2 research outputs found
Fast greedy algorithms for dictionary selection with generalized sparsity constraints
In dictionary selection, several atoms are selected from finite candidates
that successfully approximate given data points in the sparse representation.
We propose a novel efficient greedy algorithm for dictionary selection. Not
only does our algorithm work much faster than the known methods, but it can
also handle more complex sparsity constraints, such as average sparsity. Using
numerical experiments, we show that our algorithm outperforms the known methods
for dictionary selection, achieving competitive performances with dictionary
learning algorithms in a smaller running time
On Maximization of Weakly Modular Functions: Guarantees of Multi-stage Algorithms, Tractability, and Hardness
Maximization of {\it non-submodular} functions appears in various scenarios,
and many previous works studied it based on some measures that quantify the
closeness to being submodular. On the other hand, many practical non-submodular
functions are actually close to being {\it modular}, which has been utilized in
few studies. In this paper, we study cardinality-constrained maximization of
{\it weakly modular} functions, whose closeness to being modular is measured by
{\it submodularity} and {\it supermodularity ratios}, and reveal what we can
and cannot do by using the weak modularity. We first show that guarantees of
multi-stage algorithms can be proved with the weak modularity, which generalize
and improve some existing results, and experiments confirm their effectiveness.
We then show that weakly modular maximization is {\it fixed-parameter
tractable} under certain conditions; as a byproduct, we provide a new
time--accuracy trade-off for -constrained minimization. We finally
prove that, even if objective functions are weakly modular, no polynomial-time
algorithms can improve the existing approximation guarantees achieved by the
greedy algorithm