Relaxed intersection

The ''relaxed intersection'' of ''m'' sets corresponds to the classical
intersection between sets except that it is allowed to relax few sets in order to avoid an empty intersection.
This notion can be used to solve Constraints Satisfaction Problems
that are inconsistent by relaxing a small number of constraints.
When a bounded-error approach is considered for parameter estimation,
the relaxed intersection makes it possible to be robust with respect
to some outliers.

Definition

The ''q''-relaxed intersection of the ''m'' subsets
$X_,\dots ,X_$
of $R^$,
denoted by
$X^=\bigcap^X_$
is the set of all
$x \in R^$
which belong to all
$X_$
's, except
$q$
at most.
This definition is illustrated by Figure 1.

Define
$\lambda (x) =\text \left\.$

We have

X^=\lambda ^( -q,m .

Characterizing the q-relaxed intersection is a thus a set inversion problem.

Example

Consider 8 intervals:

X_= ,4

X_=\ ,4

X_= ,7

X_= ,9

X_= ,4

X_= ,7

We have

$X^ = \emptyset,$

X^= ,4

X^= ,4

X^= ,4\cup ,7

X^= ,7

X^= ,9

$X^=]-\infty ,\infty [.$

Relaxed intersection of intervals

The relaxed intersection of intervals is not necessary an interval. We thus take
the interval hull of the result. If $X_i$'s are intervals, the relaxed
intersection can be computed with a complexity of ''m''.log(''m'') by using the
Marzullo's algorithm. It suffices to
sort all lower and upper bounds of the ''m'' intervals to represent the
function $\lambda$. Then, we easily get the set

X^=\lambda^( -q,m

which corresponds to a union of intervals.
We then return the
smallest interval which contains this union.

Figure 2 shows the function
$\lambda(x)$
associated to the previous example.

Relaxed intersection of boxes

To compute the ''q''-relaxed intersection of ''m'' boxes of
$R^$, we project all ''m'' boxes with respect to the ''n'' axes.
For each of the ''n'' groups of ''m'' intervals, we compute the ''q''-relaxed intersection.
We return Cartesian product of the ''n'' resulting intervals.

Figure 3 provides an
illustration of the 4-relaxed intersection of 6 boxes. Each point of the
red box belongs to 4 of the 6 boxes.

Relaxed union

The ''q''-relaxed union of $X_1,\dots ,X_m$ is defined by

$\oversetX_=\bigcap^X_i$

Note that when ''q''=0, the relaxed union/intersection corresponds to
the classical union/intersection. More precisely, we have

$\bigcap^X_ =\bigcap X_i$

and

$\oversetX_ =\bigcup X_i$

De Morgan's law

If $\overline$ denotes the complementary set of $X_i$, we have

$\overline = \overset\overline$

$\overline=\bigcap^\overline.$

As a consequence

$\overline=\overline=\bigcap^\overline$

Relaxation of contractors

Let $C_1,\dots ,C_m$ be ''m'' contractors for the sets $X_1,\dots ,X_m$,
then

C( =\bigcap^C_i( .

is a contractor for $X^$
and

\overline( =\bigcap^\overline_i(

is a contractor for $\overline^$, where

$\overline_1,\dots,\overline_$

are contractors for

$\overline_1,\dots ,\overline_m.$

Combined with a branch-and-bound algorithm such as SIVIA (Set Inversion Via Interval Analysis), the ''q''-relaxed
intersection of ''m'' subsets of $R^$ can be computed.

Application to bounded-error estimation

The ''q''-relaxed intersection can be used for robust localization

or for tracking.

Robust observers can also be implemented using the relaxed intersections to be robust with respect to outliers.

We propose here a simple example

to illustrate the method.
Consider a model the ''i''th model output of which is given by

$f_i(p)=\frac\exp (-\frac)$

where $p\in R^$. Assume that we have

f_i(p) \in _i

where $t_i$ and _i/math> are given by the following list

$\$

The sets $\lambda^(q)$ for different $q$ are depicted on
Figure 4.

References

{{Reflist, 2

Satisfiability problems
Estimation theory