# Tensor decomposition and approximation schemes for constraint satisfaction problems

@inproceedings{Vega2005TensorDA, title={Tensor decomposition and approximation schemes for constraint satisfaction problems}, author={Wenceslas Fernandez de la Vega and Marek Karpinski and Ravi Kannan and Santosh S. Vempala}, booktitle={STOC '05}, year={2005} }

The only general class of MAX-rCSP problems for which Polynomial Time Approximation Schemes (PTAS) are known are the dense problems. In this paper, we give PTAS's for a much larger class of weighted MAX-rCSP problems which includes as special cases the dense problems and, for r = 2, all metric instances (where the weights satisfy the triangle inequality) and quasimetric instances; for r > 2, our class includes a generalization of metrics. Our algorithms are based on low-rank approximations with…

## 51 Citations

### Spectral methods for matrices and tensors

- Computer ScienceSTOC '10
- 2010

This survey describes methods for low-rank approximation which extend to tensors and finds that for any matrix, a random submatrix of rows/columns picked with probabilities proportional to the squared lengths, yields estimates of the singular values as well as an approximation to the whole matrix.

### Approximate Low-Rank Decomposition for Real Symmetric Tensors

- Computer Science, MathematicsArXiv
- 2022

Two main theorems on symmetric tensor rank with ε -room of tolerance are proved, which are based on some techniques in geometric functional analysis and rigorous complexity estimates.

### N ov 2 01 9 Matrix Decompositions and Sparse Graph Regularity ∗

- Computer Science, Mathematics
- 2019

It turns out that cut pseudorandomness unifies several important pseudorRandomness concepts in prior work: it is shown that Lp upper regularity and a version of low threshold rank are both special cases, thus implying weak and strong regularity lemmas for these graph classes where only weak ones were previously known.

### Randomized interpolative decomposition of separated representations

- MathematicsJ. Comput. Phys.
- 2015

### Tensor sparsification via a bound on the spectral norm of random tensors

- Computer Science, MathematicsArXiv
- 2010

A simple, element-wise sparsification algorithm that zeroes out all sufficiently small elements, keeps all sufficiently large elements of A, and retains some of the remaining elements with probabilities proportional to the square of their magnitudes is presented.

### Most Tensor Problems Are NP-Hard

- Computer Science, MathematicsJACM
- 2013

It is proved that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard and how computing the combinatorial hyperdeterminant is NP-, #P-, and VNP-hard.

### Matrix Decompositions and Sparse Graph Regularity

- Computer Science, MathematicsArXiv
- 2019

It turns out that cut pseudorandomness unifies several important pseudorRandomness concepts in prior work, and it is shown that upper regularity and a version of low threshold rank are both special cases, thus implying weak and strong regularity lemmas for these graph classes where only weak ones were previously known.

### The approximate rank of a matrix and its algorithmic applications: approximate rank

- Mathematics, Computer ScienceSTOC '13
- 2013

Borders are given on the ε-rank of a real matrix A, defined for any ε > 0 as the minimum rank over matrices that approximate every entry of A to within an additive ε.

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