Search for: All records

Landmark universal function approximation results for neural networks with trained weights and biases provided the impetus for the ubiquitous use of neural networks as learning models in neuroscience and Artificial Intelligence (AI). Recent work has extended these results to networks in which a smaller subset of weights (e.g., output weights) are tuned, leaving other parameters random. However, it remains an open question whether universal approximation holds when only biases are learned, despite evidence from neuroscience and AI that biases significantly shape neural responses. The current paper answers this question. We provide theoretical and numerical evidence demonstrating that feedforward neural networks with fixed random weights can approximate any continuous function on compact sets. We further show an analogous result for the approximation of dynamical systems with recurrent neural networks. Our findings are relevant to neuroscience, where they demonstrate the potential for behaviourally relevant changes in dynamics without modifying synaptic weights, as well as for AI, where they shed light on recent fine-tuning methods for large language models, like bias and prefix-based approaches. 

Recurrent neural networks (RNNs) have been successfully applied to a variety of problems involving sequential data, but their optimization is sensitive to parameter initialization, architecture, and optimizer hyperparameters. Considering RNNs as dynamical systems, a natural way to capture stability, i.e., the growth and decay over long iterates, are the Lyapunov Exponents (LEs), which form the Lyapunov spectrum. The LEs have a bearing on stability of RNN training dynamics since forward propagation of information is related to the backward propagation of error gradients. LEs measure the asymptotic rates of expansion and contraction of non-linear system trajectories, and generalize stability analysis to the time-varying attractors structuring the non-autonomous dynamics of data-driven RNNs. As a tool to understand and exploit stability of training dynamics, the Lyapunov spectrum fills an existing gap between prescriptive mathematical approaches of limited scope and computationally-expensive empirical approaches. To leverage this tool, we implement an efficient way to compute LEs for RNNs during training, discuss the aspects specific to standard RNN architectures driven by typical sequential datasets, and show that the Lyapunov spectrum can serve as a robust readout of training stability across hyperparameters. With this exposition-oriented contribution, we hope to draw attention to this under-studied, but theoretically grounded tool for understanding training stability in RNNs. 

In modern relational machine learning it is common to encounter large graphs that arise via interactions or similarities between observations in many domains. Further, in many cases the target entities for analysis are actually signals on such graphs. We propose to compare and organize such datasets of graph signals by using an earth mover’s distance (EMD) with a geodesic cost over the underlying graph. Typically, EMD is computed by optimizing over the cost of transporting one probability distribution to another over an underlying metric space. However, this is inefficient when computing the EMD between many signals. Here, we propose an unbalanced graph EMD that efficiently embeds the unbalanced EMD on an underlying graph into an L1 space, whose metric we call unbalanced diffusion earth mover’s distance (UDEMD). Next, we show how this gives distances between graph signals that are robust to noise. Finally, we apply this to organizing patients based on clinical notes, embedding cells modeled as signals on a gene graph, and organizing genes modeled as signals over a large cell graph. In each case, we show that UDEMD-based embeddings find accurate distances that are highly efficient compared to other methods.