Kernel Characterization of an Interval Function

Solving equations and optimization problems with uncertainty

“…The right-hand side of (7), v(x), is a cocycle that does not depend on k (assuming k > j). The left-hand side is a combination of coboundaries of characteristic cocycles of n-simplices ∆ n and cochains of the form v(w) for (n − 1)-cocycles w. To each ∆ and w is assigned a filtration value and we want to minimize the value r k such that v(x) can be expressed as a combination of δc's and v(w)'s such that c and w have filtration values ≤ r k , but cannot be expressed as a combination of such cochains with filtrations of c and w strictly smaller than r k .…”

“…Zero of f arithmetic and detection of zeros resistant to bounded errors is a frequent problem in this field [32,22,3,35,7]. Other instances of uncertain functions come from measurements of physical quantities, such as in medical imaging [12,33,21] or robotics [43,6].…”

“…Other instances of uncertain functions come from measurements of physical quantities, such as in medical imaging [12,33,21] or robotics [43,6]. We suppose that potential applications include robust detection of level sets f −1 (a) in medical image processing, analysing robot trajectory based on data obtained from sensors [6,7], or computing the inner approximation of reachable regions of a robotic arm [34]. The algorithm could also be exploited for analysis of functions obtained by regression (say, in machine learning), where the function is chosen to fit some given set of sampled values.…”

“…Our research is motivated by practical applications, in which the data is often known only approximately. We address the case where the input data is limited to the approximate values of a continuous function f arithmetic and detection of zeros resistant to bounded errors is a frequent problem in this field [32,22,3,35,7]. Other instances of uncertain functions come from measurements of physical quantities, such as in medical imaging [12,33,21] or robotics [43,6].…”

“…To avoid further simplicial subdivisions, we again need to assume that r > αn 1/p , i.e., that the description of f is fine-grained enough for the retrieval of the homotopy class of f | Ar ) 7. An upper bound is obtained when the dimension is at most n or n ≤ 2 for the primary obstruction and at most n + 1 for the secondary obstruction.…”