Solving underdetermined systems with error-correcting codes

Abstract

In an underdetermined system of equations $Ax=y$, where $A$ is an $m\times n$ matrix, only $u$ of the entries of $y$ with $u < m$ are known. Thus $E_jw$, called `measurements', are known for certain $j\in J \subset \{0,1,\ldots,m-1\}$ where $\{E_i, i=0,1,\ldots, m-1\}$ are the rows of $A$ and $|J|=u$. It is required, if possible, to solve the system uniquely when $x$ has at most $t$ non-zero entries with $u\geq 2t$. Here such systems are considered from an error-correcting coding point of view. The unknown $x$ can be shown to be the error vector of a code subject to certain conditions on the rows of the matrix $A$. This reduces the problem to finding a suitable decoding algorithm which then finds $x$. Decoding workable algorithms are shown to exist, from which the unknown $x$ may be determined, in cases where the known $2t$ values are evenly spaced (that is, when the elements of $J$ are in arithmetic progression) for classes of matrices satisfying certain row properties. These cases include Fourier $n\times n $ matrices where the arithmetic difference $k$ satisfies $\gcd(n,k)=1$, and classes of Vandermonde matrices $V(x_1,x_2,\ldots,x_n)$ (with $x_i\neq 0$) with arithmetic difference $k$ where the ratios $x_i/x_j$ for $i\neq j$ are not $k^{th}$ roots of unity. The decoding algorithm has complexity $O(nt)$ and in some cases, including the Fourier matrix cases, the complexity is $O(t^2)$. Matrices which have the property that the determinant of any square submatrix is non-zero are of particular interest. Randomly choosing rows of such matrices can then give $t$ error-correcting pairs to generate a `measuring' code $C^\perp=\{E_j | j\in J\}$ with a decoding algorithm which finds $x$. This has applications to signal processing and compressed sensing.