Vector Quantile Regression on Manifolds

Quantile regression (QR) is a statistical tool for distribution-free estimation of conditional quantiles of a target variable given explanatory features. QR is limited by the assumption that the target distribution is univariate and defined on an Euclidean domain. Although the notion of quantiles was recently extended to multi-variate distributions, QR for multi-variate distributions on manifolds remains underexplored, even though many important applications inherently involve data distributed on, e.g., spheres (climate measurements), tori (dihedral angles in proteins), or Lie groups (attitude in navigation). By leveraging optimal transport theory and the notion of $c$-concave functions, we meaningfully define conditional vector quantile functions of high-dimensional variables on manifolds (M-CVQFs). Our approach allows for quantile estimation, regression, and computation of conditional confidence sets. We demonstrate the approach's efficacy and provide insights regarding the meaning of non-Euclidean quantiles through preliminary synthetic data experiments.

Fast Nonlinear Vector Quantile Regression

Quantile regression (QR) is a powerful tool for estimating one or more conditional quantiles of a target variable $\mathrm{Y}$ given explanatory features $\boldsymbol{\mathrm{X}}$. A limitation of QR is that it is only defined for scalar target variables, due to the formulation of its objective function, and since the notion of quantiles has no standard definition for multivariate distributions. Recently, vector quantile regression (VQR) was proposed as an extension of QR for high-dimensional target variables, thanks to a meaningful generalization of the notion of quantiles to multivariate distributions. Despite its elegance, VQR is arguably not applicable in practice due to several limitations: (i) it assumes a linear model for the quantiles of the target $\mathrm{Y}$ given the features $\boldsymbol{\mathrm{X}}$; (ii) its exact formulation is intractable even for modestly-sized problems in terms of target dimensions, number of regressed quantile levels, or number of features, and its relaxed dual formulation may violate the monotonicity of the estimated quantiles; (iii) no fast or scalable solvers for VQR currently exist. In this work we fully address these limitations, namely: (i) We extend VQR to the non-linear case, showing substantial improvement over linear VQR; (ii) We propose vector monotone rearrangement, a method which ensures the estimates obtained by VQR relaxations are monotone functions; (iii) We provide fast, GPU-accelerated solvers for linear and nonlinear VQR which maintain a fixed memory footprint with number of samples and quantile levels, and demonstrate that they scale to millions of samples and thousands of quantile levels; (iv) We release an optimized python package of our solvers as to widespread the use of VQR in real-world applications.

Joint optimization of system design and reconstruction in MIMO radar imaging

Multiple-input multiple-output (MIMO) radar is one of the leading depth sensing modalities. However, the usage of multiple receive channels lead to relative high costs and prevent the penetration of MIMOs in many areas such as the automotive industry. Over the last years, few studies concentrated on designing reduced measurement schemes and image reconstruction schemes for MIMO radars, however these problems have been so far addressed separately. On the other hand, recent works in optical computational imaging have demonstrated growing success of simultaneous learning-based design of the acquisition and reconstruction schemes, manifesting significant improvement in the reconstruction quality. Inspired by these successes, in this work, we propose to learn MIMO acquisition parameters in the form of receive (Rx) antenna elements locations jointly with an image neural-network based reconstruction. To this end, we propose an algorithm for training the combined acquisition-reconstruction pipeline end-to-end in a differentiable way. We demonstrate the significance of using our learned acquisition parameters with and without the neural-network reconstruction.

Towards learned optimal q-space sampling in diffusion MRI

Fiber tractography is an important tool of computational neuroscience that enables reconstructing the spatial connectivity and organization of white matter of the brain. Fiber tractography takes advantage of diffusion Magnetic Resonance Imaging (dMRI) which allows measuring the apparent diffusivity of cerebral water along different spatial directions. Unfortunately, collecting such data comes at the price of reduced spatial resolution and substantially elevated acquisition times, which limits the clinical applicability of dMRI. This problem has been thus far addressed using two principal strategies. Most of the efforts have been extended towards improving the quality of signal estimation for any, yet fixed sampling scheme (defined through the choice of diffusion-encoding gradients). On the other hand, optimization over the sampling scheme has also proven to be effective. Inspired by the previous results, the present work consolidates the above strategies into a unified estimation framework, in which the optimization is carried out with respect to both estimation model and sampling design {\it concurrently}. The proposed solution offers substantial improvements in the quality of signal estimation as well as the accuracy of ensuing analysis by means of fiber tractography. While proving the optimality of the learned estimation models would probably need more extensive evaluation, we nevertheless claim that the learned sampling schemes can be of immediate use, offering a way to improve the dMRI analysis without the necessity of deploying the neural network used for their estimation. We present a comprehensive comparative analysis based on the Human Connectome Project data. Code and learned sampling designs aviliable at https://github.com/tomer196/Learned_dMRI.

3D FLAT: Feasible Learned Acquisition Trajectories for Accelerated MRI

Magnetic Resonance Imaging (MRI) has long been considered to be among the gold standards of today's diagnostic imaging. The most significant drawback of MRI is long acquisition times, prohibiting its use in standard practice for some applications. Compressed sensing (CS) proposes to subsample the k-space (the Fourier domain dual to the physical space of spatial coordinates) leading to significantly accelerated acquisition. However, the benefit of compressed sensing has not been fully exploited; most of the sampling densities obtained through CS do not produce a trajectory that obeys the stringent constraints of the MRI machine imposed in practice. Inspired by recent success of deep learning based approaches for image reconstruction and ideas from computational imaging on learning-based design of imaging systems, we introduce 3D FLAT, a novel protocol for data-driven design of 3D non-Cartesian accelerated trajectories in MRI. Our proposal leverages the entire 3D k-space to simultaneously learn a physically feasible acquisition trajectory with a reconstruction method. Experimental results, performed as a proof-of-concept, suggest that 3D FLAT achieves higher image quality for a given readout time compared to standard trajectories such as radial, stack-of-stars, or 2D learned trajectories (trajectories that evolve only in the 2D plane while fully sampling along the third dimension). Furthermore, we demonstrate evidence supporting the significant benefit of performing MRI acquisitions using non-Cartesian 3D trajectories over 2D non-Cartesian trajectories acquired slice-wise.

Deep geometric matrix completion: Are we doing it right?

We address the problem of reconstructing a matrix from a subset of its entries. Current methods, branded as geometric matrix completion, augment classical rank regularization techniques by incorporating geometric information into the solution. This information is usually provided as graphs encoding relations between rows/columns. In this work we propose a simple spectral approach for solving the matrix completion problem, via the framework of functional maps. We introduce the zoomout loss, a multiresolution spectral geometric loss inspired by recent advances in shape correspondence, whose minimization leads to state-of-the-art results on various recommender systems datasets. Surprisingly, for some datasets we were able to achieve comparable results even without incorporating geometric information. This puts into question both the quality of such information and current methods' ability to use it in a meaningful and efficient way.

PILOT: Physics-Informed Learned Optimal Trajectories for Accelerated MRI

Magnetic Resonance Imaging (MRI) has long been considered to be among "the gold standards" of diagnostic medical imaging. The long acquisition times, however, render MRI prone to motion artifacts, let alone their adverse contribution to the relative high costs of MRI examination. Over the last few decades, multiple studies have focused on the development of both physical and post-processing methods for accelerated acquisition of MRI scans. These two approaches, however, have so far been addressed separately. On the other hand, recent works in optical computational imaging have demonstrated growing success of concurrent learning-based design of data acquisition and image reconstruction schemes. In this work, we propose a novel approach to the learning of optimal schemes for conjoint acquisition and reconstruction of MRI scans, with the optimization carried out simultaneously with respect to the time-efficiency of data acquisition and the quality of resulting reconstructions. To be of a practical value, the schemes are encoded in the form of general k-space trajectories, whose associated magnetic gradients are constrained to obey a set of predefined hardware requirements (as defined in terms of, e.g., peak currents and maximum slew rates of magnetic gradients). With this proviso in mind, we propose a novel algorithm for the end-to-end training of a combined acquisition-reconstruction pipeline using a deep neural network with differentiable forward- and back-propagation operators. We also demonstrate the effectiveness of the proposed solution in application to both image reconstruction and image segmentation, reporting substantial improvements in terms of acceleration factors as well as the quality of these end tasks.

Self-supervised learning of inverse problem solvers in medical imaging

In the past few years, deep learning-based methods have demonstrated enormous success for solving inverse problems in medical imaging. In this work, we address the following question:\textit{Given a set of measurements obtained from real imaging experiments, what is the best way to use a learnable model and the physics of the modality to solve the inverse problem and reconstruct the latent image?} Standard supervised learning based methods approach this problem by collecting data sets of known latent images and their corresponding measurements. However, these methods are often impractical due to the lack of availability of appropriately sized training sets, and, more generally, due to the inherent difficulty in measuring the "groundtruth" latent image. In light of this, we propose a self-supervised approach to training inverse models in medical imaging in the absence of aligned data. Our method only requiring access to the measurements and the forward model at training. We showcase its effectiveness on inverse problems arising in accelerated magnetic resonance imaging (MRI).

Learning Fast Magnetic Resonance Imaging

Magnetic Resonance Imaging (MRI) is considered today the golden-standard modality for soft tissues. The long acquisition times, however, make it more prone to motion artifacts as well as contribute to the relatively high costs of this examination. Over the years, multiple studies concentrated on designing reduced measurement schemes and image reconstruction schemes for MRI, however, these problems have been so far addressed separately. On the other hand, recent works in optical computational imaging have demonstrated growing success of the simultaneous learning-based design of the acquisition and reconstruction schemes manifesting significant improvement in the reconstruction quality with a constrained time budget. Inspired by these successes, in this work, we propose to learn accelerated MR acquisition schemes (in the form of Cartesian trajectories) jointly with the image reconstruction operator. To this end, we propose an algorithm for training the combined acquisition-reconstruction pipeline end-to-end in a differentiable way. We demonstrate the significance of using the learned Cartesian trajectories at different speed up rates.