# Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices

@article{Levi2018SublinearTimeQM, title={Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices}, author={Amit Levi and Yuichi Yoshida}, journal={ArXiv}, year={2018}, volume={abs/1806.10626} }

We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into…

## References

SHOWING 1-10 OF 25 REFERENCES

### Quick Approximation to Matrices and Applications

- Computer ScienceComb.
- 1999

The matrix approximation is generalized to multi-dimensional arrays and from that derive approximation algorithms for all dense Max-SNP problems and the Regularity Lemma is derived.

### Sublinear Optimization for Machine Learning

- Computer Science2010 IEEE 51st Annual Symposium on Foundations of Computer Science
- 2010

Lower bounds are given which show the running times of many of the algorithms to be nearly best possible in the unit-cost RAM model and implementations of these algorithms in the semi-streaming setting, obtaining the first low pass polylogarithmic space and sub linear time algorithms achieving arbitrary approximation factor.

### The regularity lemma and approximation schemes for dense problems

- MathematicsProceedings of 37th Conference on Foundations of Computer Science
- 1996

The central point here is that the Regularity Lemma provides an explanation of why these Max-SNP hard problems turn out to be easy in dense graphs.

### Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization

- MathematicsSIAM J. Optim.
- 1995

It is argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion.

### Fast Monte-Carlo algorithms for finding low-rank approximations

- Computer ScienceProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

This paper develops an algorithm which is qualitatively faster provided the entries of the matrix are sampled according to a natural probability distribution and the algorithm takes time polynomial in k, 1//spl epsiv/, log(1//spl delta/) only, independent of m, n.

### New Results on Quadratic Minimization

- Mathematics, Computer ScienceSIAM J. Optim.
- 2003

This paper proposes a polynomial-time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint, and introduces a parameterized problem and proves the existence of a trajectory that will lead to an optimal solution.

### Cut-norms and spectra of matrices

- Mathematics
- 2009

One of the aims of this paper is to solve an open problem of Lovasz about relations between graph spectra and cut-distance. The paper starts with several inequalities between two versions of the…

### A linear-time algorithm for trust region problems

- Computer Science, MathematicsMath. Program.
- 2016

This work gives the first provable linear-time (in the number of non-zero entries of the input) algorithm for approximately solving the fundamental problem of minimizing a general quadratic function over an ellipsoidal domain.

### Hidden convexity in some nonconvex quadratically constrained quadratic programming

- Mathematics, Computer ScienceMath. Program.
- 1996

It is shown that the original problem is equivalent to a convex minimization problem with simple linear constraints, and a special problem of minimizing a concave quadratic function subject to finitely many convexquadratic constraints, which is also shown to be equivalents to a minimax convex problem.

### Interior-point polynomial algorithms in convex programming

- MathematicsSiam studies in applied mathematics
- 1994

This book describes the first unified theory of polynomial-time interior-point methods, and describes several of the new algorithms described, e.g., the projective method, which have been implemented, tested on "real world" problems, and found to be extremely efficient in practice.