Streaming Algorithms For $\ell_p$ Flows and $\ell_p$ Regression

We initiate the study of one-pass streaming algorithms for underdetermined $\ell_p$ linear regression problems of the form $$ \min_{\mathbf A\mathbf x = \mathbf b} \lVert\mathbf x\rVert_p \,, \qquad \text{where } \mathbf A \in \mathbb R^{n \times d} \text{ with } n \ll d \,, $$ which generalizes basis pursuit ($p = 1$) and least squares solutions to underdetermined linear systems ($p = 2$). We study the column-arrival streaming model, in which the columns of $\mathbf A$ are presented one by one in a stream. When $\mathbf A$ is the incidence matrix of a graph, this corresponds to an edge insertion graph stream, and the regression problem captures $\ell_p$ flows which includes transshipment ($p = 1$), electrical flows ($p = 2$), and max flow ($p = \infty$) on undirected graphs as special cases. Our goal is to design algorithms which use space much less than the entire stream, which has a length of $d$. For the task of estimating the cost of the $\ell_p$ regression problem for $p\in[2,\infty]$, we show a streaming algorithm which constructs a sparse instance supported on $\tilde O(\varepsilon^{-2}n)$ columns of $\mathbf A$ which approximates the cost up to a $(1\pm\varepsilon)$ factor, which corresponds to $\tilde O(\varepsilon^{-2}n^2)$ bits of space in general and an $\tilde O(\varepsilon^{-2}n)$ space semi-streaming algorithm for constructing $\ell_p$ flow sparsifiers on graphs. This extends to $p\in(1, 2)$ with $\tilde O(\varepsilon^{2}n^{q/2})$ columns, where $q$ is the H\"older conjugate exponent of $p$. For $p = 2$, we show that $\Omega(n^2)$ bits of space are required in general even for outputting a constant factor solution. For $p = 1$, we show that the cost cannot be estimated even to an $o(\sqrt n)$ factor in $\mathrm{poly}(n)$ space. On the other hand, if we are interested in outputting a solution $\mathbf x$, then we show that $(1+\varepsilon)$-approximations require $\Omega(d)$ space for $p > 1$, and in general, $\kappa$-approximations require $\tilde\Omega(d/\kappa^{2q})$ space for $p > 1$. We complement these lower bounds with the first sublinear space upper bounds for this problem, showing that we can output a $\kappa$-approximation using space only $\mathrm{poly}(n) \cdot \tilde O(d/\kappa^q)$ for $p > 1$, as well as a $\sqrt n$-approximation using $\mathrm{poly}(n, \log d)$ space for $p = 1$.