Rotary Positional Encodings (RoPE) have emerged as a highly effective technique for one-dimensional sequences in Natural Language Processing spurring recent progress towards generalizing RoPE to higher-dimensional data such as images and videos. The success of RoPE has been thought to be due to its positional equivariance, i.e. its status as a \textit{relative} positional encoding. In this paper, we mathematically show RoPE to be one of the most general solutions for equivariant positional embedding in one-dimensional data. Moreover, we show Mixed RoPE to be the analogously general solution for $M$-dimensional data, if we require commutative generators -- a property necessary for RoPE's equivariance. However, we question the necessity of equivariance. We propose Spherical RoPE, a method analogous to Mixed RoPE, but with the assumption of anti-commutative generators -- relaxing the equivariant condition. Empirically, we find Spherical RoPE to have the equivalent learning behavior as its equivariant analogues. This strongly suggests that relative positional embeddings are not as important as is commonly believed. We expect this discovery to facilitate future work in positional encodings for vision that are faster and generalize better by removing the preconception that they must be relative.

In mechanical structures like airplanes, cars and houses, noise is generated and transmitted through vibrations. To take measures to reduce this noise, vibrations need to be simulated with expensive numerical computations. Deep learning surrogate models present a promising alternative to classical numerical simulations as they can be evaluated magnitudes faster, while trading-off accuracy. To quantify such trade-offs systematically and foster the development of methods, we present a benchmark on the task of predicting the vibration of harmonically excited plates. The benchmark features a total of 12,000 plate geometries with varying forms of beadings, material, boundary conditions, load position and sizes with associated numerical solutions. To address the benchmark task, we propose a new network architecture, named \modelname, which predicts vibration patterns of plate geometries given a specific excitation frequency. Applying principles from operator learning and implicit models for shape encoding, our approach effectively addresses the prediction of highly variable frequency response functions occurring in dynamic systems. To quantify the prediction quality, we introduce a set of evaluation metrics and evaluate the method on our vibrating-plates benchmark. Our method outperforms DeepONets, Fourier Neural Operators and more traditional neural network architectures and can be used for design optimization. Code, dataset and visualizations: https://github.com/ecker-lab/Learning_Vibrating_Plates

Image segmentation is usually addressed by training a model for a fixed set of object classes. Incorporating additional classes or more complex queries later is expensive as it requires re-training the model on a dataset that encompasses these expressions. Here we propose a system that can generate image segmentations based on arbitrary prompts at test time. A prompt can be either a text or an image. This approach enables us to create a unified model (trained once) for three common segmentation tasks, which come with distinct challenges: referring expression segmentation, zero-shot segmentation and one-shot segmentation. We build upon the CLIP model as a backbone which we extend with a transformer-based decoder that enables dense prediction. After training on an extended version of the PhraseCut dataset, our system generates a binary segmentation map for an image based on a free-text prompt or on an additional image expressing the query. We analyze different variants of the latter image-based prompts in detail. This novel hybrid input allows for dynamic adaptation not only to the three segmentation tasks mentioned above, but to any binary segmentation task where a text or image query can be formulated. Finally, we find our system to adapt well to generalized queries involving affordances or properties. Code is available at https://eckerlab.org/code/CLIPSeg

Path planning is important for robot action execution, since a path or a motion trajectory for a particular action has to be defined first before the action can be executed. Most of the current approaches are iterative methods where the trajectory is generated by predicting the next state based on the current state. Here we propose a novel method by utilising a fully convolutional neural network, which allows generation of complete paths even for several agents with one network prediction iteration. We demonstrate that our method is able to successfully generate optimal or close to optimal paths (less than 10% longer) in more than 99% of the cases for single path predictions in 2D and 3D environments. Furthermore, we show that the network is - without specific training on such cases - able to create (close to) optimal paths in 96% of the cases for two and in 84% of the cases for three simultaneously generated paths.

The distributional hypothesis states that the meaning of a concept is defined through the contexts it occurs in. In practice, often word co-occurrence and proximity are analyzed in text corpora for a given word to obtain a real-valued semantic word vector, which is taken to (at least partially) encode the meaning of this word. Here we transfer this idea from text to images, where pre-assigned labels of other objects or activations of convolutional neural networks serve as context. We propose a simple algorithm that extracts and processes object contexts from an image database and yields semantic vectors for objects. We show empirically that these representations exhibit on par performance with state-of-the-art distributional models over a set of conventional objects. For this we employ well-known word benchmarks in addition to a newly proposed object-centric benchmark.