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Multi-parameter regularization in high-dimensional learning

高维学习中的多参数正则化

基本信息

批准号：
254193214
负责人：
Professor Dr. Massimo Fornasier
金额：
--
依托单位：
Lehrstuhl für Angewandte Numerische Analysis und Optimierung und Data Analysis (M15)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2014
资助国家：
德国
起止时间：
2013-12-31 至 2017-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/254193214?language=en
关键词：
Multi parameter regularization dimensional learning

项目摘要

Making accurate predictions is a crucial factor in many systems for cost savings, efficiency, health, safety, and organizational purposes. Inspired by the increased demand of robust predictive methods, in this joint international project we are developing a comprehensive analysis of techniques and numerical methods for performing reliable predictions from roughly measured high-dimensional data. The challenge of working with high-dimensional noisy data shall be overcome by incorporating additional information on top of the available data, through optimization by means of multi-parameter regularization, and studying different candidate core models together with additional sets of constraints. We address specifically three fundamental objectives, the first two of them have methodological nature and the last one has applicative nature. The first objective is to develop both comprehensive theoretical and numerical approaches to multi-penalty regularization in Banach spaces, which may be reproducing kernel Banach spaces or spaces of sparsely represented functions. This is motivated by the largely expected geometrical/structured features of high-dimensional data, which may not be well-represented in the framework of (typically more isotropic) Hilbert spaces. Moreover, it is a rather open research field where only preliminary results are available.The second objective will be to use multi-penalty regularization in Banach spaces in high-dimensional supervised learning. Here we focus on two main mechanisms of dimensionality reduction by assuming that our function has a special representation / format and then we recast the learning problem into the framework of multi-penalty regularization with the adaptively chosen parameters. As the last objective we shall apply the methodologies developed in the previous two tasks to meta-learning for optimal parameter choices of algorithms. Since in many algorithms, for numerical simulation purposes, but even more crucially in data analysis, certain parameters need to be tuned for optimal performances, measured in terms either of speed or of resulting (approximation) quality, this begs for the development of a fast choice rule for the parameters, possibly provided certain features of the data, which may retain nevertheless a rather high dimensionality. This rule shall be learned by training on previous applications of the algorithm. It appears that this issue has not been systematically studied in the context of high-dimensional learning.The above mentioned project directions may, in the future, serve as a solid bridge across regularization, learning, and approximation theories and can play a fundamental role for various practical applications.

在许多系统中，为了节省成本、提高效率、健康、安全和组织目的，做出准确的预测是一个关键因素。受到对稳健预测方法日益增长的需求的启发，在这个联合国际项目中，我们正在开发一种技术和数值方法的综合分析，以便根据粗略测量的高维数据进行可靠的预测。通过在可用数据之上合并附加信息，通过多参数正则化进行优化，以及研究不同的候选核心模型和附加约束集，可以克服处理高维噪声数据的挑战。我们具体解决三个基本目标，其中前两个具有方法论性质，最后一个具有应用性质。第一个目标是开发巴纳赫空间中多重惩罚正则化的综合理论和数值方法，这可能是再现核巴纳赫空间或稀疏表示函数的空间。这是由高维数据的很大程度上预期的几何/结构化特征驱动的，这些特征可能无法在（通常更各向同性）希尔伯特空间的框架中得到很好的表示。此外，这是一个相当开放的研究领域，仅提供初步结果。第二个目标是在高维监督学习中使用 Banach 空间中的多重惩罚正则化。在这里，我们通过假设我们的函数具有特殊的表示/格式来关注降维的两个主要机制，然后我们将学习问题重新构建到具有自适应选择参数的多重惩罚正则化的框架中。作为最后一个目标，我们将应用前两个任务中开发的方法来进行元学习，以实现算法的最佳参数选择。由于在许多算法中，出于数值模拟的目的，但更重要的是在数据分析中，需要调整某些参数以获得最佳性能，以速度或结果（近似）质量来衡量，这需要开发参数的快速选择规则，可能提供数据的某些特征，但可能保留相当高的维度。该规则应通过对算法的先前应用进行训练来学习。看来这个问题在高维学习的背景下还没有得到系统的研究。上述项目方向未来可能会成为跨越正则化、学习和逼近理论的坚实桥梁，并为各种实际应用发挥基础性作用。