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Continual learning is a challenge for models with static architecture, as they fail to adapt to when data distributions evolve across tasks. We introduce a mathematical framework that jointly models architecture and weights in a Sobolev…

We analytically derive the geometrical structure of the weight space in multilayer neural networks (MLN), in terms of the volumes of couplings associated to the internal representations of the training set. Focusing on the parity and…

We study the storage and retrieval of phase-coded patterns as stable dynamical attractors in recurrent neural networks, for both an analog and a integrate-and-fire spiking model. The synaptic strength is determined by a learning rule based…

As deep neural networks grow in size, from thousands to millions to billions of weights, the performance of those networks becomes limited by our ability to accurately train them. A common naive question arises: if we have a system with…

In recent years, deep neural networks have had great success in machine learning and pattern recognition. Architecture size for a neural network contributes significantly to the success of any neural network. In this study, we optimize the…

Reservoir computing is a recently introduced machine learning paradigm that has been shown to be well-suited for the processing of spatiotemporal data. Rather than training the network node connections and weights via backpropagation in…

This paper presents an empirical study on the weights of neural networks, where we interpret each model as a point in a high-dimensional space -- the neural weight space. To explore the complex structure of this space, we sample from a…

How much information does a learning algorithm extract from the training data and store in a neural network's weights? Too much, and the network would overfit to the training data. Too little, and the network would not fit to anything at…

We study a class of Hopfield models where the memories are represented by a mixture of Gaussian and binary variables and the neurons are Ising spins. We study the properties of this family of models as the relative weight of the two kinds…

We consider an ensemble of $K$ single-layer perceptrons exposed to random inputs and investigate the conditions under which the couplings of these perceptrons can be chosen such that prescribed correlations between the outputs occur. A…

We consider the problem of training a neural network to store a set of patterns with maximal noise robustness. A solution, in terms of optimal weights and state update rules, is derived by training each individual neuron to perform either…

Exact solutions for the learning problem of autoassociative networks with binary couplings are determined by a new method. The use of a branch-and-bound algorithm leads to a substantial saving of computational time compared with complete…

This paper introduces a novel theoretical framework for the analysis of vector-valued neural networks through the development of vector-valued variation spaces, a new class of reproducing kernel Banach spaces. These spaces emerge from…

The history of deep learning has shown that human-designed problem-specific networks can greatly improve the classification performance of general neural models. In most practical cases, however, choosing the optimal architecture for a…

Neural plasticity is an important functionality of human brain, in which number of neurons and synapses can shrink or expand in response to stimuli throughout the span of life. We model this dynamic learning process as an $L_0$-norm…

The information processing abilities of a multilayer neural network with a number of hidden units scaling as the input dimension are studied using statistical mechanics methods. The mapping from the input layer to the hidden units is…

Working memory (WM) is limited in its temporal length and capacity. Classic conceptions of WM capacity assume the system possesses a finite number of slots, but recent evidence suggests WM may be a continuous resource. Resource models…

When neural networks are trained to classify a dataset, one finds a set of weights from which the network produces a label for each data point. We study the algorithmic complexity of finding a collision in a single-layer neural net, where a…

Designing neural network architectures is a task that lies somewhere between science and art. For a given task, some architectures are eventually preferred over others, based on a mix of intuition, experience, experimentation and luck. For…

Characterizing the relation between weight structure and input/output statistics is fundamental for understanding the computational capabilities of neural circuits. In this work, I study the problem of storing associations between analog…