Quantization and clustering on Riemannian manifolds with an application to air traffic analysis

International audienceThe goal of quantization is to find the best approximation of a probability distribution by a discrete measure with finite support. When dealing with empirical distributions, this boils down to finding the best summary of the data by a smaller number of points, and automatically yields a k-means-type clustering. In this paper, we introduce Competitive Learning Riemannian Quantization (CLRQ), an online quantization algorithm that applies when the data does not belong to a vector space, but rather a Riemannian manifold. It can be seen as a density approximation procedure as well as a clustering method. Compared to many clustering algorihtms, it requires few distance computations, which is particularly computationally advantageous in the manifold setting. We prove its convergence and show simulated examples on the sphere and the hyperbolic plane. We also provide an application to real data by using CLRQ to create summaries of images of covariance matrices estimated from air traffic images. These summaries are representative of the air traffic complexity and yield clusterings of the airspaces into zones that are homogeneous with respect to that criterion. They can then be compared using discrete optimal transport and be further used as inputs of a machine learning algorithm or as indexes in a traffic database

Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimization

Recent research indicates that the performance of machine learning models can be improved by aligning the geometry of the latent space with the underlying data structure. Rather than relying solely on Euclidean space, researchers have proposed using hyperbolic and spherical spaces with constant curvature, or combinations thereof, to better model the latent space and enhance model performance. However, little attention has been given to the problem of automatically identifying the optimal latent geometry for the downstream task. We mathematically define this novel formulation and coin it as neural latent geometry search (NLGS). More specifically, we introduce a principled method that searches for a latent geometry composed of a product of constant curvature model spaces with minimal query evaluations. To accomplish this, we propose a novel notion of distance between candidate latent geometries based on the Gromov-Hausdorff distance from metric geometry. In order to compute the Gromov-Hausdorff distance, we introduce a mapping function that enables the comparison of different manifolds by embedding them in a common high-dimensional ambient space. Finally, we design a graph search space based on the calculated distances between candidate manifolds and use Bayesian optimization to search for the optimal latent geometry in a query-efficient manner. This is a general method which can be applied to search for the optimal latent geometry for a variety of models and downstream tasks. Extensive experiments on synthetic and real-world datasets confirm the efficacy of our method in identifying the optimal latent geometry for multiple machine learning problems

3D time series analysis of cell shape using Laplacian approaches

Background: Fundamental cellular processes such as cell movement, division or food uptake critically depend on cells being able to change shape. Fast acquisition of three-dimensional image time series has now become possible, but we lack efficient tools for analysing shape deformations in order to understand the real three-dimensional nature of shape changes. Results: We present a framework for 3D+time cell shape analysis. The main contribution is three-fold: First, we develop a fast, automatic random walker method for cell segmentation. Second, a novel topology fixing method is proposed to fix segmented binary volumes without spherical topology. Third, we show that algorithms used for each individual step of the analysis pipeline (cell segmentation, topology fixing, spherical parameterization, and shape representation) are closely related to the Laplacian operator. The framework is applied to the shape analysis of neutrophil cells. Conclusions: The method we propose for cell segmentation is faster than the traditional random walker method or the level set method, and performs better on 3D time-series of neutrophil cells, which are comparatively noisy as stacks have to be acquired fast enough to account for cell motion. Our method for topology fixing outperforms the tools provided by SPHARM-MAT and SPHARM-PDM in terms of their successful fixing rates. The different tasks in the presented pipeline for 3D+time shape analysis of cells can be solved using Laplacian approaches, opening the possibility of eventually combining individual steps in order to speed up computations

Neural latent geometry search: product manifold inference via Gromov-Hausdorff-informed Bayesian optimization

Recent research indicates that the performance of machine learning models can be improved by aligning the geometry of the latent space with the underlying data structure. Rather than relying solely on Euclidean space, researchers have proposed using hyperbolic and spherical spaces with constant curvature, or combinations thereof, to better model the latent space and enhance model performance. However, little attention has been given to the problem of automatically identifying the optimal latent geometry for the downstream task. We mathematically define this novel formulation and coin it as neural latent geometry search (NLGS). More specifically, we introduce an initial attempt to search for a latent geometry composed of a product of constant curvature model spaces with a small number of query evaluations, under some simplifying assumptions. To accomplish this, we propose a novel notion of distance between candidate latent geometries based on the Gromov-Hausdorff distance from metric geometry. In order to compute the Gromov-Hausdorff distance, we introduce a mapping function that enables the comparison of different manifolds by embedding them in a common high-dimensional ambient space. We then design a graph search space based on the notion of smoothness between latent geometries and employ the calculated distances as an additional inductive bias. Finally, we use Bayesian optimization to search for the optimal latent geometry in a query-efficient manner. This is a general method which can be applied to search for the optimal latent geometry for a variety of models and downstream tasks. We perform experiments on synthetic and real-world datasets to identify the optimal latent geometry for multiple machine learning problems

An accurate fingerprint reference point determination method based on curvature estimation of separated ridges

This paper presents an effective method for the detection of a fingerprint’s reference point by analyzing fingerprint ridges’ curvatures. The proposed approach is a multi-stage system. The first step extracts the fingerprint ridges from an image and transforms them into chains of discrete points. In the second step, the obtained chains of points are processed by a dedicated algorithm to detect corners and other points of highest curvature on their planar surface. In a series of experiments we demonstrate that the proposed method based on this algorithm allows effective determination of fingerprint reference points. Furthermore, the proposed method is relatively simple and achieves better results when compared with the approaches known from the literature. The reference point detection experiments were conducted using publicly available fingerprint databases FVC2000, FVC2002, FVC2004 and NIST