1,280 research outputs found

    Quantitative toxicity prediction using topology based multi-task deep neural networks

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    The understanding of toxicity is of paramount importance to human health and environmental protection. Quantitative toxicity analysis has become a new standard in the field. This work introduces element specific persistent homology (ESPH), an algebraic topology approach, for quantitative toxicity prediction. ESPH retains crucial chemical information during the topological abstraction of geometric complexity and provides a representation of small molecules that cannot be obtained by any other method. To investigate the representability and predictive power of ESPH for small molecules, ancillary descriptors have also been developed based on physical models. Topological and physical descriptors are paired with advanced machine learning algorithms, such as deep neural network (DNN), random forest (RF) and gradient boosting decision tree (GBDT), to facilitate their applications to quantitative toxicity predictions. A topology based multi-task strategy is proposed to take the advantage of the availability of large data sets while dealing with small data sets. Four benchmark toxicity data sets that involve quantitative measurements are used to validate the proposed approaches. Extensive numerical studies indicate that the proposed topological learning methods are able to outperform the state-of-the-art methods in the literature for quantitative toxicity analysis. Our online server for computing element-specific topological descriptors (ESTDs) is available at http://weilab.math.msu.edu/TopTox/Comment: arXiv admin note: substantial text overlap with arXiv:1703.1095

    Topology of definable Hausdorff limits

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    Let AβŠ‚Rn+rA\sub \R^{n+r} be a set definable in an o-minimal expansion Β§\S of the real field, Aβ€²βŠ‚RrA' \sub \R^r be its projection, and assume that the non-empty fibers AaβŠ‚RnA_a \sub \R^n are compact for all a∈Aβ€²a \in A' and uniformly bounded, {\em i.e.} all fibers are contained in a ball of fixed radius B(0,R).B(0,R). If LL is the Hausdorff limit of a sequence of fibers Aai,A_{a_i}, we give an upper-bound for the Betti numbers bk(L)b_k(L) in terms of definable sets explicitly constructed from a fiber Aa.A_a. In particular, this allows to establish effective complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, Gabrielov introduced the {\em relative closure} to construct the o-minimal structure \S_\pfaff generated by Pfaffian functions in a way that is adapted to complexity problems. Our results can be used to estimate the Betti numbers of a relative closure (X,Y)0(X,Y)_0 in the special case where YY is empty.Comment: Latex, 23 pages, no figures. v2: Many changes in the exposition and notations in an attempt to be clearer, references adde
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