In convex analysis, a non-negative function f: Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality
f ( θ x + ( 1 − θ ) y ) ≥ f ( x ) θ f ( y ) 1 − θ {\displaystyle f(\theta x+(1-\theta )y)\geq f(x)^{\theta }f(y)^{1-\theta }}
for all x, y ∈ dom f and 0 < θ < 1. If f is strictly positive, this is equivalent to saying that the logarithm of the function, log ∘ f, is concave; that is,
log f ( θ x + ( 1 − θ ) y ) ≥ θ log f ( x ) + ( 1 − θ ) log f ( y ) {\displaystyle \log f(\theta x+(1-\theta )y)\geq \theta \log f(x)+(1-\theta )\log f(y)}
for all x, y ∈ dom f and 0 < θ < 1. Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function. Similarly, a function is log-convex if it satisfies the reverse inequality
f ( θ x + ( 1 − θ ) y ) ≤ f ( x ) θ f ( y ) 1 − θ {\displaystyle f(\theta x+(1-\theta )y)\leq f(x)^{\theta }f(y)^{1-\theta }}
for all x, y ∈ dom f and 0 < θ < 1. For non-negative discrete functions f: Z → R+, it is log-concave if
f ( k ) 2 ≥ f ( k + 1 ) f ( k − 1 ) {\displaystyle f(k)^{2}\geq f(k+1)f(k-1)}
Properties A log-concave function is also quasi-concave. This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex. Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x2/2) which is log-concave since log f(x) = −x2/2 is a concave function of x. But f is not concave since the second derivative is positive for |x| > 1:
f ″ ( x ) = e − x 2 2 ( x 2 − 1 ) ≰ 0 {\displaystyle f''(x)=e^{-{\frac {x^{2}}{2}}}(x^{2}-1)\nleq 0}
From above two points, concavity ⇒ {\displaystyle \Rightarrow } log-concavity ⇒ {\displaystyle \Rightarrow } quasiconcavity. A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all x satisfying f(x) > 0,
f ( x ) ∇ 2 f ( x ) ⪯ ∇ f ( x ) ∇ f ( x ) T {\displaystyle f(x)\nabla ^{2}f(x)\preceq \nabla f(x)\nabla f(x)^{T}} , i.e.
f ( x ) ∇ 2 f ( x ) − ∇ f ( x ) ∇ f ( x ) T {\displaystyle f(x)\nabla ^{2}f(x)-\nabla f(x)\nabla f(x)^{T}} is negative semi-definite. For functions of one variable, this condition simplifies to
f ( x ) f ″ ( x ) ≤ ( f ′ ( x ) ) 2 {\displaystyle f(x)f''(x)\leq (f'(x))^{2}}
Operations preserving log-concavity Products: The product of log-concave functions is also log-concave. Indeed, if f and g are log-concave functions, then log f and log g are concave by definition. Therefore
log f ( x ) + log g ( x ) = log ( f ( x ) g ( x ) ) {\displaystyle \log \,f(x)+\log \,g(x)=\log(f(x)g(x))}
is concave, and hence also f g is log-concave. Marginals: if f(x, y): Rn+m → R is log-concave, then
g ( x ) = ∫ f ( x , y ) d y {\displaystyle g(x)=\int f(x,y)dy}
is log-concave (see Prékopa–Leindler inequality). This implies that convolution preserves log-concavity, since h(x, y) = f(x − y) g(y) is log-concave if f and g are log-concave, and therefore
… excerpt ends here. Continue reading the full article.
