Multiple Manifold Learning by Nonlinear Dimensionality Reduction

Abstract

Methods for nonlinear dimensionality reduction have been widely used for different purposes, but they are constrained to single manifold datasets. Considering that in real world applications, like video and image analysis, datasets with multiple manifolds are common, we propose a framework to find a low-dimensional embedding for data lying on multiple manifolds. Our approach is inspired on the manifold learning algorithm Laplacian Eigenmaps-LEM, computing the relationships among samples of different datasets based on an intra manifold comparison to unfold properly the data underlying structure. According to the results, our approach shows meaningful embeddings that outperform the results obtained by the conventional LEM algorithm and a previous close related work that analyzes multiple manifolds.

Characterizing interactions between cardiac shape and deformation by non-linear manifold learning

“…MML was applied in literature work to align datasets belonging to the same manifold with either different images of objects undergoing the same orientation changes (Valencia-Aguirre et al, 2011), or 2D MRI slices at different positions (Baumgartner et al, 2017;Clough et al, 2019). In our work, we aimed at explicitly characterizing the link between two physiologically related descriptors, namely cardiac shape and deformation, using the MML framework.…”

“…We considered the formulation of the matrix M (Valencia- Aguirre et al, 2011;Benkarim et al, 2020) where:…”

“…The low-dimensional representation is therefore estimated according to each descriptor and their correspondences. This framework has been illustrated for the alignment of different datasets (Valencia-Aguirre et al, 2011;Lee et al, 2016), of different magnetic resonance slices (Clough et al, 2019;Baumgartner et al, 2017) or to find regional associations between anatomical structures (Benkarim et al, 2020). These works slightly differ regarding the definition of the affinity matrix, and in particular its extra-diagonal blocks, formulated through permutation operators (Lee et al, 2016), correlations (Valencia-Aguirre et al, 2011Benkarim et al, 2020), or the graph structure of the datasets (Baumgartner et al, 2015;Clough et al, 2019).…”

“…Nevertheless, these works only aimed at aligning differ-ent datasets that originate from the same manifold, such as 2D MRI slices for 3D stacking purposes (Baumgartner et al, 2017;Clough et al, 2019) or images of objects represented from similar sets of viewpoints (Valencia-Aguirre et al, 2011)). Our problem is slightly different, as we consider descriptors belonging to different manifolds whose (unknown) link exists but can vary depending on the samples.…”

Learning Interactions Between Cardiac Shape and Deformation: Application to Pulmonary Hypertension

“…Methods that mix several high-dimensional heterogeneous descriptors all at once do not take into account their interactions and redundancy, which may limit the analysis. In methods reported in [8][9][10], an a nity matrix is established as composed of blocks that represent either the a nity between samples according to a single descriptor (diagonal blocks), or the interactions between two descriptors (extra diagonal blocks). Because of the nonlinearity of the methods and the unique output space returned, the reconstruction to the input space is not obvious.…”