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We demonstrate how aperiodicity and disorder can be used as quantifiable mechanisms for tuning the spectral response of plasmonic nanostructure arrays. We tune the extinction spectra of these arrays using deterministically aperiodic (quasicrystal), perturbed lattice (Bernoulli point process, frozen phonon disorder, long-range frozen phonon disorder), negatively correlated (Strauss point process), and positively correlated (Log Gaussian Cox point process) assemblies. We quantify this tuning by considering the local variance of the extinction spectra, demonstrating two orders of magnitude of tunability. Our structures have potential applications in plasmonic or waveguide-based optoelectronic devices such as photovoltaics and photosensing, where spectral tuning is critical to performance. 

AI is now a cornerstone of modern dataset analysis. In many real world applications, practitioners are concerned with controlling specific kinds of errors, rather than minimizing the overall number of errors. For example, biomedical screening assays may primarily be concerned with mitigating the number of false positives rather than false negatives. Quantifying uncertainty in AI-based predictions, and in particular those controlling specific kinds of errors, remains theoretically and practically challenging. We develop a strategy called multidimensional informed generalized hypothesis testing (MIGHT) which we prove accurately quantifies uncertainty and confidence given sufficient data, and concomitantly controls for particular error types. Our key insight was that it is possible to integrate canonical cross-validation and parametric calibration procedures within a nonparametric ensemble method. Simulations demonstrate that while typical AI based-approaches cannot be trusted to obtain the truth, MIGHT can be. We apply MIGHT to answer an open question in liquid biopsies using circulating cell-free DNA (ccfDNA) in individuals with or without cancer: Which biomarkers, or combinations thereof, can we trust? Performance estimates produced by MIGHT on ccfDNA data have coefficients of variation that are often orders of magnitude lower than other state of the art algorithms such as support vector machines, random forests, and Transformers, while often also achieving higher sensitivity. We find that combinations of variable sets often decrease rather than increase sensitivity over the optimal single variable set because some variable sets add more noise than signal. This work demonstrates the importance of quantifying uncertainty and confidence—with theoretical guarantees—for the interpretation of real-world data. 

Multiple case-controlled studies have shown that analyzing fragmentation patterns in plasma cell-free DNA (cfDNA) can distinguish individuals with cancer from healthy controls. However, there have been few studies that investigate various types of cfDNA fragmentomics patterns in individuals with other diseases. We therefore developed a comprehensive statistic, called fragmentation signatures, that integrates the distributions of fragment positioning, fragment length, and fragment end-motifs in cfDNA. We found that individuals with venous thromboembolism, systemic lupus erythematosus, dermatomyositis, or scleroderma have cfDNA fragmentation signatures that closely resemble those found in individuals with advanced cancers. Furthermore, these signatures were highly correlated with increases in inflammatory markers in the blood. We demonstrate that these similarities in fragmentation signatures lead to high rates of false positives in individuals with autoimmune or vascular disease when evaluated using conventional binary classification approaches for multicancer earlier detection (MCED). To address this issue, we introduced a multiclass approach for MCED that integrates fragmentation signatures with protein biomarkers and achieves improved specificity in individuals with autoimmune or vascular disease while maintaining high sensitivity. Though these data put substantial limitations on the specificity of fragmentomics-based tests for cancer diagnostics, they also offer ways to improve the interpretability of such tests. Moreover, we expect these results will lead to a better understanding of the process—most likely inflammatory—from which abnormal fragmentation signatures are derived. 

Convex regression is the problem of fitting a convex function to a data set consisting of input-output pairs. We present a new approach to this problem called spectrahedral regression, in which we fit a spectrahedral function to the data, i.e., a function that is the maximum eigenvalue of an affine matrix expression of the input. This method represents a significant generalization of polyhedral (also called max-affine) regression, in which a polyhedral function (a maximum of a fixed number of affine functions) is fit to the data. We prove bounds on how well spectrahedral functions can approximate arbitrary convex functions via statistical risk analysis. We also analyze an alternating minimization algorithm for the nonconvex optimization problem of fitting the best spectrahedral function to a given data set. We show that this algorithm converges geometrically with high probability to a small ball around the optimal parameter given a good initialization. Finally, we demonstrate the utility of our approach with experiments on synthetic data sets as well as real data arising in applications such as economics and engineering design.