# On linear-algebraic notions of expansion

@article{Li2022OnLN, title={On linear-algebraic notions of expansion}, author={Yinan Li and Y C Qiao and A Wigderson and Yuval Wigderson and Chuan-Hai Zhang}, journal={Electron. Colloquium Comput. Complex.}, year={2022}, volume={TR23} }

A fundamental fact about bounded-degree graph expanders is that three notions of expansion -- vertex expansion, edge expansion, and spectral expansion -- are all equivalent. In this paper, we study to what extent such a statement is true for linear-algebraic notions of expansion. There are two well-studied notions of linear-algebraic expansion, namely dimension expansion (defined in analogy to graph vertex expansion) and quantum expansion (defined in analogy to graph spectral expansion…

## References

SHOWING 1-10 OF 35 REFERENCES

### Quasirandom Quantum Channels

- MathematicsTQC
- 2020

It is shown that for irreducibly covariant quantum channels, expansion is equivalent to a natural analog of uniformity for graphs, generalizing the result of Conlon and Zhao.

### Expander Graphs and their Applications

- Mathematics
- 2006

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in…

### Quantum expanders from any classical Cayley graph expander

- MathematicsQuantum Inf. Comput.
- 2008

This work gives a simple recipe for translating walks on Cayley graphs of a group G into a quantum operation on any irrep of G, and shows that using classical constantdegree constant-gap families of Cayley expander graphs on groups such as the symmetric group can construct efficient families of quantum expanders.

### Dimension Expanders via Rank Condensers

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2014

This work constructs good (seeded) rank condensers (both lossless and lossy versions), which are a small collection of linear maps F^n to F^t for t<<n such that for every subset of F*n of small rank, its rank is preserved by at least one of the maps.

### Difference equations, isoperimetric inequality and transience of certain random walks

- Mathematics
- 1984

The difference Laplacian on a square lattice in Rn has been stud- ied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many…

### il , , lsoperimetric Inequalities for Graphs , and Superconcentrators

- Mathematics
- 1985

A general method for obtaining asymptotic isoperimetric inequalities for families of graphs is developed. Some of these inequalities have been applied to functional analysis, This method uses the…

### Classical and quantum tensor product expanders

- MathematicsQuantum Inf. Comput.
- 2009

The classical case is discussed, and it is shown that a classical two-copy expander can be used to produce a quantum expander, giving tight bounds on the expectation value of the largest nontrivial eigenvalue in the quantum case.

### Spectral Analysis of Matrix Scaling and Operator Scaling

- Mathematics, Computer Science2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
- 2019

If the input matrix or operator has a spectral gap, then a natural gradient flow haslinear convergence, which implies that a simple gradient descent algorithm also has linear convergence under the same assumption.

### Towards dimension expanders over finite fields

- Mathematics2008 23rd Annual IEEE Conference on Computational Complexity
- 2008

This paper considers the finite field version of the problem of explicitly constructing a dimension expander and gives a constant number of matrices that expand the dimension of every subspace of dimension d < n/2 by a factor of (1 + 1/logn).