Symmetric Positive Definite (SPD) matrices have received wide attention in machine learning due to their intrinsic capacity of encoding underlying structural correlation in data. To reflect the non-Euclidean geometry of SPD manifolds, many successful Riemannian metrics have been proposed. However, existing fixed metric tensors might lead to sub-optimal performance for SPD matrices learning, especially for SPD neural networks. To remedy this limitation, we leverage the idea of pullback and propose adaptive Riemannian metrics for SPD manifolds. Moreover, we present comprehensive theories for our metrics. Experiments on three datasets demonstrate that equipped with the proposed metrics, SPD networks can exhibit superior performance.
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Structured light has proven instrumental in 3D imaging, LiDAR, and holographic light projection. Metasurfaces, comprised of sub-wavelength-sized nanostructures, facilitate 180$^\circ$ field-of-view (FoV) structured light, circumventing the restricted FoV inherent in traditional optics like diffractive optical elements. However, extant metasurface-facilitated structured light exhibits sub-optimal performance in downstream tasks, due to heuristic pattern designs such as periodic dots that do not consider the objectives of the end application. In this paper, we present neural 360$^\circ$ structured light, driven by learned metasurfaces. We propose a differentiable framework, that encompasses a computationally-efficient 180$^\circ$ wave propagation model and a task-specific reconstructor, and exploits both transmission and reflection channels of the metasurface. Leveraging a first-order optimizer within our differentiable framework, we optimize the metasurface design, thereby realizing neural 360$^\circ$ structured light. We have utilized neural 360$^\circ$ structured light for holographic light projection and 3D imaging. Specifically, we demonstrate the first 360$^\circ$ light projection of complex patterns, enabled by our propagation model that can be computationally evaluated 50,000$\times$ faster than the Rayleigh-Sommerfeld propagation. For 3D imaging, we improve depth-estimation accuracy by 5.09$\times$ in RMSE compared to the heuristically-designed structured light. Neural 360$^\circ$ structured light promises robust 360$^\circ$ imaging and display for robotics, extended-reality systems, and human-computer interactions.
Error-correcting codes have an important role in data storage and transmission and in cryptography, particularly in the post-quantum era. Hermitian matrices over finite fields and equipped with the rank metric have the potential to offer enhanced security with greater efficiency in encryption and decryption. One crucial tool for evaluating the error-correcting capabilities of a code is its weight distribution and the MacWilliams Theorem has long been used to identify this structure of new codes from their known duals. Earlier papers have developed the MacWilliams Theorem for certain classes of matrices in the form of a functional transformation, developed using $q$-algebra, character theory and Generalised Krawtchouk polynomials, which is easy to apply and also allows for moments of the weight distribution to be found. In this paper, recent work by Kai-Uwe Schmidt on the properties of codes based on Hermitian matrices such as bounds on their size and the eigenvalues of their association scheme is extended by introducing a negative-$q$ algebra to establish a MacWilliams Theorem in this form together with some of its associated moments. The similarities in this approach and in the paper for the Skew-Rank metric by Friedlander et al. have been emphasised to facilitate future generalisation to any translation scheme.
Person Re-identification (Re-ID) is a crucial technique for public security and has made significant progress in supervised settings. However, the cross-domain (i.e., domain generalization) scene presents a challenge in Re-ID tasks due to unseen test domains and domain-shift between the training and test sets. To tackle this challenge, most existing methods aim to learn domain-invariant or robust features for all domains. In this paper, we observe that the data-distribution gap between the training and test sets is smaller in the sample-pair space than in the sample-instance space. Based on this observation, we propose a Generalizable Metric Network (GMN) to further explore sample similarity in the sample-pair space. Specifically, we add a Metric Network (M-Net) after the main network and train it on positive and negative sample-pair features, which is then employed during the test stage. Additionally, we introduce the Dropout-based Perturbation (DP) module to enhance the generalization capability of the metric network by enriching the sample-pair diversity. Moreover, we develop a Pair-Identity Center (PIC) loss to enhance the model's discrimination by ensuring that sample-pair features with the same pair-identity are consistent. We validate the effectiveness of our proposed method through a lot of experiments on multiple benchmark datasets and confirm the value of each module in our GMN.
Certifying the robustness of a graph-based machine learning model poses a critical challenge for safety. Current robustness certificates for graph classifiers guarantee output invariance with respect to the total number of node pair flips (edge addition or edge deletion), which amounts to an $l_{0}$ ball centred on the adjacency matrix. Although theoretically attractive, this type of isotropic structural noise can be too restrictive in practical scenarios where some node pairs are more critical than others in determining the classifier's output. The certificate, in this case, gives a pessimistic depiction of the robustness of the graph model. To tackle this issue, we develop a randomised smoothing method based on adding an anisotropic noise distribution to the input graph structure. We show that our process generates structural-aware certificates for our classifiers, whereby the magnitude of robustness certificates can vary across different pre-defined structures of the graph. We demonstrate the benefits of these certificates in both synthetic and real-world experiments.
Graph Convolutional Networks (GCNs) have been widely applied in various fields due to their significant power on processing graph-structured data. Typical GCN and its variants work under a homophily assumption (i.e., nodes with same class are prone to connect to each other), while ignoring the heterophily which exists in many real-world networks (i.e., nodes with different classes tend to form edges). Existing methods deal with heterophily by mainly aggregating higher-order neighborhoods or combing the immediate representations, which leads to noise and irrelevant information in the result. But these methods did not change the propagation mechanism which works under homophily assumption (that is a fundamental part of GCNs). This makes it difficult to distinguish the representation of nodes from different classes. To address this problem, in this paper we design a novel propagation mechanism, which can automatically change the propagation and aggregation process according to homophily or heterophily between node pairs. To adaptively learn the propagation process, we introduce two measurements of homophily degree between node pairs, which is learned based on topological and attribute information, respectively. Then we incorporate the learnable homophily degree into the graph convolution framework, which is trained in an end-to-end schema, enabling it to go beyond the assumption of homophily. More importantly, we theoretically prove that our model can constrain the similarity of representations between nodes according to their homophily degree. Experiments on seven real-world datasets demonstrate that this new approach outperforms the state-of-the-art methods under heterophily or low homophily, and gains competitive performance under homophily.
The inductive biases of graph representation learning algorithms are often encoded in the background geometry of their embedding space. In this paper, we show that general directed graphs can be effectively represented by an embedding model that combines three components: a pseudo-Riemannian metric structure, a non-trivial global topology, and a unique likelihood function that explicitly incorporates a preferred direction in embedding space. We demonstrate the representational capabilities of this method by applying it to the task of link prediction on a series of synthetic and real directed graphs from natural language applications and biology. In particular, we show that low-dimensional cylindrical Minkowski and anti-de Sitter spacetimes can produce equal or better graph representations than curved Riemannian manifolds of higher dimensions.
Unsupervised domain adaptation (UDA) methods for person re-identification (re-ID) aim at transferring re-ID knowledge from labeled source data to unlabeled target data. Although achieving great success, most of them only use limited data from a single-source domain for model pre-training, making the rich labeled data insufficiently exploited. To make full use of the valuable labeled data, we introduce the multi-source concept into UDA person re-ID field, where multiple source datasets are used during training. However, because of domain gaps, simply combining different datasets only brings limited improvement. In this paper, we try to address this problem from two perspectives, \ie{} domain-specific view and domain-fusion view. Two constructive modules are proposed, and they are compatible with each other. First, a rectification domain-specific batch normalization (RDSBN) module is explored to simultaneously reduce domain-specific characteristics and increase the distinctiveness of person features. Second, a graph convolutional network (GCN) based multi-domain information fusion (MDIF) module is developed, which minimizes domain distances by fusing features of different domains. The proposed method outperforms state-of-the-art UDA person re-ID methods by a large margin, and even achieves comparable performance to the supervised approaches without any post-processing techniques.
Graph classification aims to perform accurate information extraction and classification over graphstructured data. In the past few years, Graph Neural Networks (GNNs) have achieved satisfactory performance on graph classification tasks. However, most GNNs based methods focus on designing graph convolutional operations and graph pooling operations, overlooking that collecting or labeling graph-structured data is more difficult than grid-based data. We utilize meta-learning for fewshot graph classification to alleviate the scarce of labeled graph samples when training new tasks.More specifically, to boost the learning of graph classification tasks, we leverage GNNs as graph embedding backbone and meta-learning as training paradigm to capture task-specific knowledge rapidly in graph classification tasks and transfer them to new tasks. To enhance the robustness of meta-learner, we designed a novel step controller driven by Reinforcement Learning. The experiments demonstrate that our framework works well compared to baselines.
Existing few-shot learning (FSL) methods assume that there exist sufficient training samples from source classes for knowledge transfer to target classes with few training samples. However, this assumption is often invalid, especially when it comes to fine-grained recognition. In this work, we define a new FSL setting termed few-shot fewshot learning (FSFSL), under which both the source and target classes have limited training samples. To overcome the source class data scarcity problem, a natural option is to crawl images from the web with class names as search keywords. However, the crawled images are inevitably corrupted by large amount of noise (irrelevant images) and thus may harm the performance. To address this problem, we propose a graph convolutional network (GCN)-based label denoising (LDN) method to remove the irrelevant images. Further, with the cleaned web images as well as the original clean training images, we propose a GCN-based FSL method. For both the LDN and FSL tasks, a novel adaptive aggregation GCN (AdarGCN) model is proposed, which differs from existing GCN models in that adaptive aggregation is performed based on a multi-head multi-level aggregation module. With AdarGCN, how much and how far information carried by each graph node is propagated in the graph structure can be determined automatically, therefore alleviating the effects of both noisy and outlying training samples. Extensive experiments show the superior performance of our AdarGCN under both the new FSFSL and the conventional FSL settings.
Graph convolutional networks (GCNs) have been successfully applied in node classification tasks of network mining. However, most of these models based on neighborhood aggregation are usually shallow and lack the "graph pooling" mechanism, which prevents the model from obtaining adequate global information. In order to increase the receptive field, we propose a novel deep Hierarchical Graph Convolutional Network (H-GCN) for semi-supervised node classification. H-GCN first repeatedly aggregates structurally similar nodes to hyper-nodes and then refines the coarsened graph to the original to restore the representation for each node. Instead of merely aggregating one- or two-hop neighborhood information, the proposed coarsening procedure enlarges the receptive field for each node, hence more global information can be learned. Comprehensive experiments conducted on public datasets demonstrate the effectiveness of the proposed method over the state-of-art methods. Notably, our model gains substantial improvements when only a few labeled samples are provided.
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