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Softplus
In mathematics and machine learning, the softplus function is : f(x) = \log(1 + e^x). It is a smooth approximation (in fact, an analytic function) to the ramp function, which is known as the ''rectifier'' or ''ReLU (rectified linear unit)'' in machine learning. For large negative x it is \log(1 + e^x) = \log (1 + \epsilon) \gtrapprox \log 1 = 0, so just above 0, while for large positive x it is \log(1 + e^x) \gtrapprox \log(e^x) = x, so just above x. The names ''softplus'' and ''SmoothReLU'' are used in machine learning. The name "softplus" (2000), by analogy with the earlier softmax (1989) is presumably because it is a smooth (''soft'') approximation of the positive part of , which is sometimes denoted with a superscript ''plus'', x^+ := \max(0, x). Related functions The derivative of softplus is the logistic function: :f'(x) = \frac = \frac The logistic function or the sigmoid function is a smooth approximation of the rectifier, the Heaviside step function. LogSumExp The mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectifier (neural Networks)
In the context of Neural network (machine learning), artificial neural networks, the rectifier or ReLU (rectified linear unit) activation function is an activation function defined as the non-negative part of its argument, i.e., the ramp function: :\operatorname(x) = x^+ = \max(0, x) = \frac = \begin x & \text x > 0, \\ 0 & x \le 0 \end where x is the input to a Artificial neuron, neuron. This is analogous to half-wave rectification in electrical engineering. ReLU is one of the most popular activation functions for artificial neural networks, and finds application in computer vision and speech recognitionAndrew L. Maas, Awni Y. Hannun, Andrew Y. Ng (2014)Rectifier Nonlinearities Improve Neural Network Acoustic Models using Deep learning, deep neural nets and computational neuroscience. History The ReLU was first used by Alston Scott Householder, Alston Householder in 1941 as a mathematical abstraction of biological neural networks. Kunihiko Fukushima in 1969 used R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Softplus
In mathematics and machine learning, the softplus function is : f(x) = \log(1 + e^x). It is a smooth approximation (in fact, an analytic function) to the ramp function, which is known as the ''rectifier'' or ''ReLU (rectified linear unit)'' in machine learning. For large negative x it is \log(1 + e^x) = \log (1 + \epsilon) \gtrapprox \log 1 = 0, so just above 0, while for large positive x it is \log(1 + e^x) \gtrapprox \log(e^x) = x, so just above x. The names ''softplus'' and ''SmoothReLU'' are used in machine learning. The name "softplus" (2000), by analogy with the earlier softmax (1989) is presumably because it is a smooth (''soft'') approximation of the positive part of , which is sometimes denoted with a superscript ''plus'', x^+ := \max(0, x). Related functions The derivative of softplus is the logistic function: :f'(x) = \frac = \frac The logistic function or the sigmoid function is a smooth approximation of the rectifier, the Heaviside step function. LogSumExp The mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigmoid Function
A sigmoid function is any mathematical function whose graph of a function, graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function, which is defined by the formula :\sigma(x) = \frac = \frac = 1 - \sigma(-x). Other sigmoid functions are given in the #Examples, Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as a synonym for "logistic function". Special cases of the sigmoid function include the Gompertz curve (used in modeling systems that saturate at large values of ''x'') and the ogee curve (used in the spillway of some dams). Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (''y'' axis) in the range 0 to 1. Another commonly used range is from −1 to 1. A wide variety of sigmoid functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Function
A logistic function or logistic curve is a common S-shaped curve ( sigmoid curve) with the equation f(x) = \frac where The logistic function has domain the real numbers, the limit as x \to -\infty is 0, and the limit as x \to +\infty is L. The exponential function with negated argument (e^ ) is used to define the standard logistic function, depicted at right, where L=1, k=1, x_0=0, which has the equation f(x) = \frac and is sometimes simply called the sigmoid. It is also sometimes called the expit, being the inverse function of the logit. The logistic function finds applications in a range of fields, including biology (especially ecology), biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, statistics, and artificial neural networks. There are various generalizations, depending on the field. History The logistic function was introduced in a series of three papers by Pier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Softmax
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of real numbers into a probability distribution of possible outcomes. It is a generalization of the logistic function to multiple dimensions, and is used in multinomial logistic regression. The softmax function is often used as the last activation function of a Artificial neural network, neural network to normalize the output of a network to a probability distribution over predicted output classes. Definition The softmax function takes as input a tuple of real numbers, and normalizes it into a probability distribution consisting of probabilities proportional to the exponentials of the input numbers. That is, prior to applying softmax, some tuple components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the Interval (mathematics), interval (0, 1), and the components will add up to 1, so that they can be i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoshua Bengio
Yoshua Bengio (born March 5, 1964) is a Canadian-French computer scientist, and a pioneer of artificial neural networks and deep learning. He is a professor at the Université de Montréal and scientific director of the AI institute Montreal Institute for Learning Algorithms, MILA. Bengio received the 2018 Turing Award, ACM A.M. Turing Award, often referred to as the "List of prizes known as the Nobel of a field or the highest honors of a field, Nobel Prize of Computing", together with Geoffrey Hinton and Yann LeCun, for their foundational work on deep learning. Bengio, Geoffrey Hinton, Hinton, and Yann LeCun, LeCun are sometimes referred to as the "Godfathers of AI". Bengio is the most-cited computer scientist globally (by both total citations and by h-index, ''h''-index), and the most-cited living scientist across all fields (by total citations). In 2024, Time (magazine), ''TIME'' Magazine included Bengio in its Time 100, yearly list of the world's 100 most influential people. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Loss
In machine learning and mathematical optimization, loss functions for classification are computationally feasible loss functions representing the price paid for inaccuracy of predictions in classification problems (problems of identifying which category a particular observation belongs to). Given \mathcal as the space of all possible inputs (usually \mathcal \subset \mathbb^d), and \mathcal = \ as the set of labels (possible outputs), a typical goal of classification algorithms is to find a function f: \mathcal \to \mathcal which best predicts a label y for a given input \vec. However, because of incomplete information, noise in the measurement, or probabilistic components in the underlying process, it is possible for the same \vec to generate different y. As a result, the goal of the learning problem is to minimize expected loss (also known as the risk), defined as :I = \displaystyle \int_ V(f(\vec),y) \, p(\vec,y) \, d\vec \, dy where V(f(\vec),y) is a given loss function, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentials
Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Exponential discounting, a specific form of the discount function, used in the analysis of choice over time *Exponential growth, where the growth rate of a mathematical function is proportional to the function's current value *Exponential map (Riemannian geometry), in Riemannian geometry *Exponential map (Lie theory), in Lie theory *Exponential notation, also known as scientific notation, or standard form *Exponential object, in category theory *Exponential time, in complexity theory *in probability and statistics: **Exponential distribution, a family of continuous probability distributions **Exponentially modified Gaussian distribution, describes the sum of independent Normal distribution, normal and Exponential distribution, exponential rando ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functions And Mappings
In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: ''mapping'' the Earth surface to a sheet of paper. The term ''map'' may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term ''transformation'' can be used interchangeably, but '' transformation'' often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. Maps as functions In many branches of mathematics, the term ''map'' is used to mean a function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous function" in topology, a "linear transformation" in linear algebra, etc. So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artificial Neural Networks
In machine learning, a neural network (also artificial neural network or neural net, abbreviated ANN or NN) is a computational model inspired by the structure and functions of biological neural networks. A neural network consists of connected units or nodes called '' artificial neurons'', which loosely model the neurons in the brain. Artificial neuron models that mimic biological neurons more closely have also been recently investigated and shown to significantly improve performance. These are connected by ''edges'', which model the synapses in the brain. Each artificial neuron receives signals from connected neurons, then processes them and sends a signal to other connected neurons. The "signal" is a real number, and the output of each neuron is computed by some non-linear function of the sum of its inputs, called the '' activation function''. The strength of the signal at each connection is determined by a ''weight'', which adjusts during the learning process. Typically, neur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Regression
In statistics, a logistic model (or logit model) is a statistical model that models the logit, log-odds of an event as a linear function (calculus), linear combination of one or more independent variables. In regression analysis, logistic regression (or logit regression) estimation theory, estimates the parameters of a logistic model (the coefficients in the linear or non linear combinations). In binary logistic regression there is a single binary variable, binary dependent variable, coded by an indicator variable, where the two values are labeled "0" and "1", while the independent variables can each be a binary variable (two classes, coded by an indicator variable) or a continuous variable (any real value). The corresponding probability of the value labeled "1" can vary between 0 (certainly the value "0") and 1 (certainly the value "1"), hence the labeling; the function that converts log-odds to probability is the logistic function, hence the name. The unit of measurement for the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
