The Central Spanning Tree Problem

Spanning trees are an important primitive in many data analysis tasks, when a data set needs to be summarized in terms of its "skeleton", or when a tree-shaped graph over all observations is required for downstream processing. Popular definitions of spanning trees include the minimum spanning tree and the optimum distance spanning tree, a.k.a. the minimum routing cost tree. When searching for the shortest spanning tree but admitting additional branching points, even shorter spanning trees can be realized: Steiner trees. Unfortunately, both minimum spanning and Steiner trees are not robust with respect to noise in the observations; that is, small perturbations of the original data set often lead to drastic changes in the associated spanning trees. In response, we make two contributions when the data lies in a Euclidean space: on the theoretical side, we introduce a new optimization problem, the "(branched) central spanning tree", which subsumes all previously mentioned definitions as special cases. On the practical side, we show empirically that the (branched) central spanning tree is more robust to noise in the data, and as such is better suited to summarize a data set in terms of its skeleton. We also propose a heuristic to address the NP-hard optimization problem, and illustrate its use on single cell RNA expression data from biology and 3D point clouds of plants.

Generative Modeling of Discrete Joint Distributions by E-Geodesic Flow Matching on Assignment Manifolds

This paper introduces a novel generative model for discrete distributions based on continuous normalizing flows on the submanifold of factorizing discrete measures. Integration of the flow gradually assigns categories and avoids issues of discretizing the latent continuous model like rounding, sample truncation etc. General non-factorizing discrete distributions capable of representing complex statistical dependencies of structured discrete data, can be approximated by embedding the submanifold into a the meta-simplex of all joint discrete distributions and data-driven averaging. Efficient training of the generative model is demonstrated by matching the flow of geodesics of factorizing discrete distributions. Various experiments underline the approach's broad applicability.

On the Universality of Coupling-based Normalizing Flows

We present a novel theoretical framework for understanding the expressive power of coupling-based normalizing flows such as RealNVP. Despite their prevalence in scientific applications, a comprehensive understanding of coupling flows remains elusive due to their restricted architectures. Existing theorems fall short as they require the use of arbitrarily ill-conditioned neural networks, limiting practical applicability. Additionally, we demonstrate that these constructions inherently lead to volume-preserving flows, a property which we show to be a fundamental constraint for expressivity. We propose a new distributional universality theorem for coupling-based normalizing flows, which overcomes several limitations of prior work. Our results support the general wisdom that the coupling architecture is expressive and provide a nuanced view for choosing the expressivity of coupling functions, bridging a gap between empirical results and theoretical understanding.

Quantum State Assignment Flows

This paper introduces assignment flows for density matrices as state spaces for representing and analyzing data associated with vertices of an underlying weighted graph. Determining an assignment flow by geometric integration of the defining dynamical system causes an interaction of the non-commuting states across the graph, and the assignment of a pure (rank-one) state to each vertex after convergence. Adopting the Riemannian Bogoliubov-Kubo-Mori metric from information geometry leads to closed-form local expressions which can be computed efficiently and implemented in a fine-grained parallel manner. Restriction to the submanifold of commuting density matrices recovers the assignment flows for categorial probability distributions, which merely assign labels from a finite set to each data point. As shown for these flows in our prior work, the novel class of quantum state assignment flows can also be characterized as Riemannian gradient flows with respect to a non-local non-convex potential, after proper reparametrization and under mild conditions on the underlying weight function. This weight function generates the parameters of the layers of a neural network, corresponding to and generated by each step of the geometric integration scheme. Numerical results indicates and illustrate the potential of the novel approach for data representation and analysis, including the representation of correlations of data across the graph by entanglement and tensorization.

On the Convergence Rate of Gaussianization with Random Rotations

Gaussianization is a simple generative model that can be trained without backpropagation. It has shown compelling performance on low dimensional data. As the dimension increases, however, it has been observed that the convergence speed slows down. We show analytically that the number of required layers scales linearly with the dimension for Gaussian input. We argue that this is because the model is unable to capture dependencies between dimensions. Empirically, we find the same linear increase in cost for arbitrary input $p(x)$, but observe favorable scaling for some distributions. We explore potential speed-ups and formulate challenges for further research.

On Certified Generalization in Structured Prediction

In structured prediction, target objects have rich internal structure which does not factorize into independent components and violates common i.i.d. assumptions. This challenge becomes apparent through the exponentially large output space in applications such as image segmentation or scene graph generation. We present a novel PAC-Bayesian risk bound for structured prediction wherein the rate of generalization scales not only with the number of structured examples but also with their size. The underlying assumption, conforming to ongoing research on generative models, is that data are generated by the Knothe-Rosenblatt rearrangement of a factorizing reference measure. This allows to explicitly distill the structure between random output variables into a Wasserstein dependency matrix. Our work makes a preliminary step towards leveraging powerful generative models to establish generalization bounds for discriminative downstream tasks in the challenging setting of structured prediction.

Whitening Convergence Rate of Coupling-based Normalizing Flows

Coupling-based normalizing flows (e.g. RealNVP) are a popular family of normalizing flow architectures that work surprisingly well in practice. This calls for theoretical understanding. Existing work shows that such flows weakly converge to arbitrary data distributions. However, they make no statement about the stricter convergence criterion used in practice, the maximum likelihood loss. For the first time, we make a quantitative statement about this kind of convergence: We prove that all coupling-based normalizing flows perform whitening of the data distribution (i.e. diagonalize the covariance matrix) and derive corresponding convergence bounds that show a linear convergence rate in the depth of the flow. Numerical experiments demonstrate the implications of our theory and point at open questions.

A Nonlocal Graph-PDE and Higher-Order Geometric Integration for Image Labeling

This paper introduces a novel nonlocal partial difference equation (PDE) for labeling metric data on graphs. The PDE is derived as nonlocal reparametrization of the assignment flow approach that was introduced in \textit{J.~Math.~Imaging \& Vision} 58(2), 2017. Due to this parameterization, solving the PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with respect to a nonconvex potential. We devise an entropy-regularized difference-of-convex-functions (DC) decomposition of this potential and show that the basic geometric Euler scheme for integrating the assignment flow is equivalent to solving the PDE by an established DC programming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit higher-order information of the vector field that drives the assignment flow, in order to devise a novel accelerated DC programming scheme. A detailed convergence analysis of both numerical schemes is provided and illustrated by numerical experiments.

Self-Certifying Classification by Linearized Deep Assignment

We propose a novel class of deep stochastic predictors for classifying metric data on graphs within the PAC-Bayes risk certification paradigm. Classifiers are realized as linearly parametrized deep assignment flows with random initial conditions. Building on the recent PAC-Bayes literature and data-dependent priors, this approach enables (i) to use risk bounds as training objectives for learning posterior distributions on the hypothesis space and (ii) to compute tight out-of-sample risk certificates of randomized classifiers more efficiently than related work. Comparison with empirical test set errors illustrates the performance and practicality of this self-certifying classification method.

Multi-view Monocular Depth and Uncertainty Prediction with Deep SfM in Dynamic Environments

3D reconstruction of depth and motion from monocular video in dynamic environments is a highly ill-posed problem due to scale ambiguities when projecting to the 2D image domain. In this work, we investigate the performance of the current State-of-the-Art (SotA) deep multi-view systems in such environments. We find that current supervised methods work surprisingly well despite not modelling individual object motions, but make systematic errors due to a lack of dense ground truth data. To detect such errors during usage, we extend the cost volume based Deep Video to Depth (DeepV2D) framework \cite{teed2018deepv2d} with a learned uncertainty. Our Deep Video to certain Depth (DeepV2cD) model allows i) to perform en par or better with current SotA and ii) achieve a better uncertainty measure than the naive Shannon entropy. Our experiments show that a simple filter strategy based on the uncertainty can significantly reduce systematic errors. This results in cleaner reconstructions both on static and dynamic parts of the scene.