Search for: All records

A fundamental problem in machine learning is to understand how neural networks make accurate predictions, while seemingly bypassing the curse of dimensionality. A possible explanation is that common training algorithms for neural networks implicitly perform dimensionality reduction—a process called feature learning. Recent work [A. Radhakrishnan, D. Beaglehole, P. Pandit, M. Belkin,Science383, 1461–1467 (2024).] posited that the effects of feature learning can be elicited from a classical statistical estimator called the average gradient outer product (AGOP). The authors proposed Recursive Feature Machines (RFMs) as an algorithm that explicitly performs feature learning by alternating between 1) reweighting the feature vectors by the AGOP and 2) learning the prediction function in the transformed space. In this work, we develop theoretical guarantees for how RFM performs dimensionality reduction by focusing on the class of overparameterized problems arising in sparse linear regression and low-rank matrix recovery. Specifically, we show that RFM restricted to linear models (lin-RFM) reduces to a variant of the well-studied Iteratively Reweighted Least Squares (IRLS) algorithm. Furthermore, our results connect feature learning in neural networks and classical sparse recovery algorithms and shed light on how neural networks recover low rank structure from data. In addition, we provide an implementation of lin-RFM that scales to matrices with millions of missing entries. Our implementation is faster than the standard IRLS algorithms since it avoids forming singular value decompositions. It also outperforms deep linear networks for sparse linear regression and low-rank matrix completion. 

Kernel methods are a popular class of nonlinear predictive models in machine learning. Scalable algorithms for learning kernel models need to be iterative in nature, but convergence can be slow due to poor conditioning. Spectral preconditioning is an important tool to speed-up the convergence of such iterative algorithms for training kernel models. However computing and storing a spectral preconditioner can be expensive which can lead to large computational and storage overheads, precluding the application of kernel methods to problems with large datasets. A Nystrom approximation of the spectral preconditioner is often cheaper to compute and store, and has demonstrated success in practical applications. In this paper we analyze the trade-offs of using such an approximated preconditioner. Specifically, we show that a sample of logarithmic size (as a function of the size of the dataset) enables the Nyström-based approximated preconditioner to accelerate gradient descent nearly as well as the exact preconditioner, while also reducing the computational and storage overheads. 

Optically resonant particles are key building blocks of many nanophotonic devices such as optical antennas and metasurfaces. Because the functionalities of such devices are largely determined by the optical properties of individual resonators, extending the attainable responses from a given particle is highly desirable. Practically, this is usually achieved by introducing an asymmetric dielectric environment. However, commonly used simple substrates have limited influences on the optical properties of the particles atop. Here, we show that the multipolar scattering of silicon microspheres can be effectively modified by placing the particles on a dielectric-covered mirror, which tunes the coupling between the Mie resonances of microspheres and the standing waves and waveguide modes in the dielectric spacer. This tunability allows selective excitation, enhancement, suppression, and even elimination of the multipolar resonances and enables scattering at extended wavelengths, providing transformative opportunities in controlling light–matter interactions for various applications. We further demonstrate with experiments the detection of molecular fingerprints by single-particle mid-infrared spectroscopy and with simulations strong optical repulsive forces that could elevate the particles from a substrate. 

While neural networks are used for classification tasks across domains, a long-standing open problem in machine learning is determining whether neural networks trained using standard procedures are consistent for classification, i.e., whether such models minimize the probability of misclassification for arbitrary data distributions. In this work, we identify and construct an explicit set of neural network classifiers that are consistent. Since effective neural networks in practice are typically both wide and deep, we analyze infinitely wide networks that are also infinitely deep. In particular, using the recent connection between infinitely wide neural networks and neural tangent kernels, we provide explicit activation functions that can be used to construct networks that achieve consistency. Interestingly, these activation functions are simple and easy to implement, yet differ from commonly used activations such as ReLU or sigmoid. More generally, we create a taxonomy of infinitely wide and deep networks and show that these models implement one of three well-known classifiers depending on the activation function used: 1) 1-nearest neighbor (model predictions are given by the label of the nearest training example); 2) majority vote (model predictions are given by the label of the class with the greatest representation in the training set); or 3) singular kernel classifiers (a set of classifiers containing those that achieve consistency). Our results highlight the benefit of using deep networks for classification tasks, in contrast to regression tasks, where excessive depth is harmful. 

Matrix completion problems arise in many applications including recommendation systems, computer vision, and genomics. Increasingly larger neural networks have been successful in many of these applications but at considerable computational costs. Remarkably, taking the width of a neural network to infinity allows for improved computational performance. In this work, we develop an infinite width neural network framework for matrix completion that is simple, fast, and flexible. Simplicity and speed come from the connection between the infinite width limit of neural networks and kernels known as neural tangent kernels (NTK). In particular, we derive the NTK for fully connected and convolutional neural networks for matrix completion. The flexibility stems from a feature prior, which allows encoding relationships between coordinates of the target matrix, akin to semisupervised learning. The effectiveness of our framework is demonstrated through competitive results for virtual drug screening and image inpainting/reconstruction. We also provide an implementation in Python to make our framework accessible on standard hardware to a broad audience.