Publication

On Perfect Functional Representations

2026

Abstract

For given random variables (X, Y), functional representation lemma states that there exists a random variable Z and a deterministic function g such that Y = g(X, Z), and X is independent of Z. In other words, Z captures all the information about Y not contained in X. We refer to Z as a ‘perfect functional representation’ if it satisfies I(Y ;Z) = H(Y |X). We show that the existence of perfect representation depends solely on the conditional distribution pY|X and define the corresponding conditional distributions as perfectly representable. We provide a necessary and sufficient condition for pY|X to be perfectly representable. Finally, we cast the problem of checking perfect representability as a linear program and discuss its computational tractability.

Official source

https://infoscience.epfl.ch/entities/publication/29e9bc54-5696-4520-b7f5-bc55f54b0df4

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2026

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/29e9bc54-5696-4520-b7f5-bc55f54b0df4

About this result

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Random variable

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' can be misleading as it is not actually random nor a variable, but rather it is a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads and tails ) in a sample space (e.g., the set ) to a measurable space (e.g., in which 1 corresponding to and −1 corresponding to ), often to the real numbers.

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In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts. Therefore, the distribution of one complex random variable may be interpreted as the joint distribution of two real random variables.

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In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individual statistical unit.

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