# Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm

@article{Clarkson2008CoresetsSG, title={Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm}, author={Kenneth L. Clarkson}, journal={ACM Trans. Algorithms}, year={2008}, volume={6}, pages={63:1-63:30} }

The problem of maximizing a concave function <i>f</i>(<i>x</i>) in a simplex <i>S</i> can be solved approximately by a simple greedy algorithm. For given <i>k</i>, the algorithm can find a point <i>x</i>(<i>k</i>) on a <i>k</i>-dimensional face of <i>S</i>, such that <i>f</i>(<i>x</i>(<i>k</i>)) ≥ <i>f</i>(<i>x</i>*) - <i>O</i>(1/<i>k</i>). Here <i>f</i>(<i>x</i>*) is the maximum value of <i>f</i> in <i>S.</i> This algorithm and analysis were known before, and related to problems of statistics…

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## References

SHOWING 1-10 OF 49 REFERENCES

### Smaller core-sets for balls

- Computer ScienceSODA '03
- 2003

It is shown that any point-set has an ∊-core-set of size [2/∊], and a fast algorithm is given that finds this core-set and implies the existence of small core-sets for solving approximate approximate <i>k</i>-center clustering and related problems.

### Applications of weighted Voronoi diagrams and randomization to variance-based k-clustering: (extended abstract)

- Computer ScienceSCG '94
- 1994

The optimum solution to the k-clustering problem is characterized by the ordinary Euclidean Voronoi diagram and the weighted Vor onoi diagram with both multiplicative and additive weights.

### A probabilistic algorithm for the post office problem

- Computer ScienceSTOC '85
- 1985

The algorithm employs random sampling, so the expected time holds for any set of points, and approaches the preprocessing time required for any algorithm constructing the Voronoi diagram of the input points.

### Coresets for polytope distance

- Computer ScienceSCG '09
- 2009

The coreset framework is translated to the problems of finding the point closest to the origin inside a polytope, finding the shortest distance between two polytopes, Perceptrons, and soft- as well as hard-margin Support Vector Machines (SVM).

### A Randomized Algorithm for Closest-Point Queries

- Computer ScienceSIAM J. Comput.
- 1988

This result approaches the $\Omega (n^{\lceil {{d / 2}} \rceil } )$ worst-case time required for any algorithm that constructs the Voronoi...

### Further applications of random sampling to computational geometry

- Computer ScienceSTOC '86
- 1986

This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry by creating a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer.

### The Computational Complexity of Densest Region Detection

- Computer ScienceJ. Comput. Syst. Sci.
- 2000

A formal learning model for this task that uses a hypothesis class as it “anti-overfitting” mechanism is introduced and it is shown that for some constants, depending on the hypothesis class, these problems are NP-hard to approximate to within these constant factors.

### Rounding of Polytopes in the Real Number Model of Computation

- MathematicsMath. Oper. Res.
- 1996

It is shown that the problem of 1 + en-rounding of A can be solved in Om3.5 lnme-1 operations to a relative accuracy of e in the volume, and that bounds hold for the real number model of computation.

### Minimum-Volume Enclosing Ellipsoids and Core Sets

- Computer Science
- 2005

A modification of the Khachiyan first-order algorithm is proposed with the property that the minimum-volume enclosing ellipsoid of the point set X provides a good approximation to that of S, and the size of X depends on only the dimension d and ε, but not on the number of points n.

### Relating Data Compression and Learnability

- Computer Science
- 2003

It is demonstrated that the existence of a suitable data compression scheme is sufficient to ensure learnability and the introduced compression scheme provides a rigorous model for studying data compression in connection with machine learning.