The Geometry of Latent Spaces: Projective Invariants, Structural Identification, and Ratio-Scale Measurement
Abstract
Latent variable modeling of discrete ordinal data typically relies on Euclidean priors or continuous divergence bounds, inducing a topological mismatch that leaves models vulnerable to factor score indeterminacy, rotational unidentifiability, and Neyman-Scott incidental parameter bias. We introduce an axiomatic, coordinate-free framework that constructs continuous latent coordinate spaces directly from empirical partial orders without extrinsic probability measures. By parameterizing ordinal choice topologies via a regularized log-odds field on a continuous projective product manifold $(\mathbb{P}^1)^Q$, pairwise field differences preserve the invariants of the projective linear group $\mathrm{PGL}(2, \mathbb{R})^Q$. Spectral decomposition of the integrated invariant Gramian matrix reduces the identification fiber strictly to the one-dimensional dilation group $\mathbb{R}_{>0}^Q$. This establishes ratio-scale measurement with generalized specific objectivity, transcending the restrictive uniform-discrimination assumptions of the Rasch model, identically resolving both classical rotational ambiguity and Guttman’s factor score indeterminacy. We prove that an indicator array $P \ge 2Q+2$ is topologically necessary and sufficient to anchor a $Q$-dimensional metric space and structurally resolves incidental item parameters without joint, marginal, or conditional estimation. Finally, leveraging Hadamard differentiability and Hanson-Wright operator concentration, we establish that the projective spectral estimator is strongly consistent, asymptotically normal, and achieves optimal parametric concentration at rate $\mathcal{O}_P(P^{-1/2})$.