arXiv subjects

Fred A. Hamprecht

Publications and source records attributed to Fred A. Hamprecht.

At least 37 records · Page 2Linked to original sources

arXiv2022

Theory and Approximate Solvers for Branched Optimal Transport with Multiple Sources

Branched Optimal Transport (BOT) is a generalization of optimal transport in which transportation costs along an edge are subadditive. This subadditivity models an increase in transport efficiency when shipping mass along the same route, favoring branched transportation networks. We here study the NP-hard optimization of BOT networks connecting a finite number of sources and sinks in $\mathbb{R}^2$. First, we show how to efficiently find the best geometry of a BOT network for many sources and sinks, given a topology. Second, we argue that a topology with more than three edges meeting at a branching point is never optimal. Third, we show that the results obtained for the Euclidean plane generalize directly to optimal transportation networks on two-dimensional Riemannian manifolds. Finally, we present a simple but effective approximate BOT solver combining geometric optimization with a combinatorial optimization of the network topology.

cs.LG↗

arXiv2022

GASP, a generalized framework for agglomerative clustering of signed graphs and its application to Instance Segmentation

We propose a theoretical framework that generalizes simple and fast algorithms for hierarchical agglomerative clustering to weighted graphs with both attractive and repulsive interactions between the nodes. This framework defines GASP, a Generalized Algorithm for Signed graph Partitioning, and allows us to explore many combinations of different linkage criteria and cannot-link constraints. We prove the equivalence of existing clustering methods to some of those combinations and introduce new algorithms for combinations that have not been studied before. We study both theoretical and empirical properties of these combinations and prove that some of these define an ultrametric on the graph. We conduct a systematic comparison of various instantiations of GASP on a large variety of both synthetic and existing signed clustering problems, in terms of accuracy but also efficiency and robustness to noise. Lastly, we show that some of the algorithms included in our framework, when combined with the predictions from a CNN model, result in a simple bottom-up instance segmentation pipeline. Going all the way from pixels to final segments with a simple procedure, we achieve state-of-the-art accuracy on the CREMI 2016 EM segmentation benchmark without requiring domain-specific superpixels.

cs.CV↗

arXiv2022

CellTypeGraph: A New Geometric Computer Vision Benchmark

Classifying all cells in an organ is a relevant and difficult problem from plant developmental biology. We here abstract the problem into a new benchmark for node classification in a geo-referenced graph. Solving it requires learning the spatial layout of the organ including symmetries. To allow the convenient testing of new geometrical learning methods, the benchmark of Arabidopsis thaliana ovules is made available as a PyTorch data loader, along with a large number of precomputed features. Finally, we benchmark eight recent graph neural network architectures, finding that DeeperGCN currently works best on this problem.

cs.CV↗

arXiv2021

Extensions of Karger's Algorithm: Why They Fail in Theory and How They Are Useful in Practice

The minimum graph cut and minimum $s$-$t$-cut problems are important primitives in the modeling of combinatorial problems in computer science, including in computer vision and machine learning. Some of the most efficient algorithms for finding global minimum cuts are randomized algorithms based on Karger's groundbreaking contraction algorithm. Here, we study whether Karger's algorithm can be successfully generalized to other cut problems. We first prove that a wide class of natural generalizations of Karger's algorithm cannot efficiently solve the $s$-$t$-mincut or the normalized cut problem to optimality. However, we then present a simple new algorithm for seeded segmentation / graph-based semi-supervised learning that is closely based on Karger's original algorithm, showing that for these problems, extensions of Karger's algorithm can be useful. The new algorithm has linear asymptotic runtime and yields a potential that can be interpreted as the posterior probability of a sample belonging to a given seed / class. We clarify its relation to the random walker algorithm / harmonic energy minimization in terms of distributions over spanning forests. On classical problems from seeded image segmentation and graph-based semi-supervised learning on image data, the method performs at least as well as the random walker / harmonic energy minimization / Gaussian processes.

cs.DS↗

arXiv2021

On UMAP's true loss function

UMAP has supplanted t-SNE as state-of-the-art for visualizing high-dimensional datasets in many disciplines, but the reason for its success is not well understood. In this work, we investigate UMAP's sampling based optimization scheme in detail. We derive UMAP's effective loss function in closed form and find that it differs from the published one. As a consequence, we show that UMAP does not aim to reproduce its theoretically motivated high-dimensional UMAP similarities. Instead, it tries to reproduce similarities that only encode the shared $k$ nearest neighbor graph, thereby challenging the previous understanding of UMAP's effectiveness. Instead, we claim that the key to UMAP's success is its implicit balancing of attraction and repulsion resulting from negative sampling. This balancing in turn facilitates optimization via gradient descent. We corroborate our theoretical findings on toy and single cell RNA sequencing data.

cs.LG↗

arXiv2021

The Mutex Watershed and its Objective: Efficient, Parameter-Free Graph Partitioning

Image partitioning, or segmentation without semantics, is the task of decomposing an image into distinct segments, or equivalently to detect closed contours. Most prior work either requires seeds, one per segment; or a threshold; or formulates the task as multicut / correlation clustering, an NP-hard problem. Here, we propose an efficient algorithm for graph partitioning, the "Mutex Watershed''. Unlike seeded watershed, the algorithm can accommodate not only attractive but also repulsive cues, allowing it to find a previously unspecified number of segments without the need for explicit seeds or a tunable threshold. We also prove that this simple algorithm solves to global optimality an objective function that is intimately related to the multicut / correlation clustering integer linear programming formulation. The algorithm is deterministic, very simple to implement, and has empirically linearithmic complexity. When presented with short-range attractive and long-range repulsive cues from a deep neural network, the Mutex Watershed gives the best results currently known for the competitive ISBI 2012 EM segmentation benchmark.

cs.CV↗

arXiv2021

MultiStar: Instance Segmentation of Overlapping Objects with Star-Convex Polygons

Instance segmentation of overlapping objects in biomedical images remains a largely unsolved problem. We take up this challenge and present MultiStar, an extension to the popular instance segmentation method StarDist. The key novelty of our method is that we identify pixels at which objects overlap and use this information to improve proposal sampling and to avoid suppressing proposals of truly overlapping objects. This allows us to apply the ideas of StarDist to images with overlapping objects, while incurring only a small overhead compared to the established method. MultiStar shows promising results on two datasets and has the advantage of using a simple and easy to train network architecture.

cs.CV↗

arXiv2020

Proposal-Free Volumetric Instance Segmentation from Latent Single-Instance Masks

This work introduces a new proposal-free instance segmentation method that builds on single-instance segmentation masks predicted across the entire image in a sliding window style. In contrast to related approaches, our method concurrently predicts all masks, one for each pixel, and thus resolves any conflict jointly across the entire image. Specifically, predictions from overlapping masks are combined into edge weights of a signed graph that is subsequently partitioned to obtain all final instances concurrently. The result is a parameter-free method that is strongly robust to noise and prioritizes predictions with the highest consensus across overlapping masks. All masks are decoded from a low dimensional latent representation, which results in great memory savings strictly required for applications to large volumetric images. We test our method on the challenging CREMI 2016 neuron segmentation benchmark where it achieves competitive scores.

cs.CV↗

arXiv2020

DISCo: Deep learning, Instance Segmentation, and Correlations for cell segmentation in calcium imaging

Calcium imaging is one of the most important tools in neurophysiology as it enables the observation of neuronal activity for hundreds of cells in parallel and at single-cell resolution. In order to use the data gained with calcium imaging, it is necessary to extract individual cells and their activity from the recordings. We present DISCo, a novel approach for the cell segmentation in calcium imaging videos. We use temporal information from the recordings in a computationally efficient way by computing correlations between pixels and combine it with shape-based information to identify active as well as non-active cells. We first learn to predict whether two pixels belong to the same cell; this information is summarized in an undirected, edge-weighted grid graph which we then partition. In so doing, we approximately solve the NP-hard correlation clustering problem with a recently proposed greedy algorithm. Evaluating our method on the Neurofinder public benchmark shows that DISCo outperforms all existing models trained on these datasets.

q-bio.NC↗

arXiv2020

Instance Separation Emerges from Inpainting

Deep neural networks trained to inpaint partially occluded images show a deep understanding of image composition and have even been shown to remove objects from images convincingly. In this work, we investigate how this implicit knowledge of image composition can be leveraged for fully self-supervised instance separation. We propose a measure for the independence of two image regions given a fully self-supervised inpainting network and separate objects by maximizing this independence. We evaluate our method on two microscopy image datasets and show that it reaches similar segmentation performance to fully supervised methods.

cs.CV↗

arXiv2019

The Semantic Mutex Watershed for Efficient Bottom-Up Semantic Instance Segmentation

Semantic instance segmentation is the task of simultaneously partitioning an image into distinct segments while associating each pixel with a class label. In commonly used pipelines, segmentation and label assignment are solved separately since joint optimization is computationally expensive. We propose a greedy algorithm for joint graph partitioning and labeling derived from the efficient Mutex Watershed partitioning algorithm. It optimizes an objective function closely related to the Symmetric Multiway Cut objective and empirically shows efficient scaling behavior. Due to the algorithm's efficiency it can operate directly on pixels without prior over-segmentation of the image into superpixels. We evaluate the performance on the Cityscapes dataset (2D urban scenes) and on a 3D microscopy volume. In urban scenes, the proposed algorithm combined with current deep neural networks outperforms the strong baseline of `Panoptic Feature Pyramid Networks' by Kirillov et al. (2019). In the 3D electron microscopy images, we show explicitly that our joint formulation outperforms a separate optimization of the partitioning and labeling problems.

cs.CV↗

arXiv2019

Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning

The seeded Watershed algorithm / minimax semi-supervised learning on a graph computes a minimum spanning forest which connects every pixel / unlabeled node to a seed / labeled node. We propose instead to consider all possible spanning forests and calculate, for every node, the probability of sampling a forest connecting a certain seed with that node. We dub this approach "Probabilistic Watershed". Leo Grady (2006) already noted its equivalence to the Random Walker / Harmonic energy minimization. We here give a simpler proof of this equivalence and establish the computational feasibility of the Probabilistic Watershed with Kirchhoff's matrix tree theorem. Furthermore, we show a new connection between the Random Walker probabilities and the triangle inequality of the effective resistance. Finally, we derive a new and intuitive interpretation of the Power Watershed.

cs.DS↗

arXiv2019

Deep Active Learning with Adaptive Acquisition

Model selection is treated as a standard performance boosting step in many machine learning applications. Once all other properties of a learning problem are fixed, the model is selected by grid search on a held-out validation set. This is strictly inapplicable to active learning. Within the standardized workflow, the acquisition function is chosen among available heuristics a priori, and its success is observed only after the labeling budget is already exhausted. More importantly, none of the earlier studies report a unique consistently successful acquisition heuristic to the extent to stand out as the unique best choice. We present a method to break this vicious circle by defining the acquisition function as a learning predictor and training it by reinforcement feedback collected from each labeling round. As active learning is a scarce data regime, we bootstrap from a well-known heuristic that filters the bulk of data points on which all heuristics would agree, and learn a policy to warp the top portion of this ranking in the most beneficial way for the character of a specific data distribution. Our system consists of a Bayesian neural net, the predictor, a bootstrap acquisition function, a probabilistic state definition, and another Bayesian policy network that can effectively incorporate this input distribution. We observe on three benchmark data sets that our method always manages to either invent a new superior acquisition function or to adapt itself to the a priori unknown best performing heuristic for each specific data set.

stat.ML↗

arXiv2019

Sampling-Free Variational Inference of Bayesian Neural Networks by Variance Backpropagation

We propose a new Bayesian Neural Net formulation that affords variational inference for which the evidence lower bound is analytically tractable subject to a tight approximation. We achieve this tractability by (i) decomposing ReLU nonlinearities into the product of an identity and a Heaviside step function, (ii) introducing a separate path that decomposes the neural net expectation from its variance. We demonstrate formally that introducing separate latent binary variables to the activations allows representing the neural network likelihood as a chain of linear operations. Performing variational inference on this construction enables a sampling-free computation of the evidence lower bound which is a more effective approximation than the widely applied Monte Carlo sampling and CLT related techniques. We evaluate the model on a range of regression and classification tasks against BNN inference alternatives, showing competitive or improved performance over the current state-of-the-art.

stat.ML↗

arXiv2019

On the Spectral Bias of Neural Networks

Neural networks are known to be a class of highly expressive functions able to fit even random input-output mappings with $100\%$ accuracy. In this work, we present properties of neural networks that complement this aspect of expressivity. By using tools from Fourier analysis, we show that deep ReLU networks are biased towards low frequency functions, meaning that they cannot have local fluctuations without affecting their global behavior. Intuitively, this property is in line with the observation that over-parameterized networks find simple patterns that generalize across data samples. We also investigate how the shape of the data manifold affects expressivity by showing evidence that learning high frequencies gets \emph{easier} with increasing manifold complexity, and present a theoretical understanding of this behavior. Finally, we study the robustness of the frequency components with respect to parameter perturbation, to develop the intuition that the parameters must be finely tuned to express high frequency functions.

stat.ML↗

arXiv2019

End-to-End Learned Random Walker for Seeded Image Segmentation

We present an end-to-end learned algorithm for seeded segmentation. Our method is based on the Random Walker algorithm, where we predict the edge weights of the underlying graph using a convolutional neural network. This can be interpreted as learning context-dependent diffusivities for a linear diffusion process. Besides calculating the exact gradient for optimizing these diffusivities, we also propose simplifications that sparsely sample the gradient and still yield competitive results. The proposed method achieves the currently best results on a seeded version of the CREMI neuron segmentation challenge.

cs.CV↗

arXiv2019

Essentially No Barriers in Neural Network Energy Landscape

Training neural networks involves finding minima of a high-dimensional non-convex loss function. Knowledge of the structure of this energy landscape is sparse. Relaxing from linear interpolations, we construct continuous paths between minima of recent neural network architectures on CIFAR10 and CIFAR100. Surprisingly, the paths are essentially flat in both the training and test landscapes. This implies that neural networks have enough capacity for structural changes, or that these changes are small between minima. Also, each minimum has at least one vanishing Hessian eigenvalue in addition to those resulting from trivial invariance.

stat.ML↗

arXiv2019

LeMoNADe: Learned Motif and Neuronal Assembly Detection in calcium imaging videos

Neuronal assemblies, loosely defined as subsets of neurons with reoccurring spatio-temporally coordinated activation patterns, or "motifs", are thought to be building blocks of neural representations and information processing. We here propose LeMoNADe, a new exploratory data analysis method that facilitates hunting for motifs in calcium imaging videos, the dominant microscopic functional imaging modality in neurophysiology. Our nonparametric method extracts motifs directly from videos, bypassing the difficult intermediate step of spike extraction. Our technique augments variational autoencoders with a discrete stochastic node, and we show in detail how a differentiable reparametrization and relaxation can be used. An evaluation on simulated data, with available ground truth, reveals excellent quantitative performance. In real video data acquired from brain slices, with no ground truth available, LeMoNADe uncovers nontrivial candidate motifs that can help generate hypotheses for more focused biological investigations.

q-bio.NC↗