  Finding a small spectral approximation for a tall $n \times d$ matrix $A$ is
a fundamental numerical primitive. For a number of reasons, one often seeks an
approximation whose rows are sampled from those of $A$. Row sampling improves
interpretability, saves space when $A$ is sparse, and preserves row structure,
which is especially important, for example, when $A$ represents a graph.
  However, correctly sampling rows from $A$ can be costly when the matrix is
large and cannot be stored and processed in memory. Hence, a number of recent
publications focus on row sampling in the streaming setting, using little more
space than what is required to store the outputted approximation [KL13,
KLM+14].
  Inspired by a growing body of work on online algorithms for machine learning
and data analysis, we extend this work to a more restrictive online setting: we
read rows of $A$ one by one and immediately decide whether each row should be
kept in the spectral approximation or discarded, without ever retracting these
decisions. We present an extremely simple algorithm that approximates $A$ up to
multiplicative error $\epsilon$ and additive error $\delta$ using $O(d \log d
\log(\epsilon||A||_2/\delta)/\epsilon^2)$ online samples, with memory overhead
proportional to the cost of storing the spectral approximation. We also present
an algorithm that uses $O(d^2$) memory but only requires
$O(d\log(\epsilon||A||_2/\delta)/\epsilon^2)$ samples, which we show is
optimal.
  Our methods are clean and intuitive, allow for lower memory usage than prior
work, and expose new theoretical properties of leverage score based matrix
approximation.

Data Structures and Algorithms