Projection

1 Definitions

1.1 Distance between point and set

The distance d from point \mathbf{y} \in \mathbb{R}^n to closed set S \subset \mathbb{R}^n:

d(\mathbf{y}, S, \| \cdot \|) = \inf\{\|x - y\| \mid x \in S \}

1.2 Projection of a point on set

Projection of a point \mathbf{y} \in \mathbb{R}^n on set S \subseteq \mathbb{R}^n is a point \pi_S(\mathbf{y}) \in S:

\| \pi_S(\mathbf{y}) - \mathbf{y}\| \le \|\mathbf{x} - \mathbf{y}\|, \forall \mathbf{x} \in S

  • if a set is open, and a point is beyond this set, then its projection on this set does not exist.

  • if a point is in set, then its projection is the point itself

  • \pi_S(\mathbf{y}) = \underset{\mathbf{x}}{\operatorname{argmin}} \|\mathbf{x}-\mathbf{y}\|

  • Let S \subseteq \mathbb{R}^n - convex closed set. Let the point \mathbf{y} \in \mathbb{R}^n и \mathbf{\pi} \in S. Then if for all \mathbf{x} \in S the inequality holds:

    \langle \pi -\mathbf{y}, \mathbf{x} - \pi\rangle \ge 0,

    then \pi is the projection of the point \mathbf{y} on S, so \pi_S (\mathbf{y}) = \pi.

  • Let S \subseteq \mathbb{R}^n - affine set. Let we have points \mathbf{y} \in \mathbb{R}^n and \mathbf{\pi} \in S. Then \pi is a projection of point \mathbf{y} on S, so \pi_S (\mathbf{y}) = \pi if and only if for all \mathbf{x} \in S the inequality holds:

\langle \pi -\mathbf{y}, \mathbf{x} - \pi\rangle = 0

  • Sufficient conditions of existence of a projection. If S \subseteq \mathbb{R}^n - closed set, then the projection on set S exists for any point.
  • Sufficient conditions of uniqueness of a projection. If S \subseteq \mathbb{R}^n - closed convex set, then the projection on set S is unique for any point.
Example

Find \pi_S (y) = \pi, if S = \{x \in \mathbb{R}^n \mid \|x - x_0\| \le R \}, y \notin S

Figure 1: Projection of point to the ball
  • Build a hypothesis from the figure: \pi = x_0 + R \cdot \frac{y - x_0}{\|y - x_0\|}

  • Check the inequality for a convex closed set: (\pi - y)^T(x - \pi) \ge 0

    \left( x_0 - y + R \frac{y - x_0}{\|y - x_0\|} \right)^T\left( x - x_0 - R \frac{y - x_0}{\|y - x_0\|} \right) =

    \left( \frac{(y - x_0)(R - \|y - x_0\|)}{\|y - x_0\|} \right)^T\left( \frac{(x-x_0)\|y-x_0\|-R(y - x_0)}{\|y - x_0\|} \right) =

    \frac{R - \|y - x_0\|}{\|y - x_0\|^2} \left(y - x_0 \right)^T\left( \left(x-x_0\right)\|y-x_0\|-R\left(y - x_0\right) \right) =

    \frac{R - \|y - x_0\|}{\|y - x_0\|} \left( \left(y - x_0 \right)^T\left( x-x_0\right)-R\|y - x_0\| \right) =

    \left(R - \|y - x_0\| \right) \left( \frac{(y - x_0 )^T( x-x_0)}{\|y - x_0\|}-R \right)

  • The first factor is negative for point selection y. The second factor is also negative, which follows from the Cauchy-Bunyakovsky inequality:

    (y - x_0 )^T( x-x_0) \le \|y - x_0\|\|x-x_0\|

    \frac{(y - x_0 )^T( x-x_0)}{\|y - x_0\|} - R \le \frac{\|y - x_0\|\|x-x_0\|}{\|y - x_0\|} - R = \|x - x_0\| - R \le 0

Example

Find \pi_S (y) = \pi, if S = \{x \in \mathbb{R}^n \mid c^T x = b \}, y \notin S.

Figure 2: Projection of point to the ball
  • Build a hypothesis from the figure: \pi = y + \alpha c. Coefficient \alpha is chosen so that \pi \in S: c^T \pi = b, so:

    c^T (y + \alpha c) = b

    c^Ty + \alpha c^T c = b

    c^Ty = b - \alpha c^T c

  • Check the inequality for a convex closed set: (\pi - y)^T(x - \pi) \ge 0

    (y + \alpha c - y)^T(x - y - \alpha c) =

    \alpha c^T(x - y - \alpha c) =

    \alpha (c^Tx) - \alpha (c^T y) - \alpha^2 (c^Tc) =

    \alpha b - \alpha (b - \alpha c^T c) - \alpha^2 c^Tc =

    \alpha b - \alpha b + \alpha^2 c^T c - \alpha^2 c^Tc = 0 \ge 0

Example

Find \pi_S (y) = \pi, if S = \{x \in \mathbb{R}^n \mid Ax = b, A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^{m} \}, y \notin S.

Figure 3: Projection of point to the set of linear equations
  • Build a hypothesis from the figure: \pi = y + \sum\limits_{i=1}^m\alpha_i A_i = y + A^T \alpha. Coefficient \alpha is chosen so that \pi \in S: A \pi = b, so:

    A(y + A^T\alpha) = b

    Ay = b - A A^T\alpha

  • Check the inequality for a convex closed set: (\pi - y)^T(x - \pi) \ge 0

    (y + A^T\alpha - y)^T(x - y - A^T\alpha) =

    \alpha^T A(x - y - A^T\alpha) =

    \alpha^T (Ax) - \alpha^T (A y) - \alpha^T (AA^T \alpha) =

    \alpha^T b - \alpha^T (b - A A^T\alpha) - \alpha^T AA^T \alpha =

    \alpha^T b - \alpha^T b + \alpha^T AA^T \alpha - \alpha^T AA^T \alpha = 0 \ge 0