Level set

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Points at constant slices of x2 = f(x1).
Lines at constant slices of x3 = f(x1, x2).
Planes at constant slices of x4 = f(x1, x2, x3).
(n − 1)-dimensional level sets for functions of the form f(x1, x2, ..., xn) = a1x1 + a2x2 + ... + anxn where a1, a2, ..., an are constants, in (n + 1)-dimensional Euclidean space, for n = 1, 2, 3.
Points at constant slices of x2 = f(x1).
Contour curves at constant slices of x3 = f(x1, x2).
Curved surfaces at constant slices of x4 = f(x1, x2, x3).
(n − 1)-dimensional level sets of non-linear functions f(x1, x2, ..., xn) in (n + 1)-dimensional Euclidean space, for n = 1, 2, 3.

In mathematics, a level set of a real-valued function of n real variables f is a set of the form

 L_c(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) = c \right\}~,

that is, a set where the function takes on a given constant value c.

When the number of variables is two, a level set is generically a curve, called a level curve, contour line, or isoline. So a level curve is the set of all real-valued solutions of an equation in two variables x1 and x2. When n = 3, a level set is called a level surface (see also isosurface), and for higher values of n the level set is a level hypersurface. So a level surface is the set of all real-valued roots of an equation in three variables x1, x2 and x3, and a level hypersurface is the set of all real-valued roots of an equation in n (n > 3) variables.

A level set is a special case of a fiber.


Alternative names[edit]

Intersections of a co-ordinate function's level surfaces with a trefoil knot. Red curves are closest to the viewer, while yellow curves are farthest.

Level sets show up in great many applications, often under different names.

For example, a level curve is also called an implicit curve, emphasizing that such a curve is defined by an implicit function. The name isocontour is also used, which means a contour of equal height. In various applications, isobars, isotherm, isogons and isochrones are isocontours.

Analogously, a level surface is sometimes called an implicit surface or an isosurface.

Example[edit]

For example, given a specific radius r, the equation of a circle defines an isocontour.

r^2=x^2 + y^2

If we choose r=5 then our isovalue is c=5^2=25.

All points (x,y) that evaluate to 25 constitute the isocontour. This means that they are a member of the isocontour's level set. If a point evaluates to less than 25 the point is on the inside of the isocontour. If the result is greater than 25, it is on the outside.

Level sets versus the gradient[edit]

Consider a function f whose graph looks like a hill. The blue curves are then the level sets. The red curves follow the direction of the gradient. In other words, the cautious hiker follows the blue paths, while the bold one the red paths.

Theorem. The gradient of f at a point is perpendicular to the level set of f at that point.

To understand what this means, imagine that two hikers are at the same location on a mountain. One of them is bold, and decides to go in the direction where the slope is steepest. The other one is more cautious; he does not want to either climb or descend, choosing a path which will keep him at the same height. In our analogy, the above theorem says that the two hikers will depart in directions perpendicular to one another.

Proof. Let \mathbf x_0 be the point of interest and f(x_0)=c. The level set going through \mathbf x_0 is \Phi_c = \{ \mathbf x  : f(\mathbf x) =c \}. Consider for some \delta>0 a curve \mathbf \gamma(t):(-\delta,\delta)\to \Phi_c such that \mathbf \gamma(0)= \mathbf x_0 . We have

\forall t\in (-\delta,\delta): f(\mathbf \gamma(t)) = c .

Now let us differentiate at t = 0 by using the chain rule. We find

J_f({\mathbf x_0}) {\mathbf \gamma}'(0)=0.

Equivalently, the Jacobian of f at x0 is the gradient at x0

\nabla f({\mathbf x}_0) \cdot {\mathbf \gamma}'(0)=0.

Thus, the gradient of f at x0 is perpendicular to the tangent γ′(0) to the curve (and to the level set) at that point. Since the curve γ(t) is arbitrary, it follows that the gradient is perpendicular to the level set. Q.E.D.

A consequence of this theorem is that if a level set crosses itself (more precisely, fails to be a smooth submanifold or hypersurface) then the gradient vector must be zero at all points of crossing. Then, every point in the crossing will be a critical point of f.

Sublevel and superlevel sets[edit]

A set of the form

 L_c^-(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) \leq c \right\}

is called a sublevel set of f (or, alternatively, a lower level set or trench of f).

 L_c^+(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) \geq c \right\}

is called a superlevel set of f.[1][2] Sublevel sets are important in minimization theory. The boundness of some non-empty sublevel set and the lower-semicontinuity of the function implies that a function attains its minimum, by Weierstrass's theorem. The convexity of all the sublevel sets characterizes quasiconvex functions.[3]


See also[edit]

References[edit]

  1. ^ Voitsekhovskii, M.I. (2001), "L/l058220", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
  2. ^ Weisstein, Eric W., "Level Set", MathWorld.
  3. ^ Kiwiel, Krzysztof C. (2001). "Convergence and efficiency of subgradient methods for quasiconvex minimization". Mathematical Programming (Series A) 90 (1) (Berlin, Heidelberg: Springer). pp. 1–25. doi:10.1007/PL00011414. ISSN 0025-5610. MR 1819784.