Joint Graph Rewiring and Feature Denoising via Spectral Resonance

Graph neural networks (GNNs) take as input the graph structure and the feature vectors associated with the nodes. Both contain noisy information about the labels. Here we propose joint denoising and rewiring (JDR)--an algorithm to jointly denoise the graph structure and features, which can improve the performance of any downstream algorithm. We do this by defining and maximizing the alignment between the leading eigenspaces of graph and feature matrices. To approximately solve this computationally hard problem, we propose a heuristic that efficiently handles real-world graph datasets with many classes and different levels of homophily or heterophily. We experimentally verify the effectiveness of our approach on synthetic data and real-world graph datasets. The results show that JDR consistently outperforms existing rewiring methods on node classification tasks using GNNs as downstream models.

A spring-block theory of feature learning in deep neural networks

A central question in deep learning is how deep neural networks (DNNs) learn features. DNN layers progressively collapse data into a regular low-dimensional geometry. This collective effect of non-linearity, noise, learning rate, width, depth, and numerous other parameters, has eluded first-principles theories which are built from microscopic neuronal dynamics. Here we present a noise-non-linearity phase diagram that highlights where shallow or deep layers learn features more effectively. We then propose a macroscopic mechanical theory of feature learning that accurately reproduces this phase diagram, offering a clear intuition for why and how some DNNs are ``lazy'' and some are ``active'', and relating the distribution of feature learning over layers with test accuracy.

Ice-Tide: Implicit Cryo-ET Imaging and Deformation Estimation

We introduce ICE-TIDE, a method for cryogenic electron tomography (cryo-ET) that simultaneously aligns observations and reconstructs a high-resolution volume. The alignment of tilt series in cryo-ET is a major problem limiting the resolution of reconstructions. ICE-TIDE relies on an efficient coordinate-based implicit neural representation of the volume which enables it to directly parameterize deformations and align the projections. Furthermore, the implicit network acts as an effective regularizer, allowing for high-quality reconstruction at low signal-to-noise ratios as well as partially restoring the missing wedge information. We compare the performance of ICE-TIDE to existing approaches on realistic simulated volumes where the significant gains in resolution and accuracy of recovering deformations can be precisely evaluated. Finally, we demonstrate ICE-TIDE's ability to perform on experimental data sets.

GLIMPSE: Generalized Local Imaging with MLPs

Deep learning is the current de facto state of the art in tomographic imaging. A common approach is to feed the result of a simple inversion, for example the backprojection, to a convolutional neural network (CNN) which then computes the reconstruction. Despite strong results on 'in-distribution' test data similar to the training data, backprojection from sparse-view data delocalizes singularities, so these approaches require a large receptive field to perform well. As a consequence, they overfit to certain global structures which leads to poor generalization on out-of-distribution (OOD) samples. Moreover, their memory complexity and training time scale unfavorably with image resolution, making them impractical for application at realistic clinical resolutions, especially in 3D: a standard U-Net requires a substantial 140GB of memory and 2600 seconds per epoch on a research-grade GPU when training on 1024x1024 images. In this paper, we introduce GLIMPSE, a local processing neural network for computed tomography which reconstructs a pixel value by feeding only the measurements associated with the neighborhood of the pixel to a simple MLP. While achieving comparable or better performance with successful CNNs like the U-Net on in-distribution test data, GLIMPSE significantly outperforms them on OOD samples while maintaining a memory footprint almost independent of image resolution; 5GB memory suffices to train on 1024x1024 images. Further, we built GLIMPSE to be fully differentiable, which enables feats such as recovery of accurate projection angles if they are out of calibration.

A Theoretical Analysis of the Test Error of Finite-Rank Kernel Ridge Regression

Existing statistical learning guarantees for general kernel regressors often yield loose bounds when used with finite-rank kernels. Yet, finite-rank kernels naturally appear in several machine learning problems, e.g.\ when fine-tuning a pre-trained deep neural network's last layer to adapt it to a novel task when performing transfer learning. We address this gap for finite-rank kernel ridge regression (KRR) by deriving sharp non-asymptotic upper and lower bounds for the KRR test error of any finite-rank KRR. Our bounds are tighter than previously derived bounds on finite-rank KRR, and unlike comparable results, they also remain valid for any regularization parameters.

Implicit Reconstructions from Deformed Projections for CryoET

Cryo-electron tomography (cryoET) is a technique that captures images of biological samples at different tilts, preserving their native state as much as possible. Along with the partial tilt series and noise, one of the major challenges in estimating the accurate 3D structure of the sample is the deformations in the images incurred during the acquisition. We model these deformations as continuous operators and estimate the unknown 3D volume using implicit neural representations. This framework allows to easily incorporate the deformation and estimate jointly the deformation parameters and the volume using a standard optimization algorithm. This approach doesn't require training data and can benefit from standard prior in the optimization procedure.

Harpa: High-Rate Phase Association with Travel Time Neural Fields

Phase association groups seismic wave arrivals according to their originating earthquakes. It is a fundamental task in a seismic data processing pipeline, but challenging to perform for smaller, high-rate seismic events which carry fundamental information about earthquake dynamics, especially with a commonly assumed inaccurate wave speed model. As a consequence, most association methods focus on larger events that occur at a lower rate and are thus easier to associate, even though microseismicity provides a valuable description of the elastic medium properties in the subsurface. In this paper, we show that association is possible at rates much higher than previously reported even when the wave speed is unknown. We propose Harpa, a high-rate seismic phase association method which leverages deep neural fields to build generative models of wave speeds and associated travel times, and first solves a joint spatio--temporal source localization and wave speed recovery problem, followed by association. We obviate the need for associated phases by interpreting arrival time data as probability measures and using an optimal transport loss to enforce data fidelity. The joint recovery problem is known to admit a unique solution under certain conditions but due to the non-convexity of the corresponding loss a simple gradient scheme converges to poor local minima. We show that this is effectively mitigated by stochastic gradient Langevin dynamics (SGLD). Numerical experiments show that \harpa~efficiently associates high-rate seismicity clouds over complex, unknown wave speeds and graciously handles noisy and missing picks.

A Dynamical Graph Prior for Relational Inference

Relational inference aims to identify interactions between parts of a dynamical system from the observed dynamics. Current state-of-the-art methods fit a graph neural network (GNN) on a learnable graph to the dynamics. They use one-step message-passing GNNs -- intuitively the right choice since non-locality of multi-step or spectral GNNs may confuse direct and indirect interactions. But the \textit{effective} interaction graph depends on the sampling rate and it is rarely localized to direct neighbors, leading to local minima for the one-step model. In this work, we propose a \textit{dynamical graph prior} (DYGR) for relational inference. The reason we call it a prior is that, contrary to established practice, it constructively uses error amplification in high-degree non-local polynomial filters to generate good gradients for graph learning. To deal with non-uniqueness, DYGR simultaneously fits a ``shallow'' one-step model with shared graph topology. Experiments show that DYGR reconstructs graphs far more accurately than earlier methods, with remarkable robustness to under-sampling. Since appropriate sampling rates for unknown dynamical systems are not known a priori, this robustness makes DYGR suitable for real applications in scientific machine learning.

A Transfer Principle: Universal Approximators Between Metric Spaces From Euclidean Universal Approximators

We build universal approximators of continuous maps between arbitrary Polish metric spaces $\mathcal{X}$ and $\mathcal{Y}$ using universal approximators between Euclidean spaces as building blocks. Earlier results assume that the output space $\mathcal{Y}$ is a topological vector space. We overcome this limitation by "randomization": our approximators output discrete probability measures over $\mathcal{Y}$. When $\mathcal{X}$ and $\mathcal{Y}$ are Polish without additional structure, we prove very general qualitative guarantees; when they have suitable combinatorial structure, we prove quantitative guarantees for H\"older-like maps, including maps between finite graphs, solution operators to rough differential equations between certain Carnot groups, and continuous non-linear operators between Banach spaces arising in inverse problems. In particular, we show that the required number of Dirac measures is determined by the combinatorial structure of $\mathcal{X}$ and $\mathcal{Y}$. For barycentric $\mathcal{Y}$, including Banach spaces, $\mathbb{R}$-trees, Hadamard manifolds, or Wasserstein spaces on Polish metric spaces, our approximators reduce to $\mathcal{Y}$-valued functions. When the Euclidean approximators are neural networks, our constructions generalize transformer networks, providing a new probabilistic viewpoint of geometric deep learning.

Injectivity of ReLU networks: perspectives from statistical physics

When can the input of a ReLU neural network be inferred from its output? In other words, when is the network injective? We consider a single layer, $x \mapsto \mathrm{ReLU}(Wx)$, with a random Gaussian $m \times n$ matrix $W$, in a high-dimensional setting where $n, m \to \infty$. Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for $\alpha = \frac{m}{n}$ by studying the expected Euler characteristic of a certain random set. We adopt a different perspective and show that injectivity is equivalent to a property of the ground state of the spherical perceptron, an important spin glass model in statistical physics. By leveraging the (non-rigorous) replica symmetry-breaking theory, we derive analytical equations for the threshold whose solution is at odds with that from the Euler characteristic. Furthermore, we use Gordon's min--max theorem to prove that a replica-symmetric upper bound refutes the Euler characteristic prediction. Along the way we aim to give a tutorial-style introduction to key ideas from statistical physics in an effort to make the exposition accessible to a broad audience. Our analysis establishes a connection between spin glasses and integral geometry but leaves open the problem of explaining the discrepancies.