Graph embedding methods embed the nodes in a graph in low dimensional vector space while preserving graph topology to carry out the downstream tasks such as link prediction, node recommendation and clustering. These tasks depend on a similarity measure such as cosine similarity and Euclidean distance between a pair of embeddings that are symmetric in nature and hence do not hold good for directed graphs. Recent work on directed graphs, HOPE, APP, and NERD, proposed to preserve the direction of edges among nodes by learning two embeddings, source and target, for every node. However, these methods do not take into account the properties of directed edges explicitly. To understand the directional relation among nodes, we propose a novel approach that takes advantage of the non commutative property of vector cross product to learn embeddings that inherently preserve the direction of edges among nodes. We learn the node embeddings through a Siamese neural network where the cross-product operation is incorporated into the network architecture. Although cross product between a pair of vectors is defined in three dimensional, the approach is extended to learn N dimensional embeddings while maintaining the non-commutative property. In our empirical experiments on three real-world datasets, we observed that even very low dimensional embeddings could effectively preserve the directional property while outperforming some of the state-of-the-art methods on link prediction and node recommendation tasks

Set prediction is about learning to predict a collection of unordered variables with unknown interrelations. Training such models with set losses imposes the structure of a metric space over sets. We focus on stochastic and underdefined cases, where an incorrectly chosen loss function leads to implausible predictions. Example tasks include conditional point-cloud reconstruction and predicting future states of molecules. In this paper, we propose an alternative to training via set losses by viewing learning as conditional density estimation. Our learning framework fits deep energy-based models and approximates the intractable likelihood with gradient-guided sampling. Furthermore, we propose a stochastically augmented prediction algorithm that enables multiple predictions, reflecting the possible variations in the target set. We empirically demonstrate on a variety of datasets the capability to learn multi-modal densities and produce multiple plausible predictions. Our approach is competitive with previous set prediction models on standard benchmarks. More importantly, it extends the family of addressable tasks beyond those that have unambiguous predictions.

One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time. A time-varying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic behavior. We provide an introduction to topological summaries of time-varying metric spaces including vineyards [17], crocker plots [52], and multiparameter rank functions [34]. We then introduce a new tool to summarize time-varying metric spaces: a crocker stack. Crocker stacks are convenient for visualization, amenable to machine learning, and satisfy a desirable stability property which we prove. We demonstrate the utility of crocker stacks for a parameter identification task involving an influential model of biological aggregations [54]. Altogether, we aim to bring the broader applied mathematics community up-to-date on topological summaries of time-varying metric spaces.

We study universal consistency and convergence rates of simple nearest-neighbor prototype rules for the problem of multiclass classification in metric paces. We first show that a novel data-dependent partitioning rule, named Proto-NN, is universally consistent in any metric space that admits a universally consistent rule. Proto-NN is a significant simplification of OptiNet, a recently proposed compression-based algorithm that, to date, was the only algorithm known to be universally consistent in such a general setting. Practically, Proto-NN is simpler to implement and enjoys reduced computational complexity. We then proceed to study convergence rates of the excess error probability. We first obtain rates for the standard $k$-NN rule under a margin condition and a new generalized-Lipschitz condition. The latter is an extension of a recently proposed modified-Lipschitz condition from $\mathbb R^d$ to metric spaces. Similarly to the modified-Lipschitz condition, the new condition avoids any boundness assumptions on the data distribution. While obtaining rates for Proto-NN is left open, we show that a second prototype rule that hybridizes between $k$-NN and Proto-NN achieves the same rates as $k$-NN while enjoying similar computational advantages as Proto-NN. We conjecture however that, as $k$-NN, this hybrid rule is not consistent in general.

Despite numerous attempts sought to provide empirical evidence of adversarial regularization outperforming sole supervision, the theoretical understanding of such phenomena remains elusive. In this study, we aim to resolve whether adversarial regularization indeed performs better than sole supervision at a fundamental level. To bring this insight into fruition, we study vanishing gradient issue, asymptotic iteration complexity, gradient flow and provable convergence in the context of sole supervision and adversarial regularization. The key ingredient is a theoretical justification supported by empirical evidence of adversarial acceleration in gradient descent. In addition, motivated by a recently introduced unit-wise capacity based generalization bound, we analyze the generalization error in adversarial framework. Guided by our observation, we cast doubts on the ability of this measure to explain generalization. We therefore leave as open questions to explore new measures that can explain generalization behavior in adversarial learning. Furthermore, we observe an intriguing phenomenon in the neural embedded vector space while contrasting adversarial learning with sole supervision.

The development of autonomous vehicles requires having access to a large amount of data in the concerning driving scenarios. However, manual annotation of such driving scenarios is costly and subject to the errors in the rule-based trajectory labeling systems. To address this issue, we propose an effective non-parametric trajectory clustering framework consisting of five stages: (1) aligning trajectories and quantifying their pairwise temporal dissimilarities, (2) embedding the trajectory-based dissimilarities into a vector space, (3) extracting transitive relations, (4) embedding the transitive relations into a new vector space, and (5) clustering the trajectories with an optimal number of clusters. We investigate and evaluate the proposed framework on a challenging real-world dataset consisting of annotated trajectories. We observe that the proposed framework achieves promising results, despite the complexity caused by having trajectories of varying length. Furthermore, we extend the framework to validate the augmentation of the real dataset with synthetic data generated by a Generative Adversarial Network (GAN) where we examine whether the generated trajectories are consistent with the true underlying clusters.

Artificial intelligence nowadays plays an increasingly prominent role in our life since decisions that were once made by humans are now delegated to automated systems. A machine learning algorithm trained based on biased data, however, tends to make unfair predictions. Developing classification algorithms that are fair with respect to protected attributes of the data thus becomes an important problem. Motivated by concerns surrounding the fairness effects of sharing and few-shot machine learning tools, such as the Model Agnostic Meta-Learning framework, we propose a novel fair fast-adapted few-shot meta-learning approach that efficiently mitigates biases during meta-train by ensuring controlling the decision boundary covariance that between the protected variable and the signed distance from the feature vectors to the decision boundary. Through extensive experiments on two real-world image benchmarks over three state-of-the-art meta-learning algorithms, we empirically demonstrate that our proposed approach efficiently mitigates biases on model output and generalizes both accuracy and fairness to unseen tasks with a limited amount of training samples.

In this work we study the stability of algebraic neural networks (AlgNNs) with commutative algebras which unify CNNs and GNNs under the umbrella of algebraic signal processing. An AlgNN is a stacked layered structure where each layer is conformed by an algebra $\mathcal{A}$, a vector space $\mathcal{M}$ and a homomorphism $\rho:\mathcal{A}\rightarrow\text{End}(\mathcal{M})$, where $\text{End}(\mathcal{M})$ is the set of endomorphims of $\mathcal{M}$. Signals in each layer are modeled as elements of $\mathcal{M}$ and are processed by elements of $\text{End}(\mathcal{M})$ defined according to the structure of $\mathcal{A}$ via $\rho$. This framework provides a general scenario that covers several types of neural network architectures where formal convolution operators are being used. We obtain stability conditions regarding to perturbations which are defined as distortions of $\rho$, reaching general results whose particular cases are consistent with recent findings in the literature for CNNs and GNNs. We consider conditions on the domain of the homomorphisms in the algebra that lead to stable operators. Interestingly, we found that these conditions are related to the uniform boundedness of the Fr\'echet derivative of a function $p:\text{End}(\mathcal{M})\rightarrow\text{End}(\mathcal{M})$ that maps the images of the generators of $\mathcal{A}$ on $\text{End}(\mathcal{M})$ into a power series representation that defines the filtering of elements in $\mathcal{M}$. Additionally, our results show that stability is universal to convolutional architectures whose algebraic signal model uses the same algebra.

Comparing metric measure spaces (i.e. a metric space endowed with a probability distribution) is at the heart of many machine learning problems. This includes for instance predicting properties of molecules in quantum chemistry or generating graphs with varying connectivity. The most popular distance between such metric measure spaces is the Gromov-Wasserstein (GW) distance, which is the solution of a quadratic assignment problem. This distance has been successfully applied to supervised learning and generative modeling, for applications as diverse as quantum chemistry or natural language processing. The GW distance is however limited to the comparison of metric measure spaces endowed with a \emph{probability} distribution. This strong limitation is problematic for many applications in ML where there is no a priori natural normalization on the total mass of the data. Furthermore, imposing an exact conservation of mass across spaces is not robust to outliers and often leads to irregular matching. To alleviate these issues, we introduce two Unbalanced Gromov-Wasserstein formulations: a distance and a more computationally tractable upper-bounding relaxation. They both allow the comparison of metric spaces equipped with arbitrary positive measures up to isometries.

RDF2vec is an embedding technique for representing knowledge graph entities in a continuous vector space. In this paper, we investigate the effect of materializing implicit A-box axioms induced by subproperties, as well as symmetric and transitive properties. While it might be a reasonable assumption that such a materialization before computing embeddings might lead to better embeddings, we conduct a set of experiments on DBpedia which demonstrate that the materialization actually has a negative effect on the performance of RDF2vec. In our analysis, we argue that despite the huge body of work devoted on completing missing information in knowledge graphs, such missing implicit information is actually a signal, not a defect, and we show examples illustrating that assumption.