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Benjamini, Yadin, and Yehudayoff (2007) showed that if the maximum degree of a graph $G$ is 'sub-logarithmic,' then the typical range of random $\mathbb Z$-homomorphisms is super-constant. Furthermore, they showed that there is a sharp transition on the range of random $\mathbb Z$-homomorphisms on the graph $C_{n,k}$, the tensor product of the $n$-cycle and the complete graph on $k$ vertices with self-loops, around $k=2\log n$. We extend (to some extent) their results to random $M$-Lipschitz functions and random real-valued Lipschitz functions., Comment: 16 pages, 1 figure

Latent Bayesian optimization (LBO) approaches have successfully adopted Bayesian optimization over a continuous latent space by employing an encoder-decoder architecture to address the challenge of optimization in a high dimensional or discrete input space. LBO learns a surrogate model to approximate the black-box objective function in the latent space. However, we observed that most LBO methods suffer from the `misalignment problem`, which is induced by the reconstruction error of the encoder-decoder architecture. It hinders learning an accurate surrogate model and generating high-quality solutions. In addition, several trust region-based LBO methods select the anchor, the center of the trust region, based solely on the objective function value without considering the trust region`s potential to enhance the optimization process. To address these issues, we propose Inversion-based Latent Bayesian Optimization (InvBO), a plug-and-play module for LBO. InvBO consists of two components: an inversion method and a potential-aware trust region anchor selection. The inversion method searches the latent code that completely reconstructs the given target data. The potential-aware trust region anchor selection considers the potential capability of the trust region for better local optimization. Experimental results demonstrate the effectiveness of InvBO on nine real-world benchmarks, such as molecule design and arithmetic expression fitting tasks. Code is available at https://github.com/mlvlab/InvBO., Comment: Accepted to NeurIPS 2024

Dedekind's problem, dating back to 1897, asks for the total number $\psi(n)$ of antichains contained in the Boolean lattice $B_n$ on $n$ elements. We study Dedekind's problem using a recently developed method based on the cluster expansion from statistical physics and as a result, obtain several new results on the number and typical structure of antichains in $B_n$. We obtain detailed estimates for both $\psi(n)$ and the number of antichains of size $\beta \binom{n}{\lfloor n/2 \rfloor}$ for any fixed $\beta>0$. We also establish a sparse version of Sperner's theorem: we determine the sharp threshold and scaling window for the property that almost every antichain of size $m$ is contained in a middle layer of $B_n$.

We establish a refined version of a graph container lemma due to Galvin and discuss several applications related to the hard-core model on bipartite expander graphs. Given a graph $G$ and $\lambda>0$, the hard-core model on $G$ at activity $\lambda$ is the probability distribution $\mu_{G,\lambda}$ on independent sets in $G$ given by $\mu_{G,\lambda}(I)\propto \lambda^{|I|}$. As one of our main applications, we show that the hard-core model at activity $\lambda$ on the hypercube $Q_d$ exhibits a `structured phase' for $\lambda= \Omega( \log^2 d/d^{1/2})$ in the following sense: in a typical sample from $\mu_{Q_d,\lambda}$, most vertices are contained in one side of the bipartition of $Q_d$. This improves upon a result of Galvin which establishes the same for $\lambda=\Omega(\log d/ d^{1/3})$. As another application, we establish a fully polynomial-time approximation scheme (FPTAS) for the hard-core model on a $d$-regular bipartite $\alpha$-expander, with $\alpha>0$ fixed, when $\lambda= \Omega( \log^2 d/d^{1/2})$. This improves upon the bound $\lambda=\Omega(\log d/ d^{1/4})$ due to the first author, Perkins and Potukuchi. We discuss similar improvements to results of Galvin-Tetali, Balogh-Garcia-Li and Kronenberg-Spinka.

Large Language Models (LLMs) have demonstrated remarkable generalization and instruction-following capabilities with instruction tuning. The advancements in LLMs and instruction tuning have led to the development of Large Vision-Language Models (LVLMs). However, the competency of the LLMs and instruction tuning have been less explored in the molecular domain. Thus, we propose LLaMo: Large Language Model-based Molecular graph assistant, which is an end-to-end trained large molecular graph-language model. To bridge the discrepancy between the language and graph modalities, we present the multi-level graph projector that transforms graph representations into graph tokens by abstracting the output representations of each GNN layer and motif representations with the cross-attention mechanism. We also introduce machine-generated molecular graph instruction data to instruction-tune the large molecular graph-language model for general-purpose molecule and language understanding. Our extensive experiments demonstrate that LLaMo shows the best performance on diverse tasks, such as molecular description generation, property prediction, and IUPAC name prediction. The code of LLaMo is available at https://github.com/mlvlab/LLaMo., Comment: NeurIPS 2024

Knowledge graph-grounded dialog generation requires retrieving a dialog-relevant subgraph from the given knowledge base graph and integrating it with the dialog history. Previous works typically represent the graph using an external encoder, such as graph neural networks, and retrieve relevant triplets based on the similarity between single-vector representations of triplets and the dialog history. However, these external encoders fail to leverage the rich knowledge of pretrained language models, and the retrieval process is also suboptimal due to the information bottleneck caused by the single-vector abstraction of the dialog history. In this work, we propose Dialog generation with Generative Subgraph Retrieval (DialogGSR), which retrieves relevant knowledge subgraphs by directly generating their token sequences on top of language models. For effective generative subgraph retrieval, we introduce two key methods: (i) structure-aware knowledge graph linearization with self-supervised graph-specific tokens and (ii) graph-constrained decoding utilizing graph structural proximity-based entity informativeness scores for valid and relevant generative retrieval. DialogGSR achieves state-of-the-art performance in knowledge graph-grounded dialog generation, as demonstrated on OpenDialKG and KOMODIS datasets., Comment: EMNLP (main)

Given a connected finite graph $G$, an integer-valued function $f$ on $V(G)$ is called $M$-Lipschitz if the value of $f$ changes by at most $M$ along the edges of $G$. In 2013, Peled, Samotij, and Yehudayoff showed that random $M$-Lipschitz functions on graphs with sufficiently good expansion typically exhibit small fluctuations, giving sharp bounds on the typical range of such functions, assuming $M$ is not too large. We prove that the same conclusion holds under a relaxed expansion condition and for larger $M$, (partially) answering questions of Peled et al. Our techniques involve a combination of Sapozhenko's graph container methods and entropy methods from information theory., Comment: 24 pages

We introduce VideoMamba, a novel adaptation of the pure Mamba architecture, specifically designed for video recognition. Unlike transformers that rely on self-attention mechanisms leading to high computational costs by quadratic complexity, VideoMamba leverages Mamba's linear complexity and selective SSM mechanism for more efficient processing. The proposed Spatio-Temporal Forward and Backward SSM allows the model to effectively capture the complex relationship between non-sequential spatial and sequential temporal information in video. Consequently, VideoMamba is not only resource-efficient but also effective in capturing long-range dependency in videos, demonstrated by competitive performance and outstanding efficiency on a variety of video understanding benchmarks. Our work highlights the potential of VideoMamba as a powerful tool for video understanding, offering a simple yet effective baseline for future research in video analysis., Comment: ECCV 2024. code available at http://github.com/jinyjelly/VideoMamba

Weakly-Supervised Group Activity Recognition (WSGAR) aims to understand the activity performed together by a group of individuals with the video-level label and without actor-level labels. We propose Flow-Assisted Motion Learning Network (Flaming-Net) for WSGAR, which consists of the motion-aware actor encoder to extract actor features and the two-pathways relation module to infer the interaction among actors and their activity. Flaming-Net leverages an additional optical flow modality in the training stage to enhance its motion awareness when finding locally active actors. The first pathway of the relation module, the actor-centric path, initially captures the temporal dynamics of individual actors and then constructs inter-actor relationships. In parallel, the group-centric path starts by building spatial connections between actors within the same timeframe and then captures simultaneous spatio-temporal dynamics among them. We demonstrate that Flaming-Net achieves new state-of-the-art WSGAR results on two benchmarks, including a 2.8%p higher MPCA score on the NBA dataset. Importantly, we use the optical flow modality only for training and not for inference.

Pre-trained vision-language models have shown impressive success on various computer vision tasks with their zero-shot generalizability. Recently, prompt learning approaches have been explored to efficiently and effectively adapt the vision-language models to a variety of downstream tasks. However, most existing prompt learning methods suffer from task overfitting since the general knowledge of the pre-trained vision language models is forgotten while the prompts are finetuned on a small data set from a specific target task. To address this issue, we propose a Prompt Meta-Regularization (ProMetaR) to improve the generalizability of prompt learning for vision-language models. Specifically, ProMetaR meta-learns both the regularizer and the soft prompts to harness the task-specific knowledge from the downstream tasks and task-agnostic general knowledge from the vision-language models. Further, ProMetaR augments the task to generate multiple virtual tasks to alleviate the meta-overfitting. In addition, we provide the analysis to comprehend how ProMetaR improves the generalizability of prompt tuning in the perspective of the gradient alignment. Our extensive experiments demonstrate that our ProMetaR improves the generalizability of conventional prompt learning methods under base-to-base/base-to-new and domain generalization settings. The code of ProMetaR is available at https://github.com/mlvlab/ProMetaR., Comment: CVPR 2024

Visual Relationship Detection (VRD) has seen significant advancements with Transformer-based architectures recently. However, we identify two key limitations in a conventional label assignment for training Transformer-based VRD models, which is a process of mapping a ground-truth (GT) to a prediction. Under the conventional assignment, an unspecialized query is trained since a query is expected to detect every relation, which makes it difficult for a query to specialize in specific relations. Furthermore, a query is also insufficiently trained since a GT is assigned only to a single prediction, therefore near-correct or even correct predictions are suppressed by being assigned no relation as a GT. To address these issues, we propose Groupwise Query Specialization and Quality-Aware Multi-Assignment (SpeaQ). Groupwise Query Specialization trains a specialized query by dividing queries and relations into disjoint groups and directing a query in a specific query group solely toward relations in the corresponding relation group. Quality-Aware Multi-Assignment further facilitates the training by assigning a GT to multiple predictions that are significantly close to a GT in terms of a subject, an object, and the relation in between. Experimental results and analyses show that SpeaQ effectively trains specialized queries, which better utilize the capacity of a model, resulting in consistent performance gains with zero additional inference cost across multiple VRD models and benchmarks. Code is available at https://github.com/mlvlab/SpeaQ., Comment: CVPR 2024

Chain-of-Thought (CoT) prompting along with sub-question generation and answering has enhanced multi-step reasoning capabilities of Large Language Models (LLMs). However, prompting the LLMs to directly generate sub-questions is suboptimal since they sometimes generate redundant or irrelevant questions. To deal with them, we propose a GE-Reasoning method, which directs LLMs to generate proper sub-questions and corresponding answers. Concretely, given an input question, we first prompt the LLM to generate knowledge triplets, forming a graph representation of the question. Unlike conventional knowledge triplets, our approach allows variables as head or tail entities, effectively representing a question as knowledge triplets. Second, for each triplet, the LLM generates a corresponding sub-question and answer along with using knowledge retrieval. If the prediction confidence exceeds a threshold, the sub-question and prediction are incorporated into the prompt for subsequent processing. This approach encourages that sub-questions are grounded in the extracted knowledge triplets, reducing redundancy and irrelevance. Our experiments demonstrate that our approach outperforms previous CoT prompting methods and their variants on multi-hop question answering benchmark datasets., Comment: Preprint

In this expository article, we give a gentle introduction to the Erd\H{o}s-R\'enyi random graphs and threshold phenomena that they exhibit. We also mildly introduce the Kahn-Kalai Conjecture with several intuitive examples, mainly targeting the general audience., Comment: Section 5 is expanded in v2. 10 pages, many figures, to appear in Notices of the AMS

We give a short and self-contained argument that shows that, for any positive integers $t$ and $n$ with $t =O\Bigl(\frac{n}{\log n}\Bigr)$, the number $\alpha([t]^n)$ of antichains of the poset $[t]^n$ is at most \[\exp_2\Bigl(1+O\Bigl(\Bigl(\frac{t\log^3 n}{n}\Bigr)^{1/2}\Bigr)\Bigr)N(t,n)\,,\] where $N(t,n)$ is the size of a largest level of $[t]^n$. This, in particular, says that if $t \ll n/\log^3 n$ as $n \rightarrow \infty$, then $\log\alpha([t]^n)=(1+o(1))N(t,n)$, giving a (partially) positive answer to a question of Moshkovitz and Shapira for $t, n$ in this range. Particularly for $t=3$, we prove a better upper bound: \[\log\alpha([3]^n)\le(1+4\log 3/n)N(3,n),\] which is the best known upper bound on the number of antichains of $[3]^n$., Comment: 13 pages, 1 figure

For an odd integer $n = 2d-1$, let $\mathcal B_d$ be the subgraph of the hypercube $Q_n$ induced by the two largest layers. In this paper, we describe the typical structure of proper $q$-colorings of $V(\mathcal B_d)$ and give asymptotics on the number of them. The proofs use various tools including information theory (entropy), Sapozhenko's graph container method and a recently developed method of M. Jenssen and W. Perkins that combines Sapozhenko's graph container lemma with the cluster expansion for polymer models from statistical physics., Comment: 31 pages

We introduce Sketch-based Video Object Localization (SVOL), a new task aimed at localizing spatio-temporal object boxes in video queried by the input sketch. We first outline the challenges in the SVOL task and build the Sketch-Video Attention Network (SVANet) with the following design principles: (i) to consider temporal information of video and bridge the domain gap between sketch and video; (ii) to accurately identify and localize multiple objects simultaneously; (iii) to handle various styles of sketches; (iv) to be classification-free. In particular, SVANet is equipped with a Cross-modal Transformer that models the interaction between learnable object tokens, query sketch, and video through attention operations, and learns upon a per-frame set matching strategy that enables frame-wise prediction while utilizing global video context. We evaluate SVANet on a newly curated SVOL dataset. By design, SVANet successfully learns the mapping between the query sketches and video objects, achieving state-of-the-art results on the SVOL benchmark. We further confirm the effectiveness of SVANet via extensive ablation studies and visualizations. Lastly, we demonstrate its transfer capability on unseen datasets and novel categories, suggesting its high scalability in real-world applications

Transformers have shown superior performance on various computer vision tasks with their capabilities to capture long-range dependencies. Despite the success, it is challenging to directly apply Transformers on point clouds due to their quadratic cost in the number of points. In this paper, we present a Self-Positioning point-based Transformer (SPoTr), which is designed to capture both local and global shape contexts with reduced complexity. Specifically, this architecture consists of local self-attention and self-positioning point-based global cross-attention. The self-positioning points, adaptively located based on the input shape, consider both spatial and semantic information with disentangled attention to improve expressive power. With the self-positioning points, we propose a novel global cross-attention mechanism for point clouds, which improves the scalability of global self-attention by allowing the attention module to compute attention weights with only a small set of self-positioning points. Experiments show the effectiveness of SPoTr on three point cloud tasks such as shape classification, part segmentation, and scene segmentation. In particular, our proposed model achieves an accuracy gain of 2.6% over the previous best models on shape classification with ScanObjectNN. We also provide qualitative analyses to demonstrate the interpretability of self-positioning points. The code of SPoTr is available at https://github.com/mlvlab/SPoTr., Comment: Accepted paper at CVPR 2023

This paper presents a solution to the Weather4cast 2022 Challenge Stage 2. The goal of the challenge is to forecast future high-resolution rainfall events obtained from ground radar using low-resolution multiband satellite images. We suggest a solution that performs data preprocessing appropriate to the challenge and then predicts rainfall movies using a novel RainUNet. RainUNet is a hierarchical U-shaped network with temporal-wise separable block (TS block) using a decoupled large kernel 3D convolution to improve the prediction performance. Various evaluation metrics show that our solution is effective compared to the baseline method. The source codes are available at https://github.com/jinyxp/Weather4cast-2022, Comment: NeurIPS 2022, Weather4Cast core challenge

Question Answering (QA) is a task that entails reasoning over natural language contexts, and many relevant works augment language models (LMs) with graph neural networks (GNNs) to encode the Knowledge Graph (KG) information. However, most existing GNN-based modules for QA do not take advantage of rich relational information of KGs and depend on limited information interaction between the LM and the KG. To address these issues, we propose Question Answering Transformer (QAT), which is designed to jointly reason over language and graphs with respect to entity relations in a unified manner. Specifically, QAT constructs Meta-Path tokens, which learn relation-centric embeddings based on diverse structural and semantic relations. Then, our Relation-Aware Self-Attention module comprehensively integrates different modalities via the Cross-Modal Relative Position Bias, which guides information exchange between relevant entites of different modalities. We validate the effectiveness of QAT on commonsense question answering datasets like CommonsenseQA and OpenBookQA, and on a medical question answering dataset, MedQA-USMLE. On all the datasets, our method achieves state-of-the-art performance. Our code is available at http://github.com/mlvlab/QAT., Comment: AAAI2023 (accepted)

The effective work-function of metal electrode is one of the major factors to determine the threshold voltage of metal/oxide/semiconductor junction. In this work, we demonstrate experimentally that the effective work-function of Aluminum (Al) electrode in Al/SiO$_2$/n-Si junction increases significantly by $\sim$1.04 eV with the graphene interlayer inserted at Al/SiO$_2$ interface. We also provide the device-physical analysis of solving Poisson equation when the flat-band voltage is applied to the junction, supporting that the wide tuning of Al effective work-function originates from the electrical dipole layer formed by the overlap of electron orbitals between Al and graphene layer. Our work suggests the feasibility of constructing the dual-metal gate CMOS circuitry just by using Al electrodes with area-specific underlying graphene interlayer., Comment: 11 pages, 7 figures

Transformer-based models have recently shown success in representation learning on graph-structured data beyond natural language processing and computer vision. However, the success is limited to small-scale graphs due to the drawbacks of full dot-product attention on graphs such as the quadratic complexity with respect to the number of nodes and message aggregation from enormous irrelevant nodes. To address these issues, we propose Deformable Graph Transformer (DGT) that performs sparse attention via dynamically sampled relevant nodes for efficiently handling large-scale graphs with a linear complexity in the number of nodes. Specifically, our framework first constructs multiple node sequences with various criteria to consider both structural and semantic proximity. Then, combining with our learnable Katz Positional Encodings, the sparse attention is applied to the node sequences for learning node representations with a significantly reduced computational cost. Extensive experiments demonstrate that our DGT achieves state-of-the-art performance on 7 graph benchmark datasets with 2.5 - 449 times less computational cost compared to transformer-based graph models with full attention., Comment: 16 pages, 3 figures

For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose sum of measures is small. We prove a conjecture of Talagrand that is the analog of this result in the Bernoulli-$p$ setting, and answer a question of Talagrand on the analogous result for general positive empirical processes., Comment: Final version, 22 pages

Proving the ``expectation-threshold'' conjecture of Kahn and Kalai, we show that for any increasing property $\mathcal{F}$ on a finite set $X$, $$p_c(\mathcal{F})=O(q(\mathcal{F})\log \ell(\mathcal{F})),$$ where $p_c(\mathcal{F})$ and $q(\mathcal{F})$ are the threshold and ``expectation threshold'' of $\mathcal{F}$, and $\ell(\mathcal{F})$ is the maximum of $2$ and the maximum size of a minimal member of $\mathcal{F}$.

Graph Neural Networks (GNNs) often suffer from weak-generalization due to sparsely labeled data despite their promising results on various graph-based tasks. Data augmentation is a prevalent remedy to improve the generalization ability of models in many domains. However, due to the non-Euclidean nature of data space and the dependencies between samples, designing effective augmentation on graphs is challenging. In this paper, we propose a novel framework Metropolis-Hastings Data Augmentation (MH-Aug) that draws augmented graphs from an explicit target distribution for semi-supervised learning. MH-Aug produces a sequence of augmented graphs from the target distribution enables flexible control of the strength and diversity of augmentation. Since the direct sampling from the complex target distribution is challenging, we adopt the Metropolis-Hastings algorithm to obtain the augmented samples. We also propose a simple and effective semi-supervised learning strategy with generated samples from MH-Aug. Our extensive experiments demonstrate that MH-Aug can generate a sequence of samples according to the target distribution to significantly improve the performance of GNNs., Comment: 10 pages, 5 figures

Natural Language Video Grounding (NLVG) aims to localize time segments in an untrimmed video according to sentence queries. In this work, we present a new paradigm named Explore-And-Match for NLVG that seamlessly unifies the strengths of two streams of NLVG methods: proposal-free and proposal-based; the former explores the search space to find time segments directly, and the latter matches the predefined time segments with ground truths. To achieve this, we formulate NLVG as a set prediction problem and design an end-to-end trainable Language Video Transformer (LVTR) that can enjoy two favorable properties, which are rich contextualization power and parallel decoding. We train LVTR with two losses. First, temporal localization loss allows time segments of all queries to regress targets (explore). Second, set guidance loss couples every query with their respective target (match). To our surprise, we found that training schedule shows divide-and-conquer-like pattern: time segments are first diversified regardless of the target, then coupled with each target, and fine-tuned to the target again. Moreover, LVTR is highly efficient and effective: it infers faster than previous baselines (by 2X or more) and sets competitive results on two NLVG benchmarks (ActivityCaptions and Charades-STA). Codes are available at https://github.com/sangminwoo/Explore-And-Match., Comment: Code: https://github.com/sangminwoo/Explore-And-Match

Graph neural networks (GNNs) have significantly improved the representation power for graph-structured data. Despite of the recent success of GNNs, the graph convolution in most GNNs have two limitations. Since the graph convolution is performed in a small local neighborhood on the input graph, it is inherently incapable to capture long-range dependencies between distance nodes. In addition, when a node has neighbors that belong to different classes, i.e., heterophily, the aggregated messages from them often negatively affect representation learning. To address the two common problems of graph convolution, in this paper, we propose Deformable Graph Convolutional Networks (Deformable GCNs) that adaptively perform convolution in multiple latent spaces and capture short/long-range dependencies between nodes. Separated from node representations (features), our framework simultaneously learns the node positional embeddings (coordinates) to determine the relations between nodes in an end-to-end fashion. Depending on node position, the convolution kernels are deformed by deformation vectors and apply different transformations to its neighbor nodes. Our extensive experiments demonstrate that Deformable GCNs flexibly handles the heterophily and achieve the best performance in node classification tasks on six heterophilic graph datasets., Comment: To be appear at AAAI2022

Understanding document images (e.g., invoices) is a core but challenging task since it requires complex functions such as reading text and a holistic understanding of the document. Current Visual Document Understanding (VDU) methods outsource the task of reading text to off-the-shelf Optical Character Recognition (OCR) engines and focus on the understanding task with the OCR outputs. Although such OCR-based approaches have shown promising performance, they suffer from 1) high computational costs for using OCR; 2) inflexibility of OCR models on languages or types of document; 3) OCR error propagation to the subsequent process. To address these issues, in this paper, we introduce a novel OCR-free VDU model named Donut, which stands for Document understanding transformer. As the first step in OCR-free VDU research, we propose a simple architecture (i.e., Transformer) with a pre-training objective (i.e., cross-entropy loss). Donut is conceptually simple yet effective. Through extensive experiments and analyses, we show a simple OCR-free VDU model, Donut, achieves state-of-the-art performances on various VDU tasks in terms of both speed and accuracy. In addition, we offer a synthetic data generator that helps the model pre-training to be flexible in various languages and domains. The code, trained model and synthetic data are available at https://github.com/clovaai/donut., Comment: ECCV 2022. (v5) update table 2 and figures; add LayoutLM and update scores with the latest test script at https://github.com/clovaai/donut

We address a special case of a conjecture of M. Talagrand relating two notions of "threshold" for an increasing family $\mathcal F$ of subsets of a finite set $V$. The full conjecture implies equivalence of the "Fractional Expectation-Threshold Conjecture," due to Talagrand and recently proved by the authors and B. Narayanan, and the (stronger) "Expectation-Threshold Conjecture" of the second author and G. Kalai. The conjecture under discussion here says there is a fixed $L$ such that if, for a given $\mathcal F$, $p\in [0,1]$ admits $\lambda:2^V \rightarrow \mathbb R^+$ with \[ \mbox{$\sum_{S\subseteq F}\lambda_S\ge 1 ~~\forall F\in \mathcal F$} \] and \[ \mbox{$\sum_S\lambda_Sp^{|S|} \le 1/2$} \] (a.k.a. $\mathcal F$ is weakly $p$-small), then $p/L$ admits such a $\lambda$ taking values in $\{0,1\}$ ($\mathcal F$ is $(p/L)$-small). Talagrand showed this when $\lambda$ is supported on singletons and suggested, as a more challenging test case, proving it when $\lambda$ is supported on pairs. The present work provides such a proof., Comment: 11 pages

Let $Q_d$ be the $d$-dimensional Hamming cube and $N=|V(Q_d)|=2^d$. An independent set $I$ in $Q_d$ is called balanced if $I$ contains the same number of even and odd vertices. We show that the logarithm of the number of balanced independent sets in $Q_d$ is \[(1-\Theta(1/\sqrt d))N/2.\] The key ingredient of the proof is an improved version of "Sapozhenko's graph container lemma.", Comment: 6 pages. Comments are welcome!

In recent years, graph neural networks (GNNs) have been widely adopted in the representation learning of graph-structured data and provided state-of-the-art performance in various applications such as link prediction, node classification, and recommendation. Motivated by recent advances of self-supervision for representation learning in natural language processing and computer vision, self-supervised learning has been recently studied to leverage unlabeled graph-structured data. However, employing self-supervision tasks as auxiliary tasks to assist a primary task has been less explored in the literature on graphs. In this paper, we propose a novel self-supervised auxiliary learning framework to effectively learn graph neural networks. Moreover, this work is the first study showing that a meta-path prediction is beneficial as a self-supervised auxiliary task for heterogeneous graphs. Our method is learning to learn a primary task with various auxiliary tasks to improve generalization performance. The proposed method identifies an effective combination of auxiliary tasks and automatically balances them to improve the primary task. Our methods can be applied to any graph neural network in a plug-in manner without manual labeling or additional data. Also, it can be extended to any other auxiliary tasks. Our experiments demonstrate that the proposed method consistently improves the performance of node classification and link prediction., Comment: Submitted to TPAMI, 14pages. arXiv admin note: substantial text overlap with arXiv:2007.08294