechanisms; LMM significantly reduces reliance on hand-crafted priors on both synthetic and real-world datasets, while enabling interpretable reconstruction of manifold dimensionality, curvature, and coordinate systems.

This work addresses the problem of identifying and reconstructing orbits of compact Lie group actions from point cloud data. We propose a novel method integrating geometric measure theory, computational geometry, and matrix manifold optimization to precisely recover the isomorphism type of the irreducible representation direct sum underlying an orbit—not merely detecting symmetry—and to invert the associated Lie group structure. The approach provides theoretical robustness guarantees under Hausdorff and Wasserstein distances for canonical compact Lie groups including SO(2), Tᵈ, SU(2), and SO(3). Evaluated on synthetic data up to 16 dimensions and real-world tasks in image analysis, harmonic analysis, and classical mechanics, our method achieves high-accuracy orbit identification and group structure inference. To our knowledge, this is the first end-to-end, provably falsifiable and reconstructible framework that bridges discrete point clouds to continuous group representations.

This work investigates whether self-supervised vision foundation models learn feature representations that align with the intrinsic structure of three-dimensional Euclidean space, even without explicit 3D supervision. To this end, we introduce a novel probing methodology based on neighborhood alignment and Poincaré adapters, along with a “latent space navigation” technique that leverages the topological and geometric relationship between the feature space and the SE(3) group to enable visual odometry and localization without explicit 3D reconstruction. Our experiments demonstrate a strong correspondence between the model’s latent subspace and 3D spatial structure, achieving accurate motion estimation and localization in static scenes using only latent features—an outcome that provides the first empirical validation of implicit 3D structural awareness in self-supervised visual representations.

This work addresses how to uncover interpretable concept manifolds embedded within the stacked representations of language models. The authors propose Manifold Probe, a method that generalizes traditional linear probing to manifold probing by integrating supervised manifold learning with linear predictability analysis. This approach identifies continuous geometric structures in representation space corresponding to high-level concepts—such as time or space—and determines their encoding directions. Beyond merely detecting the presence of such concepts, the method enables causal intervention: manipulating activations along discovered manifold directions directly alters model behavior. Experiments on Llama 2-7B demonstrate that perturbing representations along the extracted temporal manifold significantly shifts the model’s generated outputs regarding the release years of cultural works, thereby validating both the interpretability and causal efficacy of the recovered manifolds.

Standard variational autoencoders employ Gaussian priors, which struggle to align with data manifolds exhibiting non-Euclidean topologies—such as periodicity or boundedness—leading to distorted representations. This work proposes a topology-aware latent space modeling framework that constructs factorized prior distributions tailored to manifolds decomposable into products of circles, intervals, and lines, along with their finite group quotients. This design enables disentangled latent representations and analytically tractable KL divergences. By integrating differentiable coordinate transformations, group-invariant decoding, and anchor-point constraints, the approach ensures smooth gradients and topological consistency. To our knowledge, this is the first method to systematically align latent variable distributions with the intrinsic topology of data manifolds, supporting reparameterizable encoder–prior pairs and significantly outperforming Gaussian-prior baselines on synthetic manifolds as well as rotation- and cyclic-translation variants of MNIST.

Existing latent variable models often suffer from under-constrained objectives, leading to non-identifiable, ambiguous, and poorly interpretable representations. This work proposes the Constrained Latent State Modeling (CLSM) framework, which systematically integrates six core constraints—namely predictive sufficiency, minimality, temporal consistency, and others—for the first time. Grounded in information theory and dynamical systems theory, CLSM formally characterizes the intrinsic couplings and trade-offs among these constraints. By reframing representation learning as a constrained optimization problem, the framework unifies diverse approaches such as variational autoencoders and state-space models, revealing that non-identifiability stems from insufficient constraints rather than technical shortcomings. CLSM thus provides a principled foundation for designing latent variable models that are interpretable, robust, and aligned with downstream tasks.