We would like to thank all the reviewers for their effort, and their thoughtful comments. Being formal, it should be "the gradient associated to the pullback of We will change it to "on which standard convergence results still apply". Thm 4.3 We will change "is equivalent" to The same can be said about higher order methods. We chose not to mention them in the main paper for simplicity. In l.138 we do mean "in almost all the manifold" in a measure-theoretical sense with respect to a measure induced by These two things indeed deserve a clarifying footnote.

Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue covering) to binary concept classes. The topological space naturally associated with a concept class is its space of realizable distributions. The loss function and the class itself induce a simplicial structure on this space, with respect to which we define a simplicial covering dimension. We prove that for finite concept classes, this simplicial covering dimension exactly characterizes the list replicability number (equivalently, global stability) in PAC learning. This connection allows us to apply tools from classical dimension theory to compute the exact list replicability number of the broad family of extremal concept classes.

Smooth, non-convex optimization problems on Riemannian manifolds occur in machine learning as a result of orthonormality, rank or positivity constraints.

In probabilistic modeling, parameter estimation is commonly formulated as a minimization problem on a parameter manifold. Optimization in such spaces requires geometry-aware methods that respect the underlying information structure. While the natural gradient leverages the Fisher information metric as a form of Riemannian gradient descent, it remains a first-order method and often exhibits slow convergence near optimal solutions. Existing second-order manifold algorithms typically rely on the Levi-Civita connection, thus overlooking the dual-connection structure that is central to information geometry. We propose the dual Riemannian Newton method, a Newton-type optimization algorithm on manifolds endowed with a metric and a pair of dual affine connections. The dual Riemannian Newton method explicates how duality shapes second-order updates: when the retraction (a local surrogate of the exponential map) is defined by one connection, the associated Newton equation is posed with its dual. We establish local quadratic convergence and validate the theory with experiments on representative statistical models. Thus, the dual Riemannian Newton method thus delivers second-order efficiency while remaining compatible with the dual structures that underlie modern information-geometric learning and inference.

Concentric tube robots (CTRs) offer dexterous motion at millimeter scales, enabling minimally invasive procedures through natural orifices. This work presents a coordinated model-based resection planner and learning-based retraction network that work together to enable semi-autonomous tissue resection using a dual-arm transurethral concentric tube robot (the Virtuoso). The resection planner operates directly on segmented CT volumes of prostate phantoms, automatically generating tool trajectories for a three-phase median lobe resection workflow: left/median trough resection, right/median trough resection, and median blunt dissection. The retraction network, PushCVAE, trained on surgeon demonstrations, generates retractions according to the procedural phase. The procedure is executed under Level-3 (supervised) autonomy on a prostate phantom composed of hydrogel materials that replicate the mechanical and cutting properties of tissue. As a feasibility study, we demonstrate that our combined autonomous system achieves a 97.1% resection of the targeted volume of the median lobe. Our study establishes a foundation for image-guided autonomy in transurethral robotic surgery and represents a first step toward fully automated minimally-invasive prostate enucleation.

Interpretable representation learning is a central challenge in modern machine learning, particularly in high-dimensional settings such as neuroimaging, genomics, and text analysis. Current methods often struggle to balance the competing demands of interpretability and model flexibility, limiting their effectiveness in extracting meaningful insights from complex data. We introduce Non-negative Stiefel Approximating Flow (NSA-Flow), a general-purpose matrix estimation framework that unifies ideas from sparse matrix factorization, orthogonalization, and constrained manifold learning. NSA-Flow enforces structured sparsity through a continuous balance between reconstruction fidelity and column-wise decorrelation, parameterized by a single tunable weight. The method operates as a smooth flow near the Stiefel manifold with proximal updates for non-negativity and adaptive gradient control, yielding representations that are simultaneously sparse, stable, and interpretable. Unlike classical regularization schemes, NSA-Flow provides an intuitive geometric mechanism for manipulating sparsity at the level of global structure while simplifying latent features. We demonstrate that the NSA-Flow objective can be optimized smoothly and integrates seamlessly with existing pipelines for dimensionality reduction while improving interpretability and generalization in both simulated and real biomedical data. Empirical validation on the Golub leukemia dataset and in Alzheimer's disease demonstrate that the NSA-Flow constraints can maintain or improve performance over related methods with little additional methodological effort. NSA-Flow offers a scalable, general-purpose tool for interpretable ML, applicable across data science domains.

Smooth, non-convex optimization problems on Riemannian manifolds occur in machine learning as a result of orthonormality, rank or positivity constraints.