Logarithmically concave function

A function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{+}}$ is logarithmically concave (or log-concave for short), if its natural logarithm ${\displaystyle \ln(f(x))}$, is concave. This means that it must be:

${\displaystyle f(\theta x+(1-\theta )y)\geq f(x)^{\theta }f(y)^{1-\theta }\qquad \theta \in [0,1]}$

Note that we allow here concave functions to take value -∞.

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

In parallel, a function is log-convex if its natural log is convex.

A log-concave function is also quasi-concave.

Properties

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

${\displaystyle f(x)=e^{-x^{2}/2}}$ 

which is log-concave since

${\displaystyle \log \,f(x)=-x^{2}/2}$ 

is a concave function of ${\displaystyle x}$ . But ${\displaystyle f}$  is not concave since the second derivative is positive for ${\displaystyle |x|>1}$ .

${\displaystyle f''(x)=e^{-{\frac {x^{2}}{2}}}(x^{2}-1)\nleq 0}$ 

• A twice differentiable function with convex ___domain is log-concave if and only if for all ${\displaystyle x\in \operatorname {dom} f}$  ${\displaystyle f(x)>0}$  and

${\displaystyle f(x)\nabla ^{2}f(x)\preceq \nabla f(x)\nabla f(x)^{T}}$ , ^[1]

i.e. ${\displaystyle f(x)\nabla ^{2}f(x)-\nabla f(x)\nabla f(x)^{T}}$  is negative semi-definite. If ${\displaystyle f:\mathbb {R} \to \mathbb {R} ^{+}}$ , this condition simplifies to

${\displaystyle f(x)f''(x)\leq (f'(x))^{2}}$ 

Operations preserving the log-concavity

• 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 ${\displaystyle f}$  and ${\displaystyle g}$  are log-concave functions, ${\displaystyle \log \,f(x)}$  and ${\displaystyle \log \,g(x)}$  are concave by definition. Concavity is preserved under non-negative weighted sums, so

${\displaystyle \log \,f(x)+\log \,g(x)=\log(f(x)g(x))}$ 

is concave, and therefore ${\displaystyle f(x)g(x)}$  is log-concave.

• Integration in special cases:

If ${\displaystyle f(x,y):{\mathcal {R}}^{n+m}\rightarrow {\mathcal {R}}}$  is log-concave, then

${\displaystyle g(x)=\int f(x,y)dy}$ 

is log-concave.

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

${\displaystyle h(x,y)=f(x-y)g(y)}$ 

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

${\displaystyle (f*g)(x)=\int f(x-y)g(y)dy=\int h(x,y)dy}$ 

is log-concave.

References

1. Stephen Boyd and Lieven Vandenberghe, Convex Optimization (PDF) p.105

• Barndorff-Nielsen, Ole (1978). Information and exponential families in statistical theory. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley \& Sons, Ltd. pp. ix+238 pp. ISBN 0-471-99545-2. MR 0489333

• Dharmadhikari, Sudhakar; Joag-Dev, Kumar (1988). Unimodality, convexity, and applications. Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc. pp. xiv+278. ISBN 0-12-214690-5. {{cite book}}: Cite has empty unknown parameter: |1= (help) MR 0954608

• Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. ISBN 3-11-01-3863-8. {{cite book}}: Cite has empty unknown parameter: |1= (help) MR 1291393

• Pečarić, Josip E.; Proschan, Frank; Tong, Y. L. (1992). Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering. Vol. 187. Boston, MA: Academic Press, Inc. pp. xiv+467 pp. ISBN 0-12-549250-2. {{cite book}}: Cite has empty unknown parameters: |1= and |2= (help) MR 1162312

See also

• log-convex function
• Convex function
• Logarithmically concave measure