Block Coordinate Descent for Sparse NMF

Nonnegative matrix factorization (NMF) has become a ubiquitous tool for data analysis. An important variant is the sparse NMF problem which arises when we explicitly require the learnt features to be sparse. A natural measure of sparsity is the L$_0$ norm, however its optimization is NP-hard. Mixed norms, such as L$_1$/L$_2$ measure, have been shown to model sparsity robustly, based on intuitive attributes that such measures need to satisfy. This is in contrast to computationally cheaper alternatives such as the plain L$_1$ norm. However, present algorithms designed for optimizing the mixed norm L$_1$/L$_2$ are slow and other formulations for sparse NMF have been proposed such as those based on L$_1$ and L$_0$ norms. Our proposed algorithm allows us to solve the mixed norm sparsity constraints while not sacrificing computation time. We present experimental evidence on real-world datasets that shows our new algorithm performs an order of magnitude faster compared to the current state-of-the-art solvers optimizing the mixed norm and is suitable for large-scale datasets

Automated elaborate resection planning for bone tumor surgery

PURPOSE: Planning for bone tumor resection surgery is a technically demanding and time-consuming task, reliant on manual positioning of planar cuts in a virtual space. More elaborate cutting approaches may be possible through the use of surgical robots or patient-specific instruments; however, methods for preparing such a resection plan must be developed. METHODS: This work describes an automated approach for generating conformal bone tumor resection plans, where the resection geometry is defined by the convex hull of the tumor, and a focal point. The resection geometry is optimized using particle swarm, where the volume of healthy bone collaterally resected with the tumor is minimized. The approach was compared to manually prepared planar resection plans from an experienced surgeon for 20 tumor cases. RESULTS: It was found that algorithm-generated hull-type resections greatly reduced the volume of collaterally resected healthy bone. The hull-type resections resulted in statistically significant improvements compared to the manual approach (paired t test, p < 0.001). CONCLUSIONS: The described approach has potential to improve patient outcomes by reducing the volume of healthy bone collaterally resected with the tumor and preserving nearby critical anatomy

On Optimization over the Efficient Set in Linear Multicriteria Programming

The efficient set of a linear multicriteria programming problem can be representedby a reverse convex constraint of the form g(z) ≤ 0, where g is a concavefunction. Consequently, the problem of optimizing some real function over the efficientset belongs to an important problem class of global optimization called reverseconvex programming. Since the concave function used in the literature is only definedon some set containing the feasible set of the underlying multicriteria programmingproblem, most global optimization techniques for handling this kind of reverse convexconstraint cannot be applied. The main purpose of our article is to present amethod for overcoming this disadvantage. We construct a concave function which isfinitely defined on the whole space and can be considered as an extension of the existingfunction. Different forms of the linear multicriteria programming problem arediscussed, including the minimum maximal flow problem as an example