Sharp Bounds for Poly-GNNs and the Effect of Graph Noise

We investigate the classification performance of graph neural networks with graph-polynomial features, poly-GNNs, on the problem of semi-supervised node classification. We analyze poly-GNNs under a general contextual stochastic block model (CSBM) by providing a sharp characterization of the rate of separation between classes in their output node representations. A question of interest is whether this rate depends on the depth of the network $k$, i.e., whether deeper networks can achieve a faster separation? We provide a negative answer to this question: for a sufficiently large graph, a depth $k > 1$ poly-GNN exhibits the same rate of separation as a depth $k=1$ counterpart. Our analysis highlights and quantifies the impact of ``graph noise'' in deep GNNs and shows how noise in the graph structure can dominate other sources of signal in the graph, negating any benefit further aggregation provides. Our analysis also reveals subtle differences between even and odd-layered GNNs in how the feature noise propagates.

Step and Smooth Decompositions as Topological Clustering

We investigate a class of recovery problems for which observations are a noisy combination of continuous and step functions. These problems can be seen as non-injective instances of non-linear ICA with direct applications to image decontamination for magnetic resonance imaging. Alternately, the problem can be viewed as clustering in the presence of structured (smooth) contaminant. We show that a global topological property (graph connectivity) interacts with a local property (the degree of smoothness of the continuous component) to determine conditions under which the components are identifiable. Additionally, a practical estimation algorithm is provided for the case when the contaminant lies in a reproducing kernel Hilbert space of continuous functions. Algorithm effectiveness is demonstrated through a series of simulations and real-world studies.

Simplifying GNN Performance with Low Rank Kernel Models

We revisit recent spectral GNN approaches to semi-supervised node classification (SSNC). We posit that many of the current GNN architectures may be over-engineered. Instead, simpler, traditional methods from nonparametric estimation, applied in the spectral domain, could replace many deep-learning inspired GNN designs. These conventional techniques appear to be well suited for a variety of graph types reaching state-of-the-art performance on many of the common SSNC benchmarks. Additionally, we show that recent performance improvements in GNN approaches may be partially attributed to shifts in evaluation conventions. Lastly, an ablative study is conducted on the various hyperparameters associated with GNN spectral filtering techniques. Code available at: https://github.com/lucianoAvinas/lowrank-gnn-kernels

LapGM: A Multisequence MR Bias Correction and Normalization Model

A spatially regularized Gaussian mixture model, LapGM, is proposed for the bias field correction and magnetic resonance normalization problem. The proposed spatial regularizer gives practitioners fine-tuned control between balancing bias field removal and preserving image contrast preservation for multi-sequence, magnetic resonance images. The fitted Gaussian parameters of LapGM serve as control values which can be used to normalize image intensities across different patient scans. LapGM is compared to well-known debiasing algorithm N4ITK in both the single and multi-sequence setting. As a normalization procedure, LapGM is compared to known techniques such as: max normalization, Z-score normalization, and a water-masked region-of-interest normalization. Lastly a CUDA-accelerated Python package $\texttt{lapgm}$ is provided from the authors for use.

Sinusoidal Sensitivity Calculation for Line Segment Geometries

Purpose: Provide a closed-form solution to the sinusoidal coil sensitivity model proposed by Kern et al. This closed-form allows for precise computations of varied, simulated bias fields for ground-truth debias datasets. Methods: Fourier distribution theory and standard integration techniques were used to calculate the Fourier transform for line segment magnetic fields. Results: A $L^1_{\rm loc}(\mathbb{R}^3)$ function is derived in full generality for arbitrary line segment geometries. Sampling criteria and equivalence to the original sinusoidal model are also discussed. Lastly a CUDA accelerated implementation $\texttt{biasgen}$ is provided by authors. Conclusion: As the derived result is influenced by coil positioning and geometry, practitioners will have access to a more diverse ecosystem of simulated datasets which may be used to compare prospective debiasing methods.

Using Machine Learning to Select High-Quality Measurements

We describe the use of machine learning algorithms to select high-quality measurements for the Mu2e experiment. This technique is important for experiments with backgrounds that arise due to measurement errors. The algorithms use multiple pieces of ancillary information that are sensitive to measurement quality to separate high-quality and low-quality measurements.