In this paper, we propose the *geometric invariance hypothesis (GIH)*, which argues that the input space curvature of a neural network remains invariant under transformation in certain architecture-dependent directions during training. We investigate a simple, non-linear binary classification problem residing on a plane in a high dimensional space and observe that—unlike MLPs—ResNets fail to generalize depending on the orientation of the plane. Motivated by this example, we define a neural network's **average geometry** and **average geometry evolution** as compact *architecture-dependent* summaries of the model's input-output geometry and its evolution during training. By investigating the average geometry evolution at initialization, we discover that the geometry of a neural network evolves according to the data covariance projected onto its average geometry. This means that the geometry only changes in a subset of the input space when the average geometry is low-rank, such as in ResNets. This causes an architecture-dependent invariance property in the input space curvature, which we dub GIH. Finally, we present extensive experimental results to observe the consequences of GIH and how it relates to generalization in neural networks.

Linear recurrent neural networks (RNNs) and state-space models (SSMs) such as Mamba have become promising alternatives to softmax-attention as sequence mixing layers in Transformer architectures. Current models, however, do not exhibit the full state-tracking expressivity of RNNs because they rely on channel-wise (i.e. diagonal) sequence mixing. In this paper, we investigate parameterizations of a large class of dense linear RNNs as fixed-points of parallelizable diagonal linear RNNs. The resulting models can naturally trade expressivity for efficiency at a fixed number of parameters and achieve state-of-the-art results on the state-tracking benchmarks $A_5$ and $S_5$, while matching performance on copying and other tasks.

The advent of the Internet has caused a significant growth in the number of opinions expressed about products or services on e-commerce websites. Aspect category detection, which is one of the challenging subtasks of aspect-based sentiment analysis, deals with categorizing a given review sentence into a set of predefined categories. Most of the research efforts in this field are devoted to English language reviews, while there are a large number of reviews in other languages that are left unexplored. In this paper, we propose a multilingual method to perform aspect category detection on reviews in different languages, which makes use of a deep convolutional neural network with multilingual word embeddings. To the best of our knowledge, our method is the first attempt at performing aspect category detection on multiple languages simultaneously. Empirical results on the multilingual dataset provided by SemEval workshop demonstrate the effectiveness of the proposed method1.