An Anchor-Point Based Image-Model for Room Impulse Response Simulation with Directional Source Radiation and Sensor Directivity Patterns

The image model method has been widely used to simulate room impulse responses and the endeavor to adapt this method to different applications has also piqued great interest over the last few decades. This paper attempts to extend the image model method and develops an anchor-point-image-model (APIM) approach as a solution for simulating impulse responses by including both the source radiation and sensor directivity patterns. To determine the orientations of all the virtual sources, anchor points are introduced to real sources, which subsequently lead to the determination of the orientations of the virtual sources. An algorithm is developed to generate room impulse responses with APIM by taking into account the directional pattern functions, factional time delays, as well as the computational complexity. The developed model and algorithms can be used in various acoustic problems to simulate room acoustics and improve and evaluate processing algorithms.

Federated Classification in Hyperbolic Spaces via Secure Aggregation of Convex Hulls

Hierarchical and tree-like data sets arise in many applications, including language processing, graph data mining, phylogeny and genomics. It is known that tree-like data cannot be embedded into Euclidean spaces of finite dimension with small distortion. This problem can be mitigated through the use of hyperbolic spaces. When such data also has to be processed in a distributed and privatized setting, it becomes necessary to work with new federated learning methods tailored to hyperbolic spaces. As an initial step towards the development of the field of federated learning in hyperbolic spaces, we propose the first known approach to federated classification in hyperbolic spaces. Our contributions are as follows. First, we develop distributed versions of convex SVM classifiers for Poincar\'e discs. In this setting, the information conveyed from clients to the global classifier are convex hulls of clusters present in individual client data. Second, to avoid label switching issues, we introduce a number-theoretic approach for label recovery based on the so-called integer $B_h$ sequences. Third, we compute the complexity of the convex hulls in hyperbolic spaces to assess the extent of data leakage; at the same time, in order to limit the communication cost for the hulls, we propose a new quantization method for the Poincar\'e disc coupled with Reed-Solomon-like encoding. Fourth, at server level, we introduce a new approach for aggregating convex hulls of the clients based on balanced graph partitioning. We test our method on a collection of diverse data sets, including hierarchical single-cell RNA-seq data from different patients distributed across different repositories that have stringent privacy constraints. The classification accuracy of our method is up to $\sim 11\%$ better than its Euclidean counterpart, demonstrating the importance of privacy-preserving learning in hyperbolic spaces.

Differentially Private Decoupled Graph Convolutions for Multigranular Topology Protection

Graph learning methods, such as Graph Neural Networks (GNNs) based on graph convolutions, are highly successful in solving real-world learning problems involving graph-structured data. However, graph learning methods expose sensitive user information and interactions not only through their model parameters but also through their model predictions. Consequently, standard Differential Privacy (DP) techniques that merely offer model weight privacy are inadequate. This is especially the case for node predictions that leverage neighboring node attributes directly via graph convolutions that create additional risks of privacy leakage. To address this problem, we introduce Graph Differential Privacy (GDP), a new formal DP framework tailored to graph learning settings that ensures both provably private model parameters and predictions. Furthermore, since there may be different privacy requirements for the node attributes and graph structure, we introduce a novel notion of relaxed node-level data adjacency. This relaxation can be used for establishing guarantees for different degrees of graph topology privacy while maintaining node attribute privacy. Importantly, this relaxation reveals a useful trade-off between utility and topology privacy for graph learning methods. In addition, our analysis of GDP reveals that existing DP-GNNs fail to exploit this trade-off due to the complex interplay between graph topology and attribute data in standard graph convolution designs. To mitigate this problem, we introduce the Differentially Private Decoupled Graph Convolution (DPDGC) model, which benefits from decoupled graph convolution while providing GDP guarantees. Extensive experiments on seven node classification benchmarking datasets demonstrate the superior privacy-utility trade-off of DPDGC over existing DP-GNNs based on standard graph convolution design.

Unlearning Nonlinear Graph Classifiers in the Limited Training Data Regime

As the demand for user privacy grows, controlled data removal (machine unlearning) is becoming an important feature of machine learning models for data-sensitive Web applications such as social networks and recommender systems. Nevertheless, at this point it is still largely unknown how to perform efficient machine unlearning of graph neural networks (GNNs); this is especially the case when the number of training samples is small, in which case unlearning can seriously compromise the performance of the model. To address this issue, we initiate the study of unlearning the Graph Scattering Transform (GST), a mathematical framework that is efficient, provably stable under feature or graph topology perturbations, and offers graph classification performance comparable to that of GNNs. Our main contribution is the first known nonlinear approximate graph unlearning method based on GSTs. Our second contribution is a theoretical analysis of the computational complexity of the proposed unlearning mechanism, which is hard to replicate for deep neural networks. Our third contribution are extensive simulation results which show that, compared to complete retraining of GNNs after each removal request, the new GST-based approach offers, on average, a $10.38$x speed-up and leads to a $2.6$% increase in test accuracy during unlearning of $90$ out of $100$ training graphs from the IMDB dataset ($10$% training ratio).

Machine Unlearning of Federated Clusters

Federated clustering is an unsupervised learning problem that arises in a number of practical applications, including personalized recommender and healthcare systems. With the adoption of recent laws ensuring the "right to be forgotten", the problem of machine unlearning for federated clustering methods has become of significant importance. This work proposes the first known unlearning mechanism for federated clustering with privacy criteria that support simple, provable, and efficient data removal at the client and server level. The gist of our approach is to combine special initialization procedures with quantization methods that allow for secure aggregation of estimated local cluster counts at the server unit. As part of our platform, we introduce secure compressed multiset aggregation (SCMA), which is of independent interest for secure sparse model aggregation. In order to simultaneously facilitate low communication complexity and secret sharing protocols, we integrate Reed-Solomon encoding with special evaluation points into the new SCMA pipeline and derive bounds on the time and communication complexity of different components of the scheme. Compared to completely retraining K-means++ locally and globally for each removal request, we obtain an average speed-up of roughly 84x across seven datasets, two of which contain biological and medical information that is subject to frequent unlearning requests.

On the Study of Sample Complexity for Polynomial Neural Networks

As a general type of machine learning approach, artificial neural networks have established state-of-art benchmarks in many pattern recognition and data analysis tasks. Among various kinds of neural networks architectures, polynomial neural networks (PNNs) have been recently shown to be analyzable by spectrum analysis via neural tangent kernel, and particularly effective at image generation and face recognition. However, acquiring theoretical insight into the computation and sample complexity of PNNs remains an open problem. In this paper, we extend the analysis in previous literature to PNNs and obtain novel results on sample complexity of PNNs, which provides some insights in explaining the generalization ability of PNNs.

Certified Graph Unlearning

Graph-structured data is ubiquitous in practice and often processed using graph neural networks (GNNs). With the adoption of recent laws ensuring the ``right to be forgotten'', the problem of graph data removal has become of significant importance. To address the problem, we introduce the first known framework for \emph{certified graph unlearning} of GNNs. In contrast to standard machine unlearning, new analytical and heuristic unlearning challenges arise when dealing with complex graph data. First, three different types of unlearning requests need to be considered, including node feature, edge and node unlearning. Second, to establish provable performance guarantees, one needs to address challenges associated with feature mixing during propagation. The underlying analysis is illustrated on the example of simple graph convolutions (SGC) and their generalized PageRank (GPR) extensions, thereby laying the theoretical foundation for certified unlearning of GNNs. Our empirical studies on six benchmark datasets demonstrate excellent performance-complexity trade-offs when compared to complete retraining methods and approaches that do not leverage graph information. For example, when unlearning $20\%$ of the nodes on the Cora dataset, our approach suffers only a $0.1\%$ loss in test accuracy while offering a $4$-fold speed-up compared to complete retraining. Our scheme also outperforms unlearning methods that do not leverage graph information with a $12\%$ increase in test accuracy for a comparable time complexity.

Provably Accurate and Scalable Linear Classifiers in Hyperbolic Spaces

Many high-dimensional practical data sets have hierarchical structures induced by graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform the required learning tasks. For hierarchical data, the space of choice is a hyperbolic space because it guarantees low-distortion embeddings for tree-like structures. The geometry of hyperbolic spaces has properties not encountered in Euclidean spaces that pose challenges when trying to rigorously analyze algorithmic solutions. We propose a unified framework for learning scalable and simple hyperbolic linear classifiers with provable performance guarantees. The gist of our approach is to focus on Poincar\'e ball models and formulate the classification problems using tangent space formalisms. Our results include a new hyperbolic perceptron algorithm as well as an efficient and highly accurate convex optimization setup for hyperbolic support vector machine classifiers. Furthermore, we adapt our approach to accommodate second-order perceptrons, where data is preprocessed based on second-order information (correlation) to accelerate convergence, and strategic perceptrons, where potentially manipulated data arrives in an online manner and decisions are made sequentially. The excellent performance of the Poincar\'e second-order and strategic perceptrons shows that the proposed framework can be extended to general machine learning problems in hyperbolic spaces. Our experimental results, pertaining to synthetic, single-cell RNA-seq expression measurements, CIFAR10, Fashion-MNIST and mini-ImageNet, establish that all algorithms provably converge and have complexity comparable to those of their Euclidean counterparts. Accompanying codes can be found at: https://github.com/thupchnsky/PoincareLinearClassification.

Highly Scalable and Provably Accurate Classification in Poincare Balls

Many high-dimensional and large-volume data sets of practical relevance have hierarchical structures induced by trees, graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform required learning tasks. For hierarchical data, the space of choice is a hyperbolic space since it guarantees low-distortion embeddings for tree-like structures. Unfortunately, the geometry of hyperbolic spaces has properties not encountered in Euclidean spaces that pose challenges when trying to rigorously analyze algorithmic solutions. Here, for the first time, we establish a unified framework for learning scalable and simple hyperbolic linear classifiers with provable performance guarantees. The gist of our approach is to focus on Poincar\'e ball models and formulate the classification problems using tangent space formalisms. Our results include a new hyperbolic and second-order perceptron algorithm as well as an efficient and highly accurate convex optimization setup for hyperbolic support vector machine classifiers. All algorithms provably converge and are highly scalable as they have complexities comparable to those of their Euclidean counterparts. Their performance accuracies on synthetic data sets comprising millions of points, as well as on complex real-world data sets such as single-cell RNA-seq expression measurements, CIFAR10, Fashion-MNIST and mini-ImageNet.