Some Notes on Applied Mathematics for Machine Learning

2004 ◽  
pp. 21-40 ◽  
Author(s):  
Christopher J. C. Burges
Keyword(s):  
Machine Learning ◽  
Applied Mathematics

scholarly journals Compressive-sensing data reconstruction for structural health monitoring: a machine-learning approach

2019 ◽  
Vol 19 (1) ◽  
pp. 293-304 ◽  
Author(s):  
Yuequan Bao ◽  
Zhiyi Tang ◽  
Hui Li
Keyword(s):  
Machine Learning ◽  
Structural Health Monitoring ◽  
Compressive Sensing ◽  
Health Monitoring ◽  
Applied Mathematics ◽  
Modal Identification ◽  
Data Reconstruction ◽  
Wireless Data ◽  
Sensing Data ◽  
Structural Health

Compressive sensing has been studied and applied in structural health monitoring for data acquisition and reconstruction, wireless data transmission, structural modal identification, and spare damage identification. The key issue in compressive sensing is finding the optimal solution for sparse optimization. In the past several years, many algorithms have been proposed in the field of applied mathematics. In this article, we propose a machine learning–based approach to solve the compressive-sensing data-reconstruction problem. By treating a computation process as a data flow, the solving process of compressive sensing–based data reconstruction is formalized into a standard supervised-learning task. The prior knowledge, i.e. the basis matrix and the compressive sensing–sampled signals, is used as the input and the target of the network; the basis coefficient matrix is embedded as the parameters of a certain layer; and the objective function of conventional compressive sensing is set as the loss function of the network. Regularized by l1-norm, these basis coefficients are optimized to reduce the error between the original compressive sensing–sampled signals and the masked reconstructed signals with a common optimization algorithm. In addition, the proposed network is able to handle complex bases, such as a Fourier basis. Benefiting from the nature of a multi-neuron layer, multiple signal channels can be reconstructed simultaneously. Meanwhile, the disassembled use of a large-scale basis makes the method memory-efficient. A numerical example of multiple sinusoidal waves and an example of field-test wireless data from a suspension bridge are carried out to illustrate the data-reconstruction ability of the proposed approach. The results show that high reconstruction accuracy can be obtained by the machine learning–based approach. In addition, the parameters of the network have clear meanings; the inference of the mapping between input and output is fully transparent, making the compressive-sensing data-reconstruction neural network interpretable.

scholarly journals PREDICTION OF THE RISK GROUP (BY ACADEMIC PERFORMANCE) AMONG FIRST COURSE STUDENTS BY USING THE DECISION TREE METHOD

2020 ◽  
pp. 121-136
Author(s):  
Natalia Nakaryakova ◽  
◽  
Sergey Rusakov ◽  
Olga Rusakova ◽  
◽  
...  
Keyword(s):  
Machine Learning ◽  
Decision Tree ◽  
Computer Science ◽  
Decision Trees ◽  
Research University ◽  
Risk Group ◽  
Applied Mathematics ◽  
Binary Classification ◽  
First Trimester ◽  
Dropout Rate

Mass education in Russian universities in specialties (direction of study) related to the exact and technical sciences is characterized by a high dropout rate, starting from the first year of study. The current level of school education, the system for selecting applicants through the USE procedure, in many cases does not guarantee that future students will be able to successfully master science-intensive specialties. An emphasis on student-centered, individual learning is possible only after students have proven themselves in the early stages of their studies. Therefore, the anticipatory identification of the ability of yesterday's applicants to study effectively is a very urgent task. In this paper, we consider methods for constructing decision trees designed to classify students, highlighting from them a lot of those (risk group) who, with a high degree of probability, will be expelled after the first academic cycle (trimester). At the same time, the minimum information about the freshmen, recorded in their personal file, is used as input data. The construction of the model was carried out according to the data on students of the applied mathematics and computer science direction of the Perm State National Research University for a five-year period of sets of 2014-2018. At the same time, the information from 2014-2017 was used for training, and the flow of 2018 was used as a test one. At the stage of machine learning, several models of decision trees were considered, which were optimized using balancing, restrictions on the maximum tree depth and the minimum number of elements in a leaf. The effectiveness of the binary classification was assessed using a matrix of inaccuracies and a number of numerical criteria obtained on its basis. As a result of machine learning, a decision tree was built, which predicted 16 out of 17 people expelled from the first trimester into the risk group. That is, for a number of reasons, they turned out to be incapable of learning in the direction of applied mathematics and computer science. In addition, it was possible to determine the level of significance of various types of initial data, showing that the results of the USE largely determine the success of students at this stage of training. The definition of the risk group provides certain guidelines for the purposeful activity of teachers and university psychologists, which ultimately can serve as a basis for improving the quality of education and reducing dropout rates. The work performed demonstrates the capabilities of data mining methods in solving poorly formalized tasks characteristic of this type of human activity.

Mathematical foundations

2019 ◽  
pp. 1-40
Author(s):  
Max A. Little
Keyword(s):  
Mathematical Modeling ◽  
Machine Learning ◽  
Signal Processing ◽  
Mathematical Models ◽  
Applied Mathematics ◽  
Statistical Machine Learning ◽  
Mathematical Concepts ◽  
The Subject ◽  
Mathematical Foundations

Statistical machine learning and signal processing are topics in applied mathematics, which are based upon many abstract mathematical concepts. Defining these concepts clearly is the most important first step in this book. The purpose of this chapter is to introduce these foundational mathematical concepts. It also justifies the statement that much of the art of statistical machine learning as applied to signal processing, lies in the choice of convenient mathematical models that happen to be useful in practice. Convenient in this context means that the algebraic consequences of the choice of mathematical modeling assumptions are in some sense manageable. The seeds of this manageability are the elementary mathematical concepts upon which the subject is built.

Bioinformatics and Computational Biology

2011 ◽  
pp. 160-165
Author(s):  
Gustavo Camps-Valls ◽  
Alistair Morgan Chalk
Keyword(s):  
Machine Learning ◽  
Computational Biology ◽  
Human Genome ◽  
Genome Sequencing ◽  
High Throughput ◽  
Applied Mathematics ◽  
Biological Research ◽  
Development Environment ◽  
Sequencing Project ◽  
Human Genome Sequencing

Bioinformatics is a new, rapidly expanding field that uses computational approaches to answer biological questions (Baxevanis, 2005). These questions are answered by means of analyzing and mining biological data. The field of bioinformatics or computational biology is a multidisciplinary research and development environment, in which a variety of techniques from computer science, applied mathematics, linguistics, physics, and, statistics are used. The terms bioinformatics and computational biology are often used interchangeably (Baldi, 1998; Pevzner, 2000). This new area of research is driven by the wealth of data from high throughput genome projects, such as the human genome sequencing project (International Human Genome Sequencing Consortium, 2001; Venter, 2001). As of early 2006, 180 organisms have been sequenced, with the capacity to sequence constantly increasing. Three major DNA databases collaborate and mirror over 100 billion base pairs in Europe (EMBL), Japan (DDBJ) and the USA (Genbank.) The advent of high throughput methods for monitoring gene expression, such as microarrays (Schena, 1995) detecting the expression level of thousands of genes simultaneously. Such data can be utilized to establish gene function (functional genomics) (DeRisi, 1997). Recent advances in mass spectrometry and proteomics have made these fields high-throughput. Bioinformatics is an essential part of drug discovery, pharmacology, biotechnology, genetic engineering and a wide variety of other biological research areas. In the context of these proceedings, we emphasize that machine learning approaches, such as neural networks, hidden Markov models, or kernel machines, have emerged as good mathematical methods for analyzing (i.e. classifying, ranking, predicting, estimating and finding regularities on) biological datasets (Baldi, 1998). The field of bioinformatics has presented challenging problems to the machine learning community and the algorithms developed have resulted in new biological hypotheses. In summary, with the huge amount of information a mutually beneficial knowledge feedback has developed between theoretical disciplines and the life sciences. As further reading, we recommend the excellent “Bioinformatics: A Machine Learning Approach” (Baldi, 1998), which gives a thorough insight into topics, methods and common problems in Bioinformatics. The next section introduces the most important subfields of bioinformatics and computational biology. We go on to discuss current issues in bioinformatics and what we see are future trends.

scholarly journals Integrating machine learning and multiscale modeling—perspectives, challenges, and opportunities in the biological, biomedical, and behavioral sciences

2019 ◽  
Vol 2 (1) ◽  
Author(s):  
Mark Alber ◽  
Adrian Buganza Tepole ◽  
William R. Cannon ◽  
Suvranu De ◽  
Salvador Dura-Bernal ◽  
...  
Keyword(s):  
Machine Learning ◽  
Differential Equations ◽  
Human Health ◽  
Multiscale Modeling ◽  
Applied Mathematics ◽  
Treatment Strategies ◽  
Behavioral Sciences ◽  
Open Questions ◽  
Challenges And Opportunities ◽  
Ill Posed

AbstractFueled by breakthrough technology developments, the biological, biomedical, and behavioral sciences are now collecting more data than ever before. There is a critical need for time- and cost-efficient strategies to analyze and interpret these data to advance human health. The recent rise of machine learning as a powerful technique to integrate multimodality, multifidelity data, and reveal correlations between intertwined phenomena presents a special opportunity in this regard. However, machine learning alone ignores the fundamental laws of physics and can result in ill-posed problems or non-physical solutions. Multiscale modeling is a successful strategy to integrate multiscale, multiphysics data and uncover mechanisms that explain the emergence of function. However, multiscale modeling alone often fails to efficiently combine large datasets from different sources and different levels of resolution. Here we demonstrate that machine learning and multiscale modeling can naturally complement each other to create robust predictive models that integrate the underlying physics to manage ill-posed problems and explore massive design spaces. We review the current literature, highlight applications and opportunities, address open questions, and discuss potential challenges and limitations in four overarching topical areas: ordinary differential equations, partial differential equations, data-driven approaches, and theory-driven approaches. Towards these goals, we leverage expertise in applied mathematics, computer science, computational biology, biophysics, biomechanics, engineering mechanics, experimentation, and medicine. Our multidisciplinary perspective suggests that integrating machine learning and multiscale modeling can provide new insights into disease mechanisms, help identify new targets and treatment strategies, and inform decision making for the benefit of human health.

scholarly journals Non-monotone DR-submodular Maximization over General Convex Sets

2020 ◽  
Author(s):  
Christoph Dürr ◽  
Nguyen Kim Thang ◽  
Abhinav Srivastav ◽  
Léo Tible
Keyword(s):  
Machine Learning ◽  
Real World ◽  
Communication Systems ◽  
Convex Sets ◽  
Convex Set ◽  
Applied Mathematics ◽  
Submodular Functions ◽  
General Convex ◽  
Real World Datasets ◽  
Approximation Guarantee

Many real-world problems can often be cast as the optimization of DR-submodular functions defined over a convex domain. These functions play an important role with applications in many areas of applied mathematics, such as machine learning, computer vision, operation research, communication systems or economics. In addition, they capture a subclass of non-convex optimization that provides both practical and theoretical guarantees. In this paper, we show that for maximizing non-monotone DR-submodular functions over a general convex set (such as up-closed convex sets, conic convex set, etc) the Frank-Wolfe algorithm achieves an approximation guarantee which depends on the convex set. To the best of our knowledge, this is the first approximation guarantee. Finally we benchmark our algorithm on problems arising in machine learning domain with the real-world datasets.

scholarly journals Capturing dynamics of time-varying data via topology

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lu Xian ◽  
Henry Adams ◽  
Chad M. Topaz ◽  
Lori Ziegelmeier
Keyword(s):  
Machine Learning ◽  
Algebraic Topology ◽  
Metric Spaces ◽  
Applied Mathematics ◽  
Topological Data Analysis ◽  
Complex Data ◽  
Time Varying ◽  
Identification Task ◽  
Static Data ◽  
Biological Aggregations

<p style='text-indent:20px;'>One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time. A time-varying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic behavior. We provide an introduction to topological summaries of time-varying metric spaces including vineyards [<xref ref-type="bibr" rid="b19">19</xref>], crocker plots [<xref ref-type="bibr" rid="b55">55</xref>], and multiparameter rank functions [<xref ref-type="bibr" rid="b37">37</xref>]. We then introduce a new tool to summarize time-varying metric spaces: a <i>crocker stack</i>. Crocker stacks are convenient for visualization, amenable to machine learning, and satisfy a desirable continuity property which we prove. We demonstrate the utility of crocker stacks for a parameter identification task involving an influential model of biological aggregations [<xref ref-type="bibr" rid="b57">57</xref>]. Altogether, we aim to bring the broader applied mathematics community up-to-date on topological summaries of time-varying metric spaces.</p>

Templates, Opacity, and Simulations

2019 ◽  
pp. 61-80
Author(s):  
Paul Humphreys
Keyword(s):  
Machine Learning ◽  
Computer Simulations ◽  
Numerical Experiments ◽  
Applied Mathematics ◽  
Philosophy Of Mathematics ◽  
Weak Sense ◽  
Computational Science ◽  
Simulation Methods ◽  
Numerical Experimentation ◽  
Definition Of

Retrospective reflections are provided on the papers “Computer Simulations,” “Computational Science and Its Effects,” “The Philosophical Novelty of Computer Simulation Methods,” and “Numerical Experimentation” by Paul Humphreys. Some major themes are that it is the broader category of computational science, including such methods as machine learning, that is of interest, rather than just the narrower field of computer simulations; that numerical experiments and simulations are only analogous in a very weak sense to laboratory experiments; that computational science is a genuine emplacement revolution; and that syntax is of primary importance in computational modeling. Remarks are made on the logical properties of simulations, on the appropriate definition of a simulation, and on the need to take applied mathematics seriously as an autonomous field of study in the philosophy of mathematics. An argument is given for the conclusion that computational transformations preserve the causal origins of data but not their referential content.

Machine Learning for Signal Processing

2019 ◽  
Author(s):  
Max A. Little
Keyword(s):  
Machine Learning ◽  
Signal Processing ◽  
Applied Mathematics ◽  
Digital Signal ◽  
Statistical Modelling ◽  
Modern World ◽  
Statistical Machine Learning ◽  
Contemporary World ◽  
Mp3 Players ◽  
Intelligent Information

Digital signal processing (DSP) is one of the ‘foundational’ engineering topics of the modern world, without which technologies such the mobile phone, television, CD and MP3 players, WiFi and radar, would not be possible. A relative newcomer by comparison, statistical machine learning is the theoretical backbone of exciting technologies such as automatic techniques for car registration plate recognition, speech recognition, stock market prediction, defect detection on assembly lines, robot guidance and autonomous car navigation. Statistical machine learning exploits the analogy between intelligent information processing in biological brains and sophisticated statistical modelling and inference. DSP and statistical machine learning are of such wide importance to the knowledge economy that both have undergone rapid changes and seen radical improvements in scope and applicability. Both make use of key topics in applied mathematics such as probability and statistics, algebra, calculus, graphs and networks. Intimate formal links between the two subjects exist and because of this many overlaps exist between the two subjects that can be exploited to produce new DSP tools of surprising utility, highly suited to the contemporary world of pervasive digital sensors and high-powered and yet cheap, computing hardware. This book gives a solid mathematical foundation to, and details the key concepts and algorithms in, this important topic.

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