9,260 research outputs found
Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications
We extend the work of Narasimhan and Bilmes [30] for minimizing set functions
representable as a difference between submodular functions. Similar to [30],
our new algorithms are guaranteed to monotonically reduce the objective
function at every step. We empirically and theoretically show that the
per-iteration cost of our algorithms is much less than [30], and our algorithms
can be used to efficiently minimize a difference between submodular functions
under various combinatorial constraints, a problem not previously addressed. We
provide computational bounds and a hardness result on the mul- tiplicative
inapproximability of minimizing the difference between submodular functions. We
show, however, that it is possible to give worst-case additive bounds by
providing a polynomial time computable lower-bound on the minima. Finally we
show how a number of machine learning problems can be modeled as minimizing the
difference between submodular functions. We experimentally show the validity of
our algorithms by testing them on the problem of feature selection with
submodular cost features.Comment: 17 pages, 8 figures. A shorter version of this appeared in Proc.
Uncertainty in Artificial Intelligence (UAI), Catalina Islands, 201
Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection
We study the problem of selecting a subset of k random variables from a large
set, in order to obtain the best linear prediction of another variable of
interest. This problem can be viewed in the context of both feature selection
and sparse approximation. We analyze the performance of widely used greedy
heuristics, using insights from the maximization of submodular functions and
spectral analysis. We introduce the submodularity ratio as a key quantity to
help understand why greedy algorithms perform well even when the variables are
highly correlated. Using our techniques, we obtain the strongest known
approximation guarantees for this problem, both in terms of the submodularity
ratio and the smallest k-sparse eigenvalue of the covariance matrix. We further
demonstrate the wide applicability of our techniques by analyzing greedy
algorithms for the dictionary selection problem, and significantly improve the
previously known guarantees. Our theoretical analysis is complemented by
experiments on real-world and synthetic data sets; the experiments show that
the submodularity ratio is a stronger predictor of the performance of greedy
algorithms than other spectral parameters
Attention and Anticipation in Fast Visual-Inertial Navigation
We study a Visual-Inertial Navigation (VIN) problem in which a robot needs to
estimate its state using an on-board camera and an inertial sensor, without any
prior knowledge of the external environment. We consider the case in which the
robot can allocate limited resources to VIN, due to tight computational
constraints. Therefore, we answer the following question: under limited
resources, what are the most relevant visual cues to maximize the performance
of visual-inertial navigation? Our approach has four key ingredients. First, it
is task-driven, in that the selection of the visual cues is guided by a metric
quantifying the VIN performance. Second, it exploits the notion of
anticipation, since it uses a simplified model for forward-simulation of robot
dynamics, predicting the utility of a set of visual cues over a future time
horizon. Third, it is efficient and easy to implement, since it leads to a
greedy algorithm for the selection of the most relevant visual cues. Fourth, it
provides formal performance guarantees: we leverage submodularity to prove that
the greedy selection cannot be far from the optimal (combinatorial) selection.
Simulations and real experiments on agile drones show that our approach ensures
state-of-the-art VIN performance while maintaining a lean processing time. In
the easy scenarios, our approach outperforms appearance-based feature selection
in terms of localization errors. In the most challenging scenarios, it enables
accurate visual-inertial navigation while appearance-based feature selection
fails to track robot's motion during aggressive maneuvers.Comment: 20 pages, 7 figures, 2 table
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