Tabular data is ubiquitous in modern computer science. However, the size of these tables can be large so computing statistics over them is inefficient in both time and space. This thesis is concerned with finding and using small summaries of large tables for scalable and accurate approximation of the data’s properties; or showing such a summary is hard to obtain in small space. This perspective yields the following results:

• We introduce projected frequency analysis over an n x d binary table. If the query columns are revealed after observing the data, then we show that space exponential in d is required for constant-factor approximation to statistics such as the number of distinct elements on columns S. We present algorithms that use smaller space than a brute-force approach, while tolerating some super constant error for the frequency estimation.

• We find small-space deterministic summaries for a variety of linear algebraic problems in all p-norms for p≥ 1. These include finding rows of high leverage, subspace embedding, regression, and low rank approximation.

• We implement and compare various summary techniques for efficient training of large-scale regression models. We show that a sparse random projection can lead to fast model training despite suboptimal theoretical guarantees than dense competitors. For ridge regression we show that a deterministic summary can reduce the number of gradient steps needed to train the model compared to random projections.

We demonstrate the practicality of our approaches through various experiments by showing that small space summaries can lead to close to optimal solutions.