Tightening the Evaluation of PAC Bounds Using Formal Verification Results

Probably Approximately Correct (PAC) bounds are widely used to derive probabilistic guarantees for the generalisation of machine learning models. They highlight the components of the model which contribute to its generalisation capacity. However, current state-of-the-art results are loose in approximating the generalisation capacity of deployed machine learning models. Consequently, while PAC bounds are theoretically useful, their applicability for evaluating a model's generalisation property in a given operational design domain is limited. The underlying classical theory is supported by the idea that bounds can be tightened when the number of test points available to the user to evaluate the model increases. Yet, in the case of neural networks, the number of test points required to obtain bounds of interest is often impractical even for small problems. In this paper, we take the novel approach of using the formal verification of neural systems to inform the evaluation of PAC bounds. Rather than using pointwise information obtained from repeated tests, we use verification results on regions around test points. We show that conditioning existing bounds on verification results leads to a tightening proportional to the underlying probability mass of the verified region.

Spherical Feature Pyramid Networks For Semantic Segmentation

Semantic segmentation for spherical data is a challenging problem in machine learning since conventional planar approaches require projecting the spherical image to the Euclidean plane. Representing the signal on a fundamentally different topology introduces edges and distortions which impact network performance. Recently, graph-based approaches have bypassed these challenges to attain significant improvements by representing the signal on a spherical mesh. Current approaches to spherical segmentation exclusively use variants of the UNet architecture, meaning more successful planar architectures remain unexplored. Inspired by the success of feature pyramid networks (FPNs) in planar image segmentation, we leverage the pyramidal hierarchy of graph-based spherical CNNs to design spherical FPNs. Our spherical FPN models show consistent improvements over spherical UNets, whilst using fewer parameters. On the Stanford 2D-3D-S dataset, our models achieve state-of-the-art performance with an mIOU of 48.75, an improvement of 3.75 IoU points over the previous best spherical CNN.

Explicit Neural Surfaces: Learning Continuous Geometry With Deformation Fields

We introduce Explicit Neural Surfaces (ENS), an efficient surface reconstruction method that learns an explicitly defined continuous surface from multiple views. We use a series of neural deformation fields to progressively transform a continuous input surface to a target shape. By sampling meshes as discrete surface proxies, we train the deformation fields through efficient differentiable rasterization, and attain a mesh-independent and smooth surface representation. By using Laplace-Beltrami eigenfunctions as an intrinsic positional encoding alongside standard extrinsic Fourier features, our approach can capture fine surface details. ENS trains 1 to 2 orders of magnitude faster and can extract meshes of higher quality compared to implicit representations, whilst maintaining competitive surface reconstruction performance and real-time capabilities. Finally, we apply our approach to learn a collection of objects in a single model, and achieve disentangled interpolations between different shapes, their surface details, and textures.