This work proposes and analyzes a compressed sensing approach to polynomial approximation of complex-valued functions in high dimensions.Of particular interest is the setting where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we present an innovative weighted 1 minimization procedure with a precise choice of weights, and a new iterative hard thresholding method, for imposing the downward closed preference. Theoretical results reveal that our computational approaches possess a provably reduced sample complexity compared to existing compressed sensing techniques presented in the literature. In addition, the recovery of the corresponding best approximation using these methods is established through an improved bound for the restricted isometry property. Our analysis represents an extension of the approach for Hadamard matrices in [5] to the general case of continuous bounded orthonormal systems, quantifies the dependence of sample complexity on the successful recovery probability, and provides an estimate on the number of measurements with explicit constants. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the novel weighted 1 minimization strategy.
We present and analyze a novel sparse polynomial technique for the simultaneous approximation of parameterized partial differential equations (PDEs) with deterministic and stochastic inputs. Our approach treats the numerical solution as a jointly sparse reconstruction problem through the reformulation of the standard basis pursuit denoising, where the set of jointly sparse vectors is infinite. To achieve global reconstruction of sparse solutions to parameterized elliptic PDEs over both physical and parametric domains, we combine the standard measurement scheme developed for compressed sensing in the context of bounded orthonormal systems with a novel mixed-norm based 1 regularization method that exploits both energy and sparsity. In addition, we are able to prove that, with minimal sample complexity, error estimates comparable to the best s-term and quasi-optimal approximations are achievable, while requiring only a priori bounds on polynomial truncation error with respect to the energy norm. Finally, we perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the superior recovery properties of the proposed approach.
15Motivation: The prediction of drug resistance and the identification of its mechanisms in bacteria 16 such as Mycobacterium tuberculosis, the etiological agent of tuberculosis, is a challenging problem. 17 Modern methods based on testing against a catalogue of previously identified mutations often yield 18 poor predictive performance. On the other hand, machine learning techniques have demonstrated 19 high predictive accuracy, but lack interpretability to aid in identifying specific mutations which lead 20 to resistance. We propose a novel technique, inspired by the group testing problem and Boolean 21 compressed sensing, which yields highly accurate predictions and interpretable results at the same 22 time. 23Results: We develop a modified version of the Boolean compressed sensing problem for identifying 24 drug resistance, and implement its formulation as an integer linear program. This allows us to 25 characterize the predictive accuracy of the technique and select an appropriate metric to optimize. 26 A simple adaptation of the problem also allows us to quantify the sensitivity-specificity trade-off of 27 our model under different regimes. We test the predictive accuracy of our approach on a variety 28 of commonly used antibiotics in treating tuberculosis and find that it has accuracy comparable to 29 that of standard machine learning models and points to several genes with previously identified 30 association to drug resistance. 31 Availability: https://github.com/WGS-TB/DrugResistance/tree/RB_learning 32 Contact: hooman_zabeti@sfu.ca 33 34 2012 ACM Subject Classification Applied computing -Life and medical sciences -Computational 35 biology -Molecular sequence analysis 36 1 Introduction 43 Drug resistance is the phenomenon by which an infectious organism (also known as pathogen) 44 develops resistance to one or more drugs that are commonly used in treatment [36]. In 45 this paper we focus our attention on Mycobacterium tuberculosis, the etiological agent of 46 tuberculosis, which is the largest infectious killer in the world today, responsible for over 10 47 million new cases and 2 million deaths every year [37]. 48 The development of resistance to common drugs used in treatment is a serious public health 49 threat, not only in low and middle-income countries, but also in high-income countries where 50 it is particularly problematic in hospital settings [39]. It is estimated that, without the urgent 51 development of novel antimicrobial drugs, the total mortality due to drug resistance will 52 exceed 10 million people a year by 2050, a number exceeding the annual mortality due to 53 cancer today [35]. 54 Existing models for predicting drug resistance from whole-genome sequence (WGS) data 55 broadly fall into two classes. The first, which we refer to as "catalogue methods," involves 56 testing the WGS data of an isolate for the presence of point mutations (typically single-57 nucleotide polymorphisms, or SNPs) associated with known drug resistance. If one or 58more such mutations is identified,...
Motivation Prediction of drug resistance and identification of its mechanisms in bacteria such as Mycobacterium tuberculosis, the etiological agent of tuberculosis, is a challenging problem. Solving this problem requires a transparent, accurate, and flexible predictive model. The methods currently used for this purpose rarely satisfy all of these criteria. On the one hand, approaches based on testing strains against a catalogue of previously identified mutations often yield poor predictive performance; on the other hand, machine learning techniques typically have higher predictive accuracy, but often lack interpretability and may learn patterns that produce accurate predictions for the wrong reasons. Current interpretable methods may either exhibit a lower accuracy or lack the flexibility needed to generalize them to previously unseen data. Contribution In this paper we propose a novel technique, inspired by group testing and Boolean compressed sensing, which yields highly accurate predictions, interpretable results, and is flexible enough to be optimized for various evaluation metrics at the same time. Results We test the predictive accuracy of our approach on five first-line and seven second-line antibiotics used for treating tuberculosis. We find that it has a higher or comparable accuracy to that of commonly used machine learning models, and is able to identify variants in genes with previously reported association to drug resistance. Our method is intrinsically interpretable, and can be customized for different evaluation metrics. Our implementation is available at github.com/hoomanzabeti/INGOT_DR and can be installed via The Python Package Index (Pypi) under ingotdr. This package is also compatible with most of the tools in the Scikit-learn machine learning library.
Deep learning (DL) is transforming whole industries as complicated decision-making processes are being automated by Deep Neural Networks (DNNs) trained on real-world data. Driven in part by a rapidly-expanding literature on DNN approximation theory showing that DNNs can approximate a rich variety of functions, these tools are increasingly being considered for problems in scientific computing. Yet, unlike more traditional algorithms in this field, relatively little is known about DNNs from the principles of numerical analysis, namely, stability, accuracy, computational efficiency and sample complexity. In this paper we introduce a computational framework for examining DNNs in practice, and use it to study their empirical performance with regard to these issues. We examine the performance of DNNs of different widths and depths on a variety of test functions in various dimensions, including smooth and piecewise smooth functions. We also compare DL against best-inclass methods for smooth function approximation based on compressed sensing. Our main conclusion is that there is a crucial gap between the approximation theory of DNNs and their practical performance, with trained DNNs performing relatively poorly on functions for which there are strong approximation results (e.g. smooth functions), yet performing well in comparison to best-in-class methods for other functions. Finally, we present a novel practical existence theorem, which asserts the existence of a DNN architecture and training procedure which offers the same performance as current best-in-class schemes. This result indicates the potential for practical DNN approximation, and the need for future research into practical architecture design and training strategies.
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