Submodular functions have been studied extensively in machine learning and data mining. In particular, the optimization of submodular functions over the integer lattice (integer submodular functions) has recently attracted much interest, because this domain relates naturally to many practical problem settings, such as multilabel graph cut, budget allocation and revenue maximization with discrete assignments. In contrast, the use of these functions for probabilistic modeling has received surprisingly little attention so far. In this work, we firstly propose the Generalized Multilinear Extension, a continuous DR-submodular extension for integer submodular functions. We study central properties of this extension and formulate a new probabilistic model which is defined through integer submodular functions. Then, we introduce a block-coordinate ascent algorithm to perform approximate inference for this class of models and finally, we demonstrate its effectiveness and viability on several real-world social connection graph datasets with integer submodular objectives.

Microplastics pollution has been recognized as a serious environmental concern, with research efforts underway to determine primary causes. Experiments typically generate bright-field images of microplastic fibers that are filtered from water. Environmental decision making in process engineering critically relies on accurate quantification of mi-croplastic fibers in these images. To satisfy the required standards, images are often analyzed manually, resulting in a highly tedious process, with thousands of fiber instances per image. While the shape of individual fibers is relatively simple, it is difficult to separate them in highly crowded scenes with significant overlap. We propose a fiber instance detection pipeline, which decomposes the fiber detection and segmentation into manageable sub-problems. Well separated instances are identified with robust image processing techniques, such as adaptive thresholding, and morphological skeleton analysis, while tangled fibers are separated by an algorithm based on deep pixel embeddings. Moreover, we present a modified Intersection-over-Union metric as a more appropriate similarity metric for elongated shapes. Our approach improves significantly on out-of-sample data, in particular for difficult cases of intersecting fibers.

We consider learning of submodular functions from data. These functions are important in machine learning and have a wide range of applications, e.g. data summarization, feature selection and active learning. Despite their combinatorial nature, submodular functions can be maximized approximately with strong theoretical guarantees in polynomial time. Typically, learning the submodular function and optimization of that function are treated separately, i.e. the function is first learned using a proxy objective and subsequently maximized. In contrast, we show how to perform learning and optimization jointly. By interpreting the output of greedy maximization algorithms as distributions over sequences of items and smoothening these distributions, we obtain a differentiable objective. In this way, we can differentiate through the maximization algorithms and optimize the model to work well with the optimization algorithm. We theoretically characterize the error made by our approach, yielding insights into the tradeoff of smoothness and accuracy. We demonstrate the effectiveness of our approach for jointly learning and optimizing on synthetic maxcut data, and on real world applications such as product recommendation and image collection summarization.