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We present a mass lumping approach based on an isogeometric Petrov-Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To discretize the test function space, our method uses an approximate dual basis, whose functions are smooth, have local support and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. The resulting mass matrix is ``close'' to the identity matrix. Specifically, a lumped version of this mass matrix preserves all relevant polynomials when utilized in a Galerkin projection. Consequently, the mass matrix can be lumped (via row-sum lumping) without compromising spatial accuracy in explicit dynamics calculations. We address the imposition of Dirichlet boundary conditions and the preservation of approximate bi-orthogonality under geometric mappings. In addition, we establish a link between the exact dual and approximate dual basis functions via an iterative algorithm that improves the approximate dual basis towards exact bi-orthogonality. We demonstrate the performance of our higher-order accurate mass lumping approach via convergence studies and spectral analyses of discretized beam, plate and shell models.

We present a comprehensive study on discrete morphological symmetries of dynamical systems, which are commonly observed in biological and artificial locomoting systems, such as legged, swimming, and flying animals/robots/virtual characters. These symmetries arise from the presence of one or more planes/axis of symmetry in the system's morphology, resulting in harmonious duplication and distribution of body parts. Significantly, we characterize how morphological symmetries extend to symmetries in the system's dynamics, optimal control policies, and in all proprioceptive and exteroceptive measurements related to the system's dynamics evolution. In the context of data-driven methods, symmetry represents an inductive bias that justifies the use of data augmentation or symmetric function approximators. To tackle this, we present a theoretical and practical framework for identifying the system's morphological symmetry group $\G$ and characterizing the symmetries in proprioceptive and exteroceptive data measurements. We then exploit these symmetries using data augmentation and $\G$-equivariant neural networks. Our experiments on both synthetic and real-world applications provide empirical evidence of the advantageous outcomes resulting from the exploitation of these symmetries, including improved sample efficiency, enhanced generalization, and reduction of trainable parameters.

We present discontinuous Galerkin (DG) methods for solving a first-order semi-linear hyperbolic system, which was originally proposed as a continuum model for a one-dimensional dimer lattice of topological resonators. We examine the energy-conserving or energy-dissipating property in relation to the choices of simple, mesh-independent numerical fluxes. We demonstrate that, with certain numerical flux choices, our DG method achieves optimal convergence in the $L^2$ norm. We provide numerical experiments that validate and illustrate the effectiveness of our proposed numerical methods.

Distributional approximation is a fundamental problem in machine learning with numerous applications across all fields of science and engineering and beyond. The key challenge in most approximation methods is the need to tackle the intractable normalization constant pertaining to the parametrized distributions used to model the data. In this paper, we present a novel Stein operator on Lie groups leading to a kernel Stein discrepancy (KSD) which is a normalization-free loss function. We present several theoretical results characterizing the properties of this new KSD on Lie groups and its minimizers namely, the minimum KSD estimator (MKSDE). Proof of several properties of MKSDE are presented, including strong consistency, CLT and a closed form of the MKSDE for the von Mises-Fisher distribution on SO(N). Finally, we present experimental evidence depicting advantages of minimizing KSD over maximum likelihood estimation.

Statistical learning and logical reasoning are two major fields of AI expected to be unified for human-like machine intelligence. Most existing work considers how to combine existing logical and statistical systems. However, there is no theory of inference so far explaining how basic approaches to statistical learning and logical reasoning stem from a common principle. Inspired by the fact that much empirical work in neuroscience suggests Bayesian (or probabilistic generative) approaches to brain function including learning and reasoning, we here propose a simple Bayesian model of logical reasoning and statistical learning. The theory is statistically correct as it satisfies Kolmogorov's axioms, is consistent with both Fenstad's representation theorem and maximum likelihood estimation and performs exact Bayesian inference with a linear-time complexity. The theory is logically correct as it is a data-driven generalisation of uncertain reasoning from consistency, possibility, inconsistency and impossibility. The theory is correct in terms of machine learning as its solution to generation and prediction tasks on the MNIST dataset is not only empirically reasonable but also theoretically correct against the K nearest neighbour method. We simply model how data causes symbolic knowledge in terms of its satisfiability in formal logic. Symbolic reasoning emerges as a result of the process of going the causality forwards and backwards. The forward and backward processes correspond to an interpretation and inverse interpretation in formal logic, respectively. The inverse interpretation differentiates our work from the mainstream often referred to as inverse entailment, inverse deduction or inverse resolution. The perspective gives new insights into learning and reasoning towards human-like machine intelligence.

Recent advances in state-of-the-art DNN architecture design have been moving toward Transformer models. These models achieve superior accuracy across a wide range of applications. This trend has been consistent over the past several years since Transformer models were originally introduced. However, the amount of compute and bandwidth required for inference of recent Transformer models is growing at a significant rate, and this has made their deployment in latency-sensitive applications challenging. As such, there has been an increased focus on making Transformer models more efficient, with methods that range from changing the architecture design, all the way to developing dedicated domain-specific accelerators. In this work, we survey different approaches for efficient Transformer inference, including: (i) analysis and profiling of the bottlenecks in existing Transformer architectures and their similarities and differences with previous convolutional models; (ii) implications of Transformer architecture on hardware, including the impact of non-linear operations such as Layer Normalization, Softmax, and GELU, as well as linear operations, on hardware design; (iii) approaches for optimizing a fixed Transformer architecture; (iv) challenges in finding the right mapping and scheduling of operations for Transformer models; and (v) approaches for optimizing Transformer models by adapting the architecture using neural architecture search. Finally, we perform a case study by applying the surveyed optimizations on Gemmini, the open-source, full-stack DNN accelerator generator, and we show how each of these approaches can yield improvements, compared to previous benchmark results on Gemmini. Among other things, we find that a full-stack co-design approach with the aforementioned methods can result in up to 88.7x speedup with a minimal performance degradation for Transformer inference.

We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.

Visual recognition is currently one of the most important and active research areas in computer vision, pattern recognition, and even the general field of artificial intelligence. It has great fundamental importance and strong industrial needs. Deep neural networks (DNNs) have largely boosted their performances on many concrete tasks, with the help of large amounts of training data and new powerful computation resources. Though recognition accuracy is usually the first concern for new progresses, efficiency is actually rather important and sometimes critical for both academic research and industrial applications. Moreover, insightful views on the opportunities and challenges of efficiency are also highly required for the entire community. While general surveys on the efficiency issue of DNNs have been done from various perspectives, as far as we are aware, scarcely any of them focused on visual recognition systematically, and thus it is unclear which progresses are applicable to it and what else should be concerned. In this paper, we present the review of the recent advances with our suggestions on the new possible directions towards improving the efficiency of DNN-related visual recognition approaches. We investigate not only from the model but also the data point of view (which is not the case in existing surveys), and focus on three most studied data types (images, videos and points). This paper attempts to provide a systematic summary via a comprehensive survey which can serve as a valuable reference and inspire both researchers and practitioners who work on visual recognition problems.

Mining graph data has become a popular research topic in computer science and has been widely studied in both academia and industry given the increasing amount of network data in the recent years. However, the huge amount of network data has posed great challenges for efficient analysis. This motivates the advent of graph representation which maps the graph into a low-dimension vector space, keeping original graph structure and supporting graph inference. The investigation on efficient representation of a graph has profound theoretical significance and important realistic meaning, we therefore introduce some basic ideas in graph representation/network embedding as well as some representative models in this chapter.

Recent years have witnessed the enormous success of low-dimensional vector space representations of knowledge graphs to predict missing facts or find erroneous ones. Currently, however, it is not yet well-understood how ontological knowledge, e.g. given as a set of (existential) rules, can be embedded in a principled way. To address this shortcoming, in this paper we introduce a framework based on convex regions, which can faithfully incorporate ontological knowledge into the vector space embedding. Our technical contribution is two-fold. First, we show that some of the most popular existing embedding approaches are not capable of modelling even very simple types of rules. Second, we show that our framework can represent ontologies that are expressed using so-called quasi-chained existential rules in an exact way, such that any set of facts which is induced using that vector space embedding is logically consistent and deductively closed with respect to the input ontology.