"Some new results for streaming principal component analysis"

Streaming PCA, also known as Oja's algorithm, with roots going back to 1949, has attracted much attention in Statistics and Computer Science in the last decade. In this talk, I will discuss two of our works that address uncertainty estimation and data dependence, which have been relatively underexplored in the past.

First, I will talk about the problem of quantifying uncertainty for the estimation error of the leading eigenvector using Oja's algorithm for streaming PCA, where the data are generated IID from some unknown distribution. Combining classical tools from the U-statistics literature with recent results on high-dimensional central limit theorems for quadratic forms of random vectors and concentration of matrix products, we establish a distributional approximation result for the error between the population eigenvector and the output of Oja's algorithm. We also propose an online multiplier bootstrap algorithm and establish conditions under which the bootstrap distribution is close to the corresponding sampling distribution with high probability.

Our second work looks at dependent data streams. While streaming PCA is typically analyzed under the IID data model, in many applications like distributed optimization, data points are sampled from a Markov chain and therefore are dependent. I will show how the data dependence leads to difficulties in the theoretical analysis and present our finite sample convergence guarantees under standard assumptions on the underlying Markov chain.