386 research outputs found
Greedy low-rank algorithm for spatial connectome regression
Recovering brain connectivity from tract tracing data is an important
computational problem in the neurosciences. Mesoscopic connectome
reconstruction was previously formulated as a structured matrix regression
problem (Harris et al., 2016), but existing techniques do not scale to the
whole-brain setting. The corresponding matrix equation is challenging to solve
due to large scale, ill-conditioning, and a general form that lacks a
convergent splitting. We propose a greedy low-rank algorithm for connectome
reconstruction problem in very high dimensions. The algorithm approximates the
solution by a sequence of rank-one updates which exploit the sparse and
positive definite problem structure. This algorithm was described previously
(Kressner and Sirkovi\'c, 2015) but never implemented for this connectome
problem, leading to a number of challenges. We have had to design judicious
stopping criteria and employ efficient solvers for the three main sub-problems
of the algorithm, including an efficient GPU implementation that alleviates the
main bottleneck for large datasets. The performance of the method is evaluated
on three examples: an artificial "toy" dataset and two whole-cortex instances
using data from the Allen Mouse Brain Connectivity Atlas. We find that the
method is significantly faster than previous methods and that moderate ranks
offer good approximation. This speedup allows for the estimation of
increasingly large-scale connectomes across taxa as these data become available
from tracing experiments. The data and code are available online
Hierarchical interpolative factorization for elliptic operators: differential equations
This paper introduces the hierarchical interpolative factorization for
elliptic partial differential equations (HIF-DE) in two (2D) and three
dimensions (3D). This factorization takes the form of an approximate
generalized LU/LDL decomposition that facilitates the efficient inversion of
the discretized operator. HIF-DE is based on the multifrontal method but uses
skeletonization on the separator fronts to sparsify the dense frontal matrices
and thus reduce the cost. We conjecture that this strategy yields linear
complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity
in 3D can be achieved by skeletonizing the compressed fronts themselves, which
amounts geometrically to a recursive dimensional reduction scheme. Numerical
experiments support our claims and further demonstrate the performance of our
algorithm as a fast direct solver and preconditioner. MATLAB codes are freely
available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math.
arXiv admin note: substantial text overlap with arXiv:1307.266
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
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