Minimizing Sum of Non-Convex but Piecewise log-Lipschitz Functions using Coresets

07/23/2018
by   Ibrahim Jubran, et al.
0

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.

READ FULL TEXT
research
02/27/2019

Provable Approximations for Constrained ℓ_p Regression

The ℓ_p linear regression problem is to minimize f(x)=||Ax-b||_p over x∈...
research
03/16/2019

k-Means Clustering of Lines for Big Data

The k-means for lines is a set of k centers (points) that minimizes the ...
research
01/10/2021

Provably Approximated ICP

The goal of the alignment problem is to align a (given) point cloud P = ...
research
03/29/2022

Efficient Convex Optimization Requires Superlinear Memory

We show that any memory-constrained, first-order algorithm which minimiz...
research
04/11/2022

Optimizing a low-dimensional convex function over a high-dimensional cube

For a matrix W ∈ℤ^m × n, m ≤ n, and a convex function g: ℝ^m →ℝ, we are ...
research
03/09/2020

Sets Clustering

The input to the sets-k-means problem is an integer k≥ 1 and a set P={P_...
research
11/18/2020

Introduction to Core-sets: an Updated Survey

In optimization or machine learning problems we are given a set of items...

Please sign up or login with your details

Forgot password? Click here to reset