There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

1 Mean-field theory of phase transitions 2 Mean-field theory of spin glasses 3 Replica symmetry breaking 4 Gauge theory of spin glasses 5 Error-correcting codes 6 Image restoration 7 Associative memory 8 Learning in perceptron 9 Optimization problems A Eigenvalues of the Hessian B Parisi equation C Channel coding theorem D Distribution and free energy of K-Sat References Index

Statistical physics of spin glasses and information processing : an introduction

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2018WR022643

A Transdisciplinary Review of Deep Learning Research and Its Relevance for Water Resources Scientists

Deep Learning is attracting much attention in object recognition and speech processing. A benefit of using the deep learning is that it provides automatic pre-training. Several proposed methods that include auto-encoder are being successfully used in various applications. Moreover, deep learning uses a multilayer network that consists of many layers, a huge number of units, and huge amount of data. Thus, executing deep learning requires heavy computation, so deep learning is usually utilized with parallel computation with many cores or many machines. Deep learning employs the gradient algorithm, however this traps the learning into the saddle point or local minima. To avoid this difficulty, a rectified linear unit (ReLU) is proposed to speed up the learning convergence. However, the reasons the convergence is speeded up are not well understood. In this paper, we analyze the ReLU by a using simpler network called the soft-committee machine and clarify the reason for the speedup. We also train the network in an on-line manner. The soft-committee machine provides a good test bed to analyze deep learning. The results provide some reasons for the speedup of the convergence of the deep learning.

Analysis of function of rectified linear unit used in deep learning

Deep learning is the state-of-the-art in fields such as visual object recognition and speech recognition. This learning uses a large number of layers, huge number of units, and connections. Therefore, overfitting is a serious problem. To avoid this problem, dropout learning is proposed. Dropout learning neglects some inputs and hidden units in the learning process with a probability, p, and then, the neglected inputs and hidden units are combined with the learned network to express the final output. We find that the process of combining the neglected hidden units with the learned network can be regarded as ensemble learning, so we analyze dropout learning from this point of view.

Analysis of Dropout Learning Regarded as Ensemble Learning

Ensemble learning of K nonlinear perceptrons, which determine their outputs by sign functions, is discussed within the framework of online learning and statistical mechanics. One purpose of statistical learning theory is to theoretically obtain the generalization error. This paper shows that ensemble generalization error can be calculated by using two order parameters, that is, the similarity between a teacher and a student, and the similarity among students. The differential equations that describe the dynamical behaviors of these order parameters are derived in the case of general learning rules. The concrete forms of these differential equations are derived analytically in the cases of three well-known rules: Hebbian learning, perceptron learning, and AdaTron (adaptive perceptron) learning. Ensemble generalization errors of these three rules are calculated by using the results determined by solving their differential equations. As a result, these three rules show different characteristics in their affinity for ensemble learning, that is "maintaining variety among students." Results show that AdaTron learning is superior to the other two rules with respect to that affinity.

Analysis of ensemble learning using simple perceptrons based on online learning theory.

This paper discusses properties of activation functions in multilayer neural network applied to pattern classification. A rule of thumb for selecting activation functions or their combination is proposed. The sigmoid, Gaussian and sinusoidal functions are selected due to their independent and fundamental space division properties. The sigmoid function is not effective for a single hidden unit. On the contrary, the other functions can provide good performance. When several hidden units are employed, the sigmoid function is useful. However, the convergence speed is still slower than the others. The Gaussian function is sensitive to the additive noise, while the others are rather insensitive. As a result, based on convergence rates, the minimum error and noise sensitivity, the sinusoidal function is most useful for both with and without additive noise. The property of each function is discussed based on the internal representation, that is the distribution of the hidden unit inputs and outputs. Although this selection depends on the input signals to be classified, the periodic function can be effectively applied to a wide range of application fields. >

Comparison of activation functions in multilayer neural network for pattern classification

We analyze ensemble learning including the noisy case where teacher or student noise is present. Linear perceptrons are used as teacher and student. First, we analyze the homogeneous correlation of...

Kazuyuki Hara

Papers

Analysis of function of rectified linear unit used in deep learning

Analysis of Dropout Learning Regarded as Ensemble Learning

Analysis of ensemble learning using simple perceptrons based on online learning theory.

Comparison of activation functions in multilayer neural network for pattern classification

Ensemble Learning of Linear Perceptrons: On-Line Learning Theory