Efficient gradient computation for optimization of hyperparameters

We are interested in learning the hyperparameters in a convex objective function in a supervised setting. The complex relationship between the input data to the convex problem and the desirable hyperparameters can be modeled by a neural network; the hyperparameters and the data then drive the convex minimization problem, whose solution is then compared to training labels. In our previous work [1], we evaluated a prototype of this learning strategy in an optimization-based sinogram smoothing plus FBP reconstruction framework. A question arising in this setting is how to efficiently compute (backpropagate) the gradient from the solution of the optimization problem, to the hyperparameters to enable end-to-end training. In this work, we first develop general formulas for gradient backpropagation for a subset of convex problems, namely the proximal mapping. To illustrate the value of the general formulas and to demonstrate how to use them, we consider the specific instance of 1-D quadratic smoothing (denoising) whose solution admits a dynamic programming (DP) algorithm. The general formulas lead to another DP algorithm for exact computation of the gradient of the hyperparameters. Our numerical studies demonstrate a 55%- 65% computation time savings by providing a custom gradient instead of relying on automatic differentiation in deep learning libraries. While our discussion focuses on 1-D quadratic smoothing, our initial results (not presented) support the statement that the general formulas and the computational strategy apply equally well to TV or Huber smoothing problems on simple graphs whose solutions can be computed exactly via DP.

Efficient Proximal Mapping Computation for Low-Rank Inducing Norms

Low-rank inducing unitarily invariant norms have been introduced to convexify problems with a low-rank/sparsity constraint. The most well-known member of this family is the so-called nuclear norm. To solve optimization problems involving such norms with proximal splitting methods, efficient ways of evaluating the proximal mapping of the low-rank inducing norms are needed. This is known for the nuclear norm, but not for most other members of the low-rank inducing family. This work supplies a framework that reduces the proximal mapping evaluation into a nested binary search, in which each iteration requires the solution of a much simpler problem. The simpler problem can often be solved analytically as demonstrated for the so-called low-rank inducing Frobenius and spectral norms. The framework also allows to compute the proximal mapping of increasing convex functions composed with these norms as well as projections onto their epigraphs.

l1/2 regularization for wavelet frames based few-view CT reconstruction

Reducing the radiation exposure in computed tomography (CT) is always a significant research topic in radiology. Image reconstruction from few-view projection is a reasonable and effective way to decrease the number of rays to lower the radiation exposure. But how to maintain high image reconstruction quality while reducing radiation exposure is a major challenge. To solve this problem, several researchers are absorbed in l0 or l1 regularization based optimization models to deal with it. However, the solution of l1 regularization based optimization model is not sparser than that of l1/2 or l0 regularization, and solving the l0 regularization is more difficult than solving the l1/2 regularization. In this paper, we develop l1/2 regularization for wavelet frames based image reconstruction model to research the few-view problem. First, the existence of the solution of the corresponding model is demonstrated. Second, an alternate direction method (ADM) is utilized to separate the original problem into two subproblems, where the former subproblem about the image is solved using the idea of the proximal mapping, the simultaneous algebraic reconstruction technique (SART) and the projection and contraction (PC) algorithm, and the later subproblem about the wavelet coefficients is solved using the half thresholding (HT) algorithm. Furthermore, the convergence analysis of our method is given by the simulated implementions. Simulated and real experiments confirm the effectiveness of our method.

P420Improved real time recordings using the third generation cryoballoon technology - detection of dual fascicle electrograms

Introduction The third-generation cryoballoon has been designed with a 40% shortened tip length compared with the former second-generation CB device. Ideally, a shorter tip could result with an improved visualization of real-time (RT) recordings in the pulmonary vein (PV) due to a more proximal positioning of the inner lumen mapping catheter. Beside the high incidence of RT recordings, for the first time, we observed dual fascicle electrograms with different isolation time for each fascicle, when using the new device. Purpose We sought to analyze the rate of visualization of RT recordings and dual fascicle electrograms in our first series of patients with the third generation cryoballoon device. Methods All consecutive patients having undergone CB ablation using the third generation technology were analyzed. Procedures were performed under conscious sedation. A single transeptal puncture was preformed guided by intracardial ultrasound and a 28 mm cryoballoon was used. A single 180 seconds freeze strategy was employed. Results A total of 80 consecutive patients (77.5% male, 60.3 ± 10.8 years) were evaluated. Real-time recordings were detected in 275 of 320 PVs (85.9%). Specifically, RT recordings could be visualized in 68 left superior PVs (85%), 69 left inferior PVs (86.2%), 73 right superior PVs (91.2%), and 65 right inferior PVs (81.2%). Furthermore, in 14 of 320 PVs (4.3%) two different fascicles electrograms were detected. A distinctive isolation time for each fascicle could be observed both in sinus rhythm and in atrial fibrillation (Picture). Most commonly, dual fascicles were observed in left superior PV (7.5%) and right inferior PV (6.3%). In the rest of the veins this phenomenon was not so frequent. In 2.5% of right superior PV and in only 1.2% of left inferior PV dual fascicles could be observed. Conclusion The rate of visualization of RT recordings is significantly high during third generation CB ablation. Real-time recordings can be visualized in 85.9% of veins with this novel cryoballoon. Furthermore, for the first time, isolation of two different fascicles during a single cryoballoon lesion could be observed. Other that more proximal mapping catheter position, there is second possible explanation for the visualization of dual fascicles with this novel balloon. Shorter tip could result with poorer cryoballoon stability and weaker contact with the atrium wall, hence non-uniform freezing and isolation of different fascicles in different times. Abstract Figure.

CONTRACTION-MAPPING ALGORITHM FOR THE EQUILIBRIUM PROBLEM OVER THE FIXED POINT SET OF A NONEXPANSIVE SEMIGROUP

In this paper, we consider the proximal mapping of a bifunction. Under the Lipschitz-type and the strong monotonicity conditions, we prove that the proximal mapping is contractive. Based on this result, we construct an iterative process for solving the equilibrium problem over the fixed point sets of a nonexpansive semigroup and prove a weak convergence theorem for this algorithm. Also, some preliminary numerical experiments and comparisons are presented.