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ManiFactor: Factor analysis for maps into manifolds
Factor analysis, principal component analysis or singular value decomposition (SVD) of a collection of points in a Euclidean space, which can be represented by a matrix, is a fundamental and omnipresent technique to reveal latent factors in the data. Factor analysis was generalized in psychometrics in the 1960s to data that consists of multiple modes, leading to the nowadays popular canonical polyadic decomposition (CPD) of tensors. Taking the equivalent viewpoint that matrices and tensors represent, respectively, linear and multilinear maps in coordinates, the aforementioned decompositions (SVD and CPD) can be viewed as approximating them by simpler, low-rank versions of these maps.In this project, motivated by concrete applications in computer vision and multiscale simulations, we will vastly generalize factor analysis to maps into manifolds. An inherent obstacle in this setting is that linear combinations of factors, yielding straight-line approximations, as used in the linear and multilinear case, are fundamentally incompatible with the curved geometry of smooth manifolds. This project will investigate several generalizations suited for combining (or joining) factors in the manifold settings, including replacing straight lines by geodesics. As in the matrix and tensor case, we say that maps that can be expressed with a small number of factors admit a low-rank ManiFactor decomposition. We will develop practical Riemannian optimization algorithms for approximating an implicitly given smooth map by such a decomposition. The functional approximation properties of the proposed model will be investigated.
Date:1 Oct 2021 → Today
Keywords:Riemannian optimization, smooth maps, tensor decompostion, principal component analysis, function approximation, factor analysis, canonical polyadic decomposition
Disciplines:Linear and multilinear algebra, matrix theory, Approximations and expansions, Differential geometry, Operations research and mathematical programming