RESUMEN
Neural ordinary differential equations have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of the role played by their architecture remains elusive. In this work, we examine the interplay between the width p and the number of transitions between layers L (corresponding to a depth of L+1). Specifically, we construct explicit controls interpolating either a finite dataset D, comprising N pairs of points in Rd, or two probability measures within a Wasserstein error margin É>0. Our findings reveal a balancing trade-off between p and L, with L scaling as 1+O(N/p) for data interpolation, and as 1+Op-1+(1+p)-1É-d for measures. In the high-dimensional and wide setting where d,p>N, our result can be refined to achieve L=0. This naturally raises the problem of data interpolation in the autonomous regime, characterized by L=0. We adopt two alternative approaches: either controlling in a probabilistic sense, or by relaxing the target condition. In the first case, when p=N we develop an inductive control strategy based on a separability assumption whose probability increases with d. In the second one, we establish an explicit error decay rate with respect to p which results from applying a universal approximation theorem to a custom-built Lipschitz vector field interpolating D.
RESUMEN
Isolated evaluation of multiparametric in vivo chemical exchange saturation transfer (CEST) MRI often requires complex computational processing for both correction of B0 and B1 inhomogeneity and contrast generation. For that, sufficiently densely sampled Z-spectra need to be acquired. The list of acquired frequency offsets largely determines the total CEST acquisition time, while potentially representing redundant information. In this work, a linear projection-based multiparametric CEST evaluation method is introduced that offers fast B0 and B1 inhomogeneity correction, contrast generation and feature selection for CEST data, enabling reduction of the overall measurement time. To that end, CEST data acquired at 7 T in six healthy subjects and in one brain tumor patient were conventionally evaluated by interpolation-based inhomogeneity correction and Lorentzian curve fitting. Linear regression was used to obtain coefficient vectors that directly map uncorrected data to corrected Lorentzian target parameters. L1-regularization was applied to find subsets of the originally acquired CEST measurements that still allow for such a linear projection mapping. The linear projection method allows fast and interpretable mapping from acquired raw data to contrast parameters of interest, generalizing from healthy subject training data to unseen healthy test data and to the tumor patient dataset. The L1-regularization method shows that a fraction of the acquired CEST measurements is sufficient to preserve tissue contrasts, offering up to a 2.8-fold reduction of scan time. Similar observations as for the 7-T data can be made for data from a clinical 3-T scanner. Being a fast and interpretable computation step, the proposed method is complementary to neural networks that have recently been employed for similar purposes. The scan time acceleration offered by the L1-regularization ("CEST-LASSO") constitutes a step towards better applicability of multiparametric CEST protocols in a clinical context.
