Label-free alignment between datasets collected at different times, locations, or by different instruments is a fundamental scientific task. Hyperbolic spaces have recently provided a fruitful foundation for the development of informative representations of hierarchical data. Here, we take a purely geometric approach for label-free alignment of hierarchical datasets and introduce hyperbolic Procrustes analysis (HPA). HPA consists of new implementations of the three prototypical Procrustes analysis components: translation, scaling, and rotation, based on the Riemannian geometry of the Lorentz model of hyperbolic space. We analyze the proposed components, highlighting their useful properties for alignment. The efficacy of HPA, its theoretical properties, stability and computational efficiency are demonstrated in simulations.

Integrated Gradients (IG), a widely used axiomatic path-based attribution method, assigns importance scores to input features by integrating model gradients along a straight path from a baseline to the input. While effective in some cases, we show that straight paths can lead to flawed attributions. In this paper, we identify the cause of these misattributions and propose an alternative approach that treats the input space as a Riemannian manifold, computing attributions by integrating gradients along geodesics. We call this method Geodesic Integrated Gradients (GIG). To approximate geodesic paths, we introduce two techniques: a k-Nearest Neighbours-based approach for smaller models and a Stochastic Variational Inference-based method for larger ones. Additionally, we propose a new axiom, Strong Completeness, extending the axioms satisfied by IG. We show that this property is desirable for attribution methods and that GIG is the only method that satisfies it. Through experiments on both synthetic and real-world data, we demonstrate that GIG outperforms existing explainability methods, including IG.

Understanding intelligence is a central pursuit in neuroscience, cognitive science, and artificial intelligence. Intelligence encompasses learning, problem-solving, creativity, and even consciousness. Recent advancements in geometric analysis have revealed new insights into high-dimensional information representation and organisation, exposing intrinsic data structures and dynamic processes within neural and artificial systems. However, a comprehensive framework that unifies the static and dynamic aspects of intelligence is still lacking. This manuscript proposes a mathematical framework based on Riemannian geometry to describe the structure and dynamics of intelligence and consciousness. Intelligence elements are conceptualised as tokens embedded in a high-dimensional space. The learned token embeddings capture the interconnections of tokens across various scenarios and tasks, forming manifolds in the intelligence space. Thought flow is depicted as the sequential activation of tokens along geodesics within these manifolds. During the navigation of geodesics, consciousness, as a self-referential process, perceives the thought flow, evaluates it against predictions, and provides feedback through prediction errors, adjusting the geodesic: non-zero prediction errors, such as learning, lead to the restructuring of the curved manifolds, thus changing the geodesic of thought flow. This dynamic interaction integrates new information, evolves the geometry and facilitates learning. The geometry of intelligence guides consciousness, and consciousness structures the geometry of intelligence. By integrating geometric concepts, this proposed theory offers a unified, mathematically framework for describing the structure and dynamics of intelligence and consciousness. Applicable to biological and artificial intelligence, this framework may pave the way for future research and empirical validation.

In this paper, we dive into the reliability concerns of Integrated Gradients (IG), a prevalent feature attribution method for black-box deep learning models. We particularly address two predominant challenges associated with IG: the generation of noisy feature visualizations for vision models and the vulnerability to adversarial attributional attacks. Our approach involves an adaptation of path-based feature attribution, aligning the path of attribution more closely to the intrinsic geometry of the data manifold. Our experiments utilise deep generative models applied to several real-world image datasets. They demonstrate that IG along the geodesics conforms to the curved geometry of the Riemannian data manifold, generating more perceptually intuitive explanations and, subsequently, substantially increasing robustness to targeted attributional attacks.

Empirical studies on the landscape of neural networks have shown that low-energy configurations are often found in complex connected structures, where zero-energy paths between pairs of distant solutions can be constructed. Here we consider the spherical negative perceptron, a prototypical non-convex neural network model framed as a continuous constraint satisfaction problem. We introduce a general analytical method for computing energy barriers in the simplex with vertex configurations sampled from the equilibrium. We find that in the over-parameterized regime the solution manifold displays simple connectivity properties. There exists a large geodesically convex component that is attractive for a wide range of optimization dynamics. Inside this region we identify a subset of atypical high-margin solutions that are geodesically connected with most other solutions, giving rise to a star-shaped geometry. We analytically characterize the organization of the connected space of solutions and show numerical evidence of a transition, at larger constraint densities, where the aforementioned simple geodesic connectivity breaks down.

Mode connectivity is a phenomenon where trained models are connected by a path of low loss. We reframe this in the context of Information Geometry, where neural networks are studied as spaces of parameterized distributions with curved geometry. We hypothesize that shortest paths in these spaces, known as geodesics, correspond to mode-connecting paths in the loss landscape. We propose an algorithm to approximate geodesics and demonstrate that they achieve mode connectivity. M Figure 1: Geodesics are shortest paths in the space of parameterized distributions M. For narrow architectures linear interpolation (dashed) fails to achieve mode connectivity, passing through a region of high loss, despite using a permutation π to'shift' θ If we instead follow the geodesic (shortest) path (solid) in the curved distribution space, this does achieve mode connectivity, appearing as a curved path in the loss landscape.

In recent years, deep learning models have revolutionized medical image interpretation, offering substantial improvements in diagnostic accuracy. However, these models often struggle with challenging images where critical features are partially or fully occluded, which is a common scenario in clinical practice. In this paper, we propose a novel curriculum learning-based approach to train deep learning models to handle occluded medical images effectively. Our method progressively introduces occlusion, starting from clear, unobstructed images and gradually moving to images with increasing occlusion levels. This ordered learning process, akin to human learning, allows the model to first grasp simple, discernable patterns and subsequently build upon this knowledge to understand more complicated, occluded scenarios. Furthermore, we present three novel occlusion synthesis methods, namely Wasserstein Curriculum Learning (WCL), Information Adaptive Learning (IAL), and Geodesic Curriculum Learning (GCL). Our extensive experiments on diverse medical image datasets demonstrate substantial improvements in model robustness and diagnostic accuracy over conventional training methodologies.

In this paper, we consider data acquired by multimodal sensors capturing complementary aspects and features of a measured phenomenon. We focus on a scenario in which the measurements share mutual sources of variability but might also be contaminated by other measurement-specific sources such as interferences or noise. Our approach combines manifold learning, which is a class of nonlinear data-driven dimension reduction methods, with the well-known Riemannian geometry of symmetric and positive-definite (SPD) matrices. Manifold learning typically includes the spectral analysis of a kernel built from the measurements. Here, we take a different approach, utilizing the Riemannian geometry of the kernels. In particular, we study the way the spectrum of the kernels changes along geodesic paths on the manifold of SPD matrices. We show that this change enables us, in a purely unsupervised manner, to derive a compact, yet informative, description of the relations between the measurements, in terms of their underlying components. Based on this result, we present new algorithms for extracting the common latent components and for identifying common and measurement-specific components.

In this paper we propose a new distance metric for signals that admit a sparse representation in a known basis or dictionary. The metric is derived as the length of the sparse geodesic path between two points, by which we mean the shortest path between the points that is itself sparse. We show that the distance can be computed via a simple formula and that the entire geodesic path can be easily generated. The distance provides a natural similarity measure that can be exploited as a perceptually meaningful distance metric for natural images. Furthermore, the distance has applications in supervised, semi-supervised, and unsupervised learning settings.

We present a geometric approach to statistical shape analysis of closed curves in images. The basic idea is to specify a space of closed curves satisfying given constraints, and exploit the differential geometry of this space to solve optimization and inference problems. We demonstrate this approach by: (i) defining and computing statistics of observed shapes, (ii) defining and learning a parametric probability model on shape space, and (iii) designing a binary hypothesis test on this space.