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This paper aims to develop a mathematical foundation to model knitting with graphs. We provide a precise definition for knit objects with a knot theoretic component and propose a simple undirected graph, a simple directed graph, and a directed multigraph model for any arbitrary knit object. Using these models, we propose natural categories related to the complexity of knitting structures. We use these categories to explore the hardness of determining whether a knit object of each class exists for a given graph. We show that while this problem is NP-hard in general, under specific cases, there are linear and polynomial time algorithms which take advantage of unique properties of common knitting techniques. This work aims to bridge the gap between textile arts and graph theory, offering a useful and rigorous framework for analyzing knitting objects using their corresponding graphs and for generating knitting objects from graphs.

Certifiable robustness gives the guarantee that small perturbations around an input to a classifier will not change the prediction. There are two approaches to provide certifiable robustness to adversarial examples: a) explicitly training classifiers with small Lipschitz constants, and b) Randomized smoothing, which adds random noise to the input to create a smooth classifier. We propose \textit{SPLITZ}, a practical and novel approach which leverages the synergistic benefits of both the above ideas into a single framework. Our main idea is to \textit{split} a classifier into two halves, constrain the Lipschitz constant of the first half, and smooth the second half via randomization. Motivation for \textit{SPLITZ} comes from the observation that many standard deep networks exhibit heterogeneity in Lipschitz constants across layers. \textit{SPLITZ} can exploit this heterogeneity while inheriting the scalability of randomized smoothing. We present a principled approach to train \textit{SPLITZ} and provide theoretical analysis to derive certified robustness guarantees during inference. We present a comprehensive comparison of robustness-accuracy tradeoffs and show that \textit{SPLITZ} consistently improves upon existing state-of-the-art approaches on MNIST and CIFAR-10 datasets. For instance, with $\ell_2$ norm perturbation budget of \textbf{$\epsilon=1$}, \textit{SPLITZ} achieves $\textbf{43.2\%}$ top-1 test accuracy on CIFAR-10 dataset compared to state-of-art top-1 test accuracy $\textbf{39.8\%}

Knowledge distillation (KD) is widely used for compressing a teacher model to a smaller student model, reducing its inference cost and memory footprint while preserving model capabilities. However, current KD methods for auto-regressive sequence models (e.g., large language models) suffer from missing a standardized objective function. Moreover, the recent use of student-generated outputs to address training-inference mismatches has significantly escalated computational costs. To tackle these issues, we introduce DistiLLM, a more effective and efficient KD framework for auto-regressive language models. DistiLLM comprises two components: (1) a novel skew Kullback-Leibler divergence loss, where we unveil and leverage its theoretical properties, and (2) an adaptive off-policy approach designed to enhance the efficiency in utilizing student-generated outputs. Extensive experiments, including instruction-following tasks, demonstrate the effectiveness of DistiLLM in building high-performing student models while achieving up to 4.3$\times$ speedup compared to recent KD methods.

Multi-modal large language models (MLLMs) have achieved remarkable performance on objective multimodal perception tasks, but their ability to interpret subjective, emotionally nuanced multimodal content remains largely unexplored. Thus, it impedes their ability to effectively understand and react to the intricate emotions expressed by humans through multimodal media. To bridge this gap, we introduce EmoBench, the first comprehensive benchmark designed specifically to evaluate the emotional capabilities of MLLMs across five popular emotional tasks, using a diverse dataset of 287k images and videos paired with corresponding textual instructions. Meanwhile, we propose EmoLLM, a novel model for multimodal emotional understanding, incorporating with two core techniques. 1) Multi-perspective Visual Projection, it captures diverse emotional cues from visual data from multiple perspectives. 2) EmoPrompt, it guides MLLMs to reason about emotions in the correct direction. Experimental results demonstrate that EmoLLM significantly elevates multimodal emotional understanding performance, with an average improvement of 12.1% across multiple foundation models on EmoBench. Our work contributes to the advancement of MLLMs by facilitating a deeper and more nuanced comprehension of intricate human emotions, paving the way for the development of artificial emotional intelligence capabilities with wide-ranging applications in areas such as human-computer interaction, mental health support, and empathetic AI systems. Code, data, and model will be released.

We propose a novel database model whose basic structure is a labeled, directed, acyclic graph with a single root, in which the nodes represent the data sets of an application and the edges represent functional relationships among the data sets. We call such a graph an application context or simply context. The query language of a context consists of two types of queries, traversal queries and analytic queries. Both types of queries are defined using a simple functional algebra whose operations are functional restriction, composition of functions, pairing of functions and Cartesian product of sets. Roughly speaking, traversal queries parallel relational algebra queries, whereas analytic queries parallel SQL Group-by queries. In other words, in our model, traversal queries and analytic queries, are both defined within the same formal framework - in contrast to the relational model, where analytic queries are defined outside the relational algebra. Therefore a distinctive feature of our model is that it supports data management and data analytics within the same formal framework. We demonstrate the expressive power of our model by showing: (a) how a relational database can be defined as a view over a context, with the context playing the role of an underlying semantic layer; (b) how an analytic query over a context can be rewritten at two orthogonal levels: at the level of the traversal queries that do the grouping and measuring, and at the level of the analytic query itself; and (c) how a context can be used as a user-friendly interface for querying relations and analysing relational data.

Generative models are trained with the simple objective of imitating the conditional probability distribution induced by the data they are trained on. Therefore, when trained on data generated by humans, we may not expect the artificial model to outperform the humans on their original objectives. In this work, we study the phenomenon of transcendence: when a generative model achieves capabilities that surpass the abilities of the experts generating its data. We demonstrate transcendence by training an autoregressive transformer to play chess from game transcripts, and show that the trained model can sometimes achieve better performance than all players in the dataset. We theoretically prove that transcendence can be enabled by low-temperature sampling, and rigorously assess this claim experimentally. Finally, we discuss other sources of transcendence, laying the groundwork for future investigation of this phenomenon in a broader setting.

Interpretability methods are developed to understand the working mechanisms of black-box models, which is crucial to their responsible deployment. Fulfilling this goal requires both that the explanations generated by these methods are correct and that people can easily and reliably understand them. While the former has been addressed in prior work, the latter is often overlooked, resulting in informal model understanding derived from a handful of local explanations. In this paper, we introduce explanation summary (ExSum), a mathematical framework for quantifying model understanding, and propose metrics for its quality assessment. On two domains, ExSum highlights various limitations in the current practice, helps develop accurate model understanding, and reveals easily overlooked properties of the model. We also connect understandability to other properties of explanations such as human alignment, robustness, and counterfactual minimality and plausibility.

Many scientific problems require to process data in the form of geometric graphs. Unlike generic graph data, geometric graphs exhibit symmetries of translations, rotations, and/or reflections. Researchers have leveraged such inductive bias and developed geometrically equivariant Graph Neural Networks (GNNs) to better characterize the geometry and topology of geometric graphs. Despite fruitful achievements, it still lacks a survey to depict how equivariant GNNs are progressed, which in turn hinders the further development of equivariant GNNs. To this end, based on the necessary but concise mathematical preliminaries, we analyze and classify existing methods into three groups regarding how the message passing and aggregation in GNNs are represented. We also summarize the benchmarks as well as the related datasets to facilitate later researches for methodology development and experimental evaluation. The prospect for future potential directions is also provided.

The design of deep graph models still remains to be investigated and the crucial part is how to explore and exploit the knowledge from different hops of neighbors in an efficient way. In this paper, we propose a novel RNN-like deep graph neural network architecture by incorporating AdaBoost into the computation of network; and the proposed graph convolutional network called AdaGCN~(AdaBoosting Graph Convolutional Network) has the ability to efficiently extract knowledge from high-order neighbors and integrate knowledge from different hops of neighbors into the network in an AdaBoost way. We also present the architectural difference between AdaGCN and existing graph convolutional methods to show the benefits of our proposal. Finally, extensive experiments demonstrate the state-of-the-art prediction performance and the computational advantage of our approach AdaGCN.

Graph convolutional networks (GCNs) have recently become one of the most powerful tools for graph analytics tasks in numerous applications, ranging from social networks and natural language processing to bioinformatics and chemoinformatics, thanks to their ability to capture the complex relationships between concepts. At present, the vast majority of GCNs use a neighborhood aggregation framework to learn a continuous and compact vector, then performing a pooling operation to generalize graph embedding for the classification task. These approaches have two disadvantages in the graph classification task: (1)when only the largest sub-graph structure ($k$-hop neighbor) is used for neighborhood aggregation, a large amount of early-stage information is lost during the graph convolution step; (2) simple average/sum pooling or max pooling utilized, which loses the characteristics of each node and the topology between nodes. In this paper, we propose a novel framework called, dual attention graph convolutional networks (DAGCN) to address these problems. DAGCN automatically learns the importance of neighbors at different hops using a novel attention graph convolution layer, and then employs a second attention component, a self-attention pooling layer, to generalize the graph representation from the various aspects of a matrix graph embedding. The dual attention network is trained in an end-to-end manner for the graph classification task. We compare our model with state-of-the-art graph kernels and other deep learning methods. The experimental results show that our framework not only outperforms other baselines but also achieves a better rate of convergence.