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Logistic regression over encrypted data from fully homomorphic encryption

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Fixed-point arithmetic

In computing, fixed-point is a method of representing fractional (non-integer) numbers by storing a fixed number of digits of their fractional part. Dollar amounts, for example, are often stored with exactly two fractional digits, representing the cents (1/100 of dollar). More generally, the term may refer to representing fractional values as integer multiples of some fixed small unit, e.g. a fractional amount of hours as an integer multiple of ten-minute intervals.

Floating-point arithmetic

In computing, floating-point arithmetic (FP) is arithmetic that represents subsets of real numbers using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. Numbers of this form are called floating-point numbers. For example, 12.345 is a floating-point number in base ten with five digits of precision: However, unlike 12.345, 12.3456 is not a floating-point number in base ten with five digits of precision—it needs six digits of precision; the nearest floating-point number with only five digits is 12.

Logistic function

A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with the equation where For values of in the domain of real numbers from to , the S-curve shown on the right is obtained, with the graph of approaching as approaches and approaching zero as approaches . 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.

Logistic regression

In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression analysis, logistic regression (or logit regression) is estimating the parameters of a logistic model (the coefficients in the linear combination).

Multinomial logistic regression

In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the probabilities of the different possible outcomes of a categorically distributed dependent variable, given a set of independent variables (which may be real-valued, binary-valued, categorical-valued, etc.).

Linear regression

In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.

Decimal floating point

Decimal floating-point (DFP) arithmetic refers to both a representation and operations on decimal floating-point numbers. Working directly with decimal (base-10) fractions can avoid the rounding errors that otherwise typically occur when converting between decimal fractions (common in human-entered data, such as measurements or financial information) and binary (base-2) fractions. The advantage of decimal floating-point representation over decimal fixed-point and integer representation is that it supports a much wider range of values.

Sigmoid function

A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve. A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: Other standard sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as an alias for the logistic function.

Logistic distribution

In probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis). The logistic distribution is a special case of the Tukey lambda distribution.

Euler method

In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).