In this paper, we investigate when system identification is statistically easy or hard, in the finite sample regime. Statistically easy to learn linear system classes have sample complexity that is polynomial with the system dimension. Most prior research in the finite sample regime falls in this category, focusing on systems that are directly excited by process noise. Statistically hard to learn linear system classes have worst-case sample complexity that is at least exponential with the system dimension, regardless of the identification algorithm. Using tools from minimax theory, we show that classes of linear systems can be hard to learn. Such classes include, for example, under-actuated or under-excited systems with weak coupling among the states. Having classified some systems as easy or hard to learn, a natural question arises as to what system properties fundamentally affect the hardness of system identifiability. Towards this direction, we characterize how the controllability index of linear systems affects the sample complexity of identification. More specifically, we show that the sample complexity of robustly controllable linear systems is upper bounded by an exponential function of the controllability index. This implies that identification is easy for classes of linear systems with small controllability index and potentially hard if the controllability index is large. Our analysis is based on recent statistical tools for finite sample analysis of system identification as well as a novel lower bound that relates controllability index with the least singular value of the controllability Gramian.

This release was all about improving and enhancing toolkits' abilities to provide insights into your data, including a brand new outlier detection assistant, an update to our Machine Learning examples showcase page, an upgrade from Python 2.x to Python 3.x and a new System Identification algorithm. Outlier detection is by far the most popular use case in the industry. We constantly seek ways to offer a simple, yet rich and accurate way of helping you find outliers in your data, evaluate it and deploy it in your Splunk environment. It is not only smart by not having prejudice against your data's statistical characteristics, but also charming with a new set of custom visualizations available. With Python 2.7 coming to its end of life, Splunk 8.0 is migrating to Python 3.7 and so is the Splunk Machine Learning Toolkit.