Recent works have characterized the function-space inductive bias of infinite-width bounded-norm single-hidden-layer neural networks as a kind of bounded-variation-type space. This novel neural network Banach space encompasses many classical multivariate function spaces, including certain Sobolev spaces and the spectral Barron spaces. Notably, this Banach space also includes functions that exhibit less classical regularity, such as those that only vary in a few directions. On bounded domains, it is well-established that the Gaussian reproducing kernel Hilbert space (RKHS) strictly embeds into this Banach space, demonstrating a clear gap between the Gaussian RKHS and the neural network Banach space. It turns out that when investigating these spaces on unbounded domains, e.g., all of $\mathbb{R}^d$, the story is fundamentally different. We establish the following fundamental result: Certain functions that lie in the Gaussian RKHS have infinite norm in the neural network Banach space. This provides a nontrivial gap between kernel methods and neural networks by exhibiting functions that kernel methods easily represent, whereas neural networks cannot.

We study the implicit bias of flatness / low (loss) curvature and its effects on generalization in two-layer overparameterized ReLU networks with multivariate inputs---a problem well motivated by the minima stability and edge-of-stability phenomena in gradient-descent training. Existing work either requires interpolation or focuses only on univariate inputs. This paper presents new and somewhat surprising theoretical results for multivariate inputs. On two natural settings (1) generalization gap for flat solutions, and (2) mean-squared error (MSE) in nonparametric function estimation by stable minima, we prove upper and lower bounds, which establish that while flatness does imply generalization, the resulting rates of convergence necessarily deteriorate exponentially as the input dimension grows. This gives an exponential separation between the flat solutions compared to low-norm solutions (i.e., weight decay), which are known not to suffer from the curse of dimensionality. In particular, our minimax lower bound construction, based on a novel packing argument with boundary-localized ReLU neurons, reveals how flat solutions can exploit a kind of "neural shattering" where neurons rarely activate, but with high weight magnitudes. This leads to poor performance in high dimensions. We corroborate these theoretical findings with extensive numerical simulations. To the best of our knowledge, our analysis provides the first systematic explanation for why flat minima may fail to generalize in high dimensions.

We consider a large class of shallow neural networks with randomly initialized parameters and rectified linear unit activation functions. We prove that these random neural networks are well-defined non-Gaussian processes. As a by-product, we demonstrate that these networks are solutions to stochastic differential equations driven by impulsive white noise (combinations of random Dirac measures). These processes are parameterized by the law of the weights and biases as well as the density of activation thresholds in each bounded region of the input domain. We prove that these processes are isotropic and wide-sense self-similar with Hurst exponent 3/2. We also derive a remarkably simple closed-form expression for their autocovariance function. Our results are fundamentally different from prior work in that we consider a non-asymptotic viewpoint: The number of neurons in each bounded region of the input domain (i.e., the width) is itself a random variable with a Poisson law with mean proportional to the density parameter. Finally, we show that, under suitable hypotheses, as the expected width tends to infinity, these processes can converge in law not only to Gaussian processes, but also to non-Gaussian processes depending on the law of the weights. Our asymptotic results provide a new take on several classical results (wide networks converge to Gaussian processes) as well as some new ones (wide networks can converge to non-Gaussian processes).

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 studying the regularization effect of weight decay in training networks with activation functions like the rectified linear unit (ReLU). This framework offers a deeper understanding of multi-output networks and their function-space characteristics. A key contribution of this work is the development of a representer theorem for the vector-valued variation spaces. This representer theorem establishes that shallow vector-valued neural networks are the solutions to data-fitting problems over these infinite-dimensional spaces, where the network widths are bounded by the square of the number of training data. This observation reveals that the norm associated with these vector-valued variation spaces encourages the learning of features that are useful for multiple tasks, shedding new light on multi-task learning with neural networks. Finally, this paper develops a connection between weight-decay regularization and the multi-task lasso problem. This connection leads to novel bounds for layer widths in deep networks that depend on the intrinsic dimensions of the training data representations. This insight not only deepens the understanding of the deep network architectural requirements, but also yields a simple convex optimization method for deep neural network compression. The performance of this compression procedure is evaluated on various architectures.

The usual explanation of the efficacy of wavelet-based methods hinges on the sparsity of many real-world objects in the wavelet domain. Yet, standard wavelet-shrinkage techniques for sparse reconstruction are not competitive in practice, one reason being that the lack of shift-invariance of the wavelet transform produces blocky artifacts. The standard remedy is cycle spinning, which results in a substantial reduction of these artifacts. In this letter, we propose a theoretical investigation of the sparsity of solutions to the cycle-spinning variant of wavelet-based resolutions of linear inverse problems. We derive a representer theorem that provides a complete characterization of the solution set. Our theorem indicates that the solutions are typically not sparse, where sparsity is measured with respect to the wavelet dictionary. This exposes that the role of sparsity in the success of wavelet-based solutions of linear inverse problems requires further investigation. We corroborate our theoretical results with numerical examples for the problem of image denoising.

We study the problem of estimating an unknown function from noisy data using shallow ReLU neural networks. The estimators we study minimize the sum of squared data-fitting errors plus a regularization term proportional to the squared Euclidean norm of the network weights. This minimization corresponds to the common approach of training a neural network with weight decay. We quantify the performance (mean-squared error) of these neural network estimators when the data-generating function belongs to the second-order Radon-domain bounded variation space. This space of functions was recently proposed as the natural function space associated with shallow ReLU neural networks. We derive a minimax lower bound for the estimation problem for this function space and show that the neural network estimators are minimax optimal up to logarithmic factors. This minimax rate is immune to the curse of dimensionality. We quantify an explicit gap between neural networks and linear methods (which include kernel methods) by deriving a linear minimax lower bound for the estimation problem, showing that linear methods necessarily suffer the curse of dimensionality in this function space. As a result, this paper sheds light on the phenomenon that neural networks seem to break the curse of dimensionality.

We study a family of inverse problems in which a continuous-domain object is reconstructed from a finite number of noisy linear measurements. We study regularization methods for solving these problems in which the regularizers promote sparsity in the frequency domain. We show that sparse superpositions of decaying sinusoids are solutions to these continuous-domain linear inverse problems, where the number of terms in the superposition is upper bounded by the number of measurements. This results in new forms of regularization for sparse reconstruction that are different from classical techniques. We numerically illustrate the efficacy of these new regularization techniques in the problem of image reconstruction.

We develop a variational framework to understand the properties of the functions learned by neural networks fit to data. We propose and study a family of continuous-domain linear inverse problems with total variation-like regularization in the Radon domain subject to data fitting constraints. We derive a representer theorem showing that finite-width, single-hidden layer neural networks are solutions to these inverse problems. We draw on many techniques from variational spline theory and so we propose the notion of polynomial ridge splines, which correspond to single-hidden layer neural networks with truncated power functions as the activation function. The representer theorem is reminiscent of the classical reproducing kernel Hilbert space representer theorem, but we show that the neural network problem is posed over a non-Hilbertian Banach space. While the learning problems are posed in the continuous-domain, similar to kernel methods, the problems can be recast as finite-dimensional neural network training problems. These neural network training problems have regularizers which are related to the well-known weight decay and path-norm regularizers. Thus, our result gives insight into functional characteristics of trained neural networks and also into the design neural network regularizers. We also show that these regularizers promote neural network solutions with desirable generalization properties.

A wide variety of activation functions have been proposed for neural networks. The Rectified Linear Unit (ReLU) is especially popular today. There are many practical reasons that motivate the use of the ReLU. This paper provides new theoretical characterizations that support the use of the ReLU, its variants such as the leaky ReLU, as well as other activation functions in the case of univariate, single-hidden layer feedforward neural networks. Our results also explain the importance of commonly used strategies in the design and training of neural networks such as “weight decay” and “path-norm” regularization, and provide a new justification for the use of “skip connections” in network architectures. These new insights are obtained through the lens of spline theory. In particular, we show how neural network training problems are related to infinite-dimensional optimizations posed over Banach spaces of functions whose solutions are well-known to be fractional and polynomial splines, where the particular Banach space (which controls the order of the spline) depends on the choice of activation function.

Integrated circuit chips fabricated using nano-scale CMOS technologies will be prone to errors caused by fluctuations in threshold voltage, supply voltage, electromigration, random dopant fluctuations, aging, timing errors and soft errors. Design of nano-scale failure-resistant systems has drawn significant interest in past few years. One common approach to reducing errors is the use of triple modular redundancy (TMR). The hardware overhead associated with TMR is significantly high. This paper presents a novel partial triple modular redundancy (PTMR) approach that achieves the same or better fault-tolerance as that of TMR but with significantly less hardware overhead. In a weighted number system, the most significant bits carry greater weight and preserving these bits is more critical than the lower significant bits. In PTMR, only the P most significant bits of the result are computed using TMR as opposed to all the W bits, where W represents the word-length of the operands. The proposed PTMR approach is illustrated in the context of a ripple-carry adder. It is shown that the hardware overhead can be reduced by 75% to 87.5% with P = 4 as the word-length varies from 16 to 32, with average error power equal to or less than that of TMR. It is shown that P = 3 or 4 is sufficient for word-lengths varying from 16 to 32.