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Moving object detection and segmentation from a single moving camera is a challenging task, requiring an understanding of recognition, motion and 3D geometry. Combining both recognition and reconstruction boils down to a fusion problem, where appearance and motion features need to be combined for classification and segmentation. In this paper, we present a novel fusion architecture for monocular motion segmentation - M3Former, which leverages the strong performance of transformers for segmentation and multi-modal fusion. As reconstructing motion from monocular video is ill-posed, we systematically analyze different 2D and 3D motion representations for this problem and their importance for segmentation performance. Finally, we analyze the effect of training data and show that diverse datasets are required to achieve SotA performance on Kitti and Davis., Comment: WICCV2023

Recent progress in scene synthesis makes standalone SLAM systems purely based on optimizing hyperprimitives with a Rendering objective possible. However, the tracking performance still lacks behind traditional and end-to-end SLAM systems. An optimal trade-off between robustness, speed and accuracy has not yet been reached, especially for monocular video. In this paper, we introduce a SLAM system based on an end-to-end Tracker and extend it with a Renderer based on recent 3D Gaussian Splatting techniques. Our framework \textbf{DroidSplat} achieves both SotA tracking and rendering results on common SLAM benchmarks. We implemented multiple building blocks of modern SLAM systems to run in parallel, allowing for fast inference on common consumer GPU's. Recent progress in monocular depth prediction and camera calibration allows our system to achieve strong results even on in-the-wild data without known camera intrinsics. Code will be available at \url{https://github.com/ChenHoy/DROID-Splat}.

We consider a Gibbs distribution over all spanning trees of an undirected, edge weighted finite graph, where, up to normalization, the probability of each tree is given by the product of its edge weights. Defining the weighted degree of a node as the sum of the weights of its incident edges, we present analytical expressions for the expectation, variance and covariance of the weighted degree of a node across the Gibbs distribution. To generalize our approach, we distinguish between two types of weight: probability weights, which regulate the distribution of spanning trees, and degree weights, which define the weighted degree of nodes. This distinction allows us to define the weighted degree of nodes independently of the probability weights. By leveraging the Matrix Tree Theorem, we show that these degree moments ultimately depend on the inverse of a submatrix of the graph Laplacian. While our focus is on undirected graphs, we demonstrate that our results can be extended to the directed setting by considering incoming directed trees instead.

This paper introduces the sigma flow model for the prediction of structured labelings of data observed on Riemannian manifolds, including Euclidean image domains as special case. The approach combines the Laplace-Beltrami framework for image denoising and enhancement, introduced by Sochen, Kimmel and Malladi about 25 years ago, and the assignment flow approach introduced and studied by the authors. The sigma flow arises as Riemannian gradient flow of generalized harmonic energies and thus is governed by a nonlinear geometric PDE which determines a harmonic map from a closed Riemannian domain manifold to a statistical manifold, equipped with the Fisher-Rao metric from information geometry. A specific ingredient of the sigma flow is the mutual dependency of the Riemannian metric of the domain manifold on the evolving state. This makes the approach amenable to machine learning in a specific way, by realizing this dependency through a mapping with compact time-variant parametrization that can be learned from data. Proof of concept experiments demonstrate the expressivity of the sigma flow model and prediction performance. Structural similarities to transformer network architectures and networks generated by the geometric integration of sigma flows are pointed out, which highlights the connection to deep learning and, conversely, may stimulate the use of geometric design principles for structured prediction in other areas of scientific machine learning., Comment: 51 pages

Density-based distances (DBDs) offer an elegant solution to the problem of metric learning. By defining a Riemannian metric which increases with decreasing probability density, shortest paths naturally follow the data manifold and points are clustered according to the modes of the data. We show that existing methods to estimate Fermat distances, a particular choice of DBD, suffer from poor convergence in both low and high dimensions due to i) inaccurate density estimates and ii) reliance on graph-based paths which are increasingly rough in high dimensions. To address these issues, we propose learning the densities using a normalizing flow, a generative model with tractable density estimation, and employing a smooth relaxation method using a score model initialized from a graph-based proposal. Additionally, we introduce a dimension-adapted Fermat distance that exhibits more intuitive behavior when scaled to high dimensions and offers better numerical properties. Our work paves the way for practical use of density-based distances, especially in high-dimensional spaces.

We introduce a novel generative model for the representation of joint probability distributions of a possibly large number of discrete random variables. The approach uses measure transport by randomized assignment flows on the statistical submanifold of factorizing distributions, which also enables to sample efficiently from the target distribution and to assess the likelihood of unseen data points. The embedding of the flow via the Segre map in the meta-simplex of all discrete joint distributions ensures that any target distribution can be represented in principle, whose complexity in practice only depends on the parametrization of the affinity function of the dynamical assignment flow system. Our model can be trained in a simulation-free manner without integration by conditional Riemannian flow matching, using the training data encoded as geodesics in closed-form with respect to the e-connection of information geometry. By projecting high-dimensional flow matching in the meta-simplex of joint distributions to the submanifold of factorizing distributions, our approach has strong motivation from first principles of modeling coupled discrete variables. Numerical experiments devoted to distributions of structured image labelings demonstrate the applicability to large-scale problems, which may include discrete distributions in other application areas. Performance measures show that our approach scales better with the increasing number of classes than recent related work.

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.

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.

We present a novel theoretical framework for understanding the expressive power of normalizing flows. Despite their prevalence in scientific applications, a comprehensive understanding of 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. We propose a distributional universality theorem for well-conditioned coupling-based normalizing flows such as RealNVP. In addition, we show that volume-preserving normalizing flows are not universal, what distribution they learn instead, and how to fix their expressivity. Our results support the general wisdom that affine and related couplings are expressive and in general outperform volume-preserving flows, bridging a gap between empirical results and theoretical understanding., Comment: Proceedings of the 41 st International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024

This paper studies a meta-simplex concept and geometric embedding framework for multi-population replicator dynamics. Central results are two embedding theorems which constitute a formal reduction of multi-population replicator dynamics to single-population ones. In conjunction with a robust mathematical formalism, this provides a toolset for analyzing complex multi-population models. Our framework provides a unifying perspective on different population dynamics in the literature which in particular enables to establish a formal link between multi-population and multi-game dynamics.

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.

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.

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.

Physical systems can be used as an information processing substrate and with that extend traditional computing architectures. For such an application the experimental platform must guarantee pristine control of the initial state, the temporal evolution and readout. All these ingredients are provided by modern experimental realizations of atomic Bose Einstein condensates. By embedding the nonlinear evolution of a quantum gas in a Machine Learning pipeline, one can represent nonlinear functions while only linear operations on classical computing of the pipeline are necessary. We demonstrate successful regression and interpolation of a nonlinear function using a quasi one-dimensional cloud of potassium atoms and characterize the performance of our system.

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., Comment: Proceedings of 36th Conference on Neural Information Processing System (NeurIPS 2022)

This paper introduces a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-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 G-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 G-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.

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.

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., Comment: 20 pages, 5 figures, 3 tables, submitted to ICPRAI 2022

Metric data labeling refers to the task of assigning one of multiple predefined labels to every given datapoint based on the metric distance between label and data. This assignment of labels typically takes place in a spatial or spatio-temporal context. Assignment flows are a class of dynamical models for metric data labeling that evolve on a basic statistical manifold, the so called assignment manifold, governed by a system of coupled replicator equations. In this paper we generalize the result of a recent paper for uncoupled replicator equations and adopting the viewpoint of geometric mechanics, relate assignment flows to critical points of an action functional via the associated Euler-Lagrange equation. We also show that not every assignment flow is a critical point and characterize precisely the class of coupled replicator equations fulfilling this relation, a condition that has been missing in recent related work. Finally, some consequences of this connection to Lagrangian mechanics are investigated including the fact that assignment flows are, up to initial conditions of measure zero, reparametrized geodesics of the so-called Jacobi metric.

The Turing mechanism describes the emergence of spatial patterns due to spontaneous symmetry breaking in reaction-diffusion processes and underlies many developmental processes. Identifying Turing mechanisms in biological systems defines a challenging problem. This paper introduces an approach to the prediction of Turing parameter values from observed Turing patterns. The parameter values correspond to a parametrized system of reaction-diffusion equations that generate Turing patterns as steady state. The Gierer-Meinhardt model with four parameters is chosen as a case study. A novel invariant pattern representation based on resistance distance histograms is employed, along with Wasserstein kernels, in order to cope with the highly variable arrangement of local pattern structure that depends on the initial conditions which are assumed to be unknown. This enables to compute physically plausible distances between patterns, to compute clusters of patterns and, above all, model parameter prediction: for small training sets, classical state-of-the-art methods including operator-valued kernels outperform neural networks that are applied to raw pattern data, whereas for large training sets the latter are more accurate. Excellent predictions are obtained for single parameter values and reasonably accurate results for jointly predicting all parameter values., Comment: 32 pages, 10 figures

We introduce a novel algorithm for estimating optimal parameters of linearized assignment flows for image labeling. An exact formula is derived for the parameter gradient of any loss function that is constrained by the linear system of ODEs determining the linearized assignment flow. We show how to efficiently evaluate this formula using a Krylov subspace and a low-rank approximation. This enables us to perform parameter learning by Riemannian gradient descent in the parameter space, without the need to backpropagate errors or to solve an adjoint equation. Experiments demonstrate that our method performs as good as highly-tuned machine learning software using automatic differentiation. Unlike methods employing automatic differentiation, our approach yields a low-dimensional representation of internal parameters and their dynamics which helps to understand how assignment flows and more generally neural networks work and perform.

The assignment flow recently introduced in the J. Math. Imaging and Vision 58/2 (2017), constitutes a high-dimensional dynamical system that evolves on an elementary statistical manifold and performs contextual labeling (classification) of data given in any metric space. Vertices of a given graph index the data points and define a system of neighborhoods. These neighborhoods together with nonnegative weight parameters define regularization of the evolution of label assignments to data points, through geometric averaging induced by the affine e-connection of information geometry. Regarding evolutionary game dynamics, the assignment flow may be characterized as a large system of replicator equations that are coupled by geometric averaging. This paper establishes conditions on the weight parameters that guarantee convergence of the continuous-time assignment flow to integral assignments (labelings), up to a negligible subset of situations that will not be encountered when working with real data in practice. Furthermore, we classify attractors of the flow and quantify corresponding basins of attraction. This provides convergence guarantees for the assignment flow which are extended to the discrete-time assignment flow that results from applying a Runge-Kutta-Munthe-Kaas scheme for numerical geometric integration of the assignment flow. Several counter-examples illustrate that violating the conditions may entail unfavorable behavior of the assignment flow regarding contextual data classification., Comment: This version of the article has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s41884-021-00060-8

We conduct a study and comparison of superiorization and optimization approaches for the reconstruction problem of superiorized/regularized least-squares solutions of underdetermined linear equations with nonnegativity variable bounds. Regarding superiorization, the state of the art is examined for this problem class, and a novel approach is proposed that employs proximal mappings and is structurally similar to the established forward-backward optimization approach. Regarding convex optimization, accelerated forward-backward splitting with inexact proximal maps is worked out and applied to both the natural splitting least-squares term/regularizer and to the reverse splitting regularizer/least-squares term. Our numerical findings suggest that superiorization can approach the solution of the optimization problem and leads to comparable results at significantly lower costs, after appropriate parameter tuning. On the other hand, applying accelerated forward-backward optimization to the reverse splitting slightly outperforms superiorization, which suggests that convex optimization can approach superiorization too, using a suitable problem splitting.

This paper extends the recently introduced assignment flow approach for supervised image labeling to unsupervised scenarios where no labels are given. The resulting self-assignment flow takes a pairwise data affinity matrix as input data and maximizes the correlation with a low-rank matrix that is parametrized by the variables of the assignment flow, which entails an assignment of the data to themselves through the formation of latent labels (feature prototypes). A single user parameter, the neighborhood size for the geometric regularization of assignments, drives the entire process. By smooth geodesic interpolation between different normalizations of self-assignment matrices on the positive definite matrix manifold, a one-parameter family of self-assignment flows is defined. Accordingly, our approach can be characterized from different viewpoints, e.g. as performing spatially regularized, rank-constrained discrete optimal transport, or as computing spatially regularized normalized spectral cuts. Regarding combinatorial optimization, our approach successfully determines completely positive factorizations of self-assignments in large-scale scenarios, subject to spatial regularization. Various experiments including the unsupervised learning of patch dictionaries using a locally invariant distance function, illustrate the properties of the approach., Comment: 42 pages, 17 figures

We study the inverse problem of model parameter learning for pixelwise image labeling, using the linear assignment flow and training data with ground truth. This is accomplished by a Riemannian gradient flow on the manifold of parameters that determine the regularization properties of the assignment flow. Using the symplectic partitioned Runge--Kutta method for numerical integration, it is shown that deriving the sensitivity conditions of the parameter learning problem and its discretization commute. A convenient property of our approach is that learning is based on exact inference. Carefully designed experiments demonstrate the performance of our approach, the expressiveness of the mathematical model as well as its limitations, from the viewpoint of statistical learning and optimal control.

Assignment flows denote a class of dynamical models for contextual data labeling (classification) on graphs. We derive a novel parametrization of assignment flows that reveals how the underlying information geometry induces two processes for assignment regularization and for gradually enforcing unambiguous decisions, respectively, that seamlessly interact when solving for the flow. Our result enables to characterize the dominant part of the assignment flow as a Riemannian gradient flow with respect to the underlying information geometry. We consider a continuous-domain formulation of the corresponding potential and develop a novel algorithm in terms of solving a sequence of linear elliptic PDEs subject to a simple convex constraint. Our result provides a basis for addressing learning problems by controlling such PDEs in future work.

Variational methods are employed in situations where exact Bayesian inference becomes intractable due to the difficulty in performing certain integrals. Typically, variational methods postulate a tractable posterior and formulate a lower bound on the desired integral to be approximated, e.g. marginal likelihood. The lower bound is then optimised with respect to its free parameters, the so called variational parameters. However, this is not always possible as for certain integrals it is very challenging (or tedious) to come up with a suitable lower bound. Here we propose a simple scheme that overcomes some of the awkward cases where the usual variational treatment becomes difficult. The scheme relies on a rewriting of the lower bound on the model log-likelihood. We demonstrate the proposed scheme on a number of synthetic and real examples, as well as on a real geophysical model for which the standard variational approaches are inapplicable., Comment: published in Pattern Analysis and Applications

This paper introduces the unsupervised assignment flow that couples the assignment flow for supervised image labeling with Riemannian gradient flows for label evolution on feature manifolds. The latter component of the approach encompasses extensions of state-of-the-art clustering approaches to manifold-valued data. Coupling label evolution with the spatially regularized assignment flow induces a sparsifying effect that enables to learn compact label dictionaries in an unsupervised manner. Our approach alleviates the requirement for supervised labeling to have proper labels at hand, because an initial set of labels can evolve and adapt to better values while being assigned to given data. The separation between feature and assignment manifolds enables the flexible application which is demonstrated for three scenarios with manifold-valued features. Experiments demonstrate a beneficial effect in both directions: adaptivity of labels improves image labeling, and steering label evolution by spatially regularized assignments leads to proper labels, because the assignment flow for supervised labeling is exactly used without any approximation for label learning., Comment: 34 pages, 13 figures, published in Journal of Mathematical Imaging and Vision (JMIV)

The assignment flow is a smooth dynamical system that evolves on an elementary statistical manifold and performs contextual data labeling on a graph. We derive and introduce the linear assignment flow that evolves nonlinearly on the manifold, but is governed by a linear ODE on the tangent space. Various numerical schemes adapted to the mathematical structure of these two models are designed and studied, for the geometric numerical integration of both flows: embedded Runge-Kutta-Munthe-Kaas schemes for the nonlinear flow, adaptive Runge-Kutta schemes and exponential integrators for the linear flow. All algorithms are parameter free, except for setting a tolerance value that specifies adaptive step size selection by monitoring the local integration error, or fixing the dimension of the Krylov subspace approximation. These algorithms provide a basis for applying the assignment flow to machine learning scenarios beyond supervised labeling, including unsupervised labeling and learning from controlled assignment flows.

A novel computational approach to log-concave density estimation is proposed. Previous approaches utilize the piecewise-affine parametrization of the density induced by the given sample set. The number of parameters as well as non-smooth subgradient-based convex optimization for determining the maximum likelihood density estimate cause long runtimes for dimensions $d \geq 2$ and large sample sets. The presented approach is based on mildly non-convex smooth approximations of the objective function and \textit{sparse}, adaptive piecewise-affine density parametrization. Established memory-efficient numerical optimization techniques enable to process larger data sets for dimensions $d \geq 2$. While there is no guarantee that the algorithm returns the maximum likelihood estimate for every problem instance, we provide comprehensive numerical evidence that it does yield near-optimal results after significantly shorter runtimes. For example, 10000 samples in $\mathbb{R}^2$ are processed in two seconds, rather than in $\approx 14$ hours required by the previous approach to terminate. For higher dimensions, density estimation becomes tractable as well: Processing $10000$ samples in $\mathbb{R}^6$ requires 35 minutes. The software is publicly available as CRAN R package fmlogcondens., Comment: 22 pages, 10 figures

We introduce a novel approach to Maximum A Posteriori inference based on discrete graphical models. By utilizing local Wasserstein distances for coupling assignment measures across edges of the underlying graph, a given discrete objective function is smoothly approximated and restricted to the assignment manifold. A corresponding multiplicative update scheme combines in a single process (i) geometric integration of the resulting Riemannian gradient flow and (ii) rounding to integral solutions that represent valid labelings. Throughout this process, local marginalization constraints known from the established LP relaxation are satisfied, whereas the smooth geometric setting results in rapidly converging iterations that can be carried out in parallel for every edge.

This paper introduces a new probabilistic architecture called Sum-Product Graphical Model (SPGM). SPGMs combine traits from Sum-Product Networks (SPNs) and Graphical Models (GMs): Like SPNs, SPGMs always enable tractable inference using a class of models that incorporate context specific independence. Like GMs, SPGMs provide a high-level model interpretation in terms of conditional independence assumptions and corresponding factorizations. Thus, the new architecture represents a class of probability distributions that combines, for the first time, the semantics of graphical models with the evaluation efficiency of SPNs. We also propose a novel algorithm for learning both the structure and the parameters of SPGMs. A comparative empirical evaluation demonstrates competitive performances of our approach in density estimation.

We consider clustering problems where the goal is to determine an optimal partition of a given point set in Euclidean space in terms of a collection of affine subspaces. While there is vast literature on heuristics for this kind of problem, such approaches are known to be susceptible to poor initializations and getting trapped in bad local optima. We alleviate these issues by introducing a semidefinite relaxation based on Lasserre's method of moments. While a similiar approach is known for classical Euclidean clustering problems, a generalization to our more general subspace scenario is not straightforward, due to the high symmetry of the objective function that weakens any convex relaxation. We therefore introduce a new mechanism for symmetry breaking based on covering the feasible region with polytopes. Additionally, we introduce and analyze a deterministic rounding heuristic., Comment: 23 pages

Monocular scene reconstruction is essential for modern applications such as robotics or autonomous driving. Although stereo methods usually result in better accuracy than monocular methods, they are more expensive and more difficult to calibrate. In this work, we present a novel second order optimal minimum energy filter that jointly estimates the camera motion, the disparity map and also higher order kinematics recursively on a product Lie group containing a novel disparity group. This mathematical framework enables to cope with non-Euclidean state spaces, non-linear observations and high dimensions which is infeasible for most classical filters. To be robust against outliers, we use a generalized Charbonnier energy function in this framework rather than a quadratic energy function as proposed in related work. Experiments confirm that our method enables accurate reconstructions on-par with state-of-the-art., Comment: Preprint. The final publication will be available at Springer

We present a new vectorial total variation method that addresses the problem of color consistent image filtering. Our approach is inspired from the double-opponent cell representation in the human visual cortex. Existing methods of vectorial total variation regularizers have insufficient (or no) coupling between the color channels and thus may introduce color artifacts. We address this problem by introducing a novel coupling between the color channels related to a pullback-metric from the opponent space to the data (RGB color) space. Our energy is a non-convex, non-smooth higher-order vectorial total variation approach and promotes color consistent image filtering via a coupling term. For a convex variant, we show well-posedness and existence of a solution in the space of vectorial bounded variation. For the higher-order scheme we employ a half-quadratic strategy, which model the non-convex energy terms as the infimum of a sequence of quadratic functions. In experiments, we elaborate on traditional image restoration applications of inpainting, deblurring and denoising. Regarding the latter, we demonstrate state of the art restoration quality with respect to structure coherence and color consistency., Comment: 30 pages