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Probabilistic Geometric Principal Component Analysis with application to neural data

ICLR 2025 · Learning theory and algorithms › Matrix and tensor methods · approach: Sparse and robust PCA · 0 citations (OpenAlex)

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Proposes Probabilistic Geometric PCA (PGPCA), a PPCA generalization whose data distribution is centred on a data-fitted nonlinear manifold with Euclidean or geometric distribution coordinates, learned by an analytical EM algorithm.

Problem

PPCA and its extensions assume linear Euclidean models around the data mean, and existing nonlinear PPCA extensions require data to lie exactly on a manifold, which fails for noisy neural data distributed around a nonlinear manifold.

Approach

Model observations as a manifold point plus noise shaped by a distribution coordinate K(z) (Euclidean or geometric, from tangent and normal vectors), with a data-driven EM algorithm where the M-step reduces to a PPCA-style eigen-decomposition of a manifold-aware matrix Gamma(q); the log-likelihood then selects between coordinate systems.

Key results

• On the mouse head-direction data (Mouse12), PGPCA (GeCOV) reached a full-rank log-likelihood of -31.758 versus -34.622 for PPCA.
• PGPCA with the correct distribution coordinate had higher log-likelihood than the alternative in simulations (paired t-test p < 1.7e-12 for the 1D loop in R^2).
• Across six mice, PGPCA GeCOV had higher log-likelihood than both EuCOV and PPCA in every mouse, e.g. Mouse28: -24.752 vs -25.089 (EuCOV) vs -29.316 (PPCA).

Measured effects reported in the paper

Surprising findings

• Geometric distribution coordinates (GeCOV) fit the neural firing-rate data better than Euclidean ones, suggesting noise is organized around the manifold rather than in Euclidean space.

Limitations

• Static dimensionality reduction that does not model temporal autocorrelation in the data.
• Assumes the data distribution is stable over time.
• Manifold must be fitted beforehand from data (e.g., via TDA, splines), so manifold estimation errors are not modeled.

Experimental setup

Datasets: Mouse head-direction ADn firing rates (6 mice, public) Simulated loop in R^2 Simulated loop in R^10 Simulated torus in R^3

Benchmarks: Simulated loops and torus Mouse head-direction firing rate data (Mouse12, Mouse28 and others)

Metrics: trial-average data log-likelihood paired t-test

Baselines: PPCA Factor Analysis PGPCA (EuCOV)

Builds on: PPCA Factor Analysis Probabilistic principal geodesic analysis Isomap

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