Spark (mathematics)

Definition

In mathematics, specifically in linear algebra, the spark of a ${\displaystyle m\times n}$ matrix ${\displaystyle A}$ is the smallest number ${\displaystyle k}$ such that there exists a set of ${\displaystyle k}$ columns in ${\displaystyle A}$ which are linearly dependent. Formally,

${\displaystyle \mathrm {spark} (A)=\min _{d\neq 0}\|d\|_{0}{\text{ s.t. }}Ad=0}$

(Eq.1)

where ${\displaystyle d}$ is a nonzero vector and ${\displaystyle \|d\|_{0}}$ denotes its number of nonzero coefficients.

If all the columns are linearly independent, ${\displaystyle \mathrm {spark} (A)}$ is usually defined to be ${\displaystyle \infty }$.

By contrast, the rank of a matrix is the largest number ${\displaystyle k}$ such that some set of ${\displaystyle k}$ columns of ${\displaystyle A}$ is linearly independent.

Example

Consider the following matrix ${\displaystyle A}$.

${\displaystyle A={\begin{bmatrix}1&2&0&1\\1&2&0&2\\1&2&0&3\\1&0&-3&4\end{bmatrix}}}$

The spark of this matrix equals 3 because:

• There is no set of 1 column of ${\displaystyle A}$ which are linearly dependent.
• There is no set of 2 columns of ${\displaystyle A}$ which are linearly dependent.
• But there is a set of 3 columns of ${\displaystyle A}$ which are linearly dependent.

The first three columns are linearly dependent because ${\displaystyle {\begin{pmatrix}1\\1\\1\\1\end{pmatrix}}-{\frac {1}{2}}{\begin{pmatrix}2\\2\\2\\0\end{pmatrix}}+{\frac {1}{3}}{\begin{pmatrix}0\\0\\0\\-3\end{pmatrix}}={\begin{pmatrix}0\\0\\0\\0\end{pmatrix}}}$.

Properties

If ${\displaystyle n\leq m}$, the following simple properties hold for the spark of a ${\displaystyle m\times n}$ matrix ${\displaystyle A}$:

• ${\displaystyle \mathrm {spark} (A)=n+1\Leftrightarrow \mathrm {rank} (A)=n}$ (the spark equals ${\displaystyle n+1}$ if and only if the matrix has full rank)
• ${\displaystyle \mathrm {spark} (A)=1}$ if and only if the matrix has a zero column
• If ${\displaystyle \mathrm {spark} (A)\neq n+1}$, then ${\displaystyle \mathrm {spark} (A)\leq \mathrm {rank} (A)+1}$

Criterion for uniqueness of sparse solutions

The spark yields a simple criterion for uniqueness of sparse solutions of linear equation systems.[1] Given a linear equation system ${\displaystyle A\mathbf {x} =\mathbf {b} }$. If this system has a solution ${\displaystyle \mathbf {x} }$ that satisfies ${\displaystyle \|\mathbf {x} \|_{0}<{\frac {\mathrm {spark} (A)}{2}}}$, then this solution is the sparsest possible solution. Here ${\displaystyle \|\mathbf {x} \|_{0}}$ denotes the number of nonzero entries of the vector ${\displaystyle \mathbf {x} }$.

Lower bound in terms of dictionary coherence

If the columns of the matrix ${\displaystyle A}$ are normalized to unit norm, we can lower bound the spark of the matrix in terms of the dictionary coherence:[2]

${\displaystyle \mathrm {spark} (A)\geq 1+{\frac {1}{\mu (A)}}}$

The dictionary coherence ${\displaystyle \mu (A)}$ is defined as the maximum correlation between any two columns, i.e.

${\displaystyle \mu (A)=\max _{m\neq n}|\langle a_{m},a_{n}\rangle |}$.

Applications

The minimum distance of a linear code equals the spark of its parity-check matrix.

The concept of the spark is also of use in the theory of compressive sensing, where requirements on the spark of the measurement matrix are used to ensure stability and consistency of various estimation techniques.[3] It is also known in matroid theory as the girth of the vector matroid associated with the columns of the matrix. The spark of a matrix is NP-hard to compute.[4]

References

1. Elad, Michael (2010). Sparse and Redundant Representations From Theory to Applications in Signal and Image Processing. pp. 24.
2. Elad, Michael (2010). Sparse and Redundant Representations From Theory to Applications in Signal and Image Processing. pp. 26.
3. Donoho, David L.; Elad, Michael (March 4, 2003), "Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ¹ minimization", Proc. Natl. Acad. Sci., 100 (5): 2197–2202, Bibcode:2003PNAS..100.2197D, doi:10.1073/pnas.0437847100, PMC 153464, PMID 16576749
4. Tillmann, Andreas M.; Pfetsch, Marc E. (November 8, 2013). "The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing". IEEE Transactions on Information Theory. 60 (2): 1248–1259. arXiv:1205.2081. doi:10.1109/TIT.2013.2290112.