49 research outputs found
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
Benchmarking energy consumption and latency for neuromorphic computing in condensed matter and particle physics
The massive use of artificial neural networks (ANNs), increasingly popular in
many areas of scientific computing, rapidly increases the energy consumption of
modern high-performance computing systems. An appealing and possibly more
sustainable alternative is provided by novel neuromorphic paradigms, which
directly implement ANNs in hardware. However, little is known about the actual
benefits of running ANNs on neuromorphic hardware for use cases in scientific
computing. Here we present a methodology for measuring the energy cost and
compute time for inference tasks with ANNs on conventional hardware. In
addition, we have designed an architecture for these tasks and estimate the
same metrics based on a state-of-the-art analog in-memory computing (AIMC)
platform, one of the key paradigms in neuromorphic computing. Both
methodologies are compared for a use case in quantum many-body physics in two
dimensional condensed matter systems and for anomaly detection at 40 MHz rates
at the Large Hadron Collider in particle physics. We find that AIMC can achieve
up to one order of magnitude shorter computation times than conventional
hardware, at an energy cost that is up to three orders of magnitude smaller.
This suggests great potential for faster and more sustainable scientific
computing with neuromorphic hardware.Comment: 7 pages, 4 figures, submitted to APL Machine Learnin
Most Frugal Explanations: Occam\u92s Razor Applied to Bayesian Abduction
What constitutes \u91Best\u92 in \u91Inference to the Best Explanation\u92 has been hotly debated. In Bayesian models the traditional interpretation is \u91Best = Most Probable\u92. We propose an alternative notion, denoted as Most Frugal Explanation (MFE), that utilizes the fact that only few variables actually are relevant for deciding upon the best explanation. We show that MFE is intractable in general, but can be tractably approximated under plausible situational constraints
Treewidth and the Computational Complexity of MAP Approximations
The problem of finding the most probable explanation to a designated set of vari-ables given partial evidence (the MAP problem) is a notoriously intractable problem in Bayesian networks, both to compute exactly and to approximate. It is known, both from theoretical considerations and from practical experience, that low tree-width is typically an essential prerequisite to efficient exact computations in Bayesian networks. In this paper we investigate whether the same holds for approximating MAP. We define four notions of approximating MAP (by value, structure, rank, and expectation) and argue that all of them are intractable in general. We prove that efficient value-approximations, structure-approximations, and rank-approximations of MAP instances with high tree-width will violate the Exponential Time Hypothesis. In contrast, we show that MAP can some-times be efficiently expectation-approximated, even in instances with high tree-width, if the most probable explanation has a high probability. We introduce the complexity class FERT, analogous to the class FPT, to capture this notion of fixed-parameter expectation-approximability. We suggest a road-map to future research that yields fixed-parameter tractable results for expectation-approximate MAP, even in graphs with high tree-width. 1