Lachlan Ewen MacDonald

Chaoyang Wang, Lachlan Ewen MacDonald, Laszlo A. Jeni et al.

In this paper we present a new method for deformable NeRF that can directly use optical flow as supervision. We overcome the major challenge with respect to the computationally inefficiency of enforcing the flow constraints to the backward deformation field, used by deformable NeRFs. Specifically, we show that inverting the backward deformation function is actually not needed for computing scene flows between frames. This insight dramatically simplifies the problem, as one is no longer constrained to deformation functions that can be analytically inverted. Instead, thanks to the weak assumptions required by our derivation based on the inverse function theorem, our approach can be extended to a broad class of commonly used backward deformation field. We present results on monocular novel view synthesis with rapid object motion, and demonstrate significant improvements over baselines without flow supervision.

Cameron Gordon, Shin-Fang Chng, Lachlan MacDonald et al.

The role of quantization within implicit/coordinate neural networks is still not fully understood. We note that using a canonical fixed quantization scheme during training produces poor performance at low-rates due to the network weight distributions changing over the course of training. In this work, we show that a non-uniform quantization of neural weights can lead to significant improvements. Specifically, we demonstrate that a clustered quantization enables improved reconstruction. Finally, by characterising a trade-off between quantization and network capacity, we demonstrate that it is possible (while memory inefficient) to reconstruct signals using binary neural networks. We demonstrate our findings experimentally on 2D image reconstruction and 3D radiance fields; and show that simple quantization methods and architecture search can achieve compression of NeRF to less than 16kb with minimal loss in performance (323x smaller than the original NeRF).

Sameera Ramasinghe, Lachlan MacDonald, Moshiur Farazi et al.

Characterizing the remarkable generalization properties of over-parameterized neural networks remains an open problem. In this paper, we promote a shift of focus towards initialization rather than neural architecture or (stochastic) gradient descent to explain this implicit regularization. Through a Fourier lens, we derive a general result for the spectral bias of neural networks and show that the generalization of neural networks is heavily tied to their initialization. Further, we empirically solidify the developed theoretical insights using practical, deep networks. Finally, we make a case against the controversial flat-minima conjecture and show that Fourier analysis grants a more reliable framework for understanding the generalization of neural networks.

Lachlan Ewen MacDonald, Jack Valmadre, Hemanth Saratchandran et al.

We introduce a general theoretical framework, designed for the study of gradient optimisation of deep neural networks, that encompasses ubiquitous architecture choices including batch normalisation, weight normalisation and skip connections. Our framework determines the curvature and regularity properties of multilayer loss landscapes in terms of their constituent layers, thereby elucidating the roles played by normalisation layers and skip connections in globalising these properties. We then demonstrate the utility of this framework in two respects. First, we give the only proof of which we are aware that a class of deep neural networks can be trained using gradient descent to global optima even when such optima only exist at infinity, as is the case for the cross-entropy cost. Second, we identify a novel causal mechanism by which skip connections accelerate training, which we verify predictively with ResNets on MNIST, CIFAR10, CIFAR100 and ImageNet.

Lachlan Ewen MacDonald, René Vidal

An important quantity in the theory of gradient descent (GD) is the \emph{sharpness}, defined as the largest eigenvalue of the objective Hessian. Classical analyses typically require the step size to be uniformly smaller than twice the reciprocal of the sharpness, but this condition is frequently violated in the training of deep neural networks. Recent work bridges this gap in the setting of overparametrised least-squares with a \emph{single scalar output}, providing a normal form for large-step GD in a neighbourhood of an \emph{isolated} flat minimum and establishing three corresponding convergence results. In this paper, we extend this theory in two directions: (1) to overparametrised least-squares with \emph{vector-valued outputs} (including regression with arbitrarily many observations), and (2) to a neighbourhood of a \emph{manifold} of flat minima (which we show is essential for applications such as matrix factorisation). We generalise both the normal form and all three convergence theorems of \cite{macdonaldeos} to this broader setting, overcoming several technical challenges, including the solution of a singular partial differential equation via a novel method that may be of independent interest. We further show that our framework applies to deep matrix factorisation under mild assumptions, yielding several new structural results. In particular, we prove that the set of flat minima forms a fibre bundle over a product of spheres, and that the sharpness is Morse-Bott along this manifold.

Lachlan Ewen MacDonald

We prove a centre manifold theorem for a map along a manifold-with-boundary of fixed points, and provide an application to the study of gradient descent with large step size on two-layer matrix factorisation problems.

Lachlan Ewen MacDonald, Hancheng Min, Leandro Palma et al.

Classical optimisation theory guarantees monotonic objective decrease for gradient descent (GD) when employed in a small step size, or ``stable", regime. In contrast, gradient descent on neural networks is frequently performed in a large step size regime called the ``edge of stability", in which the objective decreases non-monotonically with an observed implicit bias towards flat minima. In this paper, we take a step toward quantifying this phenomenon by providing convergence rates for gradient descent with large learning rates in an overparametrised least squares setting. The key insight behind our analysis is that, as a consequence of overparametrisation, the set of global minimisers forms a Riemannian manifold $M$, which enables the decomposition of the GD dynamics into components parallel and orthogonal to $M$. The parallel component corresponds to Riemannian gradient descent on the objective sharpness, while the orthogonal component is a bifurcating dynamical system. This insight allows us to derive convergence rates in three regimes characterised by the learning rate size: (a) the subcritical regime, in which transient instability is overcome in finite time before linear convergence to a suboptimally flat global minimum; (b) the critical regime, in which instability persists for all time with a power-law convergence toward the optimally flat global minimum; and (c) the supercritical regime, in which instability persists for all time with linear convergence to an orbit of period two centred on the optimally flat global minimum.

Cameron Gordon, Lachlan Ewen MacDonald, Hemanth Saratchandran et al.

Deep implicit functions have been found to be an effective tool for efficiently encoding all manner of natural signals. Their attractiveness stems from their ability to compactly represent signals with little to no offline training data. Instead, they leverage the implicit bias of deep networks to decouple hidden redundancies within the signal. In this paper, we explore the hypothesis that additional compression can be achieved by leveraging redundancies that exist between layers. We propose to use a novel runtime decoder-only hypernetwork - that uses no offline training data - to better exploit cross-layer parameter redundancy. Previous applications of hypernetworks with deep implicit functions have employed feed-forward encoder/decoder frameworks that rely on large offline datasets that do not generalize beyond the signals they were trained on. We instead present a strategy for the optimization of runtime deep implicit functions for single-instance signals through a Decoder-Only randomly projected Hypernetwork (D'OH). By directly changing the latent code dimension, we provide a natural way to vary the memory footprint of neural representations without the costly need for neural architecture search on a space of alternative low-rate structures.

Lachlan Ewen MacDonald, Jack Valmadre, Simon Lucey

We present a new approach to understanding the relationship between loss curvature and input-output model behaviour in deep learning. Specifically, we use existing empirical analyses of the spectrum of deep network loss Hessians to ground an ansatz tying together the loss Hessian and the input-output Jacobian over training samples during the training of deep neural networks. We then prove a series of theoretical results which quantify the degree to which the input-output Jacobian of a model approximates its Lipschitz norm over a data distribution, and deduce a novel generalisation bound in terms of the empirical Jacobian. We use our ansatz, together with our theoretical results, to give a new account of the recently observed progressive sharpening phenomenon, as well as the generalisation properties of flat minima. Experimental evidence is provided to validate our claims.

Hemanth Saratchandran, Shin-Fang Chng, Sameera Ramasinghe et al.

Coordinate networks are widely used in computer vision due to their ability to represent signals as compressed, continuous entities. However, training these networks with first-order optimizers can be slow, hindering their use in real-time applications. Recent works have opted for shallow voxel-based representations to achieve faster training, but this sacrifices memory efficiency. This work proposes a solution that leverages second-order optimization methods to significantly reduce training times for coordinate networks while maintaining their compressibility. Experiments demonstrate the effectiveness of this approach on various signal modalities, such as audio, images, videos, shape reconstruction, and neural radiance fields.

Sameera Ramasinghe, Lachlan MacDonald, Simon Lucey

We show that typical implicit regularization assumptions for deep neural networks (for regression) do not hold for coordinate-MLPs, a family of MLPs that are now ubiquitous in computer vision for representing high-frequency signals. Lack of such implicit bias disrupts smooth interpolations between training samples, and hampers generalizing across signal regions with different spectra. We investigate this behavior through a Fourier lens and uncover that as the bandwidth of a coordinate-MLP is enhanced, lower frequencies tend to get suppressed unless a suitable prior is provided explicitly. Based on these insights, we propose a simple regularization technique that can mitigate the above problem, which can be incorporated into existing networks without any architectural modifications.

Lachlan Ewen MacDonald, Sameera Ramasinghe, Simon Lucey

Although provably robust to translational perturbations, convolutional neural networks (CNNs) are known to suffer from extreme performance degradation when presented at test time with more general geometric transformations of inputs. Recently, this limitation has motivated a shift in focus from CNNs to Capsule Networks (CapsNets). However, CapsNets suffer from admitting relatively few theoretical guarantees of invariance. We introduce a rigourous mathematical framework to permit invariance to any Lie group of warps, exclusively using convolutions (over Lie groups), without the need for capsules. Previous work on group convolutions has been hampered by strong assumptions about the group, which precludes the application of such techniques to common warps in computer vision such as affine and homographic. Our framework enables the implementation of group convolutions over any finite-dimensional Lie group. We empirically validate our approach on the benchmark affine-invariant classification task, where we achieve 30% improvement in accuracy against conventional CNNs while outperforming most CapsNets. As further illustration of the generality of our framework, we train a homography-convolutional model which achieves superior robustness on a homography-perturbed dataset, where CapsNet results degrade.