Definition of matrix-multiplication exponent $\omega$ Colloquially, the definition of the matrix-multiplication exponent $\omega$ is the smallest value for which there is a known $n^{\omega}$ matrix-multiplication algorithm. This is not acceptable as a formal mathematical definition, so I guess the technical definition is something like the infimum over all $t$ such that there exists a matrix-multiplication algorithm in $n^t$.

In this case, we cannot say there is an algorithm for matrix-multiplication in $n^{\omega}$ or even $n^{\omega + o(1)}$, merely that for all $\epsilon &gt; 0$ there exists an algorithm in $n^{\omega + \epsilon}$. Often, however, papers and results which use matrix-multiplication will report their cost as simply $O(n^{\omega})$.

Is there some alternate definition of $\omega$ that permits this usage? Are there any results that guarantee that an algorithm of time $n^{\omega}$ or $n^{\omega + o(1)}$ must exist? Or is the usage $O(n^{\omega})$ simply sloppy?
The matrix multiplication exponent being $\omega$ does not guarantee that there is an algorithm that runs in time $O(n^\omega)$, but only that for each $\epsilon &gt; 0$, there is an algorithm that runs in $O(n^{\omega+\epsilon})$. Indeed if you can find an algorithm that runs in time $O(n^2 \mathrm{polylog}(n))$, then this shows that $\omega = 2$.

You can find the formal definition in the book Algebraic Complexity Theory by Peter Bürgisser, Michael Clausen, Amin Shokrollahi.