# Approximating Nonnegative Polynomials via Spectral Sparsification

## Abstract

### Keywords

### MSC codes

## Get full access to this article

View all available purchase options and get full access to this article.

## References

*Lecture notes for Princeton ORFE course*, Lecture 15; available at https://www.princeton.edu/~amirali/Public/Teaching/ORF523.

*A bound for the number of vertices of a polytope with applications*, Combinatorica, 33 (2013), pp. 1--10.

*A Course in Convexity*, Grad. Stud. Math. 54, American Mathematical Society, Providence, RI, 2002.

*Convex geometry of orbits*, in Combinatorial and Computational Geometry, Math. Sci. Res. Inst. Publ. 52, Cambridge University Press, Cambridge, 2005, pp. 51--77.

*Approximating $L^{\infty}$ norms by $L^{2k}$ norms for functions on orbits*, Found. Comput. Math., 2 (2002), pp. 393--412.

*Thrifty approximations of convex bodies by polytopes*, Int. Math. Res. Not., 2014 (2014), pp. 4341--4356.

*Deciding polyhedrality of spectrahedra*, SIAM J. Optim., 25 (2015), pp. 1873--1884.

*Convexity properties of the cone of nonnegative polynomials*, Discrete Comput. Geom., 32 (2004), pp. 345--371.

*Semidefinite Optimization and Convex Algebraic Geometry*, MOS-SIAM Ser. Optim. 13, SIAM, Philadelphia, PA, 2013.

*Gaussian Measures*, Math. Surveys Monogr. 62, American Mathematical Society, Providence, RI, 1998.

*Remarks on the growth of Lp-norms of polynomials*, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1745, Springer, Berlin, 2000.

*The matching polytope does not admit fully-polynomial size relaxation schemes*, in Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 2015, pp. 837--846.

*Approximation limits of linear programs (beyond hierarchies)*, Math. Oper. Res., 40 (2015), pp. 756--772.

*Geometry of Isotropic Convex Bodies*, Math. Surveys Monogr. 196, American Mathematical Society, Providence, RI, 2014.

*Approximating convex bodies and matrices through Kadison--Singer*, Int. Math. Res. Not. (2017), https://doi.org/10.1093/imrn/rnx206.

*Minimal ellipsoids and their duals*, Rend. Circ. Mat. Palermo (2), 37 (1988), pp. 35--64.

*Even symmetric sextics*, Math. Z., 195 (1987), pp. 559--580.

*Lower bounds for polynomials with simplex Newton polytopes based on geometric programming*, SIAM J. Optim., 26 (2016), pp. 1128--1146.

*Extremum problems with inequalities as subsidiary conditions*, in Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, New York, NY, 1948, pp. 187--204.

*Almost tight bounds for $\varepsilon$-nets*, Discrete Comput. Geom., 7 (1992), pp. 163--173.

*Interlacing families I: Bipartite Ramanujan graphs of all degrees*, Ann. Math., 182 (2015), pp. 307--325.

*Interlacing families II: Mixed characteristic polynomials and the Kadison--Singer problem*, Ann. Math., 182 (2015), pp. 327--350.

*Lectures on Discrete Geometry*, Grad. Texts in Math. 212, Springer, New York, 2002.

*Approximating a Convex Body with a Polytope Using the Epsilon-Net Theorem*, preprint, https://arxiv.org/pdf/1705.07754.pdf, 2017.

*Exploiting Algebraic Structure in Sum of Squares Programs*, in Positive Polynomials in Control, D. Henrion and A. Garulli, eds., Lect. Notes Control Inf. Sci. 312, Springer, Berlin, 2005, pp. 181--194.

*On polynomials in normally distributed random variables*, Probab. Theory Appl., 37 (1992), pp. 692--694.

*Optimizing $n$-variate $(n+k)$-nomials for small $k$*, Theoret. Comput. Sci., 412 (2011), pp. 1457--1469.

*Polyhedra, spectrahedra, and semidefinite programming*, in Topics in Semidefinite and Interior-Point Methods, Fields Inst. Commun. 18, American Mathematical Society, Providence, RI, 1998, pp. 27--38.

*Random vectors in the isotropic position*, J. Funct. Anal., 164 (1999), pp. 60--72.

*Contact points of convex bodies*, Israel J. Math., 101 (1997), pp. 93--124.

*Spectral Sparsification and Restricted Invertibility*, Ph.D. Dissertation, Yale University, 2010; available at https://math.berkeley.edu/~nikhil/dissertation.pdf.

*Minimum-Volume Ellipsoids: Theory and Algorithms*, MOS-SIAM Ser. Optim. 23, SIAM, Philadelphia, PA, 2016.

*High-Dimensional Probability: An Introduction with Applications in Data Sciences*, Camb. Ser. Stat. Probab. Math. 47, Cambridge University Press, Cambridge, 2018.

*Introduction to the non-asymptotic analysis of random matrices*, in Compressed Sensing: Theory and Applications, Cambridge University Press, Cambridge, 2012, pp. 210--268.

## Information & Authors

### Information

#### Published In

#### Copyright

#### History

**Submitted**: 20 March 2017

**Accepted**: 8 January 2019

**Published online**: 21 March 2019

#### Keywords

#### MSC codes

### Authors

#### Funding Information

## Metrics & Citations

### Metrics

### Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

#### Cited By

- Harmonic Hierarchies for Polynomial OptimizationSIAM Journal on Optimization, Vol. 34, No. 1 | 6 February 2024
- Approximate Real Symmetric Tensor RankArnold Mathematical Journal, Vol. 9, No. 4 | 22 August 2023

## View Options

**Access via your Institution**- Questions about how to access this content? Contact SIAM at
**[email protected]**.