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Concerning the recent notion of circular chromatic number of signed graphs, for each given integer $k$ we introduce two signed bipartite graphs, each on $2k^2-k+1$ vertices, having shortest negative cycle of length $2k$, and the circular chromatic number 4. Each of the construction can be viewed as a bipartite analogue of the generalized Mycielski graphs on odd cycles, $M_{\ell}(C_{2k+1})$. In the course of proving our result, we also obtain a simple proof of the fact that $M_{\ell}(C_{2k+1})$ and some similar quadrangulations of the projective plane have circular chromatic number 4. These proofs have the advantage that they illuminate, in an elementary manner, the strong relation between algebraic topology and graph coloring problems.

Local differential privacy is a powerful method for privacy-preserving data collection. In this paper, we develop a framework for training Generative Adversarial Networks (GANs) on differentially privatized data. We show that entropic regularization of optimal transport - a popular regularization method in the literature that has often been leveraged for its computational benefits - enables the generator to learn the raw (unprivatized) data distribution even though it only has access to privatized samples. We prove that at the same time this leads to fast statistical convergence at the parametric rate. This shows that entropic regularization of optimal transport uniquely enables the mitigation of both the effects of privatization noise and the curse of dimensionality in statistical convergence. We provide experimental evidence to support the efficacy of our framework in practice.

Over the past decade, the importance of the 1D signature which can be seen as a functional defined along a path, has been pivotal in both path-wise stochastic calculus and the analysis of time series data. By considering an image as a two-parameter function that takes values in a $d$-dimensional space, we introduce an extension of the path signature to images. We address numerous challenges associated with this extension and demonstrate that the 2D signature satisfies a version of Chen's relation in addition to a shuffle-type product. Furthermore, we show that specific variations of the 2D signature can be recursively defined, thereby satisfying an integral-type equation. We analyze the properties of the proposed signature, such as continuity, invariance to stretching, translation and rotation of the underlying image. Additionally, we establish that the proposed 2D signature over an image satisfies a universal approximation property.

With the increasing multimedia information, multimodal recommendation has received extensive attention. It utilizes multimodal information to alleviate the data sparsity problem in recommendation systems, thus improving recommendation accuracy. However, the reliance on labeled data severely limits the performance of multimodal recommendation models. Recently, self-supervised learning has been used in multimodal recommendations to mitigate the label sparsity problem. Nevertheless, the state-of-the-art methods cannot avoid the modality noise when aligning multimodal information due to the large differences in the distributions of different modalities. To this end, we propose a Multi-level sElf-supervised learNing for mulTimOdal Recommendation (MENTOR) method to address the label sparsity problem and the modality alignment problem. Specifically, MENTOR first enhances the specific features of each modality using the graph convolutional network (GCN) and fuses the visual and textual modalities. It then enhances the item representation via the item semantic graph for all modalities, including the fused modality. Then, it introduces two multilevel self-supervised tasks: the multilevel cross-modal alignment task and the general feature enhancement task. The multilevel cross-modal alignment task aligns each modality under the guidance of the ID embedding from multiple levels while maintaining the historical interaction information. The general feature enhancement task enhances the general feature from both the graph and feature perspectives to improve the robustness of our model. Extensive experiments on three publicly available datasets demonstrate the effectiveness of our method. Our code is publicly available at //github.com/Jinfeng-Xu/MENTOR.

Data for which a set of objects is described by multiple distinct feature sets (called views) is known as multi-view data. When missing values occur in multi-view data, all features in a view are likely to be missing simultaneously. This leads to very large quantities of missing data which, especially when combined with high-dimensionality, makes the application of conditional imputation methods computationally infeasible. We introduce a new imputation method based on the existing stacked penalized logistic regression (StaPLR) algorithm for multi-view learning. It performs imputation in a dimension-reduced space to address computational challenges inherent to the multi-view context. We compare the performance of the new imputation method with several existing imputation algorithms in simulated data sets. The results show that the new imputation method leads to competitive results at a much lower computational cost, and makes the use of advanced imputation algorithms such as missForest and predictive mean matching possible in settings where they would otherwise be computationally infeasible.

We analyze call center data on properties such as agent heterogeneity, customer patience and breaks. Then we compare simulation models that are different in the ways these properties are modeled. We classify them according to the extend in which they approach the actual service level and average waiting times. We obtain a theoretical understanding on how to distinguish between the model error and other aspects such as random noise. We conclude that modeling explicitly breaks and agent heterogeneity is crucial for obtaining a precise model.

In this short note we formulate a stabilizer formalism in the language of noncommutative graphs. The classes of noncommutative graphs we consider are obtained via unitary representations of compact groups, and suitably chosen operators on finite-dimensional Hilbert spaces. Furthermore, in this framework, we generalize previous results in this area for determining when such noncommutative graphs have anticliques.

In large-scale systems there are fundamental challenges when centralised techniques are used for task allocation. The number of interactions is limited by resource constraints such as on computation, storage, and network communication. We can increase scalability by implementing the system as a distributed task-allocation system, sharing tasks across many agents. However, this also increases the resource cost of communications and synchronisation, and is difficult to scale. In this paper we present four algorithms to solve these problems. The combination of these algorithms enable each agent to improve their task allocation strategy through reinforcement learning, while changing how much they explore the system in response to how optimal they believe their current strategy is, given their past experience. We focus on distributed agent systems where the agents' behaviours are constrained by resource usage limits, limiting agents to local rather than system-wide knowledge. We evaluate these algorithms in a simulated environment where agents are given a task composed of multiple subtasks that must be allocated to other agents with differing capabilities, to then carry out those tasks. We also simulate real-life system effects such as networking instability. Our solution is shown to solve the task allocation problem to 6.7% of the theoretical optimal within the system configurations considered. It provides 5x better performance recovery over no-knowledge retention approaches when system connectivity is impacted, and is tested against systems up to 100 agents with less than a 9% impact on the algorithms' performance.

We present ResMLP, an architecture built entirely upon multi-layer perceptrons for image classification. It is a simple residual network that alternates (i) a linear layer in which image patches interact, independently and identically across channels, and (ii) a two-layer feed-forward network in which channels interact independently per patch. When trained with a modern training strategy using heavy data-augmentation and optionally distillation, it attains surprisingly good accuracy/complexity trade-offs on ImageNet. We will share our code based on the Timm library and pre-trained models.

Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.