arXiv subjects

George Bird

Publications and source records attributed to George Bird.

5 recordsLinked to original sources

arXiv2026

Isotropic Activation Functions Enable Deindividuated Neurons and Adaptive Topologies

Introduced is a methodology for adapting the topology of dense neural networks, enabled by isotropic activation functions. Achieved through prescribed reparameterisation symmetries and singular-value decomposition of affine maps, this diagonalises layers into one-to-one, ordered connections. This makes it simpler to assess the impact of individual connections on the function. Low-impact neurons can be removed (neurodegeneration), and a thresholded buffer of largely inactive 'scaffold' neurons is maintained (neurogenesis). These symmetry-led diagonalisation and structural changes are function-invariant, demonstrated to be computationally identical during neurogenesis, arbitrarily well approximated during neurodegeneration, and enable asymptotic 50\% parameter sparsification of dense networks with identically preserved function. Thus, real-time restructuring of the architecture in response to task demands, task appending, removal or changes is shown. The approach is conceptually centred on primitive symmetry-prescriptions, through which isotropic functions are derived that feature explicit basis independence and a loss in the individuation of neurons implicit in typical elementwise functional forms. Hence, this allows freedom in the basis to which layers are decomposed and interpreted as individual artificial neurons, directly enabling this adaptive topology approach. Additionally, a new tunable model parameter, the 'intrinsic length', is introduced to improve this analytical invariance, alongside a generalised isotropic-perceptron architecture that enables parallel precomputation of all matrix-vector products and displays a nested functional class. Diagonalisation is suggested to offer new possibilities for interpretability and monitoring of isotropic networks.

cs.NE↗

arXiv2025

The Affine Divergence: Aligning Activation Updates Beyond Normalisation

A systematic mismatch exists between mathematically ideal and effective activation updates during gradient descent. As intended, parameters update in their direction of steepest descent. However, activations are argued to constitute a more directly impactful quantity to prioritise in optimisation, as they are closer to the loss in the computational graph and carry sample-dependent information through the network. Yet their propagated updates do not take the optimal steepest-descent step. These quantities exhibit non-ideal sample-wise scaling across affine, convolutional, and attention layers.Solutions to correct for this are trivial and, incidentally, derive normalisation from first principles despite motivational independence. Consequently, such considerations offer a fresh, conceptual reframe of normalisation's action, with auxiliary experiments bolstering this mechanistic interpretation. Moreover, this analysis makes clear a second possibility: a solution that is functionally distinct from modern normalisations, without scale invariance, yet remains empirically successful -- an alternative to the affine map. This outperforms conventional normalisers across several tests. This generalises to convolution via a new functional form, ``PatchNorm'', a compositionally inseparable normaliser. Together, these provide an alternative mechanistic framework that both adds to and counters some of the discussion of normalisation. Further, it is argued that normalisers are better decomposed into activation-function-like maps with parameterised scaling. Overall, this constitutes a theoretically principled approach that yields new functions with empirical validation and raises questions about the affine + nonlinear approach.

cs.LG↗

arXiv2025

Emergence of Quantised Representations Isolated to Anisotropic Functions

Presented is a novel methodology for determining representational structure, which builds upon the existing Spotlight Resonance method. This new tool is used to gain insight into how discrete representations can emerge and organise in autoencoder models, through a controlled ablation study that alters only the activation function. Using this technique, the validity of whether function-driven symmetries can act as implicit inductive biases on representations is determined. Representations are found to tend to discretise when the activation functions are defined through a discrete algebraic permutation-equivariant symmetry. In contrast, they remain continuous under a continuous algebraic orthogonal-equivariant definition. This confirms the hypothesis that the symmetries of network primitives can carry unintended inductive biases, leading to task-independent artefactual structures in representations. The discrete symmetry of contemporary forms is shown to be a strong predictor for the production of symmetry-organised discrete representations emerging from otherwise continuous distributions -- a quantisation effect. This motivates further reassessment of functional forms in common usage due to such unintended consequences. Moreover, this supports a general causal model for a mode in which discrete representations may form, and could constitute a prerequisite for downstream interpretability phenomena, including grandmother neurons, discrete coding schemes, general linear features and a type of Superposition. Hence, this tool and proposed mechanism for the influence of functional form on representations may provide insights into interpretability research. Finally, preliminary results indicate that quantisation of representations correlates with a measurable increase in reconstruction error, reinforcing previous conjectures that this collapse can be detrimental.

cs.LG↗

arXiv2025

The Spotlight Resonance Method: Resolving the Alignment of Embedded Activations

Understanding how deep learning models represent data is currently difficult due to the limited number of methodologies available. This paper demonstrates a versatile and novel visualisation tool for determining the axis alignment of embedded data at any layer in any deep learning model. In particular, it evaluates the distribution around planes defined by the network's privileged basis vectors. This method provides both an atomistic and a holistic, intuitive metric for interpreting the distribution of activations across all planes. It ensures that both positive and negative signals contribute, treating the activation vector as a whole. Depending on the application, several variations of this technique are presented, with a resolution scale hyperparameter to probe different angular scales. Using this method, multiple examples are provided that demonstrate embedded representations tend to be axis-aligned with the privileged basis. This is not necessarily the standard basis, and it is found that activation functions directly result in privileged bases. Hence, it provides a direct causal link between functional form symmetry breaking and representational alignment, explaining why representations have a tendency to align with the neuron basis. Therefore, using this method, we begin to answer the fundamental question of what causes the observed tendency of representations to align with neurons. Finally, examples of so-called grandmother neurons are found in a variety of networks.

cs.LG↗

arXiv2021

Backpropagation Through Time For Networks With Long-Term Dependencies

Backpropagation through time (BPTT) is a technique of updating tuned parameters within recurrent neural networks (RNNs). Several attempts at creating such an algorithm have been made including: Nth Ordered Approximations and Truncated-BPTT. These methods approximate the backpropagation gradients under the assumption that the RNN only utilises short-term dependencies. This is an acceptable assumption to make for the current state of artificial neural networks. As RNNs become more advanced, a shift towards influence by long-term dependencies is likely. Thus, a new method for backpropagation is required. We propose using the 'discrete forward sensitivity equation' and a variant of it for single and multiple interacting recurrent loops respectively. This solution is exact and also allows the network's parameters to vary between each subsequent step, however it does require the computation of a Jacobian.

cs.LG↗