Minimizing Sum of Non-Convex but Piecewise log-Lipschitz Functions using Coresets
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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 forging 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 ℓ_z Linear Regression: Given z>0, n vectors p_1,...,p_n on the plane, and a vector b∈R^n, compute a unit vector x and a permutation π:[n]→[n] that minimizes ∑_i=1^n |p_ix-b_π(i)|^z. (ii) Points-to-Lines alignment: Given n lines ℓ_1,...,ℓ_n on the plane, compute the matching π:[n]→[n] and alignment (rotation matrix R and a translation vector t) that minimize the sum of Euclidean distances ∑_i=1^n dist(Rp_i-t,ℓ_π(i))^z between each point to its corresponding line. These problems are open even if z=1 and the matching π 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.
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