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For a function $f: \mathbb{R}^n \to \mathbb{R}$, I am looking for the definition of $f$ to be unimodal.

  1. From Wikipedia

    $f$ is unimodal if there is a one to one differentiable mapping $X = G(Z)$ such that $f(G(Z))$ is convex.

  2. Also from Wikipedia:

    the super-level sets $L(f, t)$ of $f$, defined by $$ L(f, t) = \{ x \in \mathbb{R}^{n} | f(x) \geq t \}, $$ are convex subsets of $\mathbb{R}^n$ for every $t ≥ 0$. (This property is sometimes referred to as being unimodal.)

  3. Added: By the comment below, a common definition is that a function is unimodal, if it has exactly one local maximum.

    I wonder if this common definition allows existence of more than one local minimum? For example a "W" shape function defined on $\mathbb{R}$ goes to $\infty$ when approaching $\pm \infty$ in domain.

    If this definition does not allow existence of any local minimum, then for any line through the mode $m \in \mathbb{R}^n$ of $f$ in its domain, does the restriction of $f$ to this line increase on one side of $m$ and decrease on the other side of $m$?

I wonder if the first two definitions agree with each other? If not, when?

What is your definition, if possible?

Thanks and regards!

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I suspect that Definition 1 has a typo: "$f(G(Z))$ is concave" would be more appropriate in the context of unimodality. Either way, the definitions are rather different. Personally, I prefer to use "unimodal" in the sense "has exactly one point of local maximum". But you are free to use it in any sense you wish as long as you precisely state your definition. And if your definition of "unimodal" is uncommon, you should pay it extra. – user31373 May 25 '12 at 2:27
I would concur with @LeonidKovalev; albeit the sense I have seen it used (line search) was having exactly one local minimum. – copper.hat May 25 '12 at 3:39
@copper.hat: Thanks! (1) I wonder if you mean having exactly one local maximum, instead of having exactly one local minimum, for definition of unimodality? (2) If it is maximum in (1), does having exactly one local maximum allows having more than one local minimum? – Tim May 25 '12 at 13:38
My only exposure to the term was with line search algorithms for optimization. It was a convenient assumption (one local min.) to prove some convergence properties. – copper.hat May 25 '12 at 15:10
@copper.hat: Does that definition you were exposed to allow existence of local maximum(s)? – Tim May 25 '12 at 16:00

I recently became interested in this definition in trying to make rigorous my answer to this Question.

In the univariate case a notion of unimodal function is generalized from that of a unimodal distribution, which qualitatively is described as a density function "having one mode and one peak".

Project Euclid hosts a paper Multivariate Unimodality by S. W. Dharmadhikari and Kumar Jogdeo (1976) which reviews the literature to that point and presents some alternative definitions. I was interested to see these authors credit A. Khinchin with defining the unimodal case in terms of a point to the left of which the cumulative distribution function is convex and to the right concave.

In view of the comments offered previously, a focus on unique local maximum is at least "somewhat unnatural". A desirable definition should entail an isolated global maximum, "the vertex of unimodality" to borrow a phrase from the above linked paper, so that paths leading to the mode produce monotone increasing values of the function.

The paper compares three definitions from the literature and summarizes what was known at the time about their equivalence or inequivalence. A more recent treatment is this technical report from 1988.

If we generalize away from distributions to functions, there are a number of issues to make the definition more difficult, e.g. strict monotonicity or weak? In some applications it is possible to avoid difficulties by requiring e.g. radial symmetry, thereby reducing the question back to a one-dimensional aspect, or by integrating over symmetric convex sets.

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