Width-Independent Compressibility of Deep Neural Networks

Deep neural networks often contain substantial hidden-state redundancy, but most compression methods operate directly on weights, neurons, or quantised representations without explicitly characterising the dynamical role of internal states. This paper proposes a controllability-observability framework for empirical state-order reduction of deep neural networks. By viewing a trained network as a depth-indexed nonlinear dynamical system, we construct data-driven reachability, observability, and balanced Gramians from hidden-state snapshots and output Jacobians. The resulting A/B/C tests estimate layer-wise reachable, observable, and jointly reachable--observable ranks. These ranks are then used not only as diagnostic measures of hidden-state redundancy, but also as actual compressed layer widths for realised reduced networks. Experiments on MNIST and CIFAR-10 compare the proposed balanced realisation against projection-based reduction, unstructured pruning, structured pruning, low-rank SVD, dynamic INT8 quantisation, and linear baselines. On MNIST, a four-layer SiLU DNN is reduced from state order 1024 to 277, giving 72.95% state compression and 73.48% parameter compression, while maintaining 95.45% accuracy compared with 96.60% for the full model. On CIFAR-10, a larger SiLU DNN is reduced from state order 4608 to 1339, giving 70.94% state compression and 83.09% parameter compression, while preserving accuracy from 54.45% to 54.44% and reducing CUDA inference latency by approximately 3X. The results show that balanced reachable-observable ranks provide a principled empirical minimal-realisation criterion for designing compact neural architectures with little or no loss in accuracy.

We consider the infinite-width limit of a fully connected deep neural network with general weights, and we prove quantitative general bounds on the $2$-Wasserstein distance between the network and its infinite-width Gaussian limit, under appropriate regularity assumptions on the activation function. Our main tool is a Lindeberg principle for Deep Neural Networks, which we use to successively replace the weights on each layer by Gaussian random variables.

Universal approximation theorems provide a mathematical explanation for the expressive power of neural networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function classes, such as continuous functions on compact subsets of $\mathbb{R}^d$, $L^p$ spaces, or Sobolev spaces. Over the past four decades, these qualitative universality results have evolved into a rich quantitative theory addressing approximation rates, parameter efficiency, and the role of architectural features such as depth and width. This survey presents several glimpses into this theory. We review classical density results for single-hidden-layer networks, as well as quantitative bounds that relate approximation error to network size and smoothness assumptions on target functions. Particular emphasis is placed on depth--width trade-offs and on results demonstrating that deeper architectures can achieve superior parameter efficiency for structured function classes. In addition to standard feedforward neural networks, we also review recent developments on Kolmogorov--Arnold Networks (KANs), which offer an alternative architectural paradigm and whose approximation-theoretic properties have begun to attract significant theoretical attention.