Deep learning models have seen significant successes in numerous applications, but their inner workings remain elusive. The purpose of this work is to quantify the learning process of deep neural networks through the lens of a novel topological invariant called magnitude. Magnitude is an isometry invariant; its properties are an active area of research as it encodes many known invariants of a metric space. We use magnitude to study the internal representations of neural networks and propose a new method for determining their generalisation capabilities. Moreover, we theoretically connect magnitude dimension and the generalisation error, and demonstrate experimentally that the proposed framework can be a good indicator of the latter.

Gaussian mixture block models are distributions over graphs that strive to model modern networks: to generate a graph from such a model, we associate each vertex $i$ with a latent feature vector $u_i \in \mathbb{R}^d$ sampled from a mixture of Gaussians, and we add edge $(i,j)$ if and only if the feature vectors are sufficiently similar, in that $\langle u_i,u_j \rangle \ge \tau$ for a pre-specified threshold $\tau$. The different components of the Gaussian mixture represent the fact that there may be different types of nodes with different distributions over features -- for example, in a social network each component represents the different attributes of a distinct community. Natural algorithmic tasks associated with these networks are embedding (recovering the latent feature vectors) and clustering (grouping nodes by their mixture component). In this paper we initiate the study of clustering and embedding graphs sampled from high-dimensional Gaussian mixture block models, where the dimension of the latent feature vectors $d\to \infty$ as the size of the network $n \to \infty$. This high-dimensional setting is most appropriate in the context of modern networks, in which we think of the latent feature space as being high-dimensional. We analyze the performance of canonical spectral clustering and embedding algorithms for such graphs in the case of 2-component spherical Gaussian mixtures, and begin to sketch out the information-computation landscape for clustering and embedding in these models.

The goal of this paper is to learn more about how idiomatic information is structurally encoded in embeddings, using a structural probing method. We repurpose an existing English verbal multi-word expression (MWE) dataset to suit the probing framework and perform a comparative probing study of static (GloVe) and contextual (BERT) embeddings. Our experiments indicate that both encode some idiomatic information to varying degrees, but yield conflicting evidence as to whether idiomaticity is encoded in the vector norm, leaving this an open question. We also identify some limitations of the used dataset and highlight important directions for future work in improving its suitability for a probing analysis.

Understanding the host-specificity of different families of viruses sheds light on the origin of, e.g., SARS-CoV-2, rabies, and other such zoonotic pathogens in humans. It enables epidemiologists, medical professionals, and policymakers to curb existing epidemics and prevent future ones promptly. In the family Coronaviridae (of which SARS-CoV-2 is a member), it is well-known that the spike protein is the point of contact between the virus and the host cell membrane. On the other hand, the two traditional mammalian orders, Carnivora (carnivores) and Chiroptera (bats) are recognized to be responsible for maintaining and spreading the Rabies Lyssavirus (RABV). We propose Virus2Vec, a feature-vector representation for viral (nucleotide or amino acid) sequences that enable vector-space-based machine learning models to identify viral hosts. Virus2Vec generates numerical feature vectors for unaligned sequences, allowing us to forego the computationally expensive sequence alignment step from the pipeline. Virus2Vec leverages the power of both the \emph{minimizer} and position weight matrix (PWM) to generate compact feature vectors. Using several classifiers, we empirically evaluate Virus2Vec on real-world spike sequences of Coronaviridae and rabies virus sequence data to predict the host (identifying the reservoirs of infection). Our results demonstrate that Virus2Vec outperforms the predictive accuracies of baseline and state-of-the-art methods.

Equilibrium solution concepts of normal-form games, such as Nash equilibria, correlated equilibria, and coarse correlated equilibria, describe the joint strategy profiles from which no player has incentive to unilaterally deviate. They are widely studied in game theory, economics, and multiagent systems. Equilibrium concepts are invariant under certain transforms of the payoffs. We define an equilibrium-inspired distance metric for the space of all normal-form games and uncover a distance-preserving equilibrium-invariant embedding. Furthermore, we propose an additional transform which defines a better-response-invariant distance metric and embedding. To demonstrate these metric spaces we study $2\times2$ games. The equilibrium-invariant embedding of $2\times2$ games has an efficient two variable parameterization (a reduction from eight), where each variable geometrically describes an angle on a unit circle. Interesting properties can be spatially inferred from the embedding, including: equilibrium support, cycles, competition, coordination, distances, best-responses, and symmetries. The best-response-invariant embedding of $2\times2$ games, after considering symmetries, rediscovers a set of 15 games, and their respective equivalence classes. We propose that this set of game classes is fundamental and captures all possible interesting strategic interactions in $2\times2$ games. We introduce a directed graph representation and name for each class. Finally, we leverage the tools developed for $2\times2$ games to develop game theoretic visualizations of large normal-form and extensive-form games that aim to fingerprint the strategic interactions that occur within.

A central goal in cognitive science is to characterize human knowledge of concepts and their properties. Many have used human-generated feature lists as norms for establishing the structural relationship between concepts in the human mind (McRae et al., 2005; Devereux et al., 2014; De Deyne et al., 2008; Buchanan et al., 2019), but this requires extensive human labor. Large language models (LLMs) have recently shown impressive capabilities when generating properties of objects (Hansen & Hebart, 2022) or answering questions(Ouyang et al., 2022; Brown et al., 2020; Hoffmann et al., 2022; Chowdhery et al., 2022; Wei et al., 2021) and thus suggest an avenue for more efficient characterization of human knowledge structures, but even state-of-the-art models can routinely fail on many common-sense questions of fact. GTP3-davinci, for instance, will deny that alligators are green, while asserting that they can be used to suck dust up from surfaces. Thus, human effort can generate high-quality norms, but with prohibitive costs, while LLMs can produce norms with little human effort, but with considerably less accuracy. This paper considers whether human and machine effort can combine to efficiently estimate high-quality semantic feature vectors.

A key question in vision is how to represent our knowledge of previously encountered objects to classify new ones. The answer depends on how we determine the similarity of two objects. Similarity tells us how relevant each previously seen object is in determining the category to which a new object belongs. Complex notions of similar(cid:173) ity appear necessary for cognitive models and applications, while simple notions of similarity form a tractable basis for current computational ap(cid:173) proaches to classification. We explore the nature of this dichotomy and why it calls for new approaches to well-studied problems in learning. We begin this process by demonstrating new computational methods for supervised learning that can handle complex notions of similarity.

Clustering and prediction of sets of curves is an important problem in many areas of science and engineering. It is often the case that curves tend to be misaligned from each other in a continuous manner, either in space (across the measurements) or in time. We develop a probabilistic framework that allows for joint clustering and continuous alignment of sets of curves in curve space (as opposed to a fixed-dimensional feature- vector space). The proposed methodology integrates new probabilistic alignment models with model-based curve clustering algorithms. The probabilistic approach allows for the derivation of consistent EM learn- ing algorithms for the joint clustering-alignment problem. Experimental results are shown for alignment of human growth data, and joint cluster- ing and alignment of gene expression time-course data.

Lasso regression tends to assign zero weights to most irrelevant or redundant features, and hence is a promising technique for feature selection. Its limitation, however, is that it only offers solutions to linear models. Kernel machines with feature scaling techniques have been studied for feature selection with non-linear models. However, such approaches require to solve hard non-convex optimization problems. This paper proposes a new approach named the Feature Vector Machine (FVM). It reformulates the standard Lasso regression into a form isomorphic to SVM, and this form can be easily extended for feature selection with non-linear models by introducing kernels defined on feature vectors.

Motivated by recent developments in manifold-valued regression we propose a family of nonparametric kernel-smoothing estimators with metric-space valued output including a robust median type estimator and the classical Frechet mean. Depending on the choice of the output space and the chosen metric the estimator reduces to partially well-known procedures for multi-class classification, multivariate regression in Euclidean space, regression with manifold-valued output and even some cases of structured output learning. In this paper we focus on the case of regression with manifold-valued input and output. We show pointwise and Bayes consistency for all estimators in the family for the case of manifold-valued output and illustrate the robustness properties of the estimator with experiments.