Logarithmically concave function: Difference between revisions

Revision as of 08:29, 3 February 2010

Revision as of 08:43, 25 March 2010

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A function <math>f : \R^n \to \R^+</math> is '''logarithmically concave''' (or '''log-concave''' for short), if its [[natural logarithm]] <math>\ln(f(x))</math>, is [[concave function|concave]]. This means that it must be:

:<math>

f( \theta x + (1 - \theta) y )

\geq

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</math>

Note that we allow here concave functions to take value <math>-\infty</math>. Every concave function is log-concave, however the reverse does not necessarily hold: an example is the function <math>e^{-x^2}</math> which is log-concave (<math>-x^2</math> is a concave function of <math>x</math>) but is not concave for <math>|x| > 1/\sqrt{2}</math>.

Examples of log-concave functions are the [[indicator function]]s of convex sets and the [[Gaussian function]].

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A log-concave function is also [[Quasi-concave_function | quasi-concave]].

==Properties==

* Every concave function is log-concave, however the reverse does not necessarily hold. An example is the function

:<math>f(x) = e^{-x^2 / 2}</math>

which is log-concave since

:<math>\log\,f(x)=-x^2 / 2</math>

is a concave function of <math>x</math>. But <math>f</math> is not concave since the second derivative is positive for <math>|x| > 1</math>.

:<math>f''(x)=e^{-\frac{x^2}{2}} (x^2-1) \nleq 0</math>

==Operations preserving the log-concavity==

* A twice differentiable function with convex ___domain is log-concave if and only if for all <math>x\in\operatorname{dom} f</math>

* product (the product of log-concave functions is a log-concave function. Notice this is *not* true for the sum of log-convex functions)

:<math>f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T</math> <ref> Stephen Boyd and Lieven Vandenberghe, [http://www.stanford.edu/~boyd/cvxbook/ Convex Optimization] (PDF) p.105</ref>

* integration:

==Operations preserving the log-concavity==

<math>

f(x,y) : \mathcal{R}^{n+m} \rightarrow \mathcal{R} \;\; \text{is log concave} \;\;

* Product (The product of log-concave functions is a log-concave function. Notice this is *not* true for the sum of log-concave functions.) If <math>f</math> and <math>g</math> are log-concave functions, <math>\log \,f(x)</math> and <math>\log\,g(x)</math> are concave by definition. Concavity is preserved under non-negative weighted sums, so

\Rightarrow

:<math>\int log\,f(x,y) dy+ \;log\;,g(x) \text{is= \log(f(x)g(x))</math> concave}

is concave, and therefore <math>f(x)g(x)</math> is log-concave.

</math>

* Integration in special cases:

This implies that [[convolution]] is an operation preserving log-concavity.

If <math>f(x,y) : \mathcal{R}^{n+m} \rightarrow \mathcal{R}</math> \;\; \text{is log -concave}, \;\;then

:<math>g(x)=\int f(x,y) dy</math>

is log-concave.

This implies that [[convolution]] is an operation preserving log-concavity. since

:<math>h(x,y)=f(x-y)g(y)</math>

is log-concave if f and g are log-concave, and therefore

:<math>(f*g)(x)=\int f(x-y)g(y) dy = \int h(x,y) dy</math>

is log-concave.

==References==

{{Reflist}}

==See also==

*[[log-convex | log-convex function]]