Submodular Generalized Matching for Peptide Identification in Tandem Mass Spectrometry.

MOTIVATION: Identification of spectra produced by a shotgun proteomics mass spectrometry experiment is commonly performed by searching the observed spectra against a peptide database. The heart of this search procedure is a score function that evaluates the quality of a hypothesized match between an observed spectrum and a theoretical spectrum corresponding to a particular peptide sequence. Accordingly, the success of a spectrum analysis pipeline depends critically upon this peptide-spectrum score function. We develop peptide-spectrum score functions that compute the maximum value of a submodular function under $m$ m matroid constraints. We call this procedure a submodular generalized matching (SGM) since it generalizes bipartite matching. We use a greedy algorithm to compute maximization, which can achieve a solution whose objective is guaranteed to be at least $\frac{1}{1+m}$ 1 1 + m of the true optimum. The advantage of the SGM framework is that known long-range properties of experimental spectra can be modeled by designing suitable submodular functions and matroid constraints. Experiments on four data sets from various organisms and mass spectrometry platforms show that the SGM approach leads to significantly improved performance compared to several state-of-the-art methods. Supplementary information, C++ source code, and data sets can be found at https://melodi-lab.github.io/SGM.

Submodular Maximization via Gradient Ascent: The Case of Deep Submodular Functions.

We study the problem of maximizing deep submodular functions (DSFs) [13, 3] subject to a matroid constraint. DSFs are an expressive class of submodular functions that include, as strict subfamilies, the facility location, weighted coverage, and sums of concave composed with modular functions. We use a strategy similar to the continuous greedy approach [6], but we show that the multilinear extension of any DSF has a natural and computationally attainable concave relaxation that we can optimize using gradient ascent. Our results show a guarantee of max 0 < δ < 1 ( 1 - ϵ - δ - e - δ 2 Ω ( k ) ) with a running time of O(n 2 /ϵ 2 ) plus time for pipage rounding [6] to recover a discrete solution, where k is the rank of the matroid constraint. This bound is often better than the standard 1 - 1/e guarantee of the continuous greedy algorithm, but runs much faster. Our bound also holds even for fully curved (c = 1) functions where the guarantee of 1 - c/e degenerates to 1 - 1/e where c is the curvature of f [37]. We perform computational experiments that support our theoretical results.