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We take a random matrix theory approach to random sketching and show an asymptotic first-order equivalence of the regularized sketched pseudoinverse of a positive semidefinite matrix to a certain evaluation of the resolvent of the same matrix. We focus on real-valued regularization and extend previous results on an asymptotic equivalence of random matrices to the real setting, providing a precise characterization of the equivalence even under negative regularization, including a precise characterization of the smallest nonzero eigenvalue of the sketched matrix, which may be of independent interest. We then further characterize the second-order equivalence of the sketched pseudoinverse. We also apply our results to the analysis of the sketch-and-project method and to sketched ridge regression. Lastly, we prove that these results generalize to asymptotically free sketching matrices, obtaining the resulting equivalence for orthogonal sketching matrices and comparing our results to several common sketches used in practice.

Principal Component Analysis (PCA) is a popular tool in data analysis, especially when the data is high-dimensional. PCA aims to find subspaces, spanned by the so-called \textit{principal components}, that best explain the variance in the dataset. The deflation method is a popular meta-algorithm -- used to discover such subspaces -- that sequentially finds individual principal components, starting from the most important one and working its way towards the less important ones. However, due to its sequential nature, the numerical error introduced by not estimating principal components exactly -- e.g., due to numerical approximations through this process -- propagates, as deflation proceeds. To the best of our knowledge, this is the first work that mathematically characterizes the error propagation of the inexact deflation method, and this is the key contribution of this paper. We provide two main results: $i)$ when the sub-routine for finding the leading eigenvector is generic, and $ii)$ when power iteration is used as the sub-routine. In the latter case, the additional directional information from power iteration allows us to obtain a tighter error bound than the analysis of the sub-routine agnostic case. As an outcome, we provide explicit characterization on how the error progresses and affects subsequent principal component estimations for this fundamental problem.

The goal of motion understanding is to establish a reliable mapping between motion and action semantics, while it is a challenging many-to-many problem. An abstract action semantic (i.e., walk forwards) could be conveyed by perceptually diverse motions (walk with arms up or swinging), while a motion could carry different semantics w.r.t. its context and intention. This makes an elegant mapping between them difficult. Previous attempts adopted direct-mapping paradigms with limited reliability. Also, current automatic metrics fail to provide reliable assessments of the consistency between motions and action semantics. We identify the source of these problems as the significant gap between the two modalities. To alleviate this gap, we propose Kinematic Phrases (KP) that take the objective kinematic facts of human motion with proper abstraction, interpretability, and generality characteristics. Based on KP as a mediator, we can unify a motion knowledge base and build a motion understanding system. Meanwhile, KP can be automatically converted from motions and to text descriptions with no subjective bias, inspiring Kinematic Prompt Generation (KPG) as a novel automatic motion generation benchmark. In extensive experiments, our approach shows superiority over other methods. Our code and data would be made publicly available at //foruck.github.io/KP.

We present an incomplete proof synthesis method for the Calculus of Constructions which is always terminating and a complete Vernacular for the Calculus of Constructions based on this method.

Large Language Models (LLMs) have the ability to solve a variety of tasks, such as text summarization and mathematical questions, just out of the box, but they are often trained with a single task in mind. Due to high computational costs, the current trend is to use prompt instruction tuning to better adjust monolithic, pretrained LLMs for new -- but often individual -- downstream tasks. Thus, how one would expand prompt tuning to handle -- concomitantly -- heterogeneous tasks and data distributions is a widely open question. To address this gap, we suggest the use of \emph{Mixture of Prompts}, or MoPs, associated with smart gating functionality: the latter -- whose design is one of the contributions of this paper -- can identify relevant skills embedded in different groups of prompts and dynamically assign combined experts (i.e., collection of prompts), based on the target task. Additionally, MoPs are empirically agnostic to any model compression technique applied -- for efficiency reasons -- as well as instruction data source and task composition. In practice, MoPs can simultaneously mitigate prompt training "interference" in multi-task, multi-source scenarios (e.g., task and data heterogeneity across sources), as well as possible implications from model approximations. As a highlight, MoPs manage to decrease final perplexity from $\sim20\%$ up to $\sim70\%$, as compared to baselines, in the federated scenario, and from $\sim 3\%$ up to $\sim30\%$ in the centralized scenario.

Connected Medical Devices (CMDs) have a large impact on patients as they allow them to lead a more normal life. Any malfunction could not only remove the health benefits the CMDs provide, they could also cause further harm to the patient. Due to this, there are many safety regulations which must be adhered to prior to a CMD entering the market. However, while many detailed safety regulations exist, there are a fundamental lack of cybersecurity frameworks applicable to CMDs. While there are recent regulations which aim to enforce cybersecurity practices, they are vague and do not contain the concrete steps necessary to implement cybersecurity. This paper aims to fill that gap by describing a framework, CyMed, to be used by vendors and ens-users, which contains concrete measures to improve the resilience of CMDs against cyber attack. The CyMed framework is subsequently evaluated based on practical tests as well as expert interviews.

We discuss the problem of bounding partially identifiable queries, such as counterfactuals, in Pearlian structural causal models. A recently proposed iterated EM scheme yields an inner approximation of those bounds by sampling the initialisation parameters. Such a method requires multiple (Bayesian network) queries over models sharing the same structural equations and topology, but different exogenous probabilities. This setup makes a compilation of the underlying model to an arithmetic circuit advantageous, thus inducing a sizeable inferential speed-up. We show how a single symbolic knowledge compilation allows us to obtain the circuit structure with symbolic parameters to be replaced by their actual values when computing the different queries. We also discuss parallelisation techniques to further speed up the bound computation. Experiments against standard Bayesian network inference show clear computational advantages with up to an order of magnitude of speed-up.

Graph Neural Networks (GNNs) have been studied from the lens of expressive power and generalization. However, their optimization properties are less well understood. We take the first step towards analyzing GNN training by studying the gradient dynamics of GNNs. First, we analyze linearized GNNs and prove that despite the non-convexity of training, convergence to a global minimum at a linear rate is guaranteed under mild assumptions that we validate on real-world graphs. Second, we study what may affect the GNNs' training speed. Our results show that the training of GNNs is implicitly accelerated by skip connections, more depth, and/or a good label distribution. Empirical results confirm that our theoretical results for linearized GNNs align with the training behavior of nonlinear GNNs. Our results provide the first theoretical support for the success of GNNs with skip connections in terms of optimization, and suggest that deep GNNs with skip connections would be promising in practice.

Graph Neural Networks (GNNs) have recently become increasingly popular due to their ability to learn complex systems of relations or interactions arising in a broad spectrum of problems ranging from biology and particle physics to social networks and recommendation systems. Despite the plethora of different models for deep learning on graphs, few approaches have been proposed thus far for dealing with graphs that present some sort of dynamic nature (e.g. evolving features or connectivity over time). In this paper, we present Temporal Graph Networks (TGNs), a generic, efficient framework for deep learning on dynamic graphs represented as sequences of timed events. Thanks to a novel combination of memory modules and graph-based operators, TGNs are able to significantly outperform previous approaches being at the same time more computationally efficient. We furthermore show that several previous models for learning on dynamic graphs can be cast as specific instances of our framework. We perform a detailed ablation study of different components of our framework and devise the best configuration that achieves state-of-the-art performance on several transductive and inductive prediction tasks for dynamic graphs.

Many natural language processing tasks solely rely on sparse dependencies between a few tokens in a sentence. Soft attention mechanisms show promising performance in modeling local/global dependencies by soft probabilities between every two tokens, but they are not effective and efficient when applied to long sentences. By contrast, hard attention mechanisms directly select a subset of tokens but are difficult and inefficient to train due to their combinatorial nature. In this paper, we integrate both soft and hard attention into one context fusion model, "reinforced self-attention (ReSA)", for the mutual benefit of each other. In ReSA, a hard attention trims a sequence for a soft self-attention to process, while the soft attention feeds reward signals back to facilitate the training of the hard one. For this purpose, we develop a novel hard attention called "reinforced sequence sampling (RSS)", selecting tokens in parallel and trained via policy gradient. Using two RSS modules, ReSA efficiently extracts the sparse dependencies between each pair of selected tokens. We finally propose an RNN/CNN-free sentence-encoding model, "reinforced self-attention network (ReSAN)", solely based on ReSA. It achieves state-of-the-art performance on both Stanford Natural Language Inference (SNLI) and Sentences Involving Compositional Knowledge (SICK) datasets.