Parametrizing Product Shape Manifolds by Composite Networks

Parametrizations of data manifolds in shape spaces can be computed using the rich toolbox of Riemannian geometry. This, however, often comes with high computational costs, which raises the question if one can learn an efficient neural network approximation. We show that this is indeed possible for shape spaces with a special product structure, namely those smoothly approximable by a direct sum of low-dimensional manifolds. Our proposed architecture leverages this structure by separately learning approximations for the low-dimensional factors and a subsequent combination. After developing the approach as a general framework, we apply it to a shape space of triangular surfaces. Here, typical examples of data manifolds are given through datasets of articulated models and can be factorized, for example, by a Sparse Principal Geodesic Analysis (SPGA). We demonstrate the effectiveness of our proposed approach with experiments on synthetic data as well as manifolds extracted from data via SPGA

Convergent autoencoder approximation of low bending and low distortion manifold embeddings

Autoencoders, which consist of an encoder and a decoder, are widely used in machine learning for dimension reduction of high-dimensional data. The encoder embeds the input data manifold into a lower-dimensional latent space, while the decoder represents the inverse map, providing a parametrization of the data manifold by the manifold in latent space. A good regularity and structure of the embedded manifold may substantially simplify further data processing tasks such as cluster analysis or data interpolation. We propose and analyze a novel regularization for learning the encoder component of an autoencoder: a loss functional that prefers isometric, extrinsically flat embeddings and allows to train the encoder on its own. To perform the training it is assumed that for pairs of nearby points on the input manifold their local Riemannian distance and their local Riemannian average can be evaluated. The loss functional is computed via Monte Carlo integration with different sampling strategies for pairs of points on the input manifold. Our main theorem identifies a geometric loss functional of the embedding map as the $\Gamma$-limit of the sampling-dependent loss functionals. Numerical tests, using image data that encodes different explicitly given data manifolds, show that smooth manifold embeddings into latent space are obtained. Due to the promotion of extrinsic flatness, these embeddings are regular enough such that interpolation between not too distant points on the manifold is well approximated by linear interpolation in latent space as one possible postprocessing.Comment: 27 pages, 10 figures. This publication is an extended version of the previous conference proceeding presented at DiffCVML 202

An APRI+ALBI Based Multivariable Model as Preoperative Predictor for Posthepatectomy Liver Failure.

OBJECTIVE AND BACKGROUND Clinically significant posthepatectomy liver failure (PHLF B+C) remains the main cause of mortality after major hepatic resection. This study aimed to establish an APRI+ALBI, aspartate aminotransferase to platelet ratio (APRI) combined with albumin-bilirubin grade (ALBI), based multivariable model (MVM) to predict PHLF and compare its performance to indocyanine green clearance (ICG-R15 or ICG-PDR) and albumin-ICG evaluation (ALICE). METHODS 12,056 patients from the National Surgical Quality Improvement Program (NSQIP) database were used to generate a MVM to predict PHLF B+C. The model was determined using stepwise backwards elimination. Performance of the model was tested using receiver operating characteristic curve analysis and validated in an international cohort of 2,525 patients. In 620 patients, the APRI+ALBI MVM, trained in the NSQIP cohort, was compared with MVM's based on other liver function tests (ICG clearance, ALICE) by comparing the areas under the curve (AUC). RESULTS A MVM including APRI+ALBI, age, sex, tumor type and extent of resection was found to predict PHLF B+C with an AUC of 0.77, with comparable performance in the validation cohort (AUC 0.74). In direct comparison with other MVM's based on more expensive and time-consuming liver function tests (ICG clearance, ALICE), the APRI+ALBI MVM demonstrated equal predictive potential for PHLF B+C. A smartphone application for calculation of the APRI+ALBI MVM was designed. CONCLUSION Risk assessment via the APRI+ALBI MVM for PHLF B+C increases preoperative predictive accuracy and represents an universally available and cost-effective risk assessment prior to hepatectomy, facilitated by a freely available smartphone app