Provable Subspace Tracking from Missing Data and Matrix Completion
This work provides a foundational advance for applications requiring real-time data analysis with incomplete information, such as video surveillance or sensor networks, by offering the first complete theoretical guarantees for subspace tracking with missing data.
The paper tackles the problem of subspace tracking with missing data by proposing a provable solution that ensures output subspace estimates are close to true subspaces at all times, even when subspaces change piecewise constantly, backed by extensive numerical experiments.
We study the problem of subspace tracking in the presence of missing data (ST-miss). In recent work, we studied a related problem called robust ST. In this work, we show that a simple modification of our robust ST solution also provably solves ST-miss and robust ST-miss. To our knowledge, our result is the first `complete' guarantee for ST-miss. This means that we can prove that under assumptions on only the algorithm inputs, the output subspace estimates are close to the true data subspaces at all times. Our guarantees hold under mild and easily interpretable assumptions, and allow the underlying subspace to change with time in a piecewise constant fashion. In contrast, all existing guarantees for ST are partial results and assume a fixed unknown subspace. Extensive numerical experiments are shown to back up our theoretical claims. Finally, our solution can be interpreted as a provably correct mini-batch and memory-efficient solution to low-rank Matrix Completion (MC).