We suggest a new optimization technique for minimizing the sum i=1^n f_i(x) of n non-convex real functions that satisfy a property that we call piecewise log-Lipschitz. This is by URLing links between techniques in computational geometry, combinatorics and convex optimization. Example applications include the first constant-factor approximation algorithms whose running-time is polynomial in n for the following fundamental problems: (i) Constrained \ell_z Linear Regression: Given z>0, n vectors p_1,\cdots,p_n on the plane, and a vector b\in\mathbb{R^n, compute a unit vector x and a permutation \pi:}%DIFDELCMD < [%%% n%DIFDELCMD < ]\to[n] %%% that minimizes \sum_{i=1^n |p_ix-b_{\pi(i)}|^z. (ii) Points-to-Lines alignment: Given n }  lines \ell_1,\cdots,\ell_n on the plane  , compute the matching \pi:[n]\to[n] and alignment (rotation matrix R and a translation vector t) that minimize the sum of Euclidean distances %DIFDELCMD < \[ i=1^n dist(Rp_i-t,\pi(i))^z \] %%% ^n \mathrm{dist}(Rp_i-t,\ell_{\pi(i)})^z } between each point to its corresponding line. These problems are open even if z=1 and the matching \pi is given. In this case , the running time of our algorithms reduces to O(n ) using core-sets that support: streaming, dynamic, and distributed parallel computations (e.g. on the cloud) in poly-logarithmic update time. Generalizations for handling e.g. outliers or pseudo-distances such as M-estimators for these problems are also provided. Experimental results  show that our provable algorithms improve existing heuristics also in practice. A demonstration in the context of Augmented Reality show how such algorithms may be used in real-time systems.
We suggest a new optimization technique for minimizing the sum i=1^n f_i(x) of n non-convex real functions that satisfy a property that we call piecewise log-Lipschitz. This is by URLing links between techniques in computational geometry, combinatorics and convex optimization. As an example application, we provide the first constant-factor approximation algorithms whose running-time is polynomial in n for the fundamental problem ofPoints-to-Lines alignment: Given n points p_1,\cdots,p_n ^n, compute a unit vector x and a permutation \pi:}%DIFDELCMD < [%%% %DIFDELCMD < ]\to[n] %%% ^n |p_ix-b_{\pi(i)}|^z. (ii) Points-to-Lines alignment: Given n } and n lines \ell_1,\cdots,\ell_n on the plane and z>0 , compute the matching \pi:[n]\to[n] and alignment (rotation matrix R and a translation vector t) that minimize the sum of Euclidean distances %DIFDELCMD < \[ i=1^n dist(Rp_i-t,\pi(i))^z \] %%% \sum_{i=1^n \mathrm{dist}(Rp_i-t,\ell_{\pi(i)})^z } between each point to its corresponding line. This problem is non-trivial even if z=1 and the matching \pi is given. If \pi is given , the running time of our algorithms is O(n ^3), and even near-linear in n using core-sets that support: streaming, dynamic, and distributed parallel computations in poly-logarithmic update time. Generalizations for handling e.g. outliers or pseudo-distances such as M-estimators for the problem are also provided. Experimental results and open source code show that our provable algorithms improve existing heuristics also in practice. A companion demonstration video in the context of Augmented Reality shows how such algorithms may be used in real-time systems.
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