Abstract
The AI community has been paying attention to submodular functions due to their various applications (e.g., target search and 3D mapping). Learning submodular functions is a challenge since the number of a function’s outcomes of N sets is . The state-of-the-art approach is based on compressed sensing techniques, which are to learn submodular functions in the Fourier domain and then recover the submodular functions in the spatial domain. However, the number of Fourier bases is relevant to the number of sets’ sensing overlapping. To overcome this issue, this research proposed a submodular deep compressed sensing (SDCS) approach to learning submodular functions. The algorithm consists of learning autoencoder networks and Fourier coefficients. The learned networks can be applied to predict values of submodular functions. Experiments conducted with this approach demonstrate that the algorithm is more efficient than the benchmark approach.
Highlights
AI and robotics communities have been paying more attention to submodularity due to its variant applications and theoretical guarantees of solutions
The distance between two adjacent subgoals is 20, and the number of Fourier basis b of spatial Fourier sparse set (SFSS) is 9852; it is decided by the algorithms in [3]
These experiments demonstrate that submodular deep compressed sensing (SDCS) approach is able to reconstruct submodular functions using fewer Fourier bases than the SFSS approach does
Summary
AI and robotics communities have been paying more attention to submodularity (see Definition 1 and Figure 1) due to its variant applications (e.g., information collection [1], task assignment [2], and target search [3]) and theoretical guarantees of solutions. The advantage of formulating a problem as maximizing submodular functions is that greedy algorithms can give theoretical guarantees under cardinality [4], knapsack [5], and routing constraints [6]. If the objective function is under the cardinality constraint, greedy algorithms can generate solutions over (1 − 1/e) of the optimum [4]. Definition 1 (Submodularity (Nemhauser et al, 1978)). Given a finite set S={1,2,...,N}, a submodular function is a set function F : 2 N → R which satisfies the diminishing return property. For every S A , SB ⊆ S with S A ⊆ SB and every s ⊆ S, F (S A ∪ s) − F (S A ) ≥ F (SB ∪ s) − F (SB ) holds
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