Latent representations are extensively used for tasks like visualization, interpolation, or feature extraction in deep learning models. This paper demonstrates the importance of considering the inductive bias imposed by an equivariant model when using latent representations as neglecting these biases can lead to decreased performance in downstream tasks. We propose principles for choosing invariant projections of latent representations and show their effectiveness in two examples: A permutation equivariant variational auto-encoder for molecular graph generation, where an invariant projection can be designed to maintain information without loss, and for a rotation-equivariant representation in image classification, where random invariant projections proves to retain a high degree of information. In both cases, the analysis of invariant latent representations proves superior to their equivariant counterparts. Finally, we illustrate that the phenomena documented here for equivariant neural networks have counterparts in standard neural networks where invariance is encouraged via augmentation.

Image registration aims to find geometric transformations that align images. Most algorithmic and deep learning-based methods solve the registration problem by minimizing a loss function, consisting of a similarity metric comparing the aligned images, and a regularization term ensuring smoothness of the transformation. Existing similarity metrics like Euclidean Distance or Normalized Cross-Correlation focus on aligning pixel intensity values or correlations, giving difficulties with low intensity contrast, noise, and ambiguous matching. We propose a semantic similarity metric for image registration, focusing on aligning image areas based on semantic correspondence instead. Our approach learns dataset-specific features that drive the optimization of a learning-based registration model. We train both an unsupervised approach extracting features with an auto-encoder, and a semi-supervised approach using supplemental segmentation data. We validate the semantic similarity metric using both deep-learning-based and algorithmic image registration methods. Compared to existing methods across four different image modalities and applications, the method achieves consistently high registration accuracy and smooth transformation fields.

Geometric alignment appears in a variety of applications, ranging from domain adaptation, optimal transport, and normalizing flows in machine learning; optical flow and learned augmentation in computer vision and deformable registration within biomedical imaging. A recurring challenge is the alignment of domains whose topology is not the same; a problem that is routinely ignored, potentially introducing bias in downstream analysis. As a first step towards solving such alignment problems, we propose an unsupervised algorithm for the detection of changes in image topology. The model is based on a conditional variational auto-encoder and detects topological changes between two images during the registration step. We account for both topological changes in the image under spatial variation and unexpected transformations. Our approach is validated on two tasks and datasets: detection of topological changes in microscopy images of cells, and unsupervised anomaly detection brain imaging.

The fact that machine learning models, despite their advancements, are still trained on randomly gathered data is proof that a lasting solution to the problem of optimal data gathering has not yet been found. In this paper, we investigate whether a Bayesian approach to the classification problem can provide assumptions under which one is guaranteed to perform at least as good as random sampling. For a logistic regression model, we show that maximal expected information gain on model parameters is a promising criterion for selecting samples, assuming that our classification model is well-matched to the data. Our derived criterion is closely related to the maximum model change. We experiment with data sets which satisfy this assumption to varying degrees to see how sensitive our performance is to the violation of our assumption in practice.

Latent variable models (LVMs) learn probabilistic models of data manifolds lying in an ambient Euclidean space. In a number of applications, a priori known spatial constraints can shrink the ambient space into a considerably smaller manifold. Additionally, in these applications the Euclidean geometry might induce a suboptimal similarity measure, which could be improved by choosing a different metric. Euclidean models ignore such information and assign probability mass to data points that can never appear as data, and vastly different likelihoods to points that are similar under the desired metric.We propose the wrapped Gaussian process latent variable model (WGPLVM), that extends Gaussian process latent variable models to take values strictly on a given Riemannian manifold, making the model blind to impossible data points. This allows non-linear, probabilistic inference of low-dimensional Riemannian submanifolds from data. Our evaluation on diverse datasets show that we improve performance on several tasks, including encoding, visualization and uncertainty quantification.

Gaussian process (GP) regression is a powerful tool in non-parametric regression providing uncertainty estimates. However, it is limited to data in vector spaces. In fields such as shape analysis and diffusion tensor imaging, the data often lies on a manifold, making GP regression nonviable, as the resulting predictive distribution does not live in the correct geometric space. We tackle the problem by defining wrapped Gaussian processes (WGPs) on Riemannian manifolds, using the probabilistic setting to generalize GP regression to the context of manifold-valued targets. The method is validated empirically on diffusion weighted imaging (DWI) data, directional data on the sphere and in the Kendall shape space, endorsing WGP regression as an efficient and flexible tool for manifold-valued regression.

We introduce a novel framework for statistical analysis of populations of non-degenerate Gaussian processes (GPs), which are natural representations of uncertain curves. This allows inherent variation or uncertainty in function-valued data to be properly incorporated in the population analysis. Using the 2-Wasserstein metric we geometrize the space of GPs with L2 mean and covariance functions over compact index spaces. We prove uniqueness of the barycenter of a population of GPs, as well as convergence of the metric and the barycenter of their finite-dimensional counterparts. This justifies practical computations. Finally, we demonstrate our framework through experimental validation on GP datasets representing brain connectivity and climate development. A Matlab library for relevant computations will be published at https://sites.google.com/view/antonmallasto/software.

Radial kernels are well-suited for machine learning over general geodesic metric spaces, where pairwise distances are often the only computable quantity available. We have recently shown that geodesic exponential kernels are only positive definite for all bandwidths when the input space has strong linear properties. This negative result hints that radial kernel are perhaps not suitable over geodesic metric spaces after all. Here, however, we present evidence that large intervals of bandwidths exist where geodesic exponential kernels have high probability of being positive definite over finite datasets, while still having significant predictive power. From this we formulate conjectures on the probability of a positive definite kernel matrix for a finite random sample, depending on the geometry of the data space and the spread of the sample.

In large datasets, manual data verification is impossible, and we must expect the number of outliers to increase with data size. While principal component analysis (PCA) can reduce data size, and scalable solutions exist, it is well-known that outliers can arbitrarily corrupt the results. Unfortunately, state-of-the-art approaches for robust PCA are not scalable. We note that in a zero-mean dataset, each observation spans a one-dimensional subspace, giving a point on the Grassmann manifold. We show that the average subspace corresponds to the leading principal component for Gaussian data. We provide a simple algorithm for computing this Grassmann Average ( GA), and show that the subspace estimate is less sensitive to outliers than PCA for general distributions. Because averages can be efficiently computed, we immediately gain scalability. We exploit robust averaging to formulate the Robust Grassmann Average (RGA) as a form of robust PCA. The resulting Trimmed Grassmann Average ( TGA) is appropriate for computer vision because it is robust to pixel outliers. The algorithm has linear computational complexity and minimal memory requirements. We demonstrate TGA for background modeling, video restoration, and shadow removal. We show scalability by performing robust PCA on the entire Star Wars IV movie; a task beyond any current method. Source code is available online.

We consider kernel methods on general geodesic metric spaces and provide both negative and positive results. First we show that the common Gaussian kernel can only be generalized to a positive definite kernel on a geodesic metric space if the space is flat. As a result, for data on a Riemannian manifold, the geodesic Gaussian kernel is only positive definite if the Riemannian manifold is Euclidean. This implies that any attempt to design geodesic Gaussian kernels on curved Riemannian manifolds is futile. However, we show that for spaces with conditionally negative definite distances the geodesic Laplacian kernel can be generalized while retaining positive definiteness. This implies that geodesic Laplacian kernels can be generalized to some curved spaces, including spheres and hyperbolic spaces. Our theoretical results are verified empirically.

We present a fast and robust atlas-based algorithm for labeling airway trees, using geodesic distances in a geometric tree-space. Possible branch label configurations for an unlabeled airway tree are evaluated using distances to a training set of labeled airway trees. In tree-space, airway tree topology and geometry change continuously, giving a natural automatic handling of anatomical differences and noise. A hierarchical approach makes the algorithm efficient, assigning labels from the trachea and downwards. Only the airway centerline tree is used, which is relatively unaffected by pathology. The algorithm is evaluated on 80 segmented airway trees from 40 subjects at two time points, labeled by three medical experts each, testing accuracy, reproducibility and robustness in patients with chronic obstructive pulmonary disease (COPD). The accuracy of the algorithm is statistically similar to that of the experts and not significantly correlated with COPD severity. The reproducibility of the algorithm is significantly better than that of the experts, and negatively correlated with COPD severity. Evaluation of the algorithm on a longitudinal set of 8724 trees from a lung cancer screening trial shows that the algorithm can be used in large scale studies with high reproducibility, and that the negative correlation of reproducibility with COPD severity can be explained by missing branches, for instance due to segmentation problems in COPD patients. We conclude that the algorithm is robust to COPD severity given equally complete airway trees, and comparable in performance to that of experts in pulmonary medicine, emphasizing the suitability of the labeling algorithm for clinical use.

As the collection of large datasets becomes increasingly automated, the occurrence of outliers will increase -- "big data" implies "big outliers". While principal component analysis (PCA) is often used to reduce the size of data, and scalable solutions exist, it is well-known that outliers can arbitrarily corrupt the results. Unfortunately, state-of-the-art approaches for robust PCA do not scale beyond small-to-medium sized datasets. To address this, we introduce the Grassmann Average (GA), which expresses dimensionality reduction as an average of the subspaces spanned by the data. Because averages can be efficiently computed, we immediately gain scalability. GA is inherently more robust than PCA, but we show that they coincide for Gaussian data. We exploit that averages can be made robust to formulate the Robust Grassmann Average (RGA) as a form of robust PCA. Robustness can be with respect to vectors (subspaces) or elements of vectors, we focus on the latter and use a trimmed average. The resulting Trimmed Grassmann Average (TGA) is particularly appropriate for computer vision because it is robust to pixel outliers. The algorithm has low computational complexity and minimal memory requirements, making it scalable to "big noisy data." We demonstrate TGA for background modeling, video restoration, and shadow removal. We show scalability by performing robust PCA on the entire Star Wars IV movie.

While graphs with continuous node attributes arise in many applications, state-of-the-art graph kernels for comparing continuous-attributed graphs suffer from a high runtime complexity; for instance, the popular shortest path kernel scales as $\mathcal{O}(n^4)$, where $n$ is the number of nodes. In this paper, we present a class of path kernels with computational complexity $\mathcal{O}(n^2 (m + \delta^2))$, where $\delta$ is the graph diameter and $m$ the number of edges. Due to the sparsity and small diameter of real-world graphs, these kernels scale comfortably to large graphs. In our experiments, the presented kernels outperform state-of-the-art kernels in terms of speed and accuracy on classification benchmark datasets.

To develop statistical methods for shapes with a tree-structure, we construct a shape space framework for tree-shapes and study metrics on the shape space. This shape space has singularities which correspond to topological transitions in the represented trees. We study two closely related metrics on the shape space, TED and QED. QED is a quotient euclidean distance arising naturally from the shape space formulation, while TED is the classical tree edit distance. Using Gromov's metric geometry, we gain new insight into the geometries defined by TED and QED. We show that the new metric QED has nice geometric properties that are needed for statistical analysis: Geodesics always exist and are generically locally unique. Following this, we can also show the existence and generic local uniqueness of average trees for QED. TED, while having some algorithmic advantages, does not share these advantages. Along with the theoretical framework we provide experimental proof-of-concept results on synthetic data trees as well as small airway trees from pulmonary CT scans. This way, we illustrate that our framework has promising theoretical and qualitative properties necessary to build a theory of statistical tree-shape analysis.

The mean is often the most important statistic of a dataset as it provides a single point that summarizes the entire set. While the mean is readily defined and computed in Euclidean spaces, no commonly accepted solutions are currently available in more complicated spaces, such as spaces of tree-structured data. In this paper we study the notion of means, both generally in Gromov's CAT(0)-spaces (metric spaces of non-positive curvature), but also specifically in the space of tree-like shapes. We prove local existence and uniqueness of means in such spaces and discuss three different algorithms for computing means. We make an experimental evaluation of the three algorithms through experiments on three different sets of data with tree-like structure: a synthetic dataset, a leaf morphology dataset from images, and a set of human airway subtrees from medical CT scans. This experimental study provides great insight into the behavior of the different methods and how they relate to each other. More importantly, it also provides mathematically well-founded, tractable and robust “average trees”. This statistic is of utmost importance due to the ever-presence of tree-like structures in human anatomy, e.g., airways and vascularization systems.

This paper presents a new geometric framework for analysis of planar treelike shapes for applications such as shape matching, recognition and morphology, using the geometry of the space of treelike shapes. Mathematically, the shape space is given the structure of a stratified set which is a quotient of a normed vector space with a metric inherited from the vector space norm. We give examples of geodesic paths in tree-space corresponding to fundamental deformations of small trees, and discuss how these deformations are key building blocks for understanding deformations between larger trees.