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The aim of these notes is to describe how to construct canonical bundles of moving frames and differential invariants for parametrized curves in Lagrangian Grassmannians, at least in the monotonic case. Such curves appear as Jacobi curves…

The expressivity of Graph Neural Networks (GNNs) has been studied broadly in recent years to reveal the design principles for more powerful GNNs. Graph canonization is known as a typical approach to distinguish non-isomorphic graphs, yet…

Generalization is an important feature of neural network, and there have been many studies on it. Recently, with the development of quantum compu-ting, it brings new opportunities. In this paper, we studied a class of quantum neural network…

Symmetries (transformations by group actions) are present in many datasets, and leveraging them holds considerable promise for improving predictions in machine learning. In this work, we aim to understand when and how deep networks -- with…

The explicit incorporation of task-specific inductive biases through symmetry has emerged as a general design precept in the development of high-performance machine learning models. For example, group equivariant neural networks have…

Feedforward generalizable models for implicit shape reconstruction from unoriented point cloud present multiple advantages, including high performance and inference speed. However, they still suffer from generalization issues, ranging from…

While rule-based attribution methods have proven useful for providing local explanations for Deep Neural Networks, explaining modern and more varied network architectures yields new challenges in generating trustworthy explanations, since…

Recent empirical work has shown that hierarchical convolutional kernels inspired by convolutional neural networks (CNNs) significantly improve the performance of kernel methods in image classification tasks. A widely accepted explanation…

Numerous invariant (or equivariant) neural networks have succeeded in handling invariant data such as point clouds and graphs. However, a generalization theory for the neural networks has not been well developed, because several essential…

It is widely believed that engineering a model to be invariant/equivariant improves generalisation. Despite the growing popularity of this approach, a precise characterisation of the generalisation benefit is lacking. By considering the…

Natural data is often organized as a hierarchical composition of features. How many samples do generative models need in order to learn the composition rules, so as to produce a combinatorially large number of novel data? What signal in the…

Canonical correlation analysis (CCA) is a powerful technique for discovering whether or not hidden sources are commonly present in two (or more) datasets. Its well-appreciated merits include dimensionality reduction, clustering,…

The notion of a (polynomial) kernelization from parameterized complexity is a well-studied model for efficient preprocessing for hard computational problems. By now, it is quite well understood which parameterized problems do or…

Prior information can be incorporated in matrix completion to improve estimation accuracy and extrapolate the missing entries. Reproducing kernel Hilbert spaces provide tools to leverage the said prior information, and derive more reliable…

Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex…

In parameterized algorithmics, the process of kernelization is defined as a polynomial time algorithm that transforms the instance of a given problem to an equivalent instance of a size that is limited by a function of the parameter. As,…

Computer vision research has long aimed to build systems that are robust to spatial transformations found in natural data. Traditionally, this is done using data augmentation or hard-coding invariances into the architecture. However, too…

Point cloud analysis is challenging due to the irregularity of the point cloud data structure. Existing works typically employ the ad-hoc sampling-grouping operation of PointNet++, followed by sophisticated local and/or global feature…

Calibration is a fundamental property of a good predictive model: it requires that the model predicts correctly in proportion to its confidence. Modern neural networks, however, provide no strong guarantees on their calibration -- and can…

That shared features between train and test data are required for generalisation in artificial neural networks has been a common assumption of both proponents and critics of these models. Here, we show that convolutional architectures avoid…