arXiv subjects

Peter E. Latham

Publications and source records attributed to Peter E. Latham.

At least 19 recordsLinked to original sources

arXiv2026

How are linear representations learned? Exact solutions to the dynamics of abstraction

In artificial and biological neural networks, concepts are often encoded as consistent linear directions in representation space. In deep learning, this idea is known as the linear representation hypothesis and underpins many interpretability and control methods based on linear probes, from concept detection to activation steering. Yet while prior work has studied whether such directions should exist $\textit{after}$ training, the dynamics of how they emerge $\textit{during}$ training remain poorly understood. Here, we develop a framework to study the alignment of concept directions during training - a process we call "abstraction". In a minimal linear network setting, we obtain exact solutions for the full trajectory of abstraction. These solutions reveal key analytic principles governing abstraction: (i) data and target geometry jointly determine abstraction at the end-of-learning, (ii) abstraction improves with network depth, and (iii) initialization scale controls the maximum abstraction reached during training. Extending our theory to nonlinear networks, we analyze how the choice of nonlinearity affects abstraction dynamics: erf networks approximate the linear theory, while abstraction in ReLU networks depends less on target geometry and more on input geometry. Across both, we prove a striking attenuation law: both nonlinearities weaken abstraction in activations relative to preactivations. We find evidence for this law in open models (DINOv3, Gemma 4) and apply our theory to improve linear probe generalization in LLMs. Together, our results provide a dynamical theory of abstraction with implications for interpretability and control.

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arXiv2026

Optimal Learning Rate Scaling Depends on Data in Deep Scalar Linear Networks

In this short note we consider the gradient descent dynamics of deep scalar linear networks, $f(x) = \prod_{l=1}^L w_l x$, which enjoy exact time-course solutions for any integer depth. We show that even in this minimal model, the optimal depth-wise learning rate scaling depends on data, whereas data-agnostic scaling rules fail to transfer across depths. Under the data-dependent optimal scaling, the learning dynamics is independent of data and weakly dependent on depth, resulting in a constant linear convergence rate across all depths including infinity. We further show similar data-dependent effects in deep scalar linear networks with residual connections.

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arXiv2026

Mapping Connectomic Structure to Function(s) in Cerebellar-like Networks using Kernel Regression

Cerebellar-like networks, in which input activity patterns are separated by projection to a much higher-dimensional space before classification, are a recurring neurobiological motif, present in the cerebellum, dentate gyrus, insect olfactory system, and electrosensory system of the electric fish. Their relatively well-understood design presents a promising test-case for probing principles of biological learning. The circuits' expansive projections have long been modelled as random, enabling effective general purpose pattern separation. However, electron-microscopy studies have discovered interesting hints of structure in both the fly mushroom body and mouse cerebellum. Recent numerical work suggested that this non-random connectivity enables the circuit to prioritise learning of some, presumably natural, tasks over others. Here, rather than numerical results, we present a robust mathematical link between the observed connectivity patterns and the cerebellar circuit's learning ability. In particular, we extend a simplified kernel regression model of the system and use recent machine learning theory results to relate connectivity to learning. We find that the reported structure in the projection weights shapes the network's inductive bias in intuitive ways: functions are easier to learn if they depend on inputs that are oversampled, or on collections of neurons that tend to connect to the same hidden layer neurons. Our approach is analytically tractable and pleasingly simple, and we hope it continues to serve as a model for understanding the functional implications of other processing motifs in cerebellar-like networks.

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arXiv2026

Saddle-to-Saddle Dynamics Explains A Simplicity Bias Across Neural Network Architectures

Neural networks trained with gradient descent often learn solutions of increasing complexity over time, a phenomenon known as simplicity bias. Despite being widely observed across architectures, existing theoretical treatments lack a unifying framework. We present a theoretical framework that explains a simplicity bias arising from saddle-to-saddle learning dynamics for a general class of neural networks, incorporating fully-connected, convolutional, and attention-based architectures. Here, simple means expressible with few hidden units, i.e., hidden neurons, convolutional kernels, or attention heads. Specifically, we show that linear networks learn solutions of increasing rank, ReLU networks learn solutions with an increasing number of kinks, convolutional networks learn solutions with an increasing number of convolutional kernels, and self-attention models learn solutions with an increasing number of attention heads. By analyzing fixed points, invariant manifolds, and dynamics of gradient descent learning, we show that saddle-to-saddle dynamics operates by iteratively evolving near an invariant manifold, approaching a saddle, and switching to another invariant manifold. Our analysis also disentangles data-induced and initialization-induced saddle-to-saddle dynamics. In particular, the former leads to low-rank weights while the latter to sparse weights. Equipped with the theory, we predict the effects of data distribution and weight initialization on the duration and number of plateaus in learning. Overall, our theory offers a framework for understanding when and why gradient descent progressively learns increasingly complex solutions.

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arXiv2026

A Biologically Plausible Dense Associative Memory with Exponential Capacity

Krotov and Hopfield (2021) proposed a biologically plausible two-layer associative memory network with memory storage capacity exponential in the number of visible neurons. However, the capacity was only linear in the number of hidden neurons. This limitation arose from the choice of nonlinearity between the visible and hidden units, which enforced winner-take-all dynamics in the hidden layer, thereby restricting each hidden unit to encode only a single memory. We overcome this limitation by introducing a novel associative memory network with a threshold nonlinearity that enables distributed representations. In contrast to winner-take-all dynamics, where each hidden neuron is tied to an entire memory, our network allows hidden neurons to encode basic components shared across many memories. Consequently, complex patterns are represented through combinations of hidden neurons. These representations reduce redundancy and allow many correlated memories to be stored compositionally. Thus, we achieve much higher capacity: exponential in the number of hidden units, provided the number of visible units is sufficiently large relative to the number of hidden units. Exponential capacity arises because all binary states of the hidden units can become stable memory patterns. Moreover, the distributed hidden representation, which has much lower dimensionality than the visible layer, preserves class-discriminative structure, supporting efficient nonlinear decoding. These results establish a new regime for associative memory, enabling high-capacity, robust, and scalable architectures consistent with biological constraints.

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arXiv2026

Convex Efficient Coding

Why do neurons encode information the way they do? Normative answers to this question model neural activity as the solution to an optimisation problem; for example, the celebrated efficient coding hypothesis frames neural activity as the optimal encoding of information under efficiency constraints. Successful normative theories have varied dramatically in complexity, from simple linear models (Atick & Redlich '90), to complex deep neural networks (Lindsay '21). What complex models gain in flexibility, they lose in tractability and often understandability. Here, we split the difference by constructing a set of tractable but flexible normative representational theories. Instead of optimising the neural activities directly, following Sengupta et al. '18, we optimise the representational similarity, a matrix formed from the dot products of each pair of neural responses. Using this, we show that a large family of interesting optimisation problems are convex. This family includes problems corresponding to linear and some non-linear neural networks, and problems from the literature not previously recognised as convex, such as modified versions of semi-nonnegative matrix factorisation or nonnegative sparse coding. We put these findings to work in three ways. First, we provide the first necessary and sufficient identifiability result for a form of semi-nonnegative matrix factorisation. Second, we show that if neural tunings are `different enough' then they are uniquely linked to the optimal representational similarity, partially justifying the use of single neuron tuning analysis in neuroscience. Finally, we use the tractable nonlinearity of some of our problems to explain why dense retinal codes, but not sparse cortical codes, optimally split the coding of a single variable into ON & OFF channels. In sum, we identify a space of convex problems, and use them to derive neural coding results.

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arXiv2026

Optimal Learning Rate Schedule for Balancing Effort and Performance

Learning how to learn efficiently is a fundamental challenge for biological agents and a growing concern for artificial ones. To learn effectively, an agent must regulate its learning speed, balancing the benefits of rapid improvement against the costs of effort, instability, or resource use. We introduce a normative framework that formalizes this problem as an optimal control process in which the agent maximizes cumulative performance while incurring a cost of learning. From this objective, we derive a closed-form solution for the optimal learning rate, which has the form of a closed-loop controller that depends only on the agent's current and expected future performance. Under mild assumptions, this solution generalizes across tasks and architectures and reproduces numerically optimized schedules in simulations. In simple learning models, we can mathematically analyze how agent and task parameters shape learning-rate scheduling as an open-loop control solution. Because the optimal policy depends on expectations of future performance, the framework predicts how overconfidence or underconfidence influence engagement and persistence, linking the control of learning speed to theories of self-regulated learning. We further show how a simple episodic memory mechanism can approximate the required performance expectations by recalling similar past learning experiences, providing a biologically plausible route to near-optimal behaviour. Together, these results provide a normative and biologically plausible account of learning speed control, linking self-regulated learning, effort allocation, and episodic memory estimation within a unified and tractable mathematical framework.

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arXiv2025

Training Dynamics of In-Context Learning in Linear Attention

While attention-based models have demonstrated the remarkable ability of in-context learning (ICL), the theoretical understanding of how these models acquired this ability through gradient descent training is still preliminary. Towards answering this question, we study the gradient descent dynamics of multi-head linear self-attention trained for in-context linear regression. We examine two parametrizations of linear self-attention: one with the key and query weights merged as a single matrix (common in theoretical studies), and one with separate key and query matrices (closer to practical settings). For the merged parametrization, we show that the training dynamics has two fixed points and the loss trajectory exhibits a single, abrupt drop. We derive an analytical time-course solution for a certain class of datasets and initialization. For the separate parametrization, we show that the training dynamics has exponentially many fixed points and the loss exhibits saddle-to-saddle dynamics, which we reduce to scalar ordinary differential equations. During training, the model implements principal component regression in context with the number of principal components increasing over training time. Overall, we provide a theoretical description of how ICL abilities evolve during gradient descent training of linear attention, revealing abrupt acquisition or progressive improvements depending on how the key and query are parametrized.

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arXiv2025

When Are Bias-Free ReLU Networks Effectively Linear Networks?

We investigate the implications of removing bias in ReLU networks regarding their expressivity and learning dynamics. We first show that two-layer bias-free ReLU networks have limited expressivity: the only odd function two-layer bias-free ReLU networks can express is a linear one. We then show that, under symmetry conditions on the data, these networks have the same learning dynamics as linear networks. This enables us to give analytical time-course solutions to certain two-layer bias-free (leaky) ReLU networks outside the lazy learning regime. While deep bias-free ReLU networks are more expressive than their two-layer counterparts, they still share a number of similarities with deep linear networks. These similarities enable us to leverage insights from linear networks to understand certain ReLU networks. Overall, our results show that some properties previously established for bias-free ReLU networks arise due to equivalence to linear networks.

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arXiv2024

Understanding Unimodal Bias in Multimodal Deep Linear Networks

Using multiple input streams simultaneously to train multimodal neural networks is intuitively advantageous but practically challenging. A key challenge is unimodal bias, where a network overly relies on one modality and ignores others during joint training. We develop a theory of unimodal bias with multimodal deep linear networks to understand how architecture and data statistics influence this bias. This is the first work to calculate the duration of the unimodal phase in learning as a function of the depth at which modalities are fused within the network, dataset statistics, and initialization. We show that the deeper the layer at which fusion occurs, the longer the unimodal phase. A long unimodal phase can lead to a generalization deficit and permanent unimodal bias in the overparametrized regime. Our results, derived for multimodal linear networks, extend to nonlinear networks in certain settings. Taken together, this work illuminates pathologies of multimodal learning under joint training, showing that late and intermediate fusion architectures can give rise to long unimodal phases and permanent unimodal bias. Our code is available at: https://yedizhang.github.io/unimodal-bias.html.

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arXiv2023

Actionable Neural Representations: Grid Cells from Minimal Constraints

To afford flexible behaviour, the brain must build internal representations that mirror the structure of variables in the external world. For example, 2D space obeys rules: the same set of actions combine in the same way everywhere (step north, then south, and you won't have moved, wherever you start). We suggest the brain must represent this consistent meaning of actions across space, as it allows you to find new short-cuts and navigate in unfamiliar settings. We term this representation an `actionable representation'. We formulate actionable representations using group and representation theory, and show that, when combined with biological and functional constraints - non-negative firing, bounded neural activity, and precise coding - multiple modules of hexagonal grid cells are the optimal representation of 2D space. We support this claim with intuition, analytic justification, and simulations. Our analytic results normatively explain a set of surprising grid cell phenomena, and make testable predictions for future experiments. Lastly, we highlight the generality of our approach beyond just understanding 2D space. Our work characterises a new principle for understanding and designing flexible internal representations: they should be actionable, allowing animals and machines to predict the consequences of their actions, rather than just encode.

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arXiv2022

Towards Biologically Plausible Convolutional Networks

Convolutional networks are ubiquitous in deep learning. They are particularly useful for images, as they reduce the number of parameters, reduce training time, and increase accuracy. However, as a model of the brain they are seriously problematic, since they require weight sharing - something real neurons simply cannot do. Consequently, while neurons in the brain can be locally connected (one of the features of convolutional networks), they cannot be convolutional. Locally connected but non-convolutional networks, however, significantly underperform convolutional ones. This is troublesome for studies that use convolutional networks to explain activity in the visual system. Here we study plausible alternatives to weight sharing that aim at the same regularization principle, which is to make each neuron within a pool react similarly to identical inputs. The most natural way to do that is by showing the network multiple translations of the same image, akin to saccades in animal vision. However, this approach requires many translations, and doesn't remove the performance gap. We propose instead to add lateral connectivity to a locally connected network, and allow learning via Hebbian plasticity. This requires the network to pause occasionally for a sleep-like phase of "weight sharing". This method enables locally connected networks to achieve nearly convolutional performance on ImageNet and improves their fit to the ventral stream data, thus supporting convolutional networks as a model of the visual stream.

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arXiv2021

Powerpropagation: A sparsity inducing weight reparameterisation

The training of sparse neural networks is becoming an increasingly important tool for reducing the computational footprint of models at training and evaluation, as well enabling the effective scaling up of models. Whereas much work over the years has been dedicated to specialised pruning techniques, little attention has been paid to the inherent effect of gradient based training on model sparsity. In this work, we introduce Powerpropagation, a new weight-parameterisation for neural networks that leads to inherently sparse models. Exploiting the behaviour of gradient descent, our method gives rise to weight updates exhibiting a "rich get richer" dynamic, leaving low-magnitude parameters largely unaffected by learning. Models trained in this manner exhibit similar performance, but have a distribution with markedly higher density at zero, allowing more parameters to be pruned safely. Powerpropagation is general, intuitive, cheap and straight-forward to implement and can readily be combined with various other techniques. To highlight its versatility, we explore it in two very different settings: Firstly, following a recent line of work, we investigate its effect on sparse training for resource-constrained settings. Here, we combine Powerpropagation with a traditional weight-pruning technique as well as recent state-of-the-art sparse-to-sparse algorithms, showing superior performance on the ImageNet benchmark. Secondly, we advocate the use of sparsity in overcoming catastrophic forgetting, where compressed representations allow accommodating a large number of tasks at fixed model capacity. In all cases our reparameterisation considerably increases the efficacy of the off-the-shelf methods.

stat.ML↗

arXiv2021

Synaptic plasticity as Bayesian inference

Learning, especially rapid learning, is critical for survival. However, learning is hard: a large number of synaptic weights must be set based on noisy, often ambiguous, sensory information. In such a high-noise regime, keeping track of probability distributions over weights is the optimal strategy. Here we hypothesize that synapses take that strategy; in essence, when they estimate weights, they include error bars. They then use that uncertainty to adjust their learning rates, with more uncertain weights having higher learning rates. We also make a second, independent, hypothesis: synapses communicate their uncertainty by linking it to variability in PSP size, with more uncertainty leading to more variability. These two hypotheses cast synaptic plasticity as a problem of Bayesian inference, and thus provide a normative view of learning. They generalize known learning rules, offer an explanation for the large variability in the size of post-synaptic potentials, and make falsifiable experimental predictions.

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arXiv2020

Kernelized information bottleneck leads to biologically plausible 3-factor Hebbian learning in deep networks

The state-of-the art machine learning approach to training deep neural networks, backpropagation, is implausible for real neural networks: neurons need to know their outgoing weights; training alternates between a bottom-up forward pass (computation) and a top-down backward pass (learning); and the algorithm often needs precise labels of many data points. Biologically plausible approximations to backpropagation, such as feedback alignment, solve the weight transport problem, but not the other two. Thus, fully biologically plausible learning rules have so far remained elusive. Here we present a family of learning rules that does not suffer from any of these problems. It is motivated by the information bottleneck principle (extended with kernel methods), in which networks learn to compress the input as much as possible without sacrificing prediction of the output. The resulting rules have a 3-factor Hebbian structure: they require pre- and post-synaptic firing rates and an error signal - the third factor - consisting of a global teaching signal and a layer-specific term, both available without a top-down pass. They do not require precise labels; instead, they rely on the similarity between pairs of desired outputs. Moreover, to obtain good performance on hard problems and retain biological plausibility, our rules need divisive normalization - a known feature of biological networks. Finally, simulations show that our rules perform nearly as well as backpropagation on image classification tasks.

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arXiv2017

Robust information propagation through noisy neural circuits

Sensory neurons give highly variable responses to stimulation, which can limit the amount of stimulus information available to downstream circuits. Much work has investigated the factors that affect the amount of information encoded in these population responses, leading to insights about the role of covariability among neurons, tuning curve shape, etc. However, the informativeness of neural responses is not the only relevant feature of population codes; of potentially equal importance is how robustly that information propagates to downstream structures. For instance, to quantify the retina's performance, one must consider not only the informativeness of the optic nerve responses, but also the amount of information that survives the spike-generating nonlinearity and noise corruption in the next stage of processing, the lateral geniculate nucleus. Our study identifies the set of covariance structures for the upstream cells that optimize the ability of information to propagate through noisy, nonlinear circuits. Within this optimal family are covariances with "differential correlations", which are known to reduce the information encoded in neural population activities. Thus, covariance structures that maximize information in neural population codes, and those that maximize the ability of this information to propagate, can be very different.

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arXiv2016

Zipf's law arises naturally in structured, high-dimensional data

Zipf's law, which states that the probability of an observation is inversely proportional to its rank, has been observed in many domains. While there are models that explain Zipf's law in each of them, those explanations are typically domain specific. Recently, methods from statistical physics were used to show that a fairly broad class of models does provide a general explanation of Zipf's law. This explanation rests on the observation that real world data is often generated from underlying causes, known as latent variables. Those latent variables mix together multiple models that do not obey Zipf's law, giving a model that does. Here we extend that work both theoretically and empirically. Theoretically, we provide a far simpler and more intuitive explanation of Zipf's law, which at the same time considerably extends the class of models to which this explanation can apply. Furthermore, we also give methods for verifying whether this explanation applies to a particular dataset. Empirically, these advances allowed us extend this explanation to important classes of data, including word frequencies (the first domain in which Zipf's law was discovered), data with variable sequence length, and multi-neuron spiking activity.

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arXiv2015

A unifying framework for understanding state-dependent network dynamics in cortex

Activity in neocortex exhibits a range of behaviors, from irregular to temporally precise, and from weakly to strongly correlated. So far there has been no single theoretical framework that could explain all these behaviors, leaving open the possibility that they are a signature of radically different mechanisms. Here, we suggest that this is not the case. Instead, we show that a single theory can account for a broad spectrum of experimental observations, including specifics such as the fine temporal details of subthreshold cross-correlations. For the model underlying our theory, we need only assume a small number of well-established properties common to all local cortical networks. When these assumptions are combined with realistically structured input, they produce exactly the repertoire of behaviors that is observed experimentally, and lead to a number of testable predictions.

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